LOGIC, MEANING AND COMPUTATION
SYNTHESE LIBRARY STUDIES IN EPISTEMOLOGY, LOGIC, METHODOLOGY, AND PHILOSOPHY OF SCIENCE
Managing Editor:
JAAKKO HINTIKKA, Boston University, U.S.A.
Editors : DIRK VAN DALEN, University of Utrecht, The Netherlands DONALD DAVIDSON, University of California, Berkeley, U.S.A. THEO A.F. KUIPERS , University of Groningen, The Netherlands PATRICK SUPPES, Stanford University, California, U.S.A. JAN WOLENSKI, Jagiellonian University, Krakow, Poland
VOLUME 305
LOGIC, MEANING AND COMPUTATION Essays inMemoryofAlonzoChurch Edited by
C. ANTHONY ANDERSON University of California, Santa Barbara, U.S.A.
and MICHAEL ZELENY PTYX, Los Angeles, California, U.S.A.
....
"
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TABLE OF CONTENTS
C . ANTHONY ANDERSON DAVID KAPLAN
and MICHAEL
and TYLER
BURGE
I
ZELENY
I
Preface
RememberingAlonzoChurch
vii xi
PART I LOGIC PETER APOSTOLI
of arithmetic JOHN CORCORAN
I
Logic, truth and number: The elementary genesis
I
Second-order logic
3
I A representation of relation algebrasusing Routley-Meyer frames
61
J. MICHAEL DUNN
I
THOMAS FORSTER ROBIN O. GANDY
I
Church's settheorywitha universal set Axioms of infinityin Church'stypetheory
EDWARD L . KEENAN
I
SAUNDERS MAC LANE
77 109 139
Logicalobjects
149
I
181
The lambdacalculus andadjointfunctors
I Atomic Booleanalgebrasa ndclassical propositional logic
GERALD J . MASSEY
ROBERT K. MEYER
relevant logic STEWART SHAPIRO
185
I
Improved decision proceduresfor pure
I
The "triumph"of first-order languages
219
I
261
191
RAYMOND SMULLYAN
Equivalencer elations andgroups PART II COMPUTATION
HENK BARENDREGT KLAUS GRUE
I
I
Discriminatingcodedlambdaterms
A-calculus as a foundation formathematics
275 287
vi
TABLE OF CONTENTS Peano'slambdacalculus:The functional abstraction implicitin arithmetic
DANIEL LEIVANT /
RALPH LOADER /
The undecidability of >'-definability
313 331
A construction of the provable wellorderings of thetheoryof species
343
Semanticsfor first and higher orderrealizability
353
PER MARTIN-LOF /
COLIN MCLARTY /
JOHN C . SHEPHERDSON /
Languageand equalityt heoryin logic
programming
365
PART III PHILOSOPHY, MEANING , AND INTENSIONAL LOGIC C . ANTHONY ANDERSON /
Alternative (1*): A criterionof identity
forintensional entities
395
Nominalistparaphraseand ontological
JOHN P. BURGESS /
commitment
429
Peace,justiceand computation : Leibniz' programand themoral and political significance Chur of ch's theorem
445
MICHAEL DETLEFSEN /
M. RANDALL HOLMES / GARY MAR /
Tarski's theoremand NFU
Church'stheoremand randomness
ENRICO MARTINO /
The logic of sense and denotation:Extensions
and applications Analysis, synonymy and sense
NATHAN SALMON /
INDEX
479
Russelliantype theoryandsemanticalparadoxes 491
TERENCE PARSONS /
MARK RICHARD /
469
The very possibilityof language
507 545 573 597
PREFACE
AlonzoChurchwas undeniablyone oftheintellectual giantsof theTwentiethCentury . These articlesare dedicatedto hismemory and illustrate the tremendousimportancehis ideas have had in logic, mathematics,c omputer science and philosophy . Discussionsof some ofthesevariouscontributions haveappearedin The Bulletin of Symbolic Logic, and the interested r eader . Here wejusttryto give somegeneralsense is invitedto seekdetailsthere of thescope, depth,andvalue of his work. Churchis perhaps best known forthe theorem, appropriatelycalled "Church 'sT heorem", thatthereis no decisionprocedureforthelogical validity of formulas first-order of logic. A decisionprocedureforthatpartof logic would have comen earto fulfilling Leibniz's dream of a calculusthatcould be mechanically used tosettlelogicaldisputes. It was not to .beIt could not be. WhatChurchprovedpreciselyis thatthereis no lambda-definable functionthatcan in every caseprovidetherightanswer, 'yes' or 'no', tothe questionof whetheror not anyarbitrarily given formula is valid . To draw themore sweepingconclusion , thatthereis no "mechanical"procedure,no "algorithm" , no "effect ively calculable " way of decidingthequestionrequires an identification theclass offormallydefinedmethods,withthosemethods charact erized thesemore in informalways. The proposal,thehypothesisor conjecture , thatthisidentification is correctis "Church' sThesis" . "Lambda-definability" is a notiondefined usinganother ofChurch'simportantcontributions-the Lambda Calculus.C hurchbegantheconstruction of thiscalculusin the hope ofshowingthatG6del'sIncompletenessT heorems are somehow not as conclusivetheyseem. as The available analysesof the notionof a correctp roofallmade use of some calculus or other.Yet G6del hadapparently shownthateverysuchformalsystem, adequatefor acertain portionof mathematics,will fail to capturesome correctmethodsof proof andcannotbe shown to beconsistente xceptby usingassumptionsstronger thanthoseit endorses. For Churchthismust have seemedintolerable. How thencan we makethenotionof proofprecise? Must not we be able to at leastgive a convincingconsistencyprooffor a formal s ystem adequatefor mathematics ? The Lambda Calculus was hoped to be asystem thatcan be proved to be consistent(which it canand has been-by the "Church-RosserT heorem") andyet somehow escape, or a tleastmitigate, thelimitationson formalsystems G6del'sIncomplet eness Theoremsseemed to impose. (I heardChurch vii
C. AnthonyAnderson and M. Zeleny (eds.), Logic. Meaning and Computation. vii-ix. © 2001 KluwerAcademic Publishers. Printed in the Netherlands.
viii
PREFACE
say, in lateryears, thatonce youunderstand G6del'sTheorem, it's obvious (!) reallythatit is correctand thereis no escape. Buthe alsosaid his en. This deavorwas notentir elyin vain for it r esulted in theLambda Calculus calculuswas and continuesto be ofenormousimportancein computerscience. So here even C hurch's failure was a kind of succes s.} Using theLambda Calculus anda definitionof "la rnbda-definability"termsof in it,Churchproposed toidentifythisnotionor (better)its extensionwiththatof "effectively , of this computablefunction " and therebyprovidean analysis,or definition latter, informalnotion.Church'sThesis is certainly notobviouslycorrect.It seems quiteamazingthaton thebasis of acertainamountof experimentationin definingintuitively computablefunctions,Church 's intuition toldhim thatallhad been captured . Laterwork byThringand Post wouldprovide more intuitively accessiblecharacterizations of effectivec omputability,but it is nowvirtually universally agreedthatChurchwas correctand thatthe Thesis is true(whatevere xactlythatmay mean!). This much, his proposed logic does analysisof effectivec omputabilityand his proofthatfirst-order not yield in thesetermsa decisionprocedure,a lreadyensureChurcha permanentplace as amajor figure inthedevelopmentof symbolic logic. But therewas more, much more. of theoretical circuitsynChurchcontributed early-onto thefoundations thesis. He alsoformulated a theoryof weakimplicationwhich became part of thebasis for work byAlanRoss Anderson,NuelBelnap,and others,on non-classical conceptionsof implication,"ent ailment" and "relevant implication." He made animportantproposalforanalyzing t heconceptof arandom sequence.The simple theoryof types, a modificationof theramified theory of types proposed by Chwistekand Ramsey, received fromChurchits first precise syntactical formulation in an elegantversion using som e of theideas . In lateryears he would give thefirst gleanedfrom theLambda Calculus clearformulation of theramifiedtheoryof types itselfand show an importantrelationship betweenRussell's solutionto the emantical s p aradoxesand Tarski'ssolutionby means of thedistinctionof object-language and metalanguage . And heformulated a ndgave arelative consistencyprooffor a set theorythatallows for theexistenceof a universalset. One would be remiss if Introduction to Mathematical Logic were not mentionedforcontaining i mportantc ontributions to logic. Itcontains,for example, whatwas to becomethestandard axiomaticformulation ofsecond-order logic(notcomplete,alas, as follows from G6del'sIncompletenessT heorem.) And Churchgives therea correctformulation of the rule of substitution for functional variables , a matterwhichhadeludedHilbertandAckermannand others. In philosophy , besides settingan admirablestandardof rigorin philosophicalargument ation,Churchcontributed most of all to defendingand developingintensional logicand relatedmattersof generalsemantics. His own favoreda pproach,theLogic of Sense andDenotation, developsideas of
PREFACE
ix
effort to preciselyformulating GottlobFrege. Buthe also gaveconsiderable whathe consideredto be aviablealternative-Russell's intensional logic as embodied in Principles of Mathematics andin Principia Mathematica. He was an abledefenderof realismin mathematicsa ndlogicanda telling criticof variousnominalistprojects. The sharpeningof Frege's argument thatsentencesd enotetruth -values,theuse ofLangford 's Translation Testto defeatanalysesa ndargumentsa boutmeaningandpropositions,an improved formulation of Quine'scriterion of ontological commitment, anda refutation of Ayer's formulation of thepositivistcriterionof empiricalsignificance,are justa few of hisdistinctively philosophical c ontributions . Churchwroteseveralexcellent papers on varioushistorical m attersconcerninglogic, forexample,on Schroder'sp artiala nticipation of thetheoryof types and on thehistoryof thenotionof a proposition . His explicationsof thephilosophicalideas of Russell a ndFregeconstitute historical scholarship at its verybest. Chapter0 of Introduction to Mathematical Logic contains a crystalclearexpositionof Frege'sideas aboutsemanticscombined witha keen sense ofw hatis worthsavingand whatoughtto beemended. His monumentalBibliography of Symbolic Logic containsevery knownitem on thesubjectof symbolic logic fromthetime of Leibniz to 1935. In effect , he continuedto work onthe Bibliography as editorof thereviewssectionof the Journal of Symbolic Logic. The purposeof those reviews was in partto defendsymboliclogicagainstfallingintodisreputeas a resultof misuse and thefoundersof theAssociationof SymbolicLogic, abuse. And he was one of playing a large roletheever in increasingrespectabilty of thesubject. We rest our case.Church'sintellectual legacyplainlyestablishesfor him an honoredand permanentplace in logic,mathematics , computerscience, . philosophy , andscholarshipa boutthehistoryof logic C. AnthonyAnderson Michael Zeleny ACKNOWLEDGEMENTS
The editorswish toexpresstheirdeep gratitude to: ErinVier Zhu for indispensableassistancewithorganizingandsustainingthistribute , Neil Nelson forsuperlative technical typesetting thatbroughtit tocompletion , andits authorsandpublishersforpatiently b earingwiththeperipetiesof theirproject alongtheway.
REMEMBERING ALONZO CHURCH
I was a graduatestudentwhen ProgessorChurchvisitedtheUniversityof California at Los Angeles (UCLA) in1960-61. He lecturedon thelogic of senseanddenotation. I wasso captivated by his lectures thatI essentially did " in theaxioms; one could nothingelsethatyear. I had found an"anomaly prove thattherewereonly twopropositions , theNecessary and the Impossible. Clearlyt hiswas uninteded . ProfessorChurchwas veryinterested , and I remember distinctlysittingwith him as he went overt he proof line by clearly false, and he was line. The openinglines weret rue,theclosing lines t helineatwhichthingswent fromtrueto false . As he checked looking for each line he s aid "OK", "OK" , "OK", untilsuddenly,"Here it is!". Itwas an instanceof Axiom 16. I alsothoughtt hattheproblemwas thataxiom, butof course I had s pentweeks working on the material.Ultimately, I wrote my dissertation on intensional logic,combining a reformulation of Professor Church'ssystem withthe"possible worlds"semanticsCarnaphad been lecturingon (stemming from Meaning and Necessity) . 1 ProfessorChurchlater publishedhis own revision along somewhatdifferentl ines' (He was schedAssociationmeeting uled topresentthiswork attheAmericanPhilosophical in Chicago,atwhich Iwasscheduledas commentator.Whenhe became ill,I read hispaper atthemeetings andthengave my commentaryentitled"The t hatmy titleproperlyconveyed ChurchReformation".I have always thought theimportanceof a change of view by ProfessorChurch.) o therteachers I had alreadyhad theopportunityto be overawed by my Carnapand Montague . Butatthe timeProfessorChurchvisited, Carnap was mainlyinterested in InductiveLogic andMontaguein Set Theory. Logic is a rigorousdiscipline, butto see the rigor Professor of Church'sintellect, the power ofinsight, and theexactnessof hisstandardsappliedto philosophical problemswas a formativeexperienceof my graduate career . Whenattending his clases orconsulting with himprivately,I alwaysthought,"This is how it studentswho shouldbe done". I believethatI am not theonly one of his developeda reviews-writingstylethatwas initially in consciousimitationof his. 1 David Kaplan, Foundations of Intensional Logic, University Microfilms(Ann Arbor: 1964) . 2 AlonzoChur ch, "Outlin e of a Revised Formalizationof the Logic of Sense and Denotat ion ", (PartI) , Nous , vol. 7 (1973) , pp . 24-33; (P a rt II), Nous , vol. 8 (1974) , pp . 135-136.
xi C. Anthony Ander son and M . Zeleny (eds.), Logic, Meaning and Computation , xi-xii, © 2001 All Rights Reserved. Printed by Kluwer Academic Publishers, The Netherlands.
xii
REMEMBERING ALONZO CHURCH
Despite the awesome intellect, ProfessorChurchwasalwaysgood-humored and patientwith students. At a faculty meeting, afterreadingthe logic prelim of aquestion ablegraduate s tudent , he remarked,"I would be willing to pass him, if he would be willing to promise never to write article an on thephilosophical significance of Godel's Theorem." He loved to laugh, and I have fond recollections of jolly dinnerpartiesat his homeandours. (Although initially theprospectof invitinghim scaredthedaylightsout of me.) One otherremarkfrom theearly dayshas especiallystayedwith me and served me well in variety a of circumstances . I had broken sufficiently free from theChurchstyle toincludea fewsmart-alecky sentencesin a review for theJournalof Symbolic Logic . I was delightedwithmy cleverness.T he review came back with t he comment thatI couldincludeas negativean evaluation of theworkunderreviewas I wished, however , sarcasmwas not permitted . Of course I obeyed . But I latertook theopportunityto ask ProfessorChurchthereasonfor this policy . His replywas, "Evaluations can be rebutted, anderrorscan becorrected , butsarcasmcan beneitherrebutt ed nor corrected."T his has alwaysstayedwith meas a lessonaboutlife and humanrelations . ProfessorChurchwasfor me a model of how supremeintellect can combine withsupremeobjectivity w ithoutrelinquishing devotionandhumankindness. It was my greatgood fortuneto have known him.T hroughout the period thatwe were colleagues, my primarymessage toournew graduate s tudents was, "Take aclassfrom AlonzoChurch . It will change you . And even if you are notinterested in pursuingthesubjectshe teaches , you will be able to tell yourgrandchildren, 'I was a studentof AlonzoChurch'." David Kaplan Departmentof Philosophy University of California Los Angeles,California 90095-1451 USA Email:
[email protected]
REMEMBERING ALONZO CHURCH
xiii
I foundChurchto be a very calm, steady,q uiet, graciously politeman, with, but pleasantly out anyinclination to carryonnormalsocialconversation responsive whenengaged.In his earlyyears at UCLA, he would o ftencome into thedepartmentl ateat nightand departearlyin themorning. He frequentlytalkedor sangsoftly to himself. WhenI was a youngassistantprofessor, I used to drive him to departmental socialevents(which he willingly b utdid notcontribute much to). Everyonewould worry a bouthis attended . There standingin a cornersmiling at theproceedingsbutsayingnothing wereelaborate schedulingarrangements to makesurethathe wasattended studentsor byfaculty . It fell to me to duringthepartieseitherby graduate to engage him in converstaion for goodportionsofseveralsuch evenings even thoughI came to believe he would remainin goodspiritswhetheror not he was talkingwithanyone. I once asked himabouthis impressions of other , he said with greatfigures inthehistoryof logic. When we came to G6del awe and obviousa dmirationthatG6del wastheonlypracticingsolipsistthat he had ever known . I auditeda course byChurchon the G6deltheorems. It was Church's practiceto begin each class by slowly erasingthe blackboard,beginningin theupper left h andcorner , and proceedingin a deliberate, systematic,and orderlyfashion to remove every residualchalkmarkbefore th e lecture began. Church'sclassimmediatelysucceededa class inelementary logic given in the same room by DonKalish. One day Ienteredtheclassroomto find Kalish talking enthusiastically withseveralstudentsa fterhis class, havingforgotten thatit was time for C hurch'sclass to begin.Kalish'sconversation t ooktwo or threeminutesof Church'stime, whileChurchstoodquietlyin thecorner, with hishandsfolded over his midriff , swaying slowly back and forth witha benevolent, pastorial smile on his face . Finally,Kalishrealized his intrusion, hurriedly made some apologies,quicklyerasedtheboard,and bustledout of theroom. Churchmumbled somethinggraciousand, withoutmoving from his spot, watchedKalishleave.Churchstoodthere,stillsmilingbenevolently, , before withoutmoving foratleastanothert hirtyseconds afterKalish left beginninghis lecture. TylerBurge
xiii C. Anthony Anderson and M. Zeleny (eds.), Logic, Meaning and Computation, xiii. © 2001 KluwerAcademic Publishers. Printed in the Netherlands.
PART I LOGIC
PETER APOSTOL!
LOGIC , TRUTH AND NUMBER: THE ELEMENTARY GENESIS OF ARITHMETIC
Abstract.FollowingP aulGilmor e 's LK-style natural deduction bas ed set theories, we present a first orde r ogic l of type-fr ee abstract ion which employs . By adjoini ng LK bivalen ce axiom s "t rut hvaluegaps" to m a intain consiste ncy governingthe definitionsof numeric con cepts, we deduce arithmeticfrom a com binationof firstorderlogic + type-fr ee abst rac t ion.
O. FIRST-ORDER NUMBER-THEORETIC LOGICISM
0.1. Truth and Number Crispin Wright(1983) articulates a positionhe calls" number-t heoretic logicism [1] " , theclaimthatarithmetic is derivablefrom logic inthe strongsense that : it is possible to sodefinearithmeticc onceptsin termsof logical ones so thatevery statementof number-theoryhas a contentpreserving transcription in termsof a purelylogicalvocabulary , andevery axiom s cribed intoa theoremof or theoremof numbertheorycan be so tran logic. (page 137)
Several authors(Wright 1983, Boolos 1993, Demopoulos and Bell 1993) have reconstructed F'rege's Grundlagen programso thattherequired derivations of Peano's postulates are obtainedfrom aconsistentfragmentof the Grundlagen system based upon F'rege'sprincipleof equalityforthenaturalnumbers, referredto as"Hume's principle" in Boolos 1993. However, it is commonlyagreedthatthereconstruction fails to log icize number-theory in therequiredsense due totheapparently e xtra -logicalnatureof Hume's principle. This paper demonstratest hetruthof number-theoretic logicism by deriving first-order P eano Arithmeticfrom a purelylogicalfirst-order theory. Althoughthe presentprogramis not areconstruction of F'rege's, it deploys some of F'rege'scentral insightsconcerningthe role of definitionin thedevelopmentof arithmetic a nd therole oft heboundvariableof "set"I or more abstraction in theformalization of mathematical defneutrally , "relational", inition. By bringingtogethertechniquesfrom severalbranchesof modern logic, notablyanalyticp rooftheory,recursiontheory , thelambdacalculus, programminglanguagesemanticsand truththeory, weare able to deploy 3 C. Anthony Anderson and M. Zeleny [eds.], Logic. Meaning and Computation, 3-59 .
© ZOO 1 Kluwer Academic Publishers. Primed illthe Netherlands .
4
PETER APOSTOLI
relational abstraction in a first-order s ettingin such a waythatthearithmetic of thenatural numbersis derivedas a consequenceof instancesof the principleof bivalen ce appliedto thedefinitionof arithmetical concepts. The idea thatthemathematical p racticeof defining conc epts can be analyzed interms of theop erationof relational abstmction and thecognat e the bound variable of abstmction is a theme which unifies logical device of the logicistprogramsof Fregeand Russell. The analysisof mathematical definitionas relational abstraction is centralto the logicist derivationsof mathematicsbecause definitionwas one ofthetwin pillarsupon which the derivationwas to be based;mathematicswas to be formally derivedfrom thedefinitionof mathematical conceptsand logical laws by logical inference alone.Relational abstraction and conversionwere to effect theintroduction and elimination , respectively , of definitionsin theprocess of deduction,so thatclassicalt ruth-preserving inferencescould beappliedto theconceptual contents ofthosedefinitions . Although logicism is nowcommonlyregardedof at mosthistorical interest,thethemeof assimilatingmathematical definition to the device of theboundvariableof abstraction has continu ed, through t he foundational programsof Church(1941) , Curry(1930) , Fitch (1948) and Gilmore (1980, 1986) on intothe contemporaryanalys is of mathematical reasoningin computerlanguages . Schutte(1960) showedthatsecond-order P eanoArithmeticmay be interpretedin second-ord er logic based upon typed relational abstraction a nd a singledescriptivefunctionsymbol. However, itis commonly believedthat there is nocorrespondingderivationof first-order P eano Arithmeticfrom a purelylogical first -ordertheorybased on abstraction . This is a pointthat could be ofmoment withinthedebateover number theor et iclogicism, as ed by second-order som e philosophersview theexistentialc ommitmentincurr quantification on par withtheposit of extra-logical set existenceprinciples. The insufficiency of first-ord er logic to yield "pur a elylogical " derivationof arithmeticwould betakenby most philosophersto be characteristic of the family offirst-order languages for whichTarskidefined hisnotionof truth on a relational structure:it is a triviality t hattheselanguagesa dmit of finite models in the absenceof descriptivevocabularya nd extra -logicalaxioms. However, mynotionof a "first-order language"is more liberalt hanthenow proprietary sense in whichthattermapplies to theTarskianfamily of languages. By "afirst-order theory"I simply mean a logical theorywhich does not quantifyover anyobjectsotherthanindividuals . It is, afterall,theresourcesof second-orderquantification thatare commonlycreditedwiththe successfulderivation oftheinductionscheme fromtheclassical logicist definitionoftheset ofnatural numbers as theintersection of allero-successor z s ets. And, as mentioned,it is theexistential commitmentof suchquantification thatpurportsto undercutthatsuccess. The first-orderlanguageL in which this derivationof arithmetictakes place fails to belong theTarskianlanguage to family byvirtueof the type-
LOGIC, TRUTH AND NUMBER
5
freeconceptionof predicationthatunderliesit. Becausewithinthelong shadowcastby theTarskiantradition, theconceptof a first-order language over non -individuals . requiresmore thantheabstinencefrom quantification It alsorequiresallegiance to atype-theoretic restriction on thegrammarof of thetypes of objects predicationthatmirrorsa semanticregimentation thatcanstandin thesatisfaction relation.A first-order Tarskianlanguage typessyntacticcombinationso thatthesemanticfunctionperformedby the categoryof formulas, viz. , to bepredicatedof, or satisfiedby, an individual, cannotbe performedby thecategoryof singularterms. Singularterms, or rathert heindividualstheydenote , bearidentityconditionsand existential . witness, buttheycannotbe predicatedof, orsatisfiedby, otherindividuals From theperspectiveof categorial grammar, a Tarskianfirst-order languagethusassigns two quite d istinctsemantictasksor functionsto its two syntacticcategoriessingular term andformula. The semanticvalue of asingular term is a first level individual , wherethelogicalc haracteristic of a first levelindividualis its abilityto bearidentityconditionsa ndserveexistential witness, butnot to bepredicatedofotherindividuals.T hedistinctivelogical role ofsingulart ermsis realizedprimarilythroughthelogic ofident ity and quantification . In contrast , thesemanticvalue of aformula is a set, or characteristic function , of first level individuals(or tuplesthereof).T he semantic role of a formula is bearsatisfaction to conditions,to be satisfiedby first levelindividuals.By stratifying predication,Tarskianlanguages ensurethat theentitiest hatthe logic of identityandquantification treatof aredistinct from theentitiest hatcan bearrecursively definedsatisfaction conditions. first-order PeanoArithmeticin a firstIn thepresentessay, weinterpret abstraction . Alorderlanguage , L , designed toexpresstype-freerelational thoughL's surfacesyntaxindeedstratifies syntactic combination,distinguishing betweenthesyntacticcategoriesof term and formula in themannerof Tarskianlanguages,this superficialstratification masks theintendedinterpretationof L, whereuponabstraction t erms denoterelations and "e" exof satisfaction or membership holding between relations presses therelation and orderedpairs of first level individuals.Our semanticintentionsmight combinationofsingular be more faithfully represented by usingthesyntactic termsto indicatepredication , in thefashion ofapplicativegrammars, viz., AXY. R(x, y)(a, b). Hence, theabstraction termsof L playsimultaneously the : theybear distinctivesemanticroles ofT arskiansingulart ermandformulas identityconditionsa ndexistential witness,butalsoembed themeta-theoretic thatis, they aregeneralpredicatesof individuals. satisfaction relation, It is easy to seethatthesimultaneousdeploymentof thesetwo logical roles inducesnumericalexistenceprinciplesunavailable on theTarskian conof identicals,we see ception. For example, by applyingtheindescernibility that0 =df {x : x =I- x} is provablydistinctfrom 1 =df {x : x = O}, since thelatter has apropertynotsharedby theformer, namely beingsatisfiedby some object,i.e., having a member. 2=df {x : x = I} is provablydistinct
6
PETER APOSTOLI
from 0 forthesame reason,but also from ,1since it has apropertynotshared by thelatter , namelybeing satisfiedby 1. In thisway, the logic ofidentit y combines withtype-freeabstraction to generate t hedomain of natural numbers undertheencodingof 0 by theempty setandthesuccessor functionby theoperationof passing from anobjectto itssingletonset. It is crucialto thegeneration of thesequence
0, {O} , {{O}}, {{{On} thatthesatisfaction relationwe embed in L is type-freein thesense that complexsingulart ermscansatisfyothercomplexsingulart erms,since every successortermin thesequenceis satisfiedby its predecessor. 0.2. The grounding of semantics and the semantics of grounding
Church'searlyhope was tomarrytype-freefunctional abstraction w ithfull quantificational logic. In the sixtiesPaulGilmoreshowedthattype-freerelational abstraction may be consistently combined withfirst-ord er quantificationtheoryif one si willing to give up theprinciple of excludedmiddleand workwiththenotionof a "part ialr elation".In effect , Gilmorewas taking a leaf fromtherecursiontheorists ' book; a decadeor so lat er, thisleaf would es of a partial bloom in the philosophicalc ommunity in the form oftheori truthpredicate . Gilmore(1980, 1986) presentedhis theoryas a first-order G entzen-style sequentcalculus,NaDSet I. The deductiverules ofNaDSetI are precisely thoseof an LK identitycalculusw ith the additionof a pair of LK -style introduction and elimination rules which formalize a very generalnotionof relational abstraction . However, NaDSetI is based upon a proper subsetof theLK axiom sequents.WhileLK is basedupona completeset of"bivalence axioms" (axiom sequentsoftheform A I- A forany atomicsentenceA) , NaDSet I's bivalence axioms are restricted to thosetreating i dentitystatements andthoseof theform t e c I- t e c fornon-complexindividualc onstants conly. In effect,consistencyis maintainedin NaDSetI by thestrategyof evaluating onlygrounded sentences, sentenceswhose truth conditionsreduce,under logicaldecomposition,to thetruthconditionsof simple sentences which are " facts evaluated on thebasis on "non-semant ic" or"non-definition-theoretic alone. Bivalenceaxioms foridentitystatementsprovidea sufficient basis for a NaDSetI derivationof thetheory of zeroandsuccessor(using thefamiliar set-theoretic encodingofthoseoperationsmentionedin theprevioussection). However,theabsence of bivalenceaxioms of the formt E P I- t E P governing free-objectvariablesp undercutsa NaDSetI derivationof thescheme of mathematical induction . In the firstpartof this paper, we show that, although NaDSetI does not yield aninductionscheme, it containstheresourcesforinductivedefinition.In section1, we set outthe calculusG, a
LOGIC, TRUTH AND NUMBER
7
sub-logicof NaDSetI based on binaryabstraction and bivalenceaxioms for identitystatements . The significanceof binaryabstraction is shown in section2, where itunderwr ites thedefinitionof avariablebinding"fixed-point" operatorjixy, a relational version ofthe"paradoxical"Y combinatorof the lambda calculus,forming terms which compute the fixed-points of recursive functionals. We arethenable to derive L K-styledeductionrules which implementa relational version ofthesecond recursiontheoremin termsof introduction andeliminationconditionsfortheoperatorfix y • In section4 we use jixy to construct a set term ~ which represe ntsthe set IN ofnatural numbers in theweak sensethat~ enumerat es theset of allnumerals . In section6, ~ figures intheconstruct ion of set terms t f that representarbitrary primitive recursivefunctionsj : IN -+ IN in thestrong sense ofenumerating thegraphof j. Here, we usethesecondrecursiontheorem to simulateprimitiverecursion.E xistential quantification is thenused to simulateunboundedsearchover IN, allowing G to enumerat ethegraphs complete", but of allgeneralrecursivefunctions . G is thus"computationally lackstheresourcesto reason abouttherelations it computes. Specifically, although ~ is infinite, G has no t heoremto thiseffect . Similarly , G does not yieldthePeanoaxioms whichrecursively specify t+ andt x , thesettermsthat representt headditionandmultiplication relations overthenatural numbers. This limitationis entirelydue tothefailure of bivalence in G atomic for sentenceswhich featurethoseterms as predicates , and contrastswiththe bivalenceprinciples semanticsettingofsection3, wheretermmodelsvalidate for allp rimitiverecursiveterms, hence realizingthe full theoryof 0, S, + and x. Thus, we pass successively in ctions5 se and 7 toextensionsG W and G+'x of G by adjoiningbivalence axioms governing ~ and thent+ and t x . For example, G W is theextensionof G obtainedby adjoining as new axioms allsequentsof the formt e ~ f- t e ~ for arbitrary closed terms t. As a result,GW yields "theaxiom of infinity " - t hestatementthat~ is a O-successorset-as a theorem,and G+'x sustainsrelational versionsof the Peanoaxioms specifyingtheinductivedefinitionof + and x . The following foundational paradigmemerges: inthesettingof a partial classicallogic whichmaintainsunrestricted type-freeabstraction by rejecting a set excludedmiddle, theposit of bivalence axioms forprimitiverecursive termcanjunction logically as an existence postulate. This is seen graphically in thesimplicityof thederivationof theaxiom of infinity for ~ in section7. Note thatadjoiningexcludedmiddle axioms to Gmaintainslogicalpurity: G W andG+'x areobtainedfromsub-logicsoffirst-order logic byclosureunder abstraction . And thelanguageof thesecalculi isjustthelanguageL of G. For thesereasons,G W and G+'x arereferredto as logical extensionsof G; theterminology is chosenpurposelyto contrast with thestandardn otionof an extra-logical extensionof a logic. Withinthis paradigm, progressivelystrongerlogics arecountenanced by progressively e xtendingthe"classical fragment"of G through t headditionof
8
PETER APOSTOLI
bivalence axioms forselect"primitiverecursive a vocabulary " . For example, since G isbasedupona completeaxiomatization ofidentitylogic, theclassical fragmentof G containsallformulasand set terms built up from identity statements . Given thatthe axioms of the theoryof 0 and S fallwithin this fragment,it is notsurprisingtheyare derivablein G. By hypothesis, we can reasonclassically a bout0 and S in thatfragmentand, further , the characteristic propertiesof 0 and S, viz., thatS is an inje ctionfrom its domain dom(S) intodom(S) - {O} , areclassically e ntailed by thedefinitions of 0 and S. This is exactlyhow logicistreasoningis supposed to work. Passageto thelogicalextensionsGW andG+ ' x in effecte xtendstheclassical fragmentof G tocontainan image ofthelanguage of arithmetic . In section8, we extendG+'x to asystem G z by addingthebivalence axioms governingfree-objectvariablest hatwere lacking from N aDSetI. This additionallows G quantifiz tosimulateenoughof theidiom ofsecond-order cationtounderwrite theclassical impredicativedefinitionofthesetofnatural numbersas theintersection of all O-successor sets . Once thisis achieved, the derivationof thefirst-order i nductionscheme goes throughin Gz in much thesame way as insecond-order theoriesof abstraction , such asNaDSetII. 1. TERM AND FORM ULA: THE LANGUA GE OF PAIRWI SE REL ATIONAL ABSTRA CTION
1.1. The language L
We considera first-order language L containing a thr ee placepredicatesymbol
e intendedto representset theor etic membership and abinarypredicate=
intendedto representidentity. L has a countably infinite set of individual variables , listedas VI, VZ, • • •, sentential connectives1\, V, "" anduniversaland existential quantifiers V and 3, respectively.In addition , L has a variablebinding term forming operator[Vi Vj : -J intendedto representbinary,or pairwise, set abstraction. In additionto theindividualvariables,G contains infinitelym any individualparameters, listedPI, Pz, . . . . The parametersare intendedto play the role eigenvariable of thatGentzen(1935) assignedhis so-called"freeobjectvariables":theyprovide the arbitrary witnessupon whichuniversal generalization is to beperformed, butare notboundby any operator. The set ofindividualtermsandformulasare defined bym utualrecursion as follows: DEFINITION 1.1 Term and Formula . Allindividualvariablesa nd parametersare terms. If t I, tz and t3 are terms, thent I = tz and (t I , tz) e t3 are (atomic) formulas. If A is a formulaand x , yare variabl es, then [xy : AJ is a term. If A and B are formulasand x is a variable , then...,A, A 1\ B , A V B , "Ix A and 3x A are formulas.
LOGIC, TRUTH AND NUMBER
9
The connectives- -+, +-+ areintroducedby theirstandardclassical defini. As indicated,we let"x", "y", "z" etc. tionsand used whereconvenient rangeover variables;"8" , "t" , "r" rangeover terms;and "A" , "E", " C" etc. rangeover formulas. In a ddition,we let"a" , "b", "c" etc., and somemay occurwith times "p", rangeoverparameters.Allthesemeta-variables or withoutnumericalsubscripts. All logical constants,including"e", occur autonymously in themeta-language . Atomic formulas of theform iI = ta, (tl,t2) e t3 are called identity statements, respectively, e statements. The lat teris sometimes writtenas t3(tl,t2) , to emphasize thatt3 is whatwe shall callthe predicate term oftheatomicformula(tl't2) e t3' If a termt is of the form [xy : Aj, thent is called anabstraction term ; x and yareits variables of abstraction. A formula or t ermis closed iff itcontainsno freeoccurrences of anyvariable . Sentences are closed formulas containing no parameters . A closedtermis said to beconstant iff itcontainsno parameters.Hence, wedistinguishclosed thatwe areinterested in evaluating; formulas from sentences.It isthelatter the former are merely proof-theoretic intermediaries . Let ctrm denotetheset of constant t erms. For eachk ~ 1 we use the k-arysubstitution o peratoras follows : for any formulaA, distinctvariablesx = Xl,. .. ,Xk , and terms[ = tl, " " tk , A x [£] denotesthe resultof simultaneously substituting ti for Xi uniformly A, with the usual proviso thatt ; is free forx i(l ::; i ::;k). The throughout substitution o peratorwill also be used to uniformly replaceparametersby arbitrary terms, particularly in thecontextof performinguniformsubstitu tion onorderedsetsof formulas . For example, letI' = {Ail i ::; n} be a set of formulas and p = PI , ' .. ,Pk parametersand [ = tl," . ,tk terms. Then, f a[£] = df {A~[fjl i ::; n} denotesthe orderedset ofsentenceso btainedby ti for ai everywherein I'[I ::; i ::; k). An instance of uniformlysubstituting a closedL formula is anL-sentenceobtainedby replacingevery parameter occurringin the given formula byconstant a t ermof L. More formally, let P = PI,· .. ,Pm exhausttheparametersoccurringin the closed formula A, and let[= iI, ... , t k be constant t erms. Then A p[£] is an instance of A . 1.2. Syntactic complexity versus semantic complexity
Note thatin Definition1.1, termsand formulas of L aredefined bymutual recursion . As a result,thelanguage L countenances complexatomicformulas languages,the of theform (tt, t2) e [xy : A(x, y)j. In Tarskianfirst-order notionof truth on a structure is defined byinductionon theformationof sentences , a measurewhich ismonotonein thelengthof thesentencebeing evaluated . As a result,in a Tarskianlanguage , themeasureof thesyntactic complexity inducedby theformationhistoryofsentencescoincides with the measureof the semantic complexity throughwhichsentencesareevaluated . In particular, thesentenceswhich areregardedas semantically simple are
PETER APOSTOLI
10
precisely thea tomicsentences . In contrast , the assignmentof truthvalues to sentencesof L is defined by aninductiveprocess which isnotmonotone , for in theevaluation of a givensentence,e.g., in thelengthof thesentence theatomicsentenceabove, we are led to evaluate sentences , viz., A(tl ' t2), which may be longer thanthegivensentence . This is a familiarphenomenon in theoriesbased ontheoperationof type-freetermreduction,such asthe lambdacalculus or combinatorylogic. As a result , thereareatleasttwoimportantcomplexitymeasuresfor sen, inducedby Definition tencesof L . First, a sentencehas aformationhistory 1.1, andwe canassign thestageat which a given sentenceis formed according to 1.1 as a finite complexitymeasure. This measureignores theinternal structure of setabstractt erms, consideringallatomic sentencesto have 0 comcomplexity,andcountsthecomplexityof Booleanandquantificational plexes as onegreatert hanthesum of thatof theirprincipalsubformulas. Second, "grounded " sentencescan be assigned a "level" , thetransfinite stage aresemanticallyevaluated in thecanonical evaluation process at which they defined insection3. Roughlyspeaking, the level ofsentence a is theordinal stageatwhich it receives a classical truth value, if it receives one atall. 1.3. Proof theory Next, weintroducea sequentcalculusG, in thestyleof Gentzen 's (1935) LK. G is a fragmentof NaDSetI. WhereasNaDSetI takes classicalcalculus generalized abstraction as a primitiveinference rule , we usetherule of simple binaryabstraction , whichagainmay be consider ed a simple instanceof the generalized abstraction rule ofGilmore(1986). A sequent is a pairoforderedsetsof closed formulas f ,~, possiblyempty, f-, thus: I' f- ~ . The axiom sequents,or axioms, of separatedby theturnstile G are all sequentsoftheform tl = t 2 f- tl = t z and f- tl = tl . The former are referred to as "excluded middle" or "bivalence" axiomsidentity) (for because sequentsare giventheirclassical , disjunctivereadingin G. Thatis, roughly speaking, f f- ~ may be readas assertingthateithersome member of I' is False or somemember of ~ is True. As for NaDSet I, the deductionrules for G are the LK structural and operational rules and a congruencerule for identity ,= :
Structural Rules Thinning ff-8 ff-8
/ /
Contraction A, A , r f- e r f- e, A, A
A, r f- e r f- e, A
/ /
A, r f- e r f- e, A
LOGIC, TRUTH AND NUMBER Interchange 6., A, B, r f- e r f- e, A, B, 6. Cut
r f- e, A
A, r f- e
Identity =f-
/
/
6., B, A, r f- e B, A, 6.
r f- 0, /
tle 5, r f- e
r f- e, t2e 5
11
6.,
r f- e
/
Operational Rules f-A Af-
e /
f-v Vff--, -,f-
r f- e, A A, r f- e A, r fr f- e, A
-v
r
'If3ff-3
r f- e, B / r f- e, A A B fB, r fA A B, r fr f- e, A V B r f- e, B / r f- e, A V B B, r f- 8 / A V B, r f- e / rf-8,-,A
rf-8,A A, r f-
A A B,
/
e
/
-.A,
/
f- e, Ax[P] Ax[tJ, r f- 0 Ax [pJ, r f- 8 r f- e, Ax[t]
r e
e /
e
r f- e
r f- 8, VxA VxA, r f- e 3xA, r f- e r f- e, 3xA
/ /
/
where inf-V and 3f-, theparameterp does notoccurfree inI', e or A. Note thattheuse ofthesubstitution operator' If- requiresthatt; be free forx in A. Derived rules of f--+, -+f- will be assumed and used where convenient. variantsof the rules, particularly forf- A Notethatwe will often use derived and A f-. In general,implicit use ofInterchange and Contraction will allow us totreatsequentsas unordered sets. Also, the form of the identityrule=fthatwill be used is A x [t2],
r f- e
/
tl = t2,rf- 8,
whereA is an arbitrary formula with at most x free. Abstraction Rules. In addition,G has falsitypreservingand truthpreservingrules for b inaryrelational abstraction, which are very special cases of thegeneralized abstraction rules of NaDSet: I cff-c
Axy[tl, t2], r f- 6. r f- 6., A xy[tl,t2]
/ /
(tl,t2)c !xy : Al, r f- 6., r f- 6., (tl,t2) e [xy : A].
The use ofthesubstitution operatorin these schemas requires thatti be free forbothx and y in A (i = 1,2). Derivations taketheform oftheusualGentzentrees, thoughwe shall write them in linearfashion to avoid copyingsubtrees, re where possible . All the cognateprooftheoretic notions, e.g.,t hatof derivable sequent, arestandard. Notethattheaxioms and rules of tGreating t heclassicalconnectivesare classical. G fails toextendclassical logic only becausetheabsence of of the
PETER APOSTOLI
12
LK bivalence axioms, for which adequate an basis is A f- A foratomic A. G has bivalence axioms for identitystatements only. This is a tightening of the policy in Gilmore 1980, 1986 of restricting NaDSetI's bivalence axioms to thosetreating i dentityand t e c f- t e c fornon-complexindividualconstants c. In effect,t hisaxiomaticrestriction correspondsto theNaDSetI's model theoreticstrategyof evaluating only grounded sentences, sentenceswhose truth conditionsreduce,underlogicaldecomposition, tothetruth conditions on the basis of "non semantic" or of simple sentenceswhich areevaluated "nondefinition-theoretic" facts alone . Paradoxical atomicsentencessuch as R e R, where R is the "Russell" setterm [x : --, x e x], or the settheoretic T e T, where T is[x : x e x], arenotevaluated version of the" truthteller", becausetheyfail to begroundedin thissense. Thus, theconceptionofsemanticgroundinginformingNaDSet I is formally implementedby specifyingatomic sentencesof the formt 1 = t2 and t e c groundatoms to figure in as theonly "groundatoms" and allowing only bivalence axioms. G allows only the former groundatoms. to be On this basis, however, excluded middle sequentsarederivableforsentencesall of areidentitystatements.As we shall see in section whoseatomicconstituents 3, this isenoughto interpretwithinG the theoryof 0 and thesuccessor function .
1.4. The notion of a logical extension of G A logical extension of G is a calculus for L which isobtainedby adjoiningto G as new bivalence axioms sequentsof the formset f- set forcertainclosed theform set. Here set is consideredto be a new"ground atomicformulas of predicate atom" in thecalculust husobtained. Sometimes we refer to the term t of thenew groundatom as a new"groundpredicate"of the logical extension. The groundpredicatesin thelogicalextensionsstudied in thisessay are set termswhich definenumber-theoretic conceptsthatwe areinterested in thatis, underbivalence, inthegivenextensions. reasoningaboutclassically, The notionof a logical extensioncontrasts with thestandardnotionsof an extensionof a classic altheory . We considera logicalextensionof G to be aretheories , but not alltheories a logic as opposed to a theory (all logics for the "logical purity"of these are logics).T he simple-mindedjustification extensionsis the simpleobservationthattheirnew axioms areinstancesof thegeneralprincipleof bivalence . However, it must be shown why bivalence numerical for concepts counts as a purelylogicalpostulate in a system whichrejectsthe general principle of bivalence inorderto accommodateunrestricted type-freeabstraction .1 Whatis needed is acharacterization of the class of atomicsentencesfor which 1 l owe
thepresentpointto PhilKrem er.
LOGIC, TRUTH AND NUMBER
13
theposit of bivalence can be motivatedindependently of considerations of consistencyand therequirements ofnumber-theory . Of course, the classt hus thederivation characterized will have to be consistentas well asu nderwrite of number-theory . The idealsolutionis to come upwithan analysisof the paradoxeswhich at once provides for aconsistenttreatment of type-free a nd specifies a class of a tomicsentencesfor which the posit of abstraction bivalence is sufficient for number-theory . Such ananalysisis providedby a constructivistically motivatedanalysisof thenotionof semantic grounding. On thetraditional account,"grounded " sentencesare those whose truthconditionshereditarily reduceunderlogicalanalysisand conversion(either T) to thetruth conditions set-theoretic or truth-theoretic as per convention of "groundatoms"- at omicsentenceswhosetruth-conditions aredetermined independently of definition-theoretic or semanticfacts. This roughintuition c onstructed t erm models ofNaDSet I and is formalized intheinductively arithmeticmodels of Kripke (1975) intermsof a restriction on thevocabularythatmay occurin theatomic formulasthatcountas groundatoms. Kripke'srestriction is themost straightforward: groundatomsmust not contain"semantic" vocabulary , such asthetruthpredicate"T". Gilmore'srestriction is slightly more subtle:groundatomsmay contain"e", butnot with abstraction t ermsas predicates.In bothsettings , however, them otivation is the same: theoffendingexpressionsinvoke "circular fortherestrictions concepts". In Gilmore 1986 themotivationfortreating identitiesand e statements withunstructured predicatetermsas groundatomsis thatthey aresemantically primitive in thesense thattheyarenottheoutputof anapplication of anysemanticrule. In particular, theycannotbe obtainedby abstraction on thebasis of from anysentence;it is in this senset hattheyareevaluated non-definition-theoretic facts alone . So, "sem anticprimitiveness"converges primitiveness " in NaDSetI. witha certainnotionof "syntactic From the viewpointprovidedby logical extensionsof G, Gilmore's identifi cationofsyntactic simplicitywithsemanticsimplicityis too much to concede , as arguedin section0, theprocess of seto theTarskianregime. Because manticevaluation whichconstitutes theconceptionof groundingunderlying NaDSet I invokes adistinctionbetweensyntacticformationand semantic language.Once evaluation alien totheTarskianconceptionof a first-order ofsemanticcomplexitywithsyntactic comwe havebrokentheidentification plexity,t hereis no reasonto thinkof semantically simple sentencesas being syntactically simple as well. Only when this lastvestige of theT arskian complex ground stratification of predicationis rejected,and syntactically predicatesare countenanced , can first-order theoriesof type-freerelational abstraction bloom. On ourconceptionofsemanticgrounding , a givenatomicsentenc e counts as agroundatomiffeitherit is not the o utputof asemanticrule, or its truth conditionsare suchthatits semanticevaluation reduces to the evaluation of
14
PETER APOSTOLI
syntactically strictly simpler sentences.T his intuitionis formalized by identifyinggroundatoms with primitive recursive queries. Primitiverecursive notionsare preciselythoserepresentedby a syntacticstructure whose complexitymonotonically reducesunderevaluation.In section9 we associatea constantset term t f witheach primitive recursivefunctionf such thatt f enumeratest hegraphof fin G. It turnso utthatwhentermmodels of G are specified (as in se c tion3) so thatconstant t ermsareindividuated u nderstrict syntactic identity,everyatomicsentencewitht f as predicatetermreceives a classicalt ruth value, .e., i tf representsa "total" s et. So allprimitiverecursive groundedin preciselytheoriginalsense ofGilmore queriesaresemantically (1986) and Kripke (1975) . Again, themotivationfortherestrictions is the same as fortheseearliersemanticproposals: expressionslacking aprimitive recursivestructure potentially invoke "circular concepts"which lead to nont heclass of terminating r eduction sequences. More importantly , by allowing logicalextensionof G to be closed u ndertheadjunction of bivalenceaxioms for primitiverecursivequeries,we areadoptingintuitionistically acceptable instancesof bivalence . For aprimitiverecursivequeryis a paradigmaticexample of thekind of"constructive statement " for whichtheintuitionist is willing toa cceptbivalence . 2. EXTENDED SYNTAX FOR L
2.1. K -ary abstraction In thissectionwe definek-aryabstraction in terms of binaryabstraction and derivethecorrespondingdeductionrules ofk-aryabstraction forarbitrary logicalextensionsof G. DEFINITION 2.1. Our definitionsare set up with the intentionthat2-ary abstraction is binaryabstraction.
OrderedPairing
(tl, tz) =df [Vl Vz : tl = Vl A tz = vz]' (tl,' .. , tk) =df ((tl , " . , tk-l), tk)(k > 2). Vector notation :
(n denotestheterm(tl,' . . , tk)'
DefinedSetTerms: Let A be a formula :
Unary abstraction: [z : A(x)] in A(x).
=df
[xy ; A(x)], wherey is thefirstvariablenotoccurring
Unary membership: te s
=d f
(t,t)€s. So, forexample,t€[x ; A(x)]
=df
(t,t)c[x : A(x)] ,
LOGIC, TRUTH AND NUMBER
15
k-ary abstraction: Let k 2: 3 and A be a formula . Suppose we have term [Xl ' " Xk -l : A]. Then alreadydefinedthe(k -1)-aryabstraction .Xk : A] =df [zXk : 3YI •.. Yk-l (z =
[Xl "
whereYI and:
.. • Yk-l
(YI, •• • ,
Yk-l) /\ (Yl • . . Yk-d e[XI •.. Xk-l : A])],
arethefirstvariablesnotoccurringin [Xl ' " Xk-l : A]
k-ary membership: Let k 2: 3,
(tl... tk) 10 [Xl . .. Xk : A] =df ({tl,.. . , tk-l) , tk) 10 [Xl.. . Xk : A], Vector notation : [x: A(x)]
=df [Xl . . .
Vector notation : (f)e[x: A(x)]
Xk : A(XI,' .. , Xk)],
=df (tl... tk)e[Xl
... Xk :A(XI , ' " , Xk)] .
Clearly , we may deriverulesforunaryabstraction.Notethattheargument list fork 2: 3-arymembership forgoesthecomma characteristic of theargument list forbinaryabstraction . Given f = tl . . . tk-l and tk we sometimes preservethecomma, writing"(t: tk)e[X Xk : A(x, Xk)]" for({tl, .. . , tk-l) , Xk)e [Xl . • • Xk : A(XI, . . . , Xk)], whenwe wantto emphasize thatf is in factthe singleargument(tl , " " tk--l). The following c orollary is statedforan arbitrary logicalextensionof G:
2.2 The DerivedRulesof k-ary. Let k 2: 3, f = tI, .. . , tk be any terms, x = Xl, .. . ,Xk distinct variables and A a formula. Then
COROLLARY
r
f-~, Ax[f]
Ax[f] , r f- ~
/ /
r f- ~, (f) 10 [x : A], (f) 10 [x: A], r f- ~
are derived rules, where, for k 2: 3, 10k f-conforms to the restriction that for all closed term f = tI, . . . , tk, Ax[t] f- Ax[f] is a derivable sequent.
2.2. The recursion theorem We may now presentsome recursion-theoretic control s tructures which allow us to calculate fixed-pointsfor formulasin G and logicalextensions. The following is raational reconstruction of a resultd ue to theScottishlogician , Jamie Andrews,who firstdiscoveredthedistinctiverole ofbinary abstraction in thecalculation of fixed-pointtermsin relational settings: Let A = A(x, y) be a formula . Then
diagyA fixyA
=df =df
[xy: A(x, [x : (xy) 10 y])], [x : (xdiagyA) e diagyA].
16
PETER APOSTOLI
LEMMA
2.3 Andrews'RecursionTheorem. Let A = A(x, y) be a for= t 1 , • • . ,tk be terms, where ti
mula, x = Xl,... , Xk distinct variables and is free for Xj in A (1 $ i,j $k). Then,
fixr rfix
r , A(~fixyA) r 6.
rr A(~fixyA), 6.
/ /
r
r, (t)
e fixyA
r 6.,
r r (t) e fixyA , 6..
are derived rules, where, for k ~ 3, fixr conforms to the restriction that, for all closed term = t l , ... , t« , Ax[t] r Ax[f] is a derivable sequent.
r
Proof. For rfix:
r (2) r (3) r (4) r (5) r (6) r (1)
r A(~fixyA), 6. r A(~ [x: (x, diagyA) e diagyAJ), 6. r (t: diagyA) e [xy : A(x, [x: (x, y) e yJ), 6. r (~diagyA) e diagyA, 6. r (t) e [x: (x, diagyA) e diagyA], 6. r (t) c fixyA, 6.
(df. fixyA) (rck+l)
(df. diagyA) (rck)
(df. fixyA) --l
fixr is treatedin a similarmanner, using cHI randCk r. In thesequel, we usethesetwo rules to introduce a ndeliminaterecursive definitions in our derivations . 3. TERM MODELS FOR L
3.1. The classical role of 4-valued semantics In thissectionwe specify aclassofstructures forL withinwhich we can specify models for Gandlogicalextensionsthereof.In thisclass of"termmodels" , constant termsarepositedas theirowndenotations, individuated u nderstrict syntacticidentity.Term models doprovidea convenientway ofstudyingG andits extensions,a ndare herepresentedto thatend. Ourtermmodels are based on a simplificationof thosepresentedin Gilmore 1986 forNaDSetI, although we adoptan "algebraic"n otational stylefamiliarfrom thesemanticaltreatment of programminglanguages(Scott 1975), and, more recently, thetruth theoryliterature (Gupta and Belnap 1993, Fitting 1989, 1993) . Gilmore(1986) utilizest hedevice of asigned sentence, +A, -A, to record theassignmentof Truth,respectively,Falsityto asentenceA in a model of NaDSetI. Models are thentakento be sets of signed sentenceswhich interpretation of theconnectives respecttheKleenestrong"t hree-valued" andquantifiers andthesemanticrulecorresponding torelational abstraction. Thereis a subtlety involved here . It is notpartofthedefinition of a NaDSet I model thatthe partialassignmentof classicaltruthvaluesrepresentedby
LOGIC, TRUTH AND NUMBER
17
thedistributionof + and - among thesigned sentences in t hemodel be functional ; strictly speaking, "models" may containboth + A and - A for a given sentence . It is ratherin the form of a result aboutthestructures inductively definedin Gilmore 1986 thatwe knowthatin fact every model of NaDSetI so defined assignsa tmost one of the classical valu es to a given , and thepracticeoftaking sentence. Hence, the devicethesignedsentence of models to be sets of signed sentences,is really aeryearlyrepresentation v of a four-valued semantics. Because, here we have the values of classical truth and falsitytogether w ithvalues whichrepresentthesituationof a sentence being classically undefined andclassically overdefined. Of course, theavailability of thefourthvalue,overdefined, is utilizedby level, in the construction ofthree-valued NaDSetI onlyatthemeta-theoretic models. Forexample, in theanalytic(or "reductive ") style ofcompleteness proofnatural toGentzencalculia, nested sequence of sets of signed sentences is generated by a branchingprocess thatclosesbrancheswhich come to containboth+ A and - A for someatomicsentenceA. Here, although we are onlyinterested in producingthree-valued sets of signedsentences,we do so by countenancing four-valued sets in the very definition canonical of our reductionprocess. Thepresentdevelopmentis contiguousw iththatof NaDSetI by using four-valued modelsthemeta-theoretic in process ofconstructing threevalued models . It turnsoutto bemathematically convenient to first specify theclass of four-valued models, and thendefine thethree-valued models to be thestructures which only use the three"standard"values . 3.2. Information and truth on a structure
Thefollowing pr esentation borrows fromFitting 1993, and Gupta and Belnap 1993. Let 4 bethesetoftruth values{true,jalse,..1, T} . We usually refer to T -Belnap4-valued"bi-lat tice" , as "both" . Term models are based on the Dunn consideredunderits information ordering,(4, :5k), both true
false ..1
and alsounderits alethicordering , (4, :5t), true both
..1
false
Below, theconnectivesa nd quantifiers of L are interpreted as leastupper boundoperationson (4, :5t).
PETER APOSTOLI
18
An L-structure is a pair S = (ctrm, cs) wherectrm, the domain of S, is thecollection of all closedtermsand set cs : ctrm3 -+ 4. cs is intendedto be theinterpretation thatS assigns the "pairwise membership" predicatee, and is sometimes called "membership a operator" . s trongthree -valued As mentioned, theobjectof thisstudyis the Kleene interpretation of L, based upon the set of values 3 = {true, false, .L]. We consider3 undertheinformationordering,(3, $k), false
true .i.
which gives3 thestructure of a completesemi-lattice . The Kleenestrong interpretation of theconnectivesand quantifiersis theleastupper bound operationon 3 withrespectto thecorresponding alethicordering(3, $d, true -l
false
which gives3 thestructure of acompletelattice . Notethat3 is closedunder theoperationof takingl.u.b'sandg.l.b's withrespectto (4, $t) . A 3-valued L-structure is an L-structure S = (ctrm, cs) wherecs( ctrm) ~ 3. Let A be an L sentenceand tl and tz be constantt erms of L. We now proceed to define thenotionof thetruth on an L-structure by inductionon the formation of A: DEFINITION
3.1 Truth on a Structure.
(1) S(tl = tz) = true if t 1 = tz false else; (2) S((t},tz) ct3)
(3) S(AI\B)
= cS(t3,tl,tZ);
= S(A)I\S(B), S(AVB) = S(A)VS(B) andS(-.A) =
-,S(A),
where1\ is assignedthefixedinterpretation of meet on (4 , $t), V is treated dually, and -. is assignedtheoperationwhich takestrue to false , false to true, .L to .L and T to T. Let A(x) be an L-formula containingno parameters. Then, (4) S(VxA) = I\{S(Ax[tJ) (5) S(3xA)
It E ctrm} where1\ is arbitrary meet on(4,$t).
= V{S(Ax[tJ) It E
ctrm } where V isarbitrary join on(4,$d·
LOGIC, TRUTH AND NUMBER
19
We writeA -'s B if S(A) :5k S(B) and A ...... s B iffS(A) = S(B). More usefully, we write -sA r
s
A
=df
true:5k S(A),
=df
S(A)
~k
false.
We also saythatan L sentenceA diverges on S, and writeA, if S(A) = . 1. Intuitively, S(A) = .L representsthesituationthatA is "classically undefined". We say thatA converges on S, and writeAl, otherwise . Our intentionis to take as models of G the L-structures which are3valuedand validatethe rules of binaryabstraction . Since 3 is closedunder negationand theoperationof takingleastupper and greatest lowerbounds withrespectto (4,:5t), if S is a 3-valuedL-structure, thenS(A) E 3 for all sentencesA, as can be seen byinductionon theformationof A. Hence Definition .1 3 specifies the Kleene s trongthree-valued interpretation of L when S is restricted to the class of 3-valued L-structures. We modelderivability in G upona notionofthevalidity of a sequent which a 4-valuedgeneralization ofthedefinitiongiven for NaDSet I in Gilmore 1986. This generalization is obtainedby replacingtheassignmentof true from 3 in NaDSetI semanticsby ~k true from 4. Let T I- ti. be a sequent , andletii = aI, . .. ,ak exhaustall oft heparamet ermsof tersoccurringin members ofI' or ti.. Let = tl, .. . , tk be constant L. Then fa[t] I- ti.a[t] is called an instanceof f I- ti..
r
DEFINITION 3.2. A sequentI' I- ti. is valid on S iff, for everyinstanceI" of I' and ti.' of ti., eitherS(A) = false for some A E I" , or S(A) ~t true for some A E s: An L-structure S is calledacceptable iff e I- and I- e preservevalidityon S . S is a model of G, if in additionto beingacceptable,S is 3-valued . Hence, thenotionof thevalidity of a sequent is defined over the entireclass of . Still, the rolethe of acceptable L-structures, notjustthe3-valuedstructures top element of our lattice of truth values isrestricted to thatofconstructing 3-valuedL-structures (in particular, models of G). It is routineto checkthatthe axioms sequentsof G are valid on any L-structure S, andthatt herules of G,otherthancl-and l-s, preserve validity on S . Hence we have our versionthesoundnessresult of for NaDSetI: THEOREM 3.3 Soundnessof G . All sequents which are derivable in G are valid on every acceptable L-structure. The rest of this sectionpresentssome standard definitions and results from thetruththeoryliterature . Here especially , our presentation benefits from Fitting 1993:
20
PETER APOSTOL!
DEFINITION 3.4. Let e), e2 : ctnn3 ~ 4. We put e) 5k e2 iffe) (t), t2, t3) 5k e2(t),t2,t3)' Let 8) and 8 2 be L-structures.T hen, we put 8) 5k 8 2 iff es, 5k es; Note that>..xyz E ctrm. 1-, which is arepresentation of the "empty" membership operator,is 5k minimal amongthemembership operators . The space ofL-structures is a completelatticeu nder5ki thespace of 3-valued L-structures is a completesemi-latticeu nder5k. We letAo denote(ctnn, >..xyz.1-), the5k minimal L-structure. DEFINITION 3.5. Given L-structures 8 1 and 82 , defineeS I V es z : ctnn3 ~ 4 pointwise in(4,5k) by
Thenthejoin8 1 V 8 2 of 8 1 and 8 2 in the5k orderingis theL-structure (ctnn, ss. V es z)' LEMMA 3.6. Let 8 1 and 8 2 be L-etruciures. If 8 1 5k 8 2, then 8 1 (A ) 8 2(A) for all sentences A .
s,
Proof. By inductionon theformation ofA, using themonotonicityof the sentential connectivesa ndquantifiers fortheinductionstep. -I
DEFINITION 3.7 The GilmoreRevisionOperator.For eacheo : ctnn3 ~ 4, we may define«1>(eo) : ctnn3 ~ 4 as follows : For allconstantterms [xy : A],t),t2,
«1> is calledthe Gilmore Revision Operator. Then, for eachL-structure 8, let «1>(8) be theL-structure (ctnn, «1>(es)). Trivially, if 8 is 3-valued,t henso is «1>(8). THEOREM 3.8 Monotonicity o f CP. Let 8) and 8 2 be L-structures. Then
s,
s,
Proof. Suppose 8 1 8 2 , i.e., eS I esz ' Then by Lemma 3.6, (ctnn,es,)(B) 5k (ctnn,eSz)(B) for allsentencesB , so by definition«1>, 3.7, «I>(es,)(t),t2,t3) 5k «1>(es z)([xy: A(x ,y)],t),t2) for all formulas A(x ,y) and constant t ermst), ta- It followst hat
-I
21
LOGIC, TRUTH AND NUMBER THEOREM 3.9. There is an acceptable 3-valued L-structure.
Proof. By Theorem3.8, «I> is monotonewith respectto thecompletesemilatticet hatthespace of 3-valuedL-structures forms under$k. Hence, by theKnasterTarskilemma, thereis a least(and also agreatest)fixed point A of «1>. Its easy to check thatfor all formulas A(x, y)
whence101- and 1-10 preserve validity on A.
-j
Hence,thereis a modelA of G. It followst hatG is syntactically consistent , 0 I- 0, is not derivable . Suppose in the sensethatthe empty sequent o I- 0 is a theoremof G; letA be anysentenceof L. Thenby Thinningboth I- A and I- ..., A are theorems,whencebothare valid on A by 3.9. Hence, false $t A,..., A, andso A(A) = A(...,A) = both, contrary to thefactthatA is 3-valued. In theliterature, e.g., Kripke 1975, Gilmore 1984 and Feferman 1984, it is theleast fixed pointof a revisionoperator"from usual toexplicitly c onstruct below" , as the$ k extensionof (ctrm, AXYZ •..L) obtainedby closingthatinitial structure offunder«I> by ordinalrecursion . We specify aslightly more general construction which will be of use in modeling logical extensionsof G. DEFINITION 3.10. LetS be anL-structure. We construct an ordinal sequence I Q $ wI} by inductionas follows:
{S "
o. 1. 2.
So =S, S"+1 = S" V «I>(S,,), S" = V S{3 for limitordinalsQ . /3 <"
Let A WI
=df
.Awl' where.Ao =
=> S{3(A)
(ctrm, AXYZ • ..L). Note it obvioust hat{3 $ $k S,,(A) for all sentences A.
Q
$
OBSERVATION 3.11. A is theleast fixed pointoftheGilmore revision o perator «1>.
Suppose we are given a sequence L-structures of {S" I Q $ wd as per Definition 3.10 . For each sentence B thatconverges inSWI we definethe Q such thatB converges on S" . level of Bin {S" I Q $ wd to be the least For eachsentenceB thatconverges on A, we definethe level of B (simpliciter) to be the level B ofin the canonical sequence {A" I Q $ WI} ' In contexts where{S" I Q $ wd is given, we write simply --+" for-rs; andsimilarly for cognatenotation.
22
PETER APOSTOLI 3.3. Modeling logical extensions of G
Let S be an acceptableL -structure . If tl is a constant t erm,we say that8 is tl-total iff -+S'tXY((X,Y)etl V--'(X,Y) e tl) ' Hence, S is h-total justin case for all constant termsta, h, either-+S(t 2, t3)e tl or '-5 (t2, t3) e tl' Then it is clearthata tl-total L-structure 8 validates all bivalence sequentsoftheform
(S2' S3) e tl f- (S2 , S3) etl for closedtermsSI and S2. Hence, atl-total L-structure provides a model of thelogicalextensionsof G obtainedby addingtheconstant t ermtl as a new groundpredicate. In section7, for eachprimitiverecursivefunctionf, we introduce c onstant terms tf computingf and show thatA is trtotal. Hence, all"primitive recursiveterms", i.e., those oft heform t f for someprimitiverecursivef, may be consistently adjoinedas newgroundatomsto G. In particular , in sections 8 and 9 westudythe logical extensionG+ 'x of G obtainedby adjoining bivalence axiomstreatingt + and t x, theconstants et terms representing numbers,as newgroundatoms. additionand multiplication overthenatural 4. SOME SEMANTIC RECURSION THEOREMS
4.1. Recursion introduction and elimination The purpose of thissectionis to establishsemanticversions of the derived rulesekf-, f-ek andconsequently the derivedrecursionrulesfixyf- and f-fix y. Let S = 8 0 be an L-structure and {So I a ::; wd be theordinalsequence of extensionsof 8 obtainedby Definition3.10. Let B be a sentenceof the form (f) e [x: A(x)], for someconstant t ermf = t l , • •• , tk. Ourfirsttaskis to see that
simulatesk for all k ~ 2: LEMMA 4.1. LEMMA
So(A(f))
= (So)((f) e [x: A(x)])
4.2 Abstraction . For all a ::; WI , (a) (b)
-+5 A(f) 0
'-5 A(f) 0
=> =>
-+So+!(f) e [x : A(x)) , '-So+! (f) e [x : A(x)] .
Proof. By Lemma 4.1, using Definition3.10. LEMMA
--I
4.3 RecursionIntroduction . For all a ::; WI, (a) (b)
-+5 A(t: fix yA) 0
'-5 A(t:fixyA) 0
=> =>
-+SO+2(f) efixyA , '-5
0
+2
(f) efixyA.
Proof. By a "semantic"version ofAndrew's recursiontheorem.
23
LOGIC, TRUTH AND NUMBER Now werepresenttheconverse ofa bstraction and recursionrules. LEMMA
{So I 0:
4.4 Conversion.Suppose the level of B =dj (t) I:: [x : A(x)] in is 0: for some ordinal 0: < WI such that 0: =1= O. Then
::; WI}
So (t)
I:: [x:
A(x)]) = 5 0 - 1 (A(t)).
Proof Since 0: is a successorordinal,So 3.10. So by Definition3.5, S o(B)
=
1:: 0 - 1 ([x
= S o-I V
o-d by construction
: A(x)], (tl' . .. , tk-I), tk) V
By hypothesis1:: 0
- 1
([x: A(x)], (tl ,".' tk- I), tk) = .l. Hence
as claimed.
4.5 RecursionElimination . Let A = A(x, y) be a formula. Suppose the level of (t) I:: fixyA in {50 I 0: ::; wI} is a for some ordinal 0: , 2::; 0: < WI . Then LEMMA
Proof Recall(t) I:: fixyA = df (t) I:: [x : (x diagyA) I:: diagyA]. Since thelevel of (t) I:: fixyA is 0:, thelevel of(tdiagyA) I:: diagyA is 0: - 1. Then, similarly , thelevelof A(t:fixyA) is 0: - 2. Then we have S o(t) I:: fixyA) = S o(nI:: [x : (x diagyA) I:: diagyA]) = So-I (t: diagyA) I:: [xy : A(x, [x : (xy) I:: y])]) = So-2(A(t: [x : (x diagyA) I:: diagyA])) = SO-2(A(t:fixyA))
(df. fixyA) (4.4, df. diagyA) (4.4)
as required.
(df. fixyA) -1
4.2. A finite level result for positive sent ences A formulaA is positive justin case A's onlyconnectivesare ...." 1\, V and 3, and alloccurrences of ...., prefixidentitystatements in A. Ournextresult ensuresthatpositivesentenceswhich are verified in thecanonicalmodel A are of finite level. In ectsion6, we show thatpositiveformulasare sufficient forcomputational purposes.
PETER APOSTOL!
24
4.6. Let A be a positive sentence. Suppose -+oA for some Then A is of finite level.
LEMMA
Q
S
WI.
Proof By inductionon Q. 5. THE GENESIS OF NUMBER IN G
5.1. The theory of (0, S)
In thissection,we use Andrews'recursiontheoremto construct a constant theset ofnatural numbersin G. First,we define setterm~ whichenumerates
Q=
[V I : VI
f. VI],
St =
[VI: VI
= t]
(t E ctrm).
Thatis, the natural number0 is represented by thetermfortheempty set, by theoperationoftakingsingleton and thesuccessorfunctionis represented sets. Let Num, theset of Gilmore numerals, be thesmallestsubsetof ctrm thatcontainsQ and containsSt whenever it containst hetermt. Write11 for t ermcorresponding to thenatural numbern. snQ (n E IN); 11 is theconstant The following derivation of thetheoryof 0 and thesuccessorfunctionfollows thatof Gilmore(privatecorrespondence) forNaDSetI:
AI : \Ix (Sx f. Q) A2:
\lXI X2 (SXI
LEMMA
("0 is not the successor of any number") , =
SX2 --> XI
5.1. f-\Ix (Sx
=
X2)
("Successoris 1 - I").
f. Q) .
Proof
(1) f-a=a (2) f- a e [VI : VI = a] (3) f- a e Sa (4) a f. a f(5) a e [VI : VI f. VI] f(6) aeQf(7) Sa = Q f(8) f- Sa f. Q (9) f-\Ix (Sx f. Q) LEMMA
Proof
5.2. f- \lXI
X 2 (SXI
(3, 5, by =f-)
= SX2 --> Xl = X2).
LOGIC, TRUTH AND NUMBER
(2) az e [V1 : V1 = al]1-al = az (3) az e Sal I- a1 = az (4) I- az e Saz (5) Sal = Sa2 I- a1 = a2 (6) I- Sal = Saz ---+ a1 = az (7) I- 'VX1 Xz (SXl = Sxz ---+ Xl = xz)
25
(3, 4, by =1-) -l
COROLLARY 5.3 . If m =f:. n, then m = !!: I- (n E IN).
5.2. The natural numbers object DEFINITION 5.4. Let I' ~ ctrmk and t be a constant k -aryabstraction t erm. We say thatt enumerates r (in G) iff
r = {(51,... , 5k) II- (51 . .. 5k) e t is derivable}. We say thatr is G-enumerable iff thereis a constantk-ary abstraction term t such thatt enumeratesr. We say thatS ~ INk is G-enumerable iff {(n1 ... nk) I (nl, . . . , nk) E S} is G-enumerable . DEFINITION 5.5 The Natural N umbers Object. Let A(x, y) be theformula X = Q V 3z (x = Sz 1\ z e y),
andlet!:!l be theset termfixyA. The main goal ofthissectionis to showthat!:!l enumeratesIN in G. However,since!:!lis ourfirstexampleof asyntactically complexgroundpredicate , andournotionof grounding is semantically motivated,we firstestablishthat !:!l representsa "total"s etin A . LEMMA 5.6. ->Z(n+l)!!: e!:!l
(n E IN) .
Proof. By inductionon n. For thebasis, letn
(df. 3.1) (df. 3.1)
(1) ->oQ = Q (2) ->oQ = Q V 3z (x = Sz 1\ ze!:!l) (3) ->zQe!:!l
For theinductionstep, assume n = k k E IN. Thenwe have
(1) (2) (3) (4)
->z(k+1)1£e!:!l ->oS1£ = S1£ ->z(k+1)S1£= S1£ 1\ 1£ e!:!l ->Z(k+1)3z (S1£ = Sz 1\ z e!:!l)
= 0:
(4.3)
+
1 and that->z(k+l)1£ e!:!l for some
(assumption) (df. 3.1) (1,2, df. 3.1) (3, df. 3.1)
26
PETER APOSTOLI (4, df. 3.1) (4, 4.3, df. !!l)
(5) -+2(k+l)Slf. = Q V 3z (Slf. = Sz /\ z c!!l) (6) -+2(k+l)+2Slf. c!!l (7) -+2(k+2)11 c!!l
Hence, -+2(n+l)11c!!l, as requiredto completetheproofof Lemma 5.6. -l LEMMA 5.7. +-
A
Let tl be a constant term which is not a Gilmore numeml. Then
t 1 c!!l .
Proof Let the mnk, R(t) , of a term t be the maximum n ~ 0 such that sn s forsome term s. Let R(tl) = n; we show by inductionon n that
t =
+-2n
-l
tlc!!l.
COROLLARY 5.8. -+w't/x (x c!!l V --, xc!!l). THEOREM 5.9 The Omega Theorem. !!l
enumemtes IN.
Proof It suffices to showthat :
(1) I-llc!!l (nEIN),and (2) I- t c!!l => t E Num
(for allt ermst).
(2) follows fromLemma 5.7 and thesoundnessof G withrespectto A: Let t rf. Num be a term. Then t has aconstantinstancet' which is likewisenot a numeral.By 5.7, +- A t' c~; since A is 3-valued , it follows thatit is not thecase that-+At' c~. Hence, bysoundness,I- t' c!!l is notderivablein G. Since theoremsof G areclosedundertheuniformsubstitution of arbitrary closedtermsforparameters,it follows thatI- t c!!l is nota theoremof G, as required. The proofof (1) is by inductionon n . For n = 0, we have
(= ax.)
(l)I-Q=Q (2) I- Q = Q V 3z (Q = Sz /\ z c~) (3) I- Qc~
For theinductionstep, assume n = k Thenwe have
(1, I-V) (rec. thm.)
+
1 and thatI- If. e ~ has beenderived.
(1) I- If.c~
(2) I- Slf. = Slf. (3) I- Slf. = Slf./\lf. c~ (4) I- 3z (Slf. = Sz /\ ZC!!l)
(5) I- Slf. = Q V 3z (Slf. = Sz /\ zc~) (6)I-Slf.c~
It is time to considerourfirst logical extensionof G.
(assumption) (= axiom) (2, 3, I- /\)
(1-3) (v 1-) (rec. thm.) -l
27
LOGIC, TRUTH AND NUMBER 6. GW AND A DERIVATION OF THE AXIOM OF INFINITY
6.1. The axiom of infinity The purposeof thissectionis to drawattention toa connectionin a logical extensionof G betweentheprincipleof bivalencefor atomicsentencesfeaturing~ as a predicatetermand theAxiom of Infinity , assertingthat~ is in shortorder. a O-successorset. Roughlyput, theformerentailst helatter This connectionis surprising,since bivalenceis oftentakenas theparadigm of a logical principle, whereascriticsof logicism have takentheAxiom of Infinityto beunarguably extra logical. The main pointof thissectionis that thecoherenceof thedistinctionbetweenthe"logical"and the"extralogical" may be hardertodefendthanpreviouslythought . Let GW be thelogicalextensionof G obtainedby adjoiningas new bivalenceaxioms allsequentsof theform tc~l-tc~
for closedterms t . The idea here isthat,whileG is sufficientto compute all r.e . sets, it is tooweak toreason aboutthesets it computes. By adding bivalenceaxioms for atomic formulaswhose predicatetermis ~, we extend G's abilityto reasonaboutthefirstinfinitecollection. In particular , GW can provethat~ is, in fact, a O-successor set. THEOREM 6.1 The Axiom of Infinity. I- 3x
(Q s
z 1\ Vy(y e z
-t
Sy e x)) is
derivable in GW • Proof It suffices toderiveI- Q e ~ 1\ Vy(y e ~ - t Sy e ~). (1) pc~l-pc~ (2) p e ~ I- Sp = Sp 1\P e ~
(3) pc~ 1-3z(Sp = S z 1\ zc~) (4) pc~ I- Sp = Q V 3z(Sp = Sz 1\ zc~) (5) pc~1- SPE~ (6) I- pc~ - t Spc~ (7) I- Vy(y e ~ - tSy c ~) (8) I- Qc ~ (9) I- Qe ~ 1\ Vy(y e ~ - t Sy c~)
(GWaxiom)
(= axiom, 1-1\) (1-3) (I-V)
(rec. thm.) (I--t)
(I-V) (omega thm.)
(7,8,1-1\) -l
as required .
6.2. Semantics for GW We needonlyobservethatA is a modelof GW to seethatGW is consistent. THEOREM 6.2. GW is consistent.
Proof By 5.9, A is an w-totalL -structure. Hence, A is a model of GW •
-l
28
PETER APOSTOLI 7. HOW TO COMPUTE IN G
Before we can explore thespace of logical extensionsof G, we need to returnto G and finish an i mportant,t houghpedestrian , logicist task. We need to showthereaderhow tocomputein G. We associatea positive set abstraction termwith eachprimitiverecursivefunctionof thenatural numbers. Intuitively, existential quantification is used to effect the composition of primitiverecursive functions , and My is used tosimulateprimitiverecursion. Then,theexistential quantifier willsimulateunboundedsearch over IN to yieldrepresentations of allpartialrecursivefunctions . Let [x f:!!:l : A] abbreviatethe setterm [x : Xl f:!!:l/\ ... /\ Xl f:!!:l/\ A]. DEFINITION 7.1. We associatewith eachprimitive recursive function I : k k+l -ary set abstraction t erm t f' intendedto enumerate -... IN a constant
lN
thegraphof I, by inductionon theformationhistoryof I:
InitialFunctions : .Am.O . Then tf is thebinaryabstraction (1) Suppose I is the O-function term [x y : X f:!!:l/\ Y = Q]. (2) Suppose I is thesuccessorfunction.Am . m term [xy: X f:!!:l/\ Y = Sx].
+
1. Then t f is thebinary
(3) Suppose I is the projectionfunction.Ami, .' . , m k - mi for some k ~ 0 and i , 1 :5 i :5k . Then tf is thek+l-aryterm [Xl " .Xk f:!!:l: Y = Xi] ' Composition: (4) Let k ~ 0, n > O. Suppose h : lN n -+ IN and gi : lN k -+ IN areprimitive recursive functions (1:5 i :5 n) and I is .Amh(gl(m), ... ,gk(m)) , where m = ml, ,mk . Then t f is thek+l-aryterm
[xy: (3zl
zn){(X,Zt) el g ,
/\ ••• /\
(x, zn) e t gn
/\ (Zl . .
, zn,Y) eld]·
PrimitiveRecursion : (5) Suppose 9 : lN k - 1 -+ IN and h : lNk+ I : lNk -+ IN is given by
1
-...
IN areprimitiverecursive and
1(0, m2, ' . . ,mk) = g(m2 , '" ,mk),
IU + 1, m2,··
· , mk) = h(j, IU, m2 , . . ·, mk), m2,· · ·, mk,) .
Inductively , we have k -ary,k+2-ary,termst g , th representing g, h, respectively. Let x = Xl , . . • , Xk. Let A(x, y, z) be theformula (X l
= Q/\ (X2 .. . Xk , y) el g ) .V.
3uv (Xl = Su /\ (UX2 " ,Xk ,V) e Z /\ (UVX2" . Xk, Y) e th).
Then tf is thek+l-aryset term fixzA(x, y, z).
29
LOGIC, TRUTH AND NUMBER THEOREM
7.2. Let
1 : lNk
-+
IN be primitive recursive and
E IN. Then
m=
mI , .. . , mk
I- (mI.'. mk, I(m)) e t]
is derivable. Proof By inductionon theformationhistoryof I, using aninnerinduction on mi to establishthe primitiverecursioncase of theinductionstep. -l
7.3. Let 1 : lNk -+ IN be primitive recursive. Let tl, '." tk, s be constant terms such that it is not the case that there are mI, . . . , mk, n s E IN such that t, = mi (0 $ i $k),S = n s and l(ml, . . . ,mk) = n s • Then +-A(tl . . . tk,S)etf · THEOREM
Proof By inductionon theformationhistoryof f. The primitiverecursion case oftheinductionstep is establishedon the basis of the following three is by inductionon nl: claims, thefirst of which established
Claim 1: Let nl,... , nk, nk+1 E IN withl(nl, . . . ,nd =IThen +- (nl nk,nk+detJ . Claim 2:
+-
tl = Q 1\ (t2
Claim 3:
+-
3uv (ti
tk, s) e t g •
= Su 1\ (ut2 ' . . tk, v)
7.4. Let terms such that
COROLLARY
nk+l.
1 : lNk
-+
e t] 1\ (uv t2 . . . tk , s) cth) .
-l
IN be primitive recursive. Let tl, . . . , tk, s be
I- (t1 .. . tk,S)ctJ .
Then there are ml, ' .. ' mk, n s E IN such that ti l(ml, . . . ,mk)=ns .
= m i (0 $
i $ k), s = lis and
-
Proof Suppose thereare noml, . . . , mk, n s E IN such thatt, = m i (0 $ = n s • Then thereare constantinstances t~ , .. . , t~, s'Of h, . . . , tk, s which likewise fail theconclusion of the Corollary . Then,by 7.3, +- A (t~ . . . t~, s') et J' whence-+ A (t~ . .. t~, s') et J fails sinceA is 3-valued. Thenby soundness,I- (t~ . . . t~, s')etJ is not atheoremof G. Since of closedtermsfor theoremsof G are closed u ndertheuniformsubstitution parameters,it follows thatI- (t i . . . tk, s) e tJ is not atheoremof G either . -l
i $k), s = n s and l(ml, . . . , mk)
COROLLARY
ates
1 in G.
7.5. Let
1 : lNk
-+
IN be primitive recursive. Then tf enumer-
Proof By Theorem7.2 and Corollary 7.4. THEOREM
7.6. All r.e. sets 01 numbers are G enumerable.
30
PETER APOSTOLI
Proof Let P : INk ---> {O, I} be a primitiverecursivepredicateand tp is a k-t1-aryabstractt erm which enumeratest hegraphof P, as given by 7.5. Thenther.e. predicate {ml, . . . ,mk-l : 3nP(ml, ... , m k- l , n ) = O} is enumeratedby the(k - 1)-aryabstract t erm
[Xl"
. Xk- l : 3y
((Xl "
.Xk-l y.Q) e tp))].
Since every .e. r set ofnumberscan be obtainedas theexistential projection of theprimitiverecursiveKleeneT-predicate,t hisis sufficient. -l COROLLARY
7.7. Let
f : INk ---> IN be primitive recursive. Then ---+A'VX(xctf v.xctf) ·
Proof By 7.3 and 7.6. 8. TOWARDS A LOGICAL DERIVATION OF THE THEORY OF (0, S,
+,
x)
8.1. The logical extension G+'x We pass to thelogicalextensionG+'x by adjoiningto GW excludedmiddle axiom sequentsfortheset termsrepresenting theadditionandmultiplication n umbers. AlthoughDefinition7.1 provides set functionsover the natural termswhichenumeratet hegraphsof thosefunctions,in thissectionwe must reasonformally over thecontentof theseterms, so it isconvenientto define them more simply, afresh. Let t+, t x be fixuB+(x, y, z, u) and fixuB x (x, y, z, u),
respectively, where B+(x,y,z ,u) = df x = Q1\ Y = z 1\ Y e ~ . v. 3x' z' (x = Sx' 1\ z = Sz' 1\ (x' Y z') e u) ,
B X(x, y, z, u) =df X = Q 1\ z = Q 1\ Y e ~ . v. 3x' v (x = S x'
1\
(v Y z) e t+ 1\ (x' Y v) e u) .
Let G+'x be thelogicalextensionof GW obtainedby adjoiningas new bivalence axioms all sequentsof theform
for closedterms tl,tz, ts- So in G+ 'x , t+ and t x countas new groundpredicates. The reason whythesearithmeticset termsare well b ehavedin their new role ist hattheyhavethesyntactic s tructure of primitiverecursivedefinitions. Infinitedescendingreductionsequencesare ruledo utbecauseeach
LOGIC, TRUTH AND NUMBER
31
applicat ion of saemanticrule inthesemanticevaluation process monotonithesyntacticcomplexityof thesentencesrequiring evaluation. cally reduces However, absentfrom theabove definitionsof B+ and BX are conjuncts specified in 7.1 forprimitiverecursivetermswhichrestrict t hevariables"z", "y" and "z" and thequantifier"3v" to~. These conjunctsare requiredif t+, t x are tobear the falsification conditionsassertedin 7.3 for primitive recursiveterms,which underlie7.7. The consistencyof G+'x willtherefore of A in 3.1 toobtaina model in be established by modifyingtheconstruction whicht+ , txaretotal a fterall. In effect, we t radesimplicityin thestatement and derivation of thePeanoAxioms for aslightly more complexconsistency proof. Let us pause to reflect upon developmentsso far. We've seenthatG's bivalence axioms for identitystatements underwrite a "classical fragment"of L containing all closed formulas whose atomicconstituents are restricted to identitystatements.On thisbasis, G yields thetheoryof zero andsuccessor and also the Omega theoremstatingthat~ enumeratest heset of natural numbers. Thus, it is a metatheoretic factaboutG that~ is infinite. By ~ as predicate,and addingbivalence axioms for all atomicformulasfeaturing formulaswhose thusextendingt heclassicalfragmentof L to include all closed atomicconstituents feature ~ as predicate , GW embeds thisfact intheform a section,we further theoremwhichassertsthat~ is a O-successor set. In this sentenceswith"arithmetic extendtheclassicalfragmentof L to include all syntactic structure" by adjoiningbivalence axioms for atomicformulas having t+ or t x as predicateterms. We thusobtaina logicalextensionG+ ' x that underwrites the bivalence ofsublanguage a of L containing a relational image of thearithmeticlanguagebased on 0, S, + and x. This sublanguage --the so called"L fragment"-will be explicitlydefined inthenextsection. For now however , it is enough to note: 8.1. Let A be a closed L formula all of whose atomic subf orm ulas are either identity statements, or € statements with one of~, t+ or t x in predicate position. Then, A l- A is derivable in G+ 'x .
LEMMA
Proof. By inductionon theformationof A . The proof is formally identical Gentzencalculus LK can bebasedon sequents totheproofthatthe classical -I of the formA f- A for atomic A.
Thus, Lemma 8.1 justifiestheapplications of fixyf- to formulas constructed from G+'x groundatoms. 8.2. Axioms for
+
and
X
We presentthe following "relational " versions ofthePeano axioms which specify therecursivedefinitionsof additionand multiplication overthenaturalnumbers:
32 A3. A4. A5. A6.
PETER APOSTOLI V'x(Qx,x)et+. V'xyz«Sxy,Sz)et+ ...... (xy,z)et+)) .
V'x(Qx,Q)etx. V'xyz«Sxy,z)et x ...... (3v)«vy,z)et+A(xy,v)et x )) .
THEOREM 8.2. A3-A6 are derivable in C+ ,x . Proof The sequentsI- V'x(Qx ,x) e t+ and I-v« (Qx,Q) e t x are clearly derivable in C. So it suffices to show thatwe can derive A4a nd A6. Notethat the justification for applicationsof I-fix y and fix y I- belowsimply refer to t 3,t4)", "the recursion theorem ". Also, we also use prefix notation,"tl (t2 suppressing"s", in contextswhere space is at paremium:
A4.1: V'xyz (Sxy ,Sz) et+- (xy,z)et+) . (1) (2) (3) (4) (5) (6) (7) (8)
a = a', e = e', (a' b, e') e t; I- (ab, e) et+ (C+,x ax, =1-) a = a' A e = e' , (a' b,c') e t+ I- (ab ,e) e t+ (1, AI-) Sa = Sa' A Se = Se' I- a = a' A c = c' (A2,I-A) Sa = Sa' A Se = Se' A (a' b, e') e t+ I- (ab, e) e t+ (3, 2, Cut) 3x' z' (Sa = Sx' A Se = S z' A (x' b, z') e t+) I- (ab,e) e t+ (4,31-) Sa = Q I(AI) Sa=QAb=ee~1-
(6, AI-)
Sa = QAb = ee~ .V. 3x' z'(Sa = Sx' A Se = Sz' A (x' b,z') et+) I- (ab,e) e t; (5,7, VI-) (9) (Sab,Se) e e., I- (ab,e) e t; (8, rec. thm.) (10) I- (Sab, Se) e t+ - (ab, e) et+ (9, -I-) (11) l-V'xyz«Sxy,SZ)et+ - (xy ,z)et+) (1O,1-V')
Line (1) is obtainedby repeatedapplicationsof =1- takingt+ bivalenceaxioms as premises. Note the use of A2 atline 3. A4.2: V'xyz (xy, z) et+- (Sxy , Sz) e t+). (1) (ab , e) e t+ I- Sa = Sa A Se = S e A (ab, e) e t+ (C+,x ax, = ax, I-A) (2) (ab , e) e t+ I- 3x' z' (Sa = Sx' A S e = Sz' A (x' b, z') e t+) (1,1-3) (3) (ab,e) et+I- Sa = QAb = Sc e c; .V. 3x' z'(Sa=Sx'ASe=Sz'A(x'b,z')et+) (2, I-v) (3, rec.thm.) (4) (ab,e)et+ I- (Sab,Se)et+ (4, -I-) (5) I- (ab, e) e t+ - (Sab, Se) e t+ (5, I- V') (6) l-V'xyz«xy,Z)et+ - (Sxy,Sz)et+)
A6.1: V'xyz (Sxy,z) e t ; - (3v)«vy ,Z)et+A(Xy,v)et x ) ) . (1) a = a' , (d b, e) et+A (a' b, d) e t x I- (d b, e) e t; A (a b, d) e t x (8.1, =1-)
LOGIC, TRUTH AND NUMBER
33
(2) Sa = Sa' f- a = a' (A2) (3) Sa = Sa'l\ (db, c) e t.; 1\ (a' b, d) e t x f- (db , c) e t+ 1\ (ab, d) e t ; (1,2, Cut,f-I\) (4) Sa = Sa'l\ (db, c) e t.; 1\ (a' b, d) e t x f- 3v ((vb, c) ct+1\ (a b,v) e tx) (3, f-3) (5) 3x' v (Sa = SX'I\ (v b, c)et+ 1\ (x' b, v)et x) f- 3v ((vb , c)et+ 1\ (a b, v)ct x) (4,3f-) (6) Sa = Q f(AI) (7) Sa = Q 1\ c = Q 1\ b ew. f(6, I\f-) (8) Sa=Ql\c=Ql\bel!!. .V. 3x'v(Sa = Sx' I\t+(vb,c)l\tx(x'b,v)) f- 3v(t+(vb,c) 1\ tx(ab,v)) (4,6, Vf-) (9) (Sab,c)et x f-3v(vb,c)et+l\(ab,v)et x) (8,rec.thm.) (10) f-(Sab,c)et x -+3v((vb,c)et+l\(ab,v)et x) (9,f--+) (11) f-Vxyz((Sxy,z)ct x -+(3v)((vy,z)et+l\(xy,v)et x)) (lO,f-V)
A6.2: Vxyz ((3 v)((vy, z) ct+1\ (xy,v) ct x) -+ (Sxy, z) ct x).
(1) (db, c) e t+ 1\ (ab, d) e t x f- Sa = Sa 1\ (db, c) e t+ 1\ (ab, d) e t x
(8.1, = ax, f-I\) (2) (db, c) ct+1\ (ab, d) e t ; f- 3x'v (Sa = Sx' 1\ (vb, c) e t; 1\ (x' b, v) ct x ) (1, f-3) (3) 3v ((vb , c)et+ 1\ (a b, V)et x ) f- 3x'v (Sa = SX'I\ (v b, c)et+ 1\ (x' b, v)ct x) (2, 3f-) (4) 3v(t+(vb,c) 1\ tx(ab,v)) I- Sa = QI\C = QI\ bel!!..V. (3, I-V) 3x'v (Sa = Sx' 1\ t+(v b,c) 1\ t x (x' b, v)) (4, rec. thm.) (5) 3v((vb,c)et+l\(ab,v)et x) I- (Sab,c)ct x (5, f--+) (6) I- 3v ((vb, c) e t+ 1\ (ab, v) e t x) -+ (Sab, c) e t.; (6, f-V) (7) f-V'xyz((3v)((vy,Z)et+ 1\ (xy,v)ct x) -+ (Sxy,z)et x)
This concludest heproofof Theorem8.2. We must now showthatG+ ' x is not too powerful! 9. SEMANTICS FOR G+'x
9.1. The semantics of grounding in models of G+'x
By 7.7 we knowthatA is "almost" a model of G+'x. Thus, we willtakethe opportunity to extendA to a modelB of G+'x . S is +, x-total iff S is I!!.-total,t+-total and We say thatan L-structure tx-total. Evidently,+, x-total L -structures are models of G+ ' x. Since ourpresentdefinitionof t+ and t« omit some of theclausesrestricting thevariables"x" , "y", "z" and "v" to I!!. as per 7.1, thesetermsare not guaranteed to be false of triplesof non-numerals in A. They are, however ,
PETER APOSTOLI
34
false of alltriplesof numeralsrepresenting triplesof numbers not belong-
ing tothegraphsof theirrespectivenumber-theoretic functions. Hence , to construct an +, x-totalL-structure, it suffices toc onstruct an extensionof A on whicht+ and t x are "forced" to be false of alltriplescontaining nonnumerals. 9.2. Consistency We extendtheconstruction of A in Definition3.10 toproducean acceptable L-structure B ~ k A which holdst+ and t x false of allconstant t ermswhich are not Gilmore numerals.T henwe give areductivea rgumentshowingthat the resulting acceptableL-structure B is 3-valued . This argumentcan be viewed asanalogous to a"semantic"version of acuteliminationargument. DEFINITION 9.1. Defineeo : ctrm3 -+ 4 by
eo(tl, ta, t3) = false
if atleast one of thefollowing conditionshold:
(1) tl = t+ andit is notthecase thatthereare m,n,j E IN suchthat tz = (m, n) and t3 = i,
= t x and it is not the case thatthereare m, n,j E IN suchthat t2 = (m, n) and t3 = i;
(2) tl
and eo(tl, ta, t3) = 1- otherwise . Let B o =df (ctrm, eo). Define anordinalsequence {B o structures as per 3.10.
I a < wd
of L-
THEOREM 9.2. Ba(tl e f!:1) ¥ true for all constant terms tl which are not Gilmore numerals (a :<:; Wl) . inductionon a. Suppose, towardsa contradiction, that Proof. By transfinite Ba(tl e f!:1) = true for some constantt erm tl, and leta be theleast such ordinal.Evidently , a is the level of tle f!:1. Since a is a successorordinal, Bo = Bo-l V!P(Bo-d by construction 3.10. So by Definition 3.5,
By hypothesis,e o-l(f!:1, tl, td¥ true. Hence, true =
Bo(tle f!:1)
= !P(eo-d(f!:1,t l, td = (by df. !P) B
O -
l(tl'diagyA) e diagyA) ,
where A is theformula"x = Q V 3z(x = Sz 1\ z e y)", since f!:1 =df [xv: (x, diagyA) e diagyA] . It follows thata - 1 > 0, since B o makes nos-sentence
LOGIC, TRUTH AND NUMBER
35
true. Also, sincea is thelevel oft hesentencetle!:!!., a-I is thelevel of (t2' diagyA) e diagyA. So by 4.4 we have: 8 o - 2(t2 =QV3Z(t2 = SZ!\Ze!:!!.)) = true, but,since tz
f/. Num,
t2 =I- Q, whence8(t2 =I- Q) =false, so
8 o - 2(3z(t2 = Sz!\ze!:!!.)) = true. Then thereis an s E ctrm such that8 0 - 2(t2 = Ss !\ s e!:!!.) = true. Hence, t2 = Ss and8 0 - 2(s e!:!!.) = true. By theleastordinalassumption, s E Num, so t2 is also anumeral,c ontrary to theoriginalassumption. -1 THEOREM
9.3. a is coherent (a:::; wd.
Proof. By transfinite inductionon a :::; WI . Suppose, towardsa contradiction, thata is notcoherent; we may assumethata is theleastordinalwhich is not coherent, and hence, byconstruction, thata = {3+ 1 for somecoherent ordinal (3 < WI· Thenthereareconstant termstt,t z, t3 suchthate o(tl,t2, t3) = both. Butby construction, eo = e/3 V cI>(e/3). Since (3 is coherent , therange ofcI>(e/3) is containedin 3. Since neithere/3 nor cI>(e/3) assign thevalueboth, it follows that (a) e/3b(tl,t2,t3) = false (true)
and (b) cI>(e/3)(tl,t2,t 3) = true (false).
Let tl be written[xy : CI for some formula C. So (b) yields (c) cI>(e/3)(tI,ta, t3) = 8/3(Cxy[t2, t3]) = true (false) by definitionof cI>. With(a) in mind, let'"Y:::; (3 be the least ordinalsuch that e')'(tl,t2,t3) = false (true). We separatetwo cases: Case 1: '"Y> O. Thene')'(tl, t2, t3) = 8')'-1 (Cxy[t2, t3]) = false (true), contrary to (c), since8')'-1 :::;k 8/3 by construction . Case 2: '"Y = O. Then by definition of eo, 9.1 above, e')'(tl, ia, t3) = false, whencecI>(e/3)(tl,ta , t3) = true, and thereare two cases , correspondingto clauses 1and2. Case 2.1: Clause 1 holds. Thentl = t+ and it is not the case thatthereare m, n , j E IN such thatt2 = (m,11) and t3 = i. Recallthat
t+ =fixuB+(x,y,z,u) = [xyz: (xyzdiaguB+)e diaguB+], whereB+(x,y,z,u) =df x = Q!\ y = z r. y e!:!!. .V. 3x' z' (x = Sx' !\ z = Sz' !\(X'yz')eu). Also,
36
PETER APOSTOLI [xy z : (xy z diaguB+) e diaguB+1 =df [v z : :lxy (v = (x , y) /\ (xy z diaguB+) e diaguB+) 1
by Definition2.1 of k-aryabstraction . Then,
by (c) above. So we havec onstant t erms 81,82 such that-+/3t 2 = (81,82) /\ (8182 t3 diaguB+) e diaguB+. We may assume that{3 is the leastordinal verifying(8182 t3, diaguB+) e diaguB+; also, since Bo makes nos-atoms true, we know{3 > O. Thenby df. diagyB+ andthen4.4,
-+/3(8182 t3 diaguB+) e [xy z u : x = Q/\ y = z c!:!l .V. :lx'z' (x = Sx' /\ z = Sz' /\ (x'yz') e [xyz: (xyzu) cu])], -+/3-181 = Q/\ 82 = t3 c!:!l .V. :lx'Z' (81 = Sx' /\ t3 = Sz' /\ (x' 82 z') e [xy z : (xy z diagyB +) e diagyB+]).
Now, lettheleftdisjunctofthelastline,81 = Q/\82 = t3/\82c!:!l, be denoted L, andlettherightdisjunctbe R. We showthatB/3-1(L) =1= true, from which thatby Definition .1 3 thatB/3-1 (R) = true, since {3 - 1 is coherent. it follows For this, we may assume that82 = t3, since otherwiseB/3 -1(L) = false trivially. Since -+/3t2 = (81 ,S2), tz = (S1 ,82), so by assumption,it is notthe case thatbothS1 and 82 are numerals . If 81 fj. Num, then81 =1= Q withthe desiredresult . So assume that82 fj. Num . Thenby 9.2, B /3-1 (8 2 c !:!l) =1= true, whenceB/3-1 (L) =1= true, as required . Hence: -+/3- 1 :lx'Z' (81 = Sx' /\ is = Sz' /\ (x' 82 z' ) e [xy z : (xy z diagyB+) e diagyB+])
--+/3-1 Sl
= STI /\ t3 = ST2
--+/3-1 (T1 82T2) et+. So, B/3-1 ((TI 82 T2)ct+) = true. Then, since co ~k c/3-I , co(t+ , (TI' 82) , T2) false. Hence, bydefinitionof co, all ofTI , 82 , T2 belongto Num. But, since 81 = STI and t3 = ST2, all of 81 , 82, t3 E Num, contrary to theassumptionof thiscase. =1=
, by definitionof tx , Case 2.2: Clause(2) holds.Then tl = t -c - Recallthat
LOGIC, TRUTH AND NUMBER where B X(x, y, z, u) =df X t+ /\ (x' yv) e u). Then
=
Q /\ z = Q /\ Y e ~
.v. 3x'v (x
37 =
Sx' /\ (vy z ) e
-+133x y (t2 = (x, y) /\ (x Y t3 diagyB X) e diagyB X) by (c) above, since cI>(cl3)(tx ,t2,t3) = true. So we haveconstant t ermsSI,S2 such that:
We may assume that{3 is the leastordinalverifying(SI 82 t3, diaguB X) e diaguB x. Then {3 > o. Then, by definitionof diagyB X and 4.4
-+/3(81 82t3, diaguB X) e [xyzu : x = Q/\ z = Q/\ Yc~ .v. 3x'v (x = Sx' /\ (vy z) e t+ /\ (x' yv) e u)], -+13-181 =
Q /\ t3 = Q /\ 8 2 e ~
(x'
.v. 3x'v (81 8 2 v)
= Sx' /\ (V82t3) e t+ /\ e [xy z : (xy z diagyB X) c diagyB XJ) .
Now, letthe leftdisjunctof thelast line be d enotedL. We show that 8 13-1(L) i= true, from which it follows thatby Definition3.1 that8 13 - 1 verifiesthe rightdisjunct . For this, we may assume that81 = t3 = Q, since otherwise8 13-1 falsifies L trivially. Since -+l3t2 = (SI' S2), we have t2 = (81' 82) , whence, byassumption, 82 is not anumeral. But 82 rt Num yields813-1(82 c~) = false by 9.2, whence8 13-dL ) i= true, as required.T hen:
(1) -+13- 13x'v (81 = Sx' /\ (V82t3) e t+ 1\ (x' S2V) c [xy z: (xy zdiagy B X ) e diagyB xJ) (2) -+/3 -13x'v (SI = Sx' 1\ (vs2t3) e t+ 1\ (x' S2V) ct x ) (df. t x) (3) -+ 13- 1 81 = Sri /\ (r2 82 t3) e t+ /\ (rl 82 r2) e t x , for some rl , r2 E ctrm (4) -+13- 1 8 1 = Sri (5) -+ 13-1 (r2 S2 t3) e t+ (6) -+/3-1 (rl S2 r2) e t x Since tz = (81,82) , not all of 81, 82 , t3 are numerals.From line 4 above, = Sri. Sinceco $ k ee-: and{3-1 is coherent,we haveco(t+, (r2' 82), t3) i= false and co(t x, (rl,82),r 2) i= false from 5 and 6. Then by clauses 1 and 2 of df. co, all ofr l, 82, t3 E Num. Then, since Sl = Sri , SI is a numeraltoo. . --J This contradiction concludesCase 2, and thusthetheoremis established 81
LEMMA (a)
9.4. Let ml,m2 ,m 3 E IN. Then,
If ml +
(b) If ml
m2
x m2
i= m 3, then +-w (m l m 2,m 3) e t+, and i= m3, then +-w
(ml m 2, m3) e
tx·
38
PETER APOSTOLI
Proof. By inductionon mt, using (a) toestablish(b). This proofis a special case ofthatof Claim 1 in theprimitiverecursioncase oftheproofof 7.3. -l COROLLARY
9.5. B is
+,
x -total L-structure.
Proof. B is acceptableby construction.By Theorem 9.3, B is a 3-valued L-structure.B is w-totalsince A is and A ~k B. So let* E {+, x} and tt,ta, t3 any constantterms; it suffices to show thatone ofthe following holds: (1) --+/3 (tt tz h) e t.,
(2)
+-/3
(ttt2 t3) e t•.
We may suppose thatall oft l , t2 andt3 areGilmorenumerals,since otherwise (2) holds byconstruction 9.1. So thereare mt, m2, m 3 E IN such t, = m i (i = 1,2,3). If mt * m2 = m3, then(1) holds byTheorem7.2 and thesoundnessof G withrespectto acceptableL -structures . If, on theother hand, mt * m2 #- m3, then2 follows byLemma 9.4, (a) or (b), respectively as * is + or x. -j COROLLARY
9.6. G+'x is consistent.
may be derivedwithin This completesouranalysisof how mucharithmetic G+'x. To gofurther, we need a first -orderscheme ofmathematical induction. 10 . THE PRINCIPLE OF MATHEMATICAL INDUCTION FOR THE NATURAL NUMBERS
10.1. Varieties of ungroundedness As mentionedin section0, Tarskian first-order languagesobey two kinds of restrictions:a restriction on therangeof thefirst-order quantifier s, and a stratification of thelogicalgrammarof predication . These two restrictions can be seen ast alismansintendedto ward off, respectively , thetwovarietiesof definitional impredicativityt hatRussell wasreactingto when heformulated his ramifiedtheoryof types. The first kind ofimpredicativity , quantificational in nature,is featured by theclassical logicist definitionof theset INnatural numbers as theins ets. Here, thesourceof theanxietyis the tersection of all zero-successor factthatthetargetof thedefinition , viz., IN, is itselfa member of therange of theuniversalquantifier o ccurringin thedefinition.This kind ofimpredicativedefinitionis ruledoutby Russell'sstratification of therangeof the quantifiers.The secondkind ofimpredicativity is due toself-application and self-membership,or more generally, thepossibilityof non-well-founded membershiprelations.Russellruledoutthiskind ofimpredicativity by restricting
LOGIC, TRUTH AND NUMBER
39
thepermissiblelevel of an a rgumentto apropositional function , of whichthe Tarskianstratification of predicationmay be seen as adescendant . Since fixy introducest heself-application controlstructure diagy into the logical form of!!:!., thisdefinitionof thenatural numberscan be chargedwiththesecond, butnotthefirst, kind ofimpredicativity. In this section,we introducea constantterm N intendedto formalize numbersas theintersection of all Frege's classicaldefinitionof thenatural zero-successor sets. The derivations of 10.11 and 10.12 in thissectiondemonstrate t hestartling resultt hatthepositof bivalence for N is sufficient to yield a schema of first-order inductionforarithmeticformulasexpressedin terms of N. Althoughthevarietyof ungroundedness exhibitedby N can be describedin thesame generaltermsas thatexhibitedby !!:!. (bothdefinitions are liable to generateinfinitely descendingevaluation sequences), it emerges sectionthatthequantificational vain theconsistencyproofof thefollowing rietyof ungroundedness exhibited byN is of a lesstractable naturet hanthat associatedwith!!:!.. Whereasjg is totalin A , it appearsthatthereare fixed N's anti-extension pointsof in which theextensionof N is empty (although this is a propertythatN in A containsthenon-numerals;as we shall see, inheritsfrom thatdistinguishedzero-successor s et,!!:!.) . 10.2. Articulating an im age of arithmetic in L It is importantfortheconsistencyof NaDSetI that axioms sequentsof the form t ep I-- tep are disallow ed. Gilmore(1986) showed thatRussell'sparadox
could bereintroduced in theextensionof NaDSet I thatcontainssuch axioms by consideringthefollowing version theRussell of et s term R in which an existential quantifier binds a variablein predicatepositionin an e statement :
Ix : 3y (y =
x 1\ -, x e y)].
This is to beexpected,since parametersof NaDSetI are, qua eigenvariables, understood as standingfor anarbitrary constant set term;if wecountenance bivalence for atomswithparametersas predicateterms, thenintuitively, we are ipso facto countenancing bivalence for a toms featuringR as predicate term. However, it is precisely sequentsof the formt e p I-- t e p thatsecurethebivalence of N atomsandthusunderwrite theclassicalderivations oftheaxiom of infinityand theprincipleof inductionforthisimpredicativedefinitionof thenatural numbers. Gilmore (1986) therefore opted to stratify predication, moving on to asecond-order version ofNaDSetI, NaDSetII , where axioms p arametersP are introduced.In the oftheform t e P I-- t e P forsecond-order characterist ic idiom of econd-order s quantification, a second-ord er quantifier can happilybind a second-ordervariableoccurringas thepredicatetermof an e statement.
40
PETER APOSTOLI
Oursolutionavoids stratifying predicationby restricting therangeof the quantifiers instead. We pass tothelogicale xtensionG z by adjoiningto G+'x , as new bivalenceaxioms, allsequentsof theform t e p I- t e p forconstant termst andparametersp. Consistencyis maintainedby retracting therange of thequantifiers in Gz to asyntactically definedsubsetof ctrm, which, it willturnout,containst ermsrepresenting " total " objectsonly. The rangeof the G z quantifiers is theset of allconstantt ermsof the form [x : ¢], where ¢ is a formulaall of whose a tomicsubformulasa reG+ 'x groundatomsor N atoms (¢ is, as weshallsay, "arithmetic").T hus, as in G, we mayquantify over first level representations of predicatesin G z , butnotarbitrarily . The bivalenceof arithmeticformulasprovidesboththephilosophicala nd thetechnical m otivationforthusrestricting therangeof theG z quantifiers. Intuitively, we thinkof thelogicalextensionsof G asconcerninga domain of partialobjects(partial"sets" or "concepts")orderedunderan information ordering~k' In G z we intendthequantifiers to rangeovertotal o bjectsonly, thosewhich are maximal withrespectto ~k ' From theperspectiveof Gz, onlytotal o bjects-those having"sharpedges" as Frege would say -represent legitimatec onceptscapableof sustainingtheonticcommitmentincurredby quantification . Sincethecollection ofsettermswhichrepresent t otal concepts is notrecursively e numerable,theintendedrangeof theG z quantifiers is not syntactically specifiable.Hence, thenotionof anarithmetic t ermfunctionsas a syntactically specifiable approximation ofthenotionof atotal c oncept. As a result , Gz has thedesirabletechnical property,notsharedby NaDSetII, that its derivations areclosedunderthesubstitution of arbitrary closedtermsfor parameters . This propertyallows for straightforward a "analyt ic"p roofof theconsistencyof Gz in section12. Since thedependenceon thenotionof an "arit hmet ict erm" couldbe replacedby thenotionof a "primitiverecursive term", thespecificationof therangeof the G z quantifiers does notimport prior number-theoretic notionsin a waythatundercutst hepurelylogical natureof ourspecificationof thecalculus. Firstdefine ZS(p) to betheclosedformula Q e p t\ Vy (yep
--+
Sye p),
which "says" thatp is a O-successor set. We now defineset a termintended to representt hesetof natural numbersas "thesmallestO-successorset": N
=df
[x : Vz (ZS(z)
--+
x e z)l.
The firstobservation we shouldmake is thatN inheritsfrom ~ thedesirable propertyof being false of all non-numerals : OBSERVATION 10.1. Let tl be a constantt erm which is nota Gilmorenumeral.Then +- A tl eN.
41
LOGIC, TRUTH AND NUMBER
Proof By 5.7, +- A tle!:!l . By 6.1 (theaxiom of infinityfor!:!l)andthefactthat A is a model of GW , we have -+AZS(!:!l) . So, by 3.1, +-A (ZS(!:!l) --+ h e!:!l), whence +- A Vz (ZS(z) --+ tl e z) . Then by abstraction (4 .2), Vz(ZS(z)--+xez)],Le' ,+-AtleN.
+-
A tl e [x :
--l
Now, we formallydefine the "classica l"sublanguageo f L which, roughly speaking,containsa relational image of thelanguage o fPeanoArithmetic .
A of L recursively as follows : DEFINITION 10.2. We definea sublanguage O. Allparametersa re terms of A. N is a termof A. If ¢ is a formulaof A and x a variable,t hen[x : ¢] is a term of A. 1. If tl, iz and t3 aretermsof L , p is a parameterand ¢ a formulaof A, then i. tl= t2, ii. tl e [x : ¢], iii. tleN, iv. tle!:!l, andvi. (tl t2 , t3) e t x are(atomic) formulaso f A.
v. (tlt2, t3) e t+,
2. If ¢ and 1f; areformulasof A and x a variable,t hen--, ¢, ¢ V 1f;, ¢ 1\ 1f;, 3x ¢ andVx ¢ areformulasof A.
The connectives--+ and...... areintroduced i ntoA by theirusualdefinitions, as in L. A formulaor termof A is calledan arithmetic formula or term. Notethat we use ¢ , ,¢, K , X torangeover formulas of A. We let"5", "t", "r " , ... , with or withoutsubscripts,rangeovertermsof A. So thearithmeticformulasare preciselythose L formulaswhose atomic subformulasa re groundatoms of C+ , x or atomic formulasof L featuring arithmeticpredicateterms, thatis N or set terms of the form tle [x : ¢] where¢ is arithmetic.We definea notionof the complexity, comp(¢) , of a formula¢ of A inductively on theformationof f as follows :
1. If ¢ is atomicandone of theforms i, iii, iv, v or vi, thencomp(¢) = O. 2. If ¢ is atomicandof theform t 1 e [x : '¢] forsome formula'¢ of A, then
comp(¢)
= 1+
camp('¢).
3. If ¢ is non-atomic,the comp(¢) = 1+max{thepropersubformulasof '¢} .
10.3. The logical extension G 2 Let G 2 be thelogicalextensionof G+'x obtainedby adjoiningas new bivalenceaxioms allsequentsof theform t e p f- t e p, wheret is anyclosedterm and p is anyparameter,and replacingtheVf- and f-3 rulesby
PETER APOSTOLI
42
Ax[t] , r f- e r f- e, Ax[t]
/ /
r f- e, r f- e, 3x A
VxA,
respectively , wheret is an arithmetict erm. Gz's parametricbivalence axioms yield derivation a of bivalence for our impredicativerepresentative of thenatural numbers, N. First, LEMMA 10.3.
Let p be a parameter. Then ZS(P) f- ZS(p) is derivable in Gz .
Proof (1) (2) (3) (4) (5) (6) (7) (8) (9)
be p r be p Sbe p r Sb e p b e p r-r Sb e p.b e p r Sb e p b e p r-s Sb e p r b e p r-s Sb e p Vy(yep-Syep)f-bep-Sbep Vy(yep - Syep) f- Vy(yep - Syep) Qepf- Qep Qep, Vy(yep - Syep) f- Qept\ Vy(yep - Syep) Qept\ Vy(yep - Syep) f- Qept\ Vy(yep - Syep)
(Gz axiom) (Gz axiom) (1,2, _f-) (3, f--) (4, Vf-)
(5, f-V) (G z axiom) (6,7, f-t\) (7, t\f-) -I
Finally ,
Let t be a closed term . Then teN f- teN is derivable in G z .
LEMMA 10.4.
Proof (1) (2) (3) (4) (5) (6)
(7) (8)
ZS(p) f- ZS(p) t e p f- t e p ZS(p) - t e p, ZS(p) f- t e p ZS(p)-tepf-ZS(p)-tep Vz(ZS(z) - t e z) f- ZS(p) - t e p Vz(ZS(z) - t e z) f- Vz(ZS(z) - t e z) t e [x : Vz (ZS(z) - z e z)] f- t e [x : Vz (ZS(z) - x e z)] teN f- teN
(10 .3)
(Gz axiom) (1,2, -I--) (3, f--) (4, '1 2 1--) (5, f-V) (6, ef-,f-e) (7, df. N) -I
Let
LEMMA 10.5 . Xl, ... , Xk
Proof By inductionon comp(
and let
o-.
x=
Let
Sl,... , Sk
LOGIC, TRUTH AND NUMBER
43
Proof. By inductionon comp(
10.7. Let t be a closed term and [z :
<1>]
be a closed arithmetic term.
Then
t E: [x : <1>] f- t E: [x : <1>] is derivable in G 2 . Proof. By Lemma 10.6, x[t] f- x[t] is derivablein G2. Since t is free forx in <1>, two applicationsof unaryabstraction yield t E: [x : <1>] f- t E: [x : <1>], as -1 required. LEMMA
10.8. Let t be any closed arithmetic term and p a parameter. Suppose
r f- ~ is derivable in G 2 . Then so is rp[t] f- ~p[t].
Proof. By inductionon thedepthof G2 derivations. 10.4. Deriving induction from the impredicative definition of the natural numbers Thusfar we have delineated a fragmentA of L on the basis of two desiderata . First,thelanguageA containsa relational image ofthe languageof Peano repArithmetic.Second, we canreasonclassically in G2 abouttherelations resentedin A by virtueof thefactthatA is bivalentin G2. A is based upon theG+' x groundatomsandtheset termN =df [x : \/z (ZS( z) --+ X E: z)] which has been given the impredicatives yntactic s tructure oftheclassical definition ofthenatural numbersas "t heleastO-successor et s ". Becauseparametersa re notgroundpredicateterms of G+'x , N is deductively i nertin thatcalculus. Howeverwiththeintroduction of parametersas groundpredicates, G2 can simulateenoughof theidiom of second-order quantification to reasonclassicallyaboutN. In particular, an axiom of infinityand afirst-order scheme of induct ion for N are forthcoming. Notethatby Lemma 10.4 N is "total"in G2 in the sensethatall closed atomicformulasw ithN as predicatetermsare bivalent. Given theterm modelsemanticsof section3, thereadermay objectto readingN as "theleast O-successor set" since the quantifier\/ z does not rangeoversetsat all,etlalone O-successor sets. Indeed,in G2 thequantifiers rangeover closedarithmeticterms , of theform [x : (x)] where(x) is a A is given byextendingthe nominalisticviewpoint formula. The justification presupposed by our term model semanticsto thereadingof N. Here, we consider(x) to betheA-representation of anarbitrary formulaF(x) of the languageof arithmetic.We may thinkof quantifi cationover thecountably many closedarithmetict erms [x : (x)] as simulatingquantification overthe countably many formulasF(x) of arithmetic.So, in general , in G2 a sentence involvingquantification can in fact specify an infinite first-order schemain F
44
PETER APOSTOLI
forthelanguage of arithmetic.A more accurate r eadingof N would be"the set of objectssatisfyingevery O-successorformula", where a "O-successor formula " is a formulaof arithmeticwith at most one freevariablewhich holdsof 0 and thesuccessorof anyobjectit holds of.T his readingsuffices to underwrite a first - orderprincipleof inductionforN. First,we establishthatN itselfrepresentsa O-successorset: LEMMA
10.9. f- Q € N
1\ Vy (y e N --+
Sy e N) is derivable in G z.
Proof. First, we show f- Q e N: . (1) Q€pf- Q€P (2) Q e P 1\ Vy (y € P --+ Sy e p) l- Q e P (3) ZS(p) f- Qe P (4) f- ZS(p) --+ Q€P (5) f-Vz(ZS(z)--+Qcz) (6) f-Q.€[x:Vz(ZS(z)--+X€z)] (7) f- Q€N
Now we showvvt» e N --+ Sy EN) :
(Gz axiom) (1) Sb e P f- Sb e P (Gz axiom) (2) bcpf- b e p (1,2, --+f-) (3) b e p, b e P --+ Sb e P f- Sb e P (4) b e p, Vy(y€ X --+ Sy€p) f- Sb e p (3, Vzf-) (4, f-I\) (5) b e p, Q e P 1\ Vy (y e P --+ Sy e p) f- Sb e P (df. ZS) (6) bcp,ZS(p)f-Sb€p (Lemma 10.3) (7) ZS(p) f- ZS(p) (6,7, --+f-) (8) ZS(p) --+ be p, ZS(p) l- Sb € P (8, f---+) (9) ZS(p) --+ be p f- ZS(p) --+ Sb c p (9, Vzf-) (10) Vz (ZS(z) --+ be z) f- ZS(p) --+ Sbe P (10, f-V) (11) Vz (ZS(z) --+ be z) f- Vz (ZS(z) --+ Sbe z) (11 , l-s) (12) Vz (ZS(z) --+ be z) f- Sbe [x : Vz (ZS(z) --+ Xe z)] (12 , €f-) (13) be [x: Vz (ZS(z) --+ x 0)] f- Sbe [x : Vz (ZS(z) --+ x e z)] (13, df. N) (14) b€Nf-Sb€N (15) f- beN --+ Sb€N (14, f---+) (15, f-V) (16) f- v» (y e N --+ Sy e N) Hence, N is itselfan arithmetict ermwhich representsa O-successorset, and so fallsw ithintherangeof thequantifier Vy thatoccursin its owndefinition bindinga variablein predicateposition. -1
LOGIC, TRUTH AND NUMBER LEMMA
45
10.10. The axiom of induction for N, I-Vx(ZS(x) -> (VyeN)yex),
is derivable in G2. Proof. (1) b e p r b e p (2) ZS(P) I- ZS(p) (3) ZS(p) , ZS(p) -> be pI- be p (4) ZS(p), Vz(ZS(z) -> be z) I- b ep (5) ZS(p), be [x: Vz (ZS(z) -> z s z)] I- be p (6) ZS(p),beNl-bep (7) ZS(p)l-beN->bep (8) ZS(P)I-Vy(yeN ->yep) (9) I-ZS(p)->Vy(yeN->yep) (10) I- Vx(ZS(x) -> Vy(yeN -> y e x)) (11) I- Vx(ZS(x) -> (Vy e N)y e x)
(G2 axiom) (Lemma 10.3) (1,2, ->1-) (3, V21-) (4, el-)
(5, df. N) (6, 1--» (7, I-V) (8, 1--»
(9, I-V) (10, df, Vye N) 4
10.11 The Schema of Pre-Induction f or N. Let ¢(x) be an arithmetic formula, r, D. sets of formulas , and p a parameter such that the sequent ¢(p), r I- D. is derivable in G 2 • Then,
THEOREM
¢(Q) 1\ Vy(¢(y) -> ¢(Sy)), p e N, r I- D. is derivable in G 2 • Proof. Since ¢(x) is in A, by Lemma 10.5 so aretheclosedL2 formulas¢(Q), ¢(p) and ¢(Sp). Then (1) ¢m) I- ¢(Q) (2) ¢m) I- Qe [x: ¢(x)]
(3) (4) (5) (6) (7) (8)
(10.4) (1,l-e)
¢(p) I- ¢(p) (10.5) ¢(Sp) I- ¢(Sp) (10.5) ¢(p) -> ¢(Sp) , ¢(p) I- ¢(Sp) (3,4, ->1-) ¢(p) -> ¢(Sp), p e [x: ¢(x)] I- Sp e [x: ¢(x)] (5, el-, l-e) ¢(p) -> ¢(Sp) I- PE: [x : ¢(x)] -> Sp e [x: ¢(x)] (6, ->1-) Vy(¢(y) -> ¢(Sy)) I- p e [x : ¢(x)] -> SpE: [x: ¢(x)] (7, V21-) (9) Vy(¢(y) -> ¢(Sy)) I- Vy(y e [x : ¢(x)] -> Sye [x : ¢(x)]) (8, I-V) (10) ¢(Q) 1\ Vy(¢(y) -> ¢(Sy)) I- Vy(y e [x : ¢(x)] -> Sye [x : ¢(x)]) (9 , 1\1-) (11) ¢(Q) 1\ Vy(¢(y) -> ¢(Sy)) I- Qe [x: ¢(x)] (2,1\1-)
46
PETER APOSTOLI
(12) ¢(Q) 1\ Vy(¢(y) ---+ ¢(Sy)) f-Qe [x: ¢(x)].I\. Vy(y e [x : ¢(x)]---+ Sye [x : ¢(x)]) (10,11 , f-I\) (12, df. ZS) (13) ¢(Q) 1\ Vy(¢(y) ---+ ¢(Sy)) f- ZS([x : ¢(x)]) (14) ¢(p), r f- D.. (assumption) (15) p e [x : ¢(x)], r f- D.. (14, ef-) (16) ¢(Q) 1\ Vy(¢(y) ---+ ¢(Sy)) , ZS([x: ¢(x)]) ---+ p e [x : ¢(x)], r f- D.. (13, 15, ---+f-) (17) ¢(Q) 1\ Vy(¢(y) ---+ ¢(Sy)) , Vz(ZS(z) ---+ p e z), r f- D.. (16, V2f-) (18) ¢(Q) 1\ Vy(¢(y) ---+ ¢(Sy)), p e [x: Vz(ZS(z) ---+ Xe z)], r f- D.. (17, ef-) (19) ¢(Q) 1\ Vy(¢(y) ---+ ¢(Sy)), p e N, r f- D.. (18, df. N) --j
10.12 The Schema of First-Order I nductionfor N. Let ¢(x) be an arithmetic formula . Then
THEOREM
f-¢(Q) 1\ vv (¢(y)
-+
¢(Sy))
.-+.
("lyeN)¢(y)
is derivable in G 2 . Proof Since¢(x) is in A, by Lemma 10.5 so aretheclosed formulas ¢(Q) and ¢(p). By Lemma 10.6, ¢(p) f- ¢(p) is derivable in G2.Then, takingr to be the empty set of formulas D.. and to be {¢(p)}, by theschemaof pre-induction(10.11), thesequent (1) ¢(Q) 1\ Vy(¢(y)
-+
¢(Sy)), p e N f- ¢(p)
is derivable in 2G. So, we have (2) ¢(Q) 1\ Vy (¢(y) ---+ ¢(Sy)) f- peN -+ ¢(p) (3) ¢(Q) 1\ Vy(¢(y) ---+ ¢(Sy)) f- vu (yeN -+ ¢(y))
(4) f-¢(Q) 1\ Vy(¢(y) ---+ ¢(Sy)) (5) f- ¢(Q) 1\ Vy(¢(y) ---+ ¢(Sy))
.-+. .-+.
Vy(yeN -+ ¢(y)) ("lyeN)¢(y)
(1, f--+) (2, f-V) (3, f----+) (4, df."lyeN) --I
as required . 11. PEANO ARITHMETIC
11.1. Interpreting Peano 's axioms in A It is apparentthatA7 and A8 arestatedin a form more general thanthat requiredto sustainan interpretation of PeanoArithmeticin G+'x. Because, therange oft hequantifiers occurringin those axioms includes constant set terms which are not Gilmore numeralsand aretherefore not intendedto denotenatural numbers. When giving aninterpretation of afirst-order theory
47
LOGIC, TRUTH AND NUMBER
T I in anothersuch theoryT 2, in addition to providingan ~ formulato interpret each predicat e of LI and an ~ termor formula to interpret each functionsymbol of L I , we provide an ~ formulaF(x) to carveout the intendeddomain of theinterpre tationand thenwe restrictt hequantifiers occurringin the ~ image of L I to F(x) . In the followinginterpretation of PeanoArithmeticin G2, N will play the rolethequantifier of r estriction formula.Then, theresulting versions of A 7 andA8 may be derived usingthe schemaof inductionforN to formalize proofs by inductionoverthenatural numbers. Ournew versionsof thePeanoAxioms areobtainedby simply restricting theuniversal quantifiers in AI-A8 to N, as follows: PAL ("Ix e N)(Sx =F Q). PA2. (VXI X2 e N)(SxI = SX2 ---->
XI
= X2).
H
(xy, z) ct+)) .
PA3. (VxeN)(Qx,x)et+ .
PA4. (Vxy z e N)((Sxy , Sz) e t+ PA5. (VxeN)(Qx,Q)ct x .
PA6. (Vxy z e N)((Sxy , z) e t ; ...... (3ve N)((vy , z) e t+
1\
(xy ,v) ct x )) .
PA7. (Vxy e N)(3!z e N)(xy z) e t+ .
PA8. (Vxy e N)(3! z e N)( xy z) e t x . The previousG+ ' x derivations of AI-A5 may be trivially transformed intoG 2 y, usingjustThinningand ---->1- . However derivationsof PAI-PA6, respectivel PA6 is not derivablein G2 given thecurrentdefinitionof theset term t+ . This is becausethequantifi er (3ve N) thateffectsthecomposit ion ofx with + in A6 is notmatched by an N-restriction on thecorrespondingexistent ially quantified variablet hateffects compositionin thematrixB X (x, y , z, u) of t « . We redefineB x thus:
EX (x, y , z , u) =df X = Q1\ z = Q 1\ Ye ~ . V. 3x' (3v e N){ x = Sx'
1\
(v Yz) ct+ 1\ (x' Yv) e u) .
Now thederivation of PA6 goesthroughnicely . Here, we makeessentialuse of thefact,guaranteed by 10.4, thatatomsof theform peN are bivalentin G2. We startatline 3 ofthederivation of A6.I: (1) Sa = Sa'/\ (db, c) e t+ 1\ (a' b,d) e t ; I- (db,c) e t+ 1\ (ab,d) e t x (line 3,A6.I) (2) deNl-deN (10.3) (3) deN, Sa = Sa' I\(db, C)et+ 1\ (a' b,d)ct x I- deNI\(db, c)ct+ 1\ (ab, d)ct x (1, 2, 1-1\)
48
PETER APOSTOLI
(4) deN/\Sa = Sa' /\(db,c)et+/\(a' b,d)et x
r deN/\(db,c) et+/\(ab,d)et x
(3, /\r) (5) deN /\ Sa = Sa' /\ (db, c) et+ /\ (a' b,d) e t x r (3ve N)((vb,c) e t+ /\ (ab,v) e tx ) (4, r3z) (6) (3v e N)(Sa = Sa' /\ (v b, c) e t+ /\ (a' b,v) e tx ) r (3v e N)( (v b, c) et+ /\ (a b, v) e t x) (5, 3r) (7) 3x' (3v e N)(Sa = Sx' /\ (v b, c) e t+ /\ (x' b,v) e tx) r (3ve N)( (v b, c) e t+ /\ (a b,v) e t x) (6, 3r) (8) Sa=Qr (AI) (9) Sa = Q/\ c = Q/\ be ~ r (8, /\r) (10) Sa = Q/\ c = Q/\ be~ .v. 3x' (3ve N)(Sa = Sx' /\ t+(vb, c) /\ t x (x' bv)) r (3ve N)(t+(vb , c) /\ tx(abv)) (4,6, Vr) (11) (Sab, c) e t ; r (3ve N)((v b,c) et+ /\ (a b,v) et x) (10, rec.thm.) (12) r (Sa b, c) e t ; --+ (3ve N)((v b, c) et+ /\ (a b,v)et x) (11, --+r) (13) a e N, be Nce N r (Sab, c) e tx --+ (3v e N)((vb, c) e t+ /\ (ab, v) et x ) (12, Thin) (14) r (Vxy z e N)((Sxy , z) e t x --+ (3ve N)((v y, z) e t+ /\ (xy, v) e tx ) ) (13, --+r, HI) Here, line11 is an applicationof the recursion theoremusing thealternative definition of t x as fixuB x (x, y, z, u) . We similarlymodify thederivationof A6.2 toobtain:
r (VxyzeN)((3veN)((vy , z)et+ /\(xy,v)etx) --+ (Sxy,z)et x) , as requiredto completethederivation of PA6. 11.2. Deriving the functionality axioms in G 2
We now proceed with G z's main task, thatof yieldingderivations of PA7and PA8. LEMMA 11.1.
r (Vy e N)(3!z e N)(Qy, z) e t+ .
Proof First,we derivebeN
r (3z e N)(Q b, z) e t+ :
(1) beNrbe~ (9.11) (2) beNrQ=Q/\b=b/\be~ (1,=ax,r/\) (3) beN r Q = Q/\ b = b/\ be ~ .V. 3x' z' (Q = Sx' /\ b = Sz' /\ (x' b, z') et+) (2, Vr) (3, rec.thm.) (4) beNr(Qb,b)et+ (4, Thin) (5) beN, beNr (Qb ,b)et+ (5, r--+) (6) beN r beN --+ (Qb, b) e t.;
LOGIC, TRUTH AND NUMBER
(7) beNf-3z(zeN--+(Qb,b)et+) (8) beN f- (3z e N)(Qb,z) e t+
49 (6 , 3f-)
(df. 3z e N)
Next,f- '1Z1 Z2 ((Qb, zI) et + 1\(Qb , Z2 ) et+ .--+. Zl = Z2):
b = a1 f- b = a1 (= ax) Q=Ql\b=a1I\bef!lf-b=a1 (1,1\f-) Q = Sq1 f(4.1) Q = Sq11\ a1 = Sq2 1\(q1b, q2) e t+ f(3,1\ f-) 3x' z' (Q = Sx' 1\a1 = Sz'l\ (x' b, z') e t+) f(4, 3 f-) Q = Ql\b = a1l\bef!l .V. 3x' z' (Q = SX'I\a1 = Sz'l\(x' b, Z')et+) f- b = a1 (2,5, Vf-) (7) (Q b,a1) e t+ f- b = a1 (6, rec. thm.)
(1) (2) (3) (4) (5) (6)
By a similarderivation,we obtain:
(8) (9) (10) (11)
(Qb, a2) e t+ f- b = a2 (Qb,aI) e t+ 1\ (Qb, a2) e t+ f- b = a11\ b = a2 b = a1 1\b = a2 f- a1 = a2 (Qb,aI) et+ 1\ (Qb, a2) et+ f- a1 = a2 (12) f- (Q b, a1) e t+ 1\(Q b, a2) e t+) . --+. a1 = a2 (13) f- '1Z 1 Z2 ((Qb,zI) e t+ 1\ (Qb, Z2 ) e t+ .--+. Zl = Z2)
(7[ad a 2]) (7,8, f-I\, 1\ f-) (= logic) (9, 10, Cut) (11, f---+) (12, f- V)
Puttingthesetwo resultst ogether
be N f- (3z e N)(Qb, z ) e t+ 1\'1Z1 Z2 ((Qb, Zl) e t+ 1\ (Qb, Z2) e t+ .--+. Zl = Z2) f- be N --+ ((3ze N)(Qb, z) e t+1\'1Z1 Z2 ((Q b, zI) ct+ 1\(Q b, Z2) e t; . -r-e , Zl = Z2)) f- vv (y eN --+ (3z e N)(Qy, z ) e e., 1\ '1Z1Z2 ((Qy, Zl ) e t+ 1\ (Qy, Z2 ) e t+ .--+. Zl = Z2)) f- ('1Y eN)((3zeN)(Qy , z )et+1\'1Z1 Z2 ((Q y , zI) et+ 1\(Qy, Z2 )et+ . --+. Zl = Z2 )) f- ('1YeN)(3!zeN)(Qy , z)et+ (df. 3!) as required . LEMMA
-I
11.2 . Let a and b be any parameters of L. Then,
is a theorem of G 2 . Proof. Let c be a parameter distinctfrom a and b.
(1) (ab ,e)et+ f- (Sab ,Se)et+ (2) c e N f- Se e N (3) e eN, (ab, e) et+ f- Se e N 1\(Sab , Se) e t+
(A4.2) (10.8) (1,2 f-I\)
50
PETER APOSTOLI (3, f-A)
(4) CeNA(ab,c)et+f-SceNA(Sab,Sc)et+
(5) C e N A (a b, c) e t+ f- 3z (z e N A (Sa b, z) e t+) (6) 3z (z e N A (ab, z) et+) f- 3z (z e N A (Sab, z) et+) (7) (3zeN)(ab,z)et+ f- (3zeN)(Sab,z)et+
(4, f-3 2 ) (5 ,3f-)
(6, df. 3z e N) --I
LEMMA
11.3.
f- \fx ((\fy e N)(3!z e N)(x y, z) s z., --> (\fy e N)(3!z e N)(Sxy , z) e t+) .
Proof Let a, band C be anyparametersof L . Let
By a lengthy derivation,we can showthat¢(a, b) f- ¢(Sa, b). Thenfrom 11.2 by f-A and Af-, (3z e N)(ab, z) e t+ A ¢(a, b) f- (3z e N)(Sab, z) e t+ A ¢(Sa, b)
from which the desired sequentis obtainedby beN f- beN (lOA) using-->f-, f--->, f-\f and f-\f. --I COROLLARY
11.4. A7 is derivable in G2 .
Proof Let ¢(x) be thearithmeticformula(\fYe N)(3!z e N)(xy, z) e t+ . By
ILl and 11.3, f- ¢(Q) A \fx (¢(x)
-+
¢(Sx))
is a theoremof G2 • By 10.12, theinductionschema forN, f- (\fx e N)¢(x) is derivable in 2G• Thatis, f- (\fx e N)(\fy e N)(3!z e N)(xy, z) e t+
is a theoremof G2 , as required . LEMMA
11.5. f- (\fy e N)(3!z e N)(Qy, z) et x •
Proof Similar tothetreatment of theO-clausefor A7 in 11.1. LEMMA
--I
11.6. Let d, b, Cl , C2 be distinct parameters. Then
is a theorem of G 2 •
--I
51
LOGIC, TRUTH AND NUMBER
Proof Let ¢(x) be the A formulaVZ 1 Z2 ((xbzd e t+ I\. (xbz 2) c t+ = Z2». We canderive
.->.
Zl
, using G2 bivalenceaxioms. By 10.11, theschemaof pre-induction
is a theoremof G2 • Now, by line (13) oft hederivationof Lemma 11.1 and thelengthy d erivationmentionedin theproofof Lemma 11.3, I- ¢(Q) I\. "Ix (¢(x) -> ¢(Sx»
is a theoremof G2. Hence, we obtaina G2 derivationof thedesiredsequent -1 by Cut. LEMMA
11.7. Let a and b be arbitrary parameters of L.
(Vzl z2)((abzd e t x I\. (abz 2) e t x .->. Zl = Z2) I- (Vz1 z2)((Sabz l) c tx I\. (Sabz 2) e tx .->.
Zl
= Z2)'
Proof Let ¢(x, y, Zl, Z2) =df (xy zd e t x I\. (x y Z2) e t ; V¢ =df (Vzl z2)¢(a, b, Zl, Z2).
.->. Zl
= Z2,
dlcN, (dlb,cdct+, (dlb,C2)ct+I-Cl =C2 (11.6) (2) dl = d2 , (d2 b, C2) ct+ I- (d l b, C2) e t+ (G+'x ax, =1-) (3) d, eN, d l = d2, (d l bcd e t+, (d2 bC2) c t+ I- Cl = C2 (2, 3 Cut) (4) Sa = Sp, Sa = Sp' I- p = a I\. p' = a (A2,1-1\.) (5) p = a I\. p' = a, (pbd l) c t x, (p' bd 2) ctx I- (abdd e t x I\. (abd 2) e t ; (G+'x ax, =1-,1-1\.) (6) Sa = Sp, (pbddct x, Sa = Sp', (p' bd2)et x I- (abdl)ct x I\. (abd 2)ct x
(1)
(4,5, Cut,Inter) (7) ¢(a,b,d l,d2), d 1 eN, Sa = Sp, t+(d l bcd, tx(pbdd, Sa = Sp', t+(d 2 b C2), t x (p' bd 2) I- ci = C2 (3, 6, =1-) (8) V¢, d l e N, Sa = Sp, t+(d l bcd, t x (pbdd, Sa = Sp', t+(d 2 bC2), t x (p' bd2 ) I- Cl = C2 (7, "121-) (9) V¢, dlcN,Sa = Sp, t+(d l bCl), t x (pbd l), d2cNI\.Sa = Sp' I\.t+(d2 bC2)1\. t x (p' bd 2) I- Cl = C2 (8,1\.1-) (10) V¢, dl eN, Sa = Sp, t+(d l bcd, t x (pbdd, 3x' (3v e N)(Sa = Sx' I\. t+(v bC2) I\. tx(x' bv» I- Cl = C2 (9, 31-) (11) Sa = Q I(A1) (12) Sa = QI\. C2 = QI\. bc~ I(11,1\.1-)
PETER APOSTOLI
52
tx(pbd 1), B X(Sa,b,c2,t x) f- Cl = C2 (10, 12, Vf-) (14) Vf, d, eN, Sa = Sp, t+(d1bcd, tx(pbdd, tx(Sabc2) f- Cl = C2 (13, rec. thm.) (15) V¢, d1 eN 1\ Sa = Sp 1\ t+(d 1bcd 1\ t x (pbdd, t x (Sabc2) f- Cl = C2 (13) V¢, d1 eN, Sa
= Sp, t+(d1bcd,
(14,1\f-)
(16) V¢, 3x'(3veN)(Sa
(17) (18) (19) (20) (21) (22)
= Sx'l\ t+(VbCl) 1\ t x(x' bv)),
t x(SabC2) f- ci
Sa=Ql\cl =Ql\be~fV¢, BX(Sa,b,cl ,t x), tx(Sabc2) f- Cl = C2 V¢, (Sabcd e t«, (Sabc2) e t x f- Cl = C2 V¢, (Sabcd e t« 1\ (Sabc2) e t x f- Cl = C2 V¢ f- (Sabcd e t x 1\ (Sabc2) e t x . -+. Cl = C2 (VZ 1z2)¢(a,b,zl,Z2) f- (Sabcl)ctx 1\ (Sabc2)ct x .-+.
= C2
(15,3f-) (Al,I\f-) (16,17, Vf-) (18, rec. thm.) (19,1\f-) (20, f--+) Cl
=C2
(21, df. V¢) (23) (Vz1 z2)¢(a, b, Zl, Z2) f- (Vz1 z2)((Sa bzdct x 1\ (Sa bZ2)ct x .-+. Zl = Z2) (22, f-V) (24) (Vz1 z2)((abzd e t x 1\ (abz 2) e t x .-+. Zl = Z2 ) f- (Vz1 z2)((Sabz 1) e t+ 1\ (Sabz 2) c t x .-+. Z l = Z2)
Note thatB X(Sa,b ,c2,t x) atline13 abbreviates
withsimilarremarksforB X (Sa, b, ci , tx) at 18. LEMMA
11.8.
C d~,
beN f- (3z e N)( C b, z) et+ is a theorem of
G 2.
Proof Let ¢(x) =d f (Vye N)(3z e N)(xy, z) c t+. The G 2 bivalence axioms, 10.4 anddf. 3z e N yield: (Vye N)(3z e N)(cy , z) t: t+, beN f- (Bze N)(cb, z) e t+,
thatis, ¢(c), be N l- (3z e N)(cb, z) ct+, so by 10.11, theschemaof pre-induction,
¢(Q)
1\ Vy (¢(y)
-+ ¢(Sy)),c c: N,b c N f- (3z e N)(cb, z) e t+.
From 11.2, we have
(1) (3z cN)(ab, z) e i i. f- (3zeN)(Sab , z) ct+ (2) f- (3z e N)(ab, z ) et + -+ (3zc N)(Sab, z) et + (3) f-Vx((3zeN)( xb,z)et+ -+ (3zcN)(Sxb,z)et+)
(11.2) (1, -+f-) (2, f-V)
LOGIC, TRUTH AND NUMBER
53
And thefirstderivationin Lemma 11.1 yields, beN I- (3z e N)(Q b, z) et+ , so 1-1\ yields a G2 derivation of beN I- ¢(Q) 1\ vv (¢(y) -+ ¢(Sy)). Hence, we obtainthedesiredG 2 derivation of . c e N, be N I- (3ze N)(cb , z) e t+
by Cut. LEMMA 11.9. I- Vx (Vy e N)(3! z e N)(x y, z) e t x -+ (VYeN)(3!z e N)(Sxy , z) et x ) .
Proof. Let a, bandc be anyparametersof L. Let
¢(x, y)
=df
(Vz1 Z2)(xy zd e t ;
1\
(xy Z2) e t « .-+.
Zl =
Z2).
Note Lemma 11.7 is a G2 derivat ion of ¢(a, b) I- ¢(Sa , b). (G+'x ax, 1-1\) (1) (cb, d) e t +, (ab, c) e t x I- (cb, d) e t+ 1\ (ab, c) e t x (2) (cb, d) e t+ , (ab, c) e t x I- Sa = Sa 1\ (cb, d) e t+ 1\ (a/3, c) e t x (1, = ax, 1-1\) (3) (cb, d) et+, (ab, c) e t x I- 3x'v (Sa = Sx' 1\ (v b, d) E: t+ 1\ (x' b, v) e t ; (2, 1-32 ) (4) (cb,d)E:t+, (ab,c)E:t x I-Sa=Ql\d=Ql\bE:f:!l. .V. 3x'v(Sa = Sx' 1\ (vb ,d) d+ 1\ (x' b,v) et x ) (3,I-V) (5) (cb,d)et+ ,(ab,c)d x I- (Sab,d)et x (4, ree. thm.) (10.3) (6) deN I- deN (5, 6, 1-1\) (7) deN 1\ (cb, d) et+ , (a b, c) e t ; I- deN 1\ (Sa b, d) e t ; (8) deNI\(cb,d)E:t+ , (ab,c)et x I- (3zeN)(Sab,z)et x (7, 1-32 , df. 3zE:N) (9) (3zE:N)(cb, z)E:t+, (ab,c)E:t x I- (3zeN)(Sab, z)et x (8,31-, df. 3z E:N) (10) CE:N, bE:NI- (3zE:N)(cb, z)et+ (11.7) (11) be N, c E: N 1\ (ab, c) e t.; I- (3ze N)(Sab, z) E: t x (9, 10, Cut, 1\1-) (12) beN, (3ZE: N)(ab,z) s t ; I- (3zeN)(Sab,z) E:t x (11,31-, df. 3zeN) (13) ¢(a,b) I- ¢(Sa ,b) (11.6) (14) be NA, (3z E: N)(a b, z) e t ; 1\ ¢(a, b) I- (3ze N)(Sa b, z) e t ; 1\ ¢(Sa, b) (12, 13, 1-1\) (15) b E:N , (3!zE:N)(ab ,z)et x I- (3!zE:N)(Sab , z)E:t x (14, df. 3!zeN) (16) beN I- beN (10.3) (17) bE:N -+ (3!zE:N)(ab,z)E:t x , bE: N I- (3!zeN)(Sab,z)et x (15,16, -+1-) (18) beN -+ (3!zeN)(ab,z) E:t x I- bE:N -+ (3!zeN)(Sab,z)et x (17,1--+)
54
PETER APOSTOL!
(19) Vy(yeN -+(3!zeN)(ay,z)et xp-beN -+(3!zeN)(Sab,z)et x (18 , V2f-)
(20) Vy (ye N -+ (3!z e N)(ay ,z) e t x ) f- Vy(ye N -+ (3!z e N)(Say,z) etx ) (19, f-V)
(21) (Vy e N)(3!z e N)(ay, z) e t x f- (Vy e N)(3!z e N)(Say, z) e t x (10, df. Vy e N) (22) f- (VyeN)(3!zcN)(ay ,z)et x -+ (VyeN)(3!zcN)(Say,z)€t x (21, f--+) (23) f-Vx«VyeN)(3!zeN)(xy,z)et x -+ (VyeN)(3!zeN)(Sxy,z)et x ) (22, f-V) ---l COROLLARY
11.10. A6 is derivable in G2 .
Proof. Let ¢(x) be thearithmeticformula(Vy e N)(3!z e N)(xy z) e t x . By 11.5 and 11.9, f- ¢(Q) 1\ Vx (¢(x) -+ ¢(Sx)) is a theoremof G2 • By the 10.12, inductionschema forN, f- (Vx e N)¢(x) is derivablein G2 • Thatis, f- (Vx e N)(Vy e N)(3!z e N)(xy z) e t x
is a theoremof G2 , as required . This completesour interpretation of Peano Arithmeticin G2 • Now we must show thatG 2 is not too powerful. 12. SEMANTICS FOR G2
12.1. Term models for G 2 In this sectionwe modify theDefinition3.1 of truth in an L-structure to accommodatethenewsyntactic features of G2 • Evidently,thechallenge here is thevalidation of G2's bivalenceaxioms t e p f- t e p, whichrequirest hatthe givenstructure be t-total for all t ermst comprisingtherangeof theG 2 variables,viz., theconstant a rithmetic t erms. Now, the+, x-total L -structure B definedby 9.1 comes very close to p rovidingsuch astructure . Ourstrategy is to modifytheDefinition3.1 of truthon a structure to accommodatethe of the G 2 quantifiersa nd variablesand thenshow intuitiveinterpretation thatB can beextendedto producean +, x -total L -structure on which every instanceof an arithmeticformulaconverges. DEFINITION 12.1. The G 2 inductivedefinitionof truth on an L-structure is given byclauses1-4 ofDefinition3.1 and, fortheG 2 quantifiers , by replacing clauses4 and5 by
LOGIC, TRUTH AND NUMBER S(V'xA) S(3x A)
= I\{S(A x[t]) I t = V{S(A x[t]) I t
55
is a constant a rithmetict erm}, is a constant a rithmetict erm}
respectively, where, as in those clauses,1\ is arbitrary meet andVis arbitrary join withrespectto (4,::::t). Hence, thequantifiers conformin G2 semantics to thesame ::::k-monotone interpretive scheme as thequantifiers in G. Allcognatesemanticdefinitionsa rethesame as those ofsection3, except forthedefinitionof an instance of a sequent,which needs to be modified to accordwithourintention t hattherangeof permissiblesubstitutends forthe G 2 parametersa reconstanta rithmetic termsonly. Aninstance of a closed L-formulais an L-sentenceobtainedby uniformlyreplacingeveryparameter occurring in thegiven formula by constant a t ermof A. If r f- D. is a sequent, a= aI, . . . ,ak exhaustall oft heparametersoccurringin members of r or D., and i = t l , . . . , tk areconstant t ermsof A, thenra[f]f- D.a[f] is an instance of r f- D.. Since G2semanticsis ::::k-monotone, it is cleart hatthedefinition of theGilmorerevisionoperatore xtendsto yield::::k-monotone revisionsof L-structures . 12.2. Consistency
Ourstrategy is to modifytheconstruction of B in Definition9.1 to produce C ?k B which totalizes ourimpredicativerepresentation of an L-structure thenatural numbers,N. Thenas in theproofof Theorem9.3, we will show thatC is in fact 3-valued. DEFINITION 12.2.
Definefo : ctrm3 --+ 4 by:
1. fO(tl,i z, t3) = false if atleastone ofthefollowing two conditionshold:
(i) tl = t+ and it is notthe casethattherearem, n, j E IN such thatt 2 = (m , n.) and t3 = i, (ii) tl = t x and it is notthecase thatthereare m, n , j E IN such thattz = (m, n.) and ts = i,
otherwise, 2. fO(tl,ta, t3)
= true
if tl= Nand t2 E Num, otherwise,
3. fO(tl,ta, t3) = ..L Let Co =df (ctrm,fo) . Using ournew ~,define an ordinalsequence{Co I WI} of L-structures as per 3.10. We writefa forfCa ' For therest ofthis section,we writesimply -+0 for...... Ca • Also, by "thelevel of saentenceB" we mean thelevel ofB in {Co I Q :::: wd. Q
::::
THEOREM 12.3. Let
A be any arithmetic sentence. Then A converges on C.
PETER APOSTOLI
56
Proof We show byinductionon comp(A) thatA!. For the basis, assume A is of one oftheforms i. tl = t2, iii. tl e N,
iv. tl c~,
V. (tlt2, t3) e t+, or
vi. (tlta, t3) e t «
as per Definition10.2. Allidentitiesconvergeby claus e 2 of Definit ion12.2 andObservation10.1 together withthefactthatA $k C, tlc N! . Of course, tlc~ similarlyconvergeson C since A is w-total and A $k C. By thesame argumentas in 9.3 and9.4, Cis + , x-tot al , so (tlta, t3)ct+ and(tlta, t3)ctx converge on C . For theinduction , suppose A is of form1O.2.ii: tle[x : 4>] for some arithmetic formula4> withatmost x free. By Lemma 10.5, the sentence4>x ltd is arithmetica nd of thesame complexityas 4>i comp(4)x[td) = comp(A) - l. Then by thehypothesisof induction , 4>x[td convergeson C, whenceA converges on C bydefinitionof (or abstraction, 4.2). If, on theotherhand, A is a Booleancompound, thenby thehypothesisof induction,A's principle subformulasconvergeon C, so A convergeson C since theinterpretation of theconnectivespreservesconvergence on an L-structure. Finally , suppose A is of theform 'r/x 4> forsome arithmeticformula4> withatmost x free. Let t be any arithmeticc onstant t erm. By 10.5, 4>x[t] is an arithm eticsentenceof thesame complexityas 4>, so by thehypothesisof induction,4>x[tJ!. Thenby Definition12.1, AL as required . This completestheproofof Theorem12.3. --I
THEOREM 12.4.
0
is coherent (0 $
wd.
Proof Suppose, towardsa contradiction , that0 is notcoherent. Then as in theproofof Theorem 9.3, thereis a coherentf3 < 0 and constantt erms t I , ta, t3 such thatboth of the following hold: Let I $ f3 be the leastordinalsuch thatC')' (tl , t2, t3) = false (true) . As in 9.3, thecase I > 0 is excluded . Then, if CO(tl,t2, t3) = false, one ofclauses 1 or 2 ofthedefinitionof co (12.2) holds andtl E {t+. tx}. This case is handledexactlyin themannerof the proofof Theorem 9.3 (in particular, subcase 2.2 is not affectedby thefact thatwe are now operatingwith a modified definitionof the matrix E X(x, Y, z, u) of thatterm). So suppose cO(tI, t2, t 3) = true. Then by clause2 of Definition12.2, tl is N and thereis an m E IN suchthatt2 = m . By (b) and definitionof .
so there is a constant arithmeticterm s suchthatCI3(ZS(s) -> t2cS) = false. Since f3 is coherent , it followst hatZS(s) is true, but t2 e s is false, on CI3' But given C13(ZS(s» = true, it is easy to show byinductionon n thatCI3(n c s) = true (n E IN), which contradic ts CI3(t2 e s) = false. This contradiction establish es Theorem 12.4. --I
LOGIC, TRUTH AND NUMBER LEMMA
57
12.5. C is a model of G 2 •
Proof C is acceptable by construction and3-valuedby 12.4. So it suffices to show thatC validates theG2 bivalenceaxioms fore atomshavingparameters as predicateterms. So suppose we a given anarbitrary such axiomt ep I- tep for some closedterm t and parameterp. Let t' e s I- t' e s be an arbitrary instanceof thegiven axiom sequent,wheres a constant a rithmetic t ermand t' is an instanceof t obtainedby uniformlyreplacingallparametersin t by constant a rithmetict erms. Then t' e s is anarithmetic s enten ce by Definition 10.2.2, andso convergeson C byTheorem12.3. Since C is3-valued , it follows that C(t' e s) E true,!alse.
This is sufficient . COROLLARY
12.6. G 2 is consistent. REFERENCES
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PETER APOSTOLI
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Hindley, J. R., andJ . P. Seldin 1986 Introduction to combinators and lambda-calculus, London Mathematical Society Texts,l . Jech, T. 1974 (editor),Axiomaticset theory,Proceedings of symposia in pure mathe matics, vol. 13, PartII, AmericanMathematicsSociety. Kripke, S. 1975 Outlineof a theoryof truth,The Journal of Philosophy, vo!. 72, pp . 690-716. Schiitte,K. 1977 Proof theory (Beweisetheorie),Springer-Verlag, Berlin. Scott,D . S. 1971 Models for various type-freecalculi,Logic and Methodology and Philosophy of Science IV (P. Suppes and others,editors),North-Holland , Amsterdam. 1975 Combinatorsand classes,Lambda-calculus and computer science, Lecture , Berlin, pp. 1-26. Notes inComputerScience, vol.37, Springer-Verlag Szabo,M. E. 1969 (editor),The collected papers of Gerhard Gentzen, North-Holland, Amsterdam. Seldin, J. P., and J. R. Hindley 1980 (editors),To: H. B. Curry : Essays in combinatory logic, lambda calculus . and formalism, Academic Press, New York Smullyan , R. 1986 First-order logic, Springer-Verlag, Berlin. Wright, C. 1983 Frege's conceptionof numbers as objects, Scots Philosophical Monographs, vol. 2,AberdeenUniversityPress, Aberdeen. Visser, A. 1984 Four valuedsemanticsand theLiar paradox,in Gabbay and Guenthner 1983, vol. 4.
JOHN CORCORAN
SECOND-ORDER LOGIC
Abstract.This expositoryarticlefocuses onthe fundamental differences betweenfirst-order logicand second-orderlogic. It employs second-order propositions and second-ord er reasoning in a natural way to illustrat e the fact that second-orderlogic is actuall y a familiarpartof our traditional intuit ive logical frameworkandthatit is notan artificial formalismcreatedby sp ecialists for technical purposes. To illustratesome of the main relationship s between first-order logic and second-ord er logic, this paper introdu ces basic logic, a kind of zeroorderlogic, whichis m or erudimentarythanfirst-ord er and which is transc ended by first-orderin the same way thatfirst-orderis transc ended by second-order . The heuristiceffectiv en ess and the historicalimportance of second-order logic are reviewedin theconte xt of thecontemporarydebate over the legitimacyof second-order logic . Rejectionof second-orderlogic is viewed as involvingradical repudiationof partof our scient ifictradition . Buteven if genuine logic comes to r , which is a real possibility, its be regardedas excludingsecond-o rde r easoning effect iv eness as a heuris tic instrumentwillremain and its importan ce for underst a ndingthehistoryof logicand m athematics willnot be diminished. Secondorde r logi c may some day be gon e, but it will ever n be forgott en. Technical formalisms have been avoidedent ire ly in an e ffortto reachan inte rdsciplinary i audience, butevery eff orthas been made to limit the inevitabl e sacri fice of rigor.
No matterwhathumanactionyou consider,if everyonedoes it to everyone t hem by everyone to whom doing it tothem, theneveryonehas it done to theydo it. For example, if everyoneteacheseveryonewho teaches them, theneveryone istaughtby everyonetheyteach.Likewise, if everyonehelps everyonewho helpsthem, theneveryoneis helpedby everyonetheyhelp. The same holds for"encourages" , "hinders","supports", "opposes", "ignores" , and therest. Each oftheabovepropositionsis actually a tautology, a propositionimplied by its ownnegation . In fact, each of themcan be proved to be trueby logical reasoningalone; .eg., by deducingthem from theirown negations. Since every propositionin thesame form as atautology is againa tautology , a discourseformallysimilarto thatexpressedabove obtainsin every universe of discourse,notjustin theuniverseof humans. In metalogic,forexample, we often discuss the universeof propositionsin so far as various logical relations areconcerned . By a logical relation I mean relations such asimplication , consequence , contradiction , compatibility , independence,etc. More specifically , I mean whatare called b inaryrelations on theuniverse ofpropositions.If R indicatessuch arelation and ifa and b are eachindividualpropositions, thenaRb canbe used toexpressthepropo61
C. Anthony Anderson and M. Zeleny (eds.), Logic. Meaning and Compu tation, 61-75 . © 2001 Kluwer Academic Publishers. Printed in the Netherlands.
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sit ionthatthefirstpropositiona is relatedby R to the secondproposition b. It is notexcluded,of course,thata and b are thesame proposition.For example, everypropositionimplies itselfa ndsome butnoteveryproposition contradicts itself. Now logical relations arecertainly notactions . Saccheri's Postulate contradicts theParallel Postulate butthereis no actionthatSaccheri's Postulate couldperform. Nevertheless , as we havejustseen, relation verbs function grammatically in certaincontextsin a mannersimilarto thefunctionof action verbs. The relation verbs significantin theuniverseof humansinclude the following:outweighs,outlives,succeeds (inseveralsenses), precedes (in several senses), equals (in many senses), and many others . The actionverbs t heuniverse ofhumansinclude the following: calls atleast (in significant in one sense), serves (inat leastone sense), teaches,commands, obeys, and many others . In normalEnglishsome of thelogicalrelations areexpressedby relation verbs, as we have seen. For example, implicationis expressedby 'implies' and contradiction is expressedby 'contradicts'. However, some them of are expressedby relation nouns. For example, consequenceis expressedby the of consequenceis that relation noun 'consequence'.The law oftransitivity everyconsequenceof aconsequenceof apropositionis againa consequenceof thatproposition. Moreover,t hereare logical relations expressedby relation adjectives. Compatibilityand independenceare expressedby 'compat ible' and 'independent'.Aristotle 's fundamental law ofcompatibilityof truthis thatevery twotruepropositionsarecompatiblewith eachother . Using 'independent'in themost widelyacceptedsense we can saythateveryproposition which isindependentof a givenpropositionis neitherimplied by norcontradictedby the givenproposition, and conversely , everypropositionwhich is neitherimplied by norcontradicted by a givenpropositionis independentof the givenproposition. thatlogicalrelationsare exOne reasonfor reviewing the various ways pressed inEnglishis to pointoutwhatallcreativew ritersa lreadyknow, viz. thatknowledge of the conventional rules of English shouldenhancebutnot inhibitEnglishwriting. For example, my very firstsentenceuses the pluwith 'everyone',which issingular . Even ralpronoun'them' as coreferential rules, is my use oft hefiller worse, fromthepoint of view ofconventional 'you consider'.The propositionbeing expressedis not apredictionof what willhappen if you considersomething.The propositionis notaboutyou per se atall.The sentenceexpressesa general proposition predicatinga certain complex propertyof everyactionon the universe humans of . The phrase'no matterwhathumanactionyou consider'is justa heuristically effective way ofexpressinga universal quantifier . From a logical pointof view the following would dojustas well : 'everyhumanactionis one suchthat ' , 'everyhuman actionis one where ', 'wit h everyhumanaction', etc. At anyrate,a sentence thatviolatesthe conventional rules of English applicableto the expression
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of a givenpropositionis sometimes nevertheless a perfectlya cceptablea nd effective way of expressingthatveryproposition . Whethera sentenceis an acceptableand effectiveexpressionof a given propositionis a matterof howreaderst akeit, andnot amatterofconventions establishedin thepast. Now wearereadyto presenta discourseobtaining in theuniverseof propositionsandformallysimilarto theone whichbegan thisessay. No matterwhatlogicalrelation you consider,if every givenproposition bears it to ev ery proposition bearing it tothegiven proposition,thenevery givenpropositionis borneit by everypropositionthegiven proposition bearsit to. For example, if everypropositioncontradicts everyproposition contradicting it, theneverypropositionis contradicted by everyproposition it contradicts . Likewise, if everypropositionimplies everypropositionimplying it,theneverypropositionis implied by everypropositionit implies. The same holds for "is a consequenceof', "is compatiblewith", "is logically equivalent to", "is independento f', "is a contradictory oppositeof', and the rest. The propositionsexpressedin theabove paragrapharealltautologies and theyare all laws of logic. The propositionsin thefirstparagraph of thisessay are alltautologies but none ofthem are laws of logic becausetheyare not abouta logicalsubject-matter . The proposition"Everypropositionimplies everypropositionimplyingit" isabouta logical s ubject-matter butit is not a law of logic becauseit is false . For example, "Everypropositionis true" implies "Everyfalsepropositionis true", butnot conversely. The proposition "Everypropositioncontradicts every propositioncontr adictingit" is a law of logic, ofcourse,butit is not a tautology becauseit is inthesame form . By a law of as a propositionconsideredjustabove and found to be false logic Imean a truepropositionabouta logicals ubject - matter , e.g., about propositions,a boutarguments,a boutargumentations , etc. The twoproperties,being tautologous andbeing a law of logic , are orthog. onal in thesense thateach ofthefourcombinationsof thetwo is exemplified is a law of logic and We have seen abovet hatsome butnoteverytautology thatsome butnoteverynon-tautology is a law of logic. Thereis much confusionconcerningthiselementary point. Some butnotall oftheconfusion is more or lessd eliberately nurtured in theserviceof variousdogmas which, happily,are waning inpopularity. The proposition,"Every propositioncontradicts every propositioncontradicting i t", is thelaw ofsymmetry (or reciprocity)of contradiction and "Some propositionimplies some propositionnot implying it in return"is thelaw of non -symmetry(or non-reciprocity) of implication . "No contradic toryoppositeof a contradictory oppositeof a propositionis a contradictory opposite of thatproposition"is the law of antit ransitivity of contradictory opposition. A contradictory opposite of a propositionis, of course, apropositionlogically equivalent to thenegationof thatproposition.For example,
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"Some truepropositionis nottautologous" and "Not every t rueproposition oppositesof "Everytrueproposition is tautologous" are bothcontradictory is tautologous". In orderto avoid confusion should it be notedthat,a lthough every twopropositionsthatarecontradictory oppositesof eachothercontradict each other,not every two propositionsthatcontradict each otherare contradictory opposites. To takean extremeexample, "No propositionimplies itself'c ontradicts "Some propositionimplies everyproposition". The same exampleillustrates another p ointthatclarifiest hingsand helps to avoid confusion, viz . thatalthough no truepropositioncontradicts a trueproposition, some falsepropositionscontradict falsepropositions. In fact, some no twocontradictfalsepropositionscontradict themselves . Thus although , some twocontradicting propositionsareboth ing propositionsarebothtrue false. In such cases,.e.,i when twocontradicting propositionsare bothfalse, theyare notcontradictory oppositesbecauseevery twocontradictory oppositeshave different truth-values. -contradiction) is a propositionthatcontradicts A contmdiction (or a self is a contra itself, i.e ., thatimplies its ownnegation. Every contradiction dictoryopposite of a tautology and everytautology is a contradictory oppositeof a contradiction. A propositionis said to becontmdictory (or selfcontradictory) if it is acontradiction. Everytwocontradictory propositions propositionsare contradiccontradict each otherbut no twocontradictory toryopposites of eachother. The expression'two contradictory propositions'means "twopropositionseach of which is self-contradictory" whereas 'twocontradict ing propositions'means "twopropositionscontradicting each other"which, in view of the symmetricalnatureof "contradicts " , amounts to "twopropositionsone of whichcontradicts theother". t o' and 'is a contrary to' are In ordinarytechnicalEnglish,'is contrary ambiguous. Sometimes, "contradicts" is meantand sometimes "is a contradictoryopposite of' is meant. Surprisingly,the ambiguitydoes not seem to betroublesome.However, in former times logicians had attached a third technical meaningthatdid lead toconfusion . Two propositionswere said to eachotherbuttheir negabe contraries (sc of eachother)if theycontradict tionsdo notcontradict eachother . For example, "Everynumberis prime" and "Every numberis non-prime" are contraries . It is easy to provethat every twocontradicting propositionsthatarenotcontradictory oppositesare contraries, and vice versa. In modernlogic,'contrary' is rarelyused inthe obsoletetechnical sense. We have had occasion justnow tostateseveral laws of logic and to men. As indicatedabove, by a law of tion(or talkabout)several laws of logic logic I mean atruepropositionabouta logical s ubject-matter (propositions, arguments,argumentations, etc.). The most basic laws of logic are thelaws : "Every of excludedmiddle, non-contradiction , and truthand consequence propositionis eithertrueor false", "Nopropositionis bothtrueand false " and "Everypropositionimplied by atruepropositionis true".
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The laws of logic, in fact propositions all aboutlogicalsubject - matterare in some sense second-level (or meta-level) propositionsin thesense thatthey , of course, nonare aboutthingsthatare themselvesaboutthings(usually logicalthings). Some people, eitherignorantof or inoppositionto logical tradit ion, call suchpropositions'second-order' .This is not how 'secondorder'is used in this essay a lthough some second-level propositionsare also second-order . A propositionis classified asbasic, first-order, second-order, etc., not on the basis of whatit is aboutbutrather on thebasis of its logical structure . The very first p ropositionof this essay is second-order . Thesecond propositionis first-order.Every propositionto the effect thatone named propositionis in a mentionedlogicalrelation to anothernamed proposition c ontradicts the Parallel Postula te". It is basic, e.g., "Saccheri'sPostulate will become obvioust hatthetwo properties,being second-level and being second-order,areorthogonal. The basic propositions, veryroughlyspeaking, are thosewithoutcommon nouns. It is perhapseasiest todescribethe basic propositions of arithmetic insteadof describingthe BPA outright, it is convenientto (BPA) . Actually, describethebasic sentences of arithmetic (BSA) and thento saythatthebasic propositionsofarithmetic arethepropositionsexpressedby the basic sen. tenceswhen thesentencesare understoodin theirintendedinterpretations ely Now, thesubstantivesof thebasic sentencesof arithmeticare exclusiv numerals (number-names)in thewide sense: 'zero','one', 'two', 't hree',. .. , 'zeroplus one', 'zero plustwo', . .. , 'two plus (zero times one)', . ... Among thenumeralsI alsointend: 'two-squared', 'two-cube d', etc. The atomic sen, all so-called equations : 'one tencesof arithmeticinclude, inthefirst place plus one is two', 'one plus twoone', is etc., in otherwords anysentencein the patternnumeral is numeral. The 'is' here, of course, is intendedto express numericalidentitywhich is oftenimproperlycalledequality and expressedby a qualityto a 'equals'. Next we havethesentencesthatnormallya ttribute number, e.g., 'one iseven','one is odd', 'two isprime', 'five isperfect';and so r elate one numberto another , on. Next we have the sentencest hatnormally e.g., 'two exceedsthree','threedivides two', etc.T hese includetheidentities (orequalities, equations)a lreadymentioned. Next we have the sentences thatnormally indicatethatthreenumbersare in aternary relation, e.g., 'two is betweenone andthree' , etc. Thereare alsoquaternary relational sentences,e.g., 'one is tothreeas threeis to nine'. And so on. Anythingofthissortis countenanced as long as thereare no common nouns . Even common nouns are allowed as long they as are understoodas nominalizedadjectives(e.g., 'two is aprime' means "two is prime") or asnominalizedrelatives(e.g., 'two is a divisor of four' means "two divides four") , etc. Once theatomicbasic sentencesofarithmetichave truthbeen determined,the basicsentencescan be defined as the so-called functional combinationsofatomicsentences,theatomicsentencesplus what can beobtainedfrom atomic sentencesby negations,conjunctions , disjune-
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tions, conditionals,bi-conditionals , etc. It should be explicitly mentioned in thisconnectionthatpassives (or converses) of binaryrelation verbs are againbinaryrelational verbs and thussentencessuch as 'two isdivided by four', 'two is exceeded by four ', etc. are included . Likewise includedare sentencesinvolving the so-called modified relation verbs : 'properlydivides', 'immediatelyprecedes', 'immediately exceeds', etc. Basic logic is the logic of basic propositions. Basic logic isconcernedfundamentally with thequestionof how wedeterminethevalidity or invalidity of anargumentwhose premisesand conclusion are exclusively basic propositions. As you know, anargumentis determinedto be valid by giving a derivation(or adeduction)of its conclusion from its premises . This means giving anextendeddiscourse,normallymuch longert hanthepremises-plusconclusionwhich showsstep-by-stephow theconclusion can be seen to be truewere the premises true. The rules for making up these derivationsare obtainedby lookingat whatpeople dowithbasic propositions when they are reasoning correctly.orderto In deduce from any set of basic premises any basic conclusion thatactually follows, it is sufficient to use rules from a theusual rules of propositional logic,therule very small set. These include of substitution of identities,t herule of conversion (the active and passive are interdeducible) and the logical axioms identity("one of is one", etc.). To show thata given basic conclusion does not followfrom a given basic premise-set, it is sufficient to producea counterargument, i.e., a conclusion anda premise-settogether in thesame form and having false conclusion and truepremises. For example, to show t hattheargumenton the left below is thattheargumenton theright is inthesame invalid itis sufficient to notice form and has truepremises and false conclusion. Two is notthree . Threeis not two plus two . ? Two is not two plus two.
One is not two. Two is not one times one . ? One is not one times on e.
The argumenton therightis obtainedin threesteps fromtheargument everywhere for 'two ' on theleft . Thenin on theleft.First'one' is substituted the "new " leftargument(in which'two'no longer occurs), 'two' issubstituted everywherefor't hree'. Then in the "second new" left argument,'times' is substituted for'plus'. Strictly speakingan argument (more properly,premise-conclusion argument) is a twopartsystem composed of a set ofpropositionscalledthe premise-set and a singlepropositioncalled theconclusion. To represent or express anargumentwe use anargument-text which is a list of sentences (not propositions)followed by a single sentence somehow markedas the conclusion-sentence. Some logic books use a line above the conclusionsentence,butit is easier and less messy to usequestion-mark a as above. The methodoutlinedabove oftransforming one argument-text intoanother
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argument-text in such a waythattheargumentrepresented by thesecondis by thefirst works only when the in thesame form astheargumentr epresented argument-texts arewrittenin a so-calledlogically perfect language in which theoutergrammatical form ofthesentencesm irrorsexactlyt heinnerlogicai important,t helanguagein form ofthepropositions. Whenlogical issues are questionis regimented (normalized) so thatit becomes logically perfect(or approximately so). This is why I write'two plus (zero times one) ' insteadof 'twoplus zerotimes one'. Logicianstypically go immediatelyto asymbolic language carefully constructed to be logically perfectbutformany purposes, especiallythatofexposition,thismethod,thoughvirtually essentialfor some purposes,can becounter-productive . argument Basic logiccanbe calledfinite logic becauseevery finite invalid of basic logic isrefutable by a counterargument whose propositionshave reference only to a finite numberof individuals.By a finiteargumentI mean an argumenthaving only a finite numberof premises and by referenceonly to a finiten umberof individualsI mean not onlythatthepropositionsrefer only to finitely manyindividuals(which is obvious)butalsothatthefunctions referredto are all defined on one thesame and finiteuniverseof discourse. By theway,thisincludestheso-calledzero-premisearguments(arguments havingthenullpremise-set)which are valid when and only when theconclusion is a tautology. Some examplesfollow . ? One is one.
? If one is twothentwo is one. t henone is two)thenone is two. ? If (if one is not two ? If one exceeds two t hentwo is exceeded by one. It follows from w hatwassaid abovethatevery basicpropositionthatis not a propositionhaving a contradiction is in thesame logical form as truebasic referenceonly to a finite number of objects. This means thatamong the basic propositionsthereare noso-calledinfinity propositions. In orderfor a propositionto be aninfinity proposition it isnecessaryandsufficientt hatit be non-contradictory andfor everypropositionin thesame formhavingreference only to a finite numberof individualsto be false. In o therwords aninfinity propositionis a propositionexpressedby a sentencewhich is "satisfiable" only in infinite universesdiscourse. of Basic logic covers most of thearithmeticreasoningdone by school children,all oft helogic "done" bycomputers(thoughin a sensecomputerscan simulate finitestretches ofhigherlogics),andmuch ofthelogic onthenormal aptitudetest. In a sense,first-order logic (FOL) begins when wegeneralize basic propoare sitions. In fact, it is not stretching t hingsto saythatbasic tautologies tautologies because theyare instancesof first-order tautologies.When you prove a basictautology you feelthatyou have not e xhaustedyourreasoning
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in thatdirection . To illustrate thisI will give a basic t autology andthengive four first -ordergeneralizations . ? If threeexceeds twothentwo is exceeded byt hree. ? Everynumberexceedingtwo is anumberthattwo isexceededby.
? Everynumberthatthreeexceeds is exceeded byt hree. ? Everynumberexceedinga givennumberis a numberthegiven number is exceeded by. ? Everynumberthata givennumberexceeds is exceeded bythegiven number. is logically derivedfrom We areinclinedto thinkthatthebasic tautology its generalization , e.g., that"if threeexceeds twothentwo isexceededby three"is truebecause "everynumberexceedingtwo is onethattwo is exceeded by" istrue... thusemphasizingthefactthattheformeris no peculiarity ofthree.Likewise weareinclined to t hinkthatthelatter generalization is truebecauseof thetruth of its generalization, viz. "Everynumberexceeding an arbitrary n umber is one thatthearbitrary n umber is exceededby" ... thusemphasizingthatno peculiarity of two is involved. The first order sentences of arithmetic (FOSA) are thesentenc es obtainand takingtruth -functional able fromthebasic sentencesby quantification combinations . It is importantt hattheseoperations aretakenrecursively,.g., e a nd thencombined withothergenerala basic sentencecan be generalized izationsby truth-functional combinationsand thengeneralized a gainbefore takingfurther t ruth-functional combinations . The first-order propositions of arithmetic (FOPA) arethepropositionsexpressedby thefirst-order sentences . Below is anexampleof one ofthesimplestvalid interpreted in theusual way argumentsin first-order logic. Every numberis eithereven or odd. No numberis botheven andodd. Everynumberwhich isodd is one whosesquareis odd. ? Everynumberwhosesquareis even isitselfeven. Thereis a radicalincreasein expressivepower offirst-order languagesas comparedto basiclanguages.Forexample,eventhefirstpremise in theabove argumentimplies infinitely m any basic consequencesb utit is not implied by any numberof its basicconsequences,not even by all themtogether of . The idea thata generalization is logicallyequivalent to theset of itssingular instancesis butone ofthefallaciesthatis to beconfronted by thoseseeking to reducefirst-orderlogic to basic logic . Below ar e a few ofthesingular instancesof thepropositionunderdiscussion.
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One is eitherodd or even. Two is eitherodd or even. Threeis eitherodd or even. As mentionedabove basic logic issometimes called finite logic because each of itsconsistent (or non-contradictory) propositionsis finitelysatisfiable. This is no longertrueof first-orderlogic. Indeed, theconjunct ion of the following two propositionsis notsatisfiable in any finite universeof discourse. Zero isnotthesuccessor of any n umber. of distinctnumEvery twonumberswhicharesuccessorsrespectively bers arethemselvesd istinct . propositionwhich isconsistent It is known , however,t hateveryfirst-order is satisfiablein a countable universe of discourse. In fact , everyconsistent first-order propositionthatis notsatisfiablein a finiteuniverseof discourse numbers. is, like the above conjunction, satisfiablein theuniverseof natural For thisreason,first-order logic can be called countable logic. Just as every validbasic argumentis deducibleusing a small set of axioms and rules of inference , thesame is trueof validfirst-order arguments. validityof first-order a rgumentsis This means thatas faras knowledge of concerned,humanknowing faculties are equal to thetask.Theso-called printhateverytruepropositioncan be known to ciple of sufficiency reason,viz. of truth are not be true,can be shown to be false. Human faculties of knowing -truth o utrunsknowledge . With validity equal tothetaskof knowingtruth of first -order aryuments, reasonis sufficient --everyvalidfirst-order argument can be known to be valid. Whetherevery validargument(whatevertheorder) can be known to be valid is question a of considerable complexityand exposition. well beyondt hescope of thiselementary Thereis another much lessimportantfactaboutfirst-order and basic logic thatis worthmentioning . Forthiswe have to divide the logical conceptsinto positive and negative . Withoutgoing into the details , let me saythatthere are nosurpriseshere. "Every", "Some" , "Is", "And", "Or", "If ' , etc. are positive. "Not", "No", "Distinct","Nor", etc. are negative . The resultis thateverycontradictory first-order propositioninvolves atleastone negative logicalconcept. Justas we motivatedthetransition from basic logic tofirst-order logic by thefactthatthe reasoningused toestablisha basic tautology reflecting on seems strongerthanneeded forthatpurpose and indeed is sufficient (or we use virtually so) to establishallgeneralizations of thebasic tautology, first-order logic.Considerthe following the same sortof insightto transcend first-order propositions .
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JOHN CORCORAN . No numberdivides exactlyt henumbersthatdo not dividethemselves No numberprecedesexactlythenumbers thatdo notprecedethemselves. No numberexceedsexactly t henumbersthatdo not exceed themselves . No numberperfectsexactly thenumbersthatdo notperfectthemselves.
The relation of perfectingarises inconnectionwith theso-calledperfect numbers. Every numberhavingproperdivisors isperfectedonly bythesuccessor of the sum of its properdivisors. The othernumbers, viz. zero, one and the primenumbers,are notperfectedby any numbersat all.Thus four is perfectedby threesince two istheonlyproperdivisor of four.B utsix is perfectedby itself. In fact, as you may have seen already,perfect every number perfectsitselfand, conversely, every numberperfectingitselfis perfect. i ntroducing theperfectingr elation is to give anexample Now, thereason for butnot as of a tautology in thesame form asthefirstthreeof the above set propositionsis mathematically mathematically trivial.T he first of the above does not divide itself, is trivialbecause zero, which theonlynumberthat is divided by everyothernumber. The second istrivialbecauseeverynumber precedesothernumbersbutnotitself.The thirdis trivialforsimilarreasons. Now, as you know , each of the above can be deduced from theirown respectivenegationsby familiar(but intricate)reasoning.The fact isthat thefollowing premise-conclusion a rgumentis valid. Some numberperfectsexactlythe numbersthatdo notperfectthemselves. ? No numberperfectsexactlythe numbersthatdo notperfectthemselves.
A deductionof thisargument,i.e., a deductionof itsconclusionfrom its premise, can easily bet ransformed into anindirectproof of its conclusion. The reasonthata deductionof a conclusionfrom thenull-premiseset is a proof (i.e., a deductionwhose premisesareknown to betrue)is becauseuniversalpropositionswith null "subjects"are vacuously t rue. Everymember be . . therebeing nocounterex of thenull set of premises is known totrue. amples. Once one ofthesetautologies has been proved to be trueby a deduction from thenull set of premises theothersare alsovirtually proved to betrue also. The reason for this is the principleof form fordeductions:every argumentation in the same form as d aeductionis againa deduction . Thus a proofof, say,thefourthcan be obtainedfrom a proofof, say, the first by substituting in thelatter the conc ept "perfects " for theconcept"divides". So it is cleart hatthereasoningestablishing one ofthefourvirtually establishes much more.
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Now we move tothe second-ordergeneralization of theabove. Actually, thefollowing second-orderpropositionis atonce ageneralization of each of theabove four first ordertautologies and, in a certainreasonable sense, the onlygeneralization. No matterwhich numericalrelation you consider, no numberbearsit to exactlythenumbersthatdo notbearit tothemselves . t hatthisis trueyou will feel thatit is theground of Once you have seen thetruth of thepreviousfourpropositions,e.g., thatthetruth of thefourth of themdependson nopeculiarity of theperfectingrelation. My main pointin this essay isthatthereasoningin a given logic achieves more thancan beexpressedin thatlogicandthatthetranscending of a given logic by going to higherorderis a one way ofreapingthefullfruitof one's reasoningin a given logic . This vagueprincipleappliesnotjustto first-order in relation to basic logica nd to second-orderlogic inrelat ion to first-order butin generalto any logic in r elation to thenext lower order. In basic sentences,t hereareno common nouns. In first-order sentences , thereare common nouns, butno "second-ordernouns" such as'property', 'relat ion', 'function' , etc. The presenceofnounsinevitably a ndautomatically becausenounsrequire articles andarticles entailst hepresenceof quantifiers express quantifiers . For example, thefollowing sentencesexpress thesame proposition . Every falsepropositionimplies a trueproposition. Everypropositionwhich is false implies some propositionwhich istrue. For everypropositionwhich is falset hereexistsa propositionwhich is trueand which isimplied by thefalseproposition. The same phenomenoncan be exemplified intheuniverseof natural numbers(beginningwithzero). Every oddnumberexceeds an evennumber. Every numberwhich isodd exceeds somenumberwhich is even. For everynumberwhich isodd thereexistsa numberwhich is evenand which is exceeded byt heodd number. Whenwe move tosecond-orderby addingsecond-order nounswe alsoadd second-order adjectiveswhose rangesof significance are the second-order objectsdenotedby thesecond-order nouns.Examplesofsecond-order adjectives are thefamiliartermsindicatingpropertiesof relations:reflexive,symmetrical,transitive, dense, etc. The following are typicalsecond-order sentences involving such expressions.
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JOHN CORCORAN Everyreflexiver elation relateseveryobjectto itself. t hatrelates everyobjectto itselfis reflexive. Everyrelation Everysymmetricrelation r elates to eachotherevery twoobjectsone of which itrelatesto theother. Everyrelation t hatrelates to eachotherevery twoobjectsone of which it relatesto theotheris symmetric.
Orthogonality is a second-orderrelation betweenproperties.In orderfor one propertyto beorthogonal to anotherit is necessaryand sufficientthat therebe fourobjects,one havingbothproperties,one havingthefirstbut lackingthe second, onelackingthe first but havingthe second, and one lackingboth. Theseexamplesshow thatmuch ofthisessay has beenwritten using asecond-order l anguage . Since basic logic is finite and since first-order logic iscountable,neither is adequateto axiomatizetheorieswhose universes of discourseareuncountable. The most familiarexamplesof suchtheoriesarecalculus andgeometry. Now justas first-order logic is not finite, second-order logic is notcountable . Thereare consistentsecond-orderpropositionswhich are not satisfiablein any countableuniverse . One example is from Hilbert'saxiom set forthe theoryof realnumbers(which isfoundational forcalculus).Anotheris from Veblen's axiom set forE uclideangeometry.Naturally, second-order logic can be calleduncountable logic. First-orderlogic isnoteven adequateto axiomatizetheorieswhose universes ofdiscoursearecountably infinite . The paradigmcase ofsucha theory numbers, which requiresthe is numbertheory, ort hearithmeticof natural principle of mathematical induction (PM!) . Everypropertybelongingto zeroandto thesuccessorof everynumber to which it belongs also belongs to every numberwithoutexception. In orderfor apropertyto belong to every number it is sufficient for thatpropertyto belong to t hesuccessorof everynumberhavingit and alsothatzero have it. Mathematical inductionis thesecond-ordergeneralization of each ofthe following propositionswhich areamong its first-order instances. If zero is evenand thesuccessorof every evennumber is even, then everynumberis even. If zero isperfectand thesuccessorof everyperfectnumberis perfect, then everynumberis perfect.
In first-order axiomatizations of arithmeticPMI, induction,is replacedby theinfinite set of its first-order instances,a set which is insufficient to imply
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mathematical induction.In fact, nosetof truefirst-order propositionsis sufficient to imply P MI andtherefore no first-order axiomatization ofarithmetic arithmetic . Moreover,t hegroundof our adequately codifiesourknowledge of knowledge of t heinstancesis ourknowledge ofP MI itself. Thus infinitely manyof the so-called axioms offirst-order arithmetic a renotaxiomaticin the traditional sense. Nevertheless, thereare able logicians and mathematicians who reject the traditional second-order a xiomatizations due toDedekindand Peanoin favor offirst-order axiomatizations whichdatefrom the1930's. Justas second-order logic isnecessaryto fully exploitfirst-order reasoning as well as to understand thegroundoffirst-order tautologies, likewise secondorderlogic isnecessaryto fully exploitfirst-order knowledge inarithmetic as well as tounderstandt hegroundof acceptanceof first-orderaxiomatizationsof arithmetic . Even logicianswho rejectsecond-ordera xiomatizations of arithmetica dmit theirhistoricimportanceand make heuristicand pedagogical use of such axiomatizations.By theway, thesame thingmay be detailsinvolved in set said ofaxiomatizations ofsettheory,b utthetechnical theoryrequiredistinctionsa nd principleswhich gobeyondthescope ofthis essay. In thecase of basic logic, as well that as of first-order logic, a small set of simple rules of inferencessuffices toenableevery validargumentto be deduced. This is no longerthecase withsecond-orderlogic. In fact, it is a corollary tothefamous G6delIncompletenessT heoremthatno simple set of rules is sufficient for thispurpose. This means thattheprincipleof sufficiency of reasonwhenappliedto second-order validityis false. To beexplicit , there are finite valid argumentsin second-order logic whoseconclusionscan not be deduced(in a finitenumberof steps using simple rules) from t heirpremisesets. This resultis known asthe incompleteness of second-order logic. Thereare logicians who feel thathumanreasoningmust be equalto the taskof determiningthevalidityof validarguments.In most cases such logicians areempiricistically o rientedand are fully willingacceptthefact to thattherearetruepropositionsaboutthematerialuniversethatcannotbe amenableto known to betrue. But theyfeelthatvalidityis intrinsically analytic a priori methodsand, in particular, thatevery validargumentmust be deducible. One way o utof this quandaryis to denythatsecond-order logic is really logic . Incidentally, second-order a xiomatizations do not evadet heincompletability ofarithmetic . First-order axiomatizations are deficient becausefirst-order languagesare too weak to expressourknowledge of arithmeticeven though validity . Withinsecond-order first-order reasoningis adequateto first-order thesituationis reversed.Second-order a xiomatizations are deficient because second-orderreasoningis too weak to deduce all the of consequencesof second-orderaxioms even thoughsecond-orderlanguageis strongenough to expressourknowledge of arithmetic . In fact,oursecond-order a rithmetic a rithmeticproposition knowledge impliesa bsolutely everytruesecond-order
74
JOHN CORCORAN
even thosethatwe are powerless to deduce(using any givensimple set of rules fixed in advance). Anotherphenomenonthatgives some logiciansd oubtsaboutsecond-order logic isexistenceof contradictory propositionsdevoid ofnegativelogical concepts. Recall t hatin first-order logic everycontradictory propositioninvolves at least one negativelogicalconcept. Below are two second-orderpropositionsthefirst of which tautological is andthesecond of which is contradictory, neitherof which involve negativelogicalconcepts. Everyobjecthas atleastone property . Everypropertybelongsto atleastone object. The reasonthatthesecondpropositionis self-contradictory is thatit contradictst hefollowing tautology. No objecthas thepropertyof beingdistinctfrom itself. We have seenthatsecond-orderlogic differsradicallyfrom first-order. First-orderis a logic ofcountability; second-orderis a logic ofuncountability. First-orderis deductively complete; second-orderis deductively incomplete. Infirst-order everycontradiction is negative;in second-order t hereare self-contradictory propositionswhich are exclusively positive. The abovementionedhistoricexamplesofaxiomatized sciences remind us thathigherorderreasoningis not arecentinnovationbut rathera featureof human thoughthavinga longhistory . Moreover, it isn otthecase thatlogicians startedo utstudyingbasic logicand thenmoved on tofirst-order a nd then tosecond-order, etc. Inthefirst place,Aristotle's logic is afragmentof firstaspectsof basic logic were not to discoveredfor be orderand fundamental m oderntimes higher-order logics some centurieslater.In thesecond place, in werestudiedbefore first-order logic wasisolatedas a system worthyofstudy in its ownright. AfterAristotle 's logic had been assimilatedby laterthinkers,people emergedwho couldnot accepttheidea thatAristotle's logic was not comprehensive . These conservative logiciansattemptedto "reduce"all logically cogentreasoningto Aristotle'ssyllogisticlogic. Likewise , afterfirst-order logic had beenisolatedand had beenassimilatedby thelogiccommunity, people emergedwho couldnotaccepttheidea thatfirst-order logic was not comprehensive.Theselogicians can be viewed notconservatives as who want to reinstate an outmodedtradition b utratheras radicalswho wantto overthrowan establishedtradition . It remainsto be seenwhetherhigher - order logic will ever regain thedegree ofacceptancet hatit enjoyedbetween1910 and 1930. Buttherehas never been saeriousdoubtconcerningits heuristic andhistoricimportance . In fact, people who do not know second-order logic t hemoderndebateover itslegitimacyand theyarecutcan notunderstand off fromtheheuristicadvantagesof second-orderlogic. And, whatmay be
SECOND-ORDER LOGIC
75
worse,theyare cut-off from an understanding of thehistoryof logicandthe historyof mathematics , andthusareconstrained to havedistortedviews of thenatureofthetwosubjects.As Aristotle first said , we do notunderstand a disciplineuntilwe have seen itsdevelopment . It is atruismthata person's conceptions of whata discipline is and of whatit can become are predicated on aconceptionofwhatit has been . ACKNOWLEDGEMENTS This essayis based on atutorial thatI led at the Ohio UniversityInference Conference,October9-11,1986. I am indebtedtoProfessorRichardButrick not only for organizingtheconferencebut also fororganizingme . In the yearpriorto theconference he suppliedme, by phone and in writing,with dozens ofquestions,hypotheses,suggestionsand requestsfrom whichthebasic contenta ndgoals of mytutorial emerged. James Gasser, Woosuk Park and Ronald Rudnicki helped with theeditingand proof-reading . I am also indebtedto thestudentsa nd colleagues withwhom I studiedsecond-order logic over the years, especially George Weaver, StewartShapiro, Michael Scanlanand EdwardKeenan. Verylittle in this essay is original. Most of whatis here isalreadyin thewritingsof Alfred Tarski, Leon Henkin, Alonzo Church, Georg Kreisel and George Boolos. Almost every humanistically oriented essay on modernlogic isindebtedto Tarski,Church,a ndQuine. Their technicalcompetence,theirobjectivity,theircreativity and, above all, their constant a ttention to thehumanimportanceof logic are largely responsible t hattracesback forpreserving,transforming and revitalizing a richtradition to Aristotle . One sign ofthevitality of thetradition is thefactthatnot one of theabove-mentioned logicians agrees with everything w ritten in this essay , even when the heuristicover-simplifications are emended.
J. MICHAEL DUNN
A REPRESENTATION OF RELATION ALGEBRAS USING ROUTLEY-MEYER FRAMES*
Abstract.A represe ntationof relationalgebras is given by modifying the Routle y-Meyer semanticsfor re levancelogic. This semantics is similar to the Kripkesema nt ics forintutionistic logic, butits frames use a ternary accessibility relationinsteadof a binaryone. The representation is foreshadowedin thework ion of Booleanalgebraswithop erators of Lyndon as well as bythe representat by Jonsson and Tarski, but the aim hereis to make it explicitand to connect it to the relatedrepresentations of alge bras arising in the studyof relevance ," e.g. , linearlogic. A philosophical logic as w ellas other"subst ruct ura l logics interpretati on is givenof thereprese ntation, showingthatan element of aelation r algebracan be understoodas a set of relations , rathertha n as the intended interpretation as a singlerelation . It is shown how aRoutley-Mey er frame can be representedso it s statesare relations , and an interpretation is provided in databas e" . Connectionsa re mentioned to re centwork by terms of a "relational Jon Barwiseon "informationchannels"between"sites" and by J . M. Dunnand R. K . Meyeron a ternary f ramesemanticsfor combinatorylogic. DEDICATION
This paper was originally dedicatedto AlonzoChurchon theoccasionof sixtihis ninetiethbirthdayand toRobertK. Meyer on the occasion of his ethbirthday . The fundamental inspirationforthispaper comes from these two. The most directdebt is to Meyer or, more obviously, to Meyer and RichardRoutley(who less obviously changedhis name toRichardSylvan). I shall here be extendingthe "Routley -Meyer"semanticsfor relevance logic (Routley and Meyer 1973) to provide arepresentation forrelation algebras. an influence Church is . His "weak Less directlybut no lessimportantly theoryof implication " (Church 1951 was notonly a seminalp aper in the developmentof relevance logic, but more precisely , his notionthatthereis an appropriatedeductiontheoremwhich holds for "relevant deducibility"in(Dunn 1966) of thealgebraof relevance logicbased as fluenced mytreatment on residuation , as did also hisdefinitionof a non-extensional "conjunct ion" (variouslylabeled"intensional conjunction","consistency","cotenability ", and "fusion" in thesubsequentrelevance logic literature) .
* A versionof thispaperwas presentedatthe Australasi anAssociation for Logic's annua l , July, 1992, and essentially t hepresent versionwas distributedin the meeting, Canberra IndianaUniversity Logic Group PreprintSeries in August, 1993 (IULG-93-28). I wish , Jon Barwise, Dirk van Gucht,Anil Gupta, Roger Maddux, and to thankGerryAllwein VaughanPrattfor helpfulcomm entsand/orinformation . 77 C. Anthony Anderson and M. Zeleny (eds.), Logic, Meaning and Computat ion . 77-108 . © 2001 Kluwer Acade mic Publishers . Printed in the Netherlands .
78
J. MICHAEL DUNN
in a At theend of thispaper, Church,Meyer, and Routleycome together kind ofgrandfinale, where describehow I theideas oftherepresentation of relation algebrashave been modified (in raecentpaper I havewrittenwith Meyer) so as toprovideRoutley-Meyer frame models for Church'sA-calculus . bothChurchand Sylvan have Since theoriginaldedication,r egrettably died. I nowdedicatethis paper to boththeirmemories, and of course to Meyer inhappiercircumstances. 1. INTRODUCTION
The calculus of relatives (relations) originated in thelasthalf of t he19th centuryandwas developed by DeMorgan,Peirce, and Schroder . Good sources forbothmathematical and historical information are Turski and Givant 1985, and Maddux 1991. Cf. alsoPratt 1990 and van Benthem 1991. Tarski(1941) 1 and askedwhether every provideda set ofpostulates for a"relation algebra" on some domain. relation algebracould berepresentedas a set of relations Relationalgebrashave received Lyndon(1950) provideda counter-example. renewedattention in recentyearsbecauseof aninterestin a moredynamic conceptionof logic which incorporates "actions ", i.e., relations betweenstates (cf., e.g., Pratt 1991 and van Benthem 1991). Sometimes this isunderstood as "informationbased" withrelations servingas "channels"forinformation interestto database (cf. Barunse 1993) . Relationalgebrashave also been of theorists(cr. K .-D. Kmegeloh and P. C. Lockemann 1976, and M. Gyssens, L. V. Saxton, and D. van Gucht 1993). Routleyand Meyer (1973) provideda "Kripke-style"semanticsfor rele vance logic using ternary a accessibilityrelation.De Morganmonoids had been introducedin Dunn 1966 to provide an algebraiccounterpart of the paradigm relevance logic R, and Routleyand Meyer (1973) give arepresentation of De Morgan monoids using theirternaryrelation . Routleyand Meyer (1973,cr. alsoRoutley and Meyer 1972 and Meyer and Routley 1972) indicatethatone can vary features oftheirframes andcorresponding features a lgebrais of eitherthe logicR or its DeMorganmonoid models. Arelation verysimilarto a DeMorganmonoid, and we show herethatrelation algebras can berepresented with onlyslightmodificationsof theRoutley-Meyer R. frames used for We do, however, have to modify theconstruction ofRoutley-Meyer (1973), leads only to immediaterepresentations ofwhatthey since thatconstruction call "prime" DeMorganmonoids.? with"representations " ofgeneral De Morgan monoidsobtainedas subdirectproductsof theserepresentations . The same would hold t rueof relation algebras.We, however , obtaina directrep1 And oth er setsof axioms producedby Tarskiand hisst udents wh ich arediscussed by Maddux(1991) . 2 A De Morganmonoid is prime, when fortheidentityelemente, e ~ a V b implies e ~ a or e ~ b.
A REPRESENTATION OF RELATION ALGEBRAS
79
resentation of allrelation algebras(andthesame can be done for De Morgan on aRoutley-Meyer framethatthere monoids) bydroppingtherequirement " 0, and replacingit with a setZ thatmay be a distinguished"zerostate containmany "zerostates". In Dunn 1933a thereare parallel discussions thathelp provide acontextforunderstanding this andotheraspectsof the current p aper. I discovered only a fterworkingouttherepresentation resultsof this paper thatthese resultswere foreshadowed by Lyndon, and Jonsson by and here is moreexplicitand alsohas thepoint of Tarski, thoughthetreatment locatingrelation algebrasand theirrepresentation withinthegeographyof substructural logics. Lyndon(1950) showed how every finite relation algebracan beanalyzed atomswithcompositionand so thateachelementis decomposedinto a join of conversion defined on theatoms. In this representation, a natural ternary relationarises ontheatoms. It is well -known thattherepresentation of a finite Boolean algebraas a field ofsetscorrespondsto itsdecompositioninto joins ofatoms(let each such joinbe replacedwith thesetofatomsoccurring in it), and thatatoms correspondto themaximal filterst hatappearin the representation of arbitrary Booleanalgebrasdue toStone(1936) . Relations on such filters thencorrespondto relations on atoms. Cf. alsoMaddux 1991.3 Anotherforeshadow isc astby Jonssonand Tarski{1951-52). The first partof theirpaper shows, in general,how torepresenta Booleanalgebra . Theseoperatorsmust meet withn-aryoperatorsusing n+1 placedrelations theirpositionsand the conditionsthatthey distributeover join in each of thatif anyoperandis the Boolean0 thentheoperatormust outputO. In thesecond partof theirpaper theypoint out thatrelationalgebrasmeet theseconditions(with relativep roductand converseas the operators)a nd go on toestablisha numberofresults concerning relation algebras . Curiously , they never detailhowtheirgeneralr epresentation relates to theseresultson statefirst-order conditionson therelarelation algebras , and in fact, never t heabstract o perationscorrespondingto relative tionsused inrepresenting forrelation productand converse to assurethattheysatisfythe postulates algebras . These foreshadows fall "aft erthe fact" interms of thedevelopmentof thepresentpaper, whichhas as its immediatecause my own work on "gaggle theframework of Jonssonand theory"(Dunn 1991, 1993a), whichgeneralizes of a nd Tarski{1951-52). Relationalgebrasare gaggles because residuation the factthateach oftheiroperationsd istributesover meet or join in some way oranother(and notjustpreservingjoin, as was requiredby Jonsson and Tarski). While the work here has been heavily influenced themore by generalgaggle-theoretic approach,it is presentedin an independentway. 3Madduxreplacesthe ternary r elation a ob ~ cof Lyndonwithaob ~ c, which is more in synchronywiththeRoutl ey-Meyerrelat ion on filters .
80
J. MICHAEL DUNN
Also itshouldbe pointedoutthattheofficialgaggle-theory representation (as well ast hatof Jonssonand Tarski) would employ abinaryrelationin thealgebraico perationcorrespondingto converse,whereasthe interpreting representation here usesinsteada unaryoperat ion. The relationship between thesetwo styles of representations is exploredin Dunn 1993b. Relationalgebrascan bethought of as a kind of s ubstructural logicrelated to thecalculus of Lambek (1958) by theadditionof theBooleanoperators and converse." Cf.P. Schroder-Heister and K. Doseii 1992 forthegeneral frameworkofsubstructural logics, whichincludesrelevancelogic, linearlogic, and Lambek calculusa mong others.The presentrepresentation of relation for algebrasplacesthem firmly inthegeneralframeworkof representations substructural logics. The Routley-Meyer ternarysemanticsfor thefirst of thesehas been modifiedand extendedto linearlogic by Allwein and Dunn by Dosen (1992). Cf. alsoDunn 1993a. (1993) and to Lambek calculus Relationalgebrashave beenproposedas models fornon-commutative linear logic by severalpeople, withlinearn egationunderstood as complementof the converse.P To thebestof my knowledget hefirst to do t hiswas Vaughan Prattat theEdinburghJumelagein 1989.6 2. THE ALGEBRAS
We willintroducea standarddefinitionof relation algebras afterfirstintroducinga few moregeneral definitionswiththeaim ofsettingrelation algebras in anabstract contextt hatmakes cleartheir relationship tothealgebrast hat arise instudyingsubstructural logics. DEFINITION 2.1. A lattic e-ordered groupoid is a structure (L , /\ , V, 0), where (L , /\ , V) is a lattice, and 0 is a binaryoperat ion on L sa tisfying:
x
(y V z ) = (x
0
(y V z)
0
0
y) V (x
x = (y 0 x)
V
0
z) ,
(z 0 x) .
(1)
(2)
When 0 is associative,i.e.,
xo(yoz) = (xoy)oz,
(3)
we speak of a lattice-ordered semi-group. 4This Gentzensyste m was introdu ced by Lambek as a way of formaliz ing cert a in linguistic const ruct ions. 5W hile wr i tingmy di ssertati on in 1966 I consideredrelation algebras as m ode ls forthe syst em R of relevance logic, but theywere too weak. Not only is relative productnot com mutative, it al s o does not satisfy squa re in creasingness (R 0 R ~ R) , the algebrai c the logic alprincipl e of controction (c/> -> (c/> -+ 1/1» -+ (c/> -+ t/J), rejec te d counte r part of by linearlogic, and some relevan celogics, buta principl e of the sys te m R . I was able to obtaina representation of De Morganlattices (t he ext ensional or, in "linea rese ," additive fragmentof bothR and linear logic) in terms of setsof relation s closed under inte rsec t ion, union ,and comp lement of converse. See Dunn 1982. 6Cf. also Brown and GU'T'T 1995.
A REPRESENTATION OF RELATION ALGEBRAS
81
It follows from (1) and (2) that0 is isotonein each of itsa rguments . DEFINITION 2.2. (L,/\, V,o,e) is a lattice-ordered monoid when (L,/\, V,o) is a lattice-ordered semigroupand e is an "identity element" satisfying
(4)
e ox = x oe = x .
DEFINITION 2.3. Rightand leftresidualsare characterized as follows :" x
° y ::; z
iff Y ::; x
-+
z iff x ::; z
+-
y.
(5)
DEFINITION 2.4 PeirceGroupoid. A Peirce groupoid (semi-group, monoid) is a distributive-lattice orderedgroupoid(semi-group, monoid) thatis doubly residuated, i.e., has bothrightand leftresiduals . Peircegroupoidsarise naturally in thestructure of substructural logics, since residuationcorrespondsnaturally to theDeductionTheorem and its converse, withthe two residualsx -+ y and y +- x read as implications. Thereare twoimplicationsbecausethe"premiss bunching"operator ° is not necessarilycommutative.fWhen Gentzensequentsare understoodin the usual way as formed by anadicconstructions rathert hanpairing, associativityis a natural p roperty. The identityelemente is a way ofindicating "theorems",w ithe ::; a readas "a is a theorem". Cf. Dunn 1993a andother papers in Schroder-Heister and Dosen 1992. Peircemonoids correspondto the(associative)L ambek calculus whichpermitsempty left-hand sides, but Peirce monoids have t headditional connectives ofconjunction a nddisjunction withtherequirement t hattheydistribute over eachother . This requirement makes thename "Peircegroupoid" even moreappropriatebecauseof the emphasis thatPeirceplacedupon thedistributive laws. De Morganand Peirceobservedin thelastcenturythatthestructures we call"Peirce monoids" arise naturally in thecontextof the algebraof relations (cf., e.g., Maddux 1991). We thinkof relations in theusualway as sets oforderedpairs. Given a set Xand two binaryrelations Rand S on X, the relative product R ° S = {(x,y) : 3z«x,z) E R & (z,y) E R)} is an ( restricted to X). Then associativeoperation,a nd e is theidentityrelation R -+ S = {(x, y) : 'v'z«z,x) E R implies (z, y) E R)}, and S +- R = {(x, y) : 'v'z«x, z) E R implies (y, z) E R)} . DEFINITION 2.5 ProtoDe MorganMonoid. (L, /\, v, 0, -+, +- , rv, e) is a proto De Morgan monoid justwhen (L , /\, V, 0, -+, +-, e) is a Peircemonoid, and rv is a unaryoperationsatisfying:
a ° b ::; c iff b rv rv
a
=a
°
c::; rv a, (period two) rv
7ef. Ward and Dilworth 1939. 8The ideaof "premiss bunching " can be found inMeyer and McRobbie 1982.
(6) (7)
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J. MICHAEL DUNN
The readercaneasily seethatin thepresenceof (7) we mayrestate(6) in symmetricforms as follows. PROPOSITION 2.6. (6) may be replaced in the definition of a proto De Moryan monoid by (8) below, and also by (9):
a 0 b ~ c implies a 0 b ~ c iff b
0
rv
C
~
rv
rv
c0 a
a iff
~ rv
rv
(8)
b,
c0 a
~
"" b.
(9)
Note thatrv is a dual automorphism, i.e., 1-1, which follows from ,(7) and orderinverting,since bysubstituting e forb we obtain a
~
c iff
rv
c~
rv
a.
(10)
Dual automorphismsof period two are inthe relevance logic literature called DeM organcomplementssince theysatisfytheDe Morganlaws: ""(a V b) =
rv
""(a 1\ b) =
rva
a
1\
rv
b,
V < b,
(11)
(12)
A De Morganmonoid is justa protoDe Morganmonoid withthefollowing additional properties: (13) x 0 y = y 0 x (commutativity), x
~
x
0
x (square - increasingness) .
(14)
In the presence of commutativity t hetwoarrowscollapsetojustone, say-, and this is thealgebraiccounterpart to relevant implicationin thesystem R. 9 A protoDe Morganmonoid can also bedescribedas a distributive, nonassociative,n on-commutativeGirard monoid (cf. Allwein and Dunn 1993) since thealgebraicmodels forthemultiplicative-additive fragmentof linear R lackingdistribution logic differ from t hosefortherelevance logic justby andsquare-increasingness. t hinkof as "conGiven alattice,a unaryoperationX-I (which we shall verse") is anautomorphism of period two when it is a -1, 1 orderpreserving, and (x- 1 ) -1 = x (period two). (15) An alternative definitionis that-1 is of periodtwoandsatisfies: (XVy) -1 = X-I Vy-l,
(16)
(x 1\ y)-I = X-I
(17)
1\ y-I.
Notethatif L is a Booleanalgebrait can also be shownthat (_X)-1 = _(X-I).
(18)
(This follows easily from thewell-known fact thatin a Booleanalgebracomplementsareunique.) 9See Dunn 1966.
A REPRESENTATION OF RELATION ALGEBRAS
83
Meyerand Routley(1973, 1974) studytheresultof adding to relevance ". logic a"Booleannegation"in additionto theusual"De Morgannegation Meyer (1979) extendsthis evenfurtherby studying"Boolean-valued mod els" for relevance logic, in whichunderlying the l attices areBooleanalgebras (in place ofthe usual DeMorganlattices).In additionto the usual operationsof relevancelogic he alsointroducesan explicitunaryconnective correspondingto the"st aroperator ", and from it he defines De Morgan notation X -I in place of x · .) Meyer negation : ""x = -(x·) . (We will use the and Routleyshow thatBooleannegationcan beaddedconservatively in the sense thattheseBooleanalgebraicmodelsvalidatet hesame theoremsin the "relevant" vocabulary (obviouslyexcludingBooleannegationexceptin the context- (x·)) as do the De Morganmonoids.!" These resultsof Meyer and Routleycan be easilygeneralized to protoDe Morganmonoids andthe following algebras,which we name inhonorof Meyer:II DEFINITION 2.7 MeyerMonoid. A structure (L ,/\, V,o,-+,+-,-, -I,e) is a Meyer monoid justwhen (L, /\, V, -) is a Booleanalgebra,
(19)
is an automorphismof periodtwo,
(20)
(L, /\, V, 0, -+, +-, e) is a Peircemonoid,
(21)
X-I
a-+b= ""(",,boa) , b+-a= ""(a 0 ""b) when "-'x is defined as _(X-I) .
(22)
A Meyermonoid is essentially a protoDe Morganmonoid wheretheunderl at ticeis a Booleanalgebraa ndwhere"-' has beenreplaced lyingdistributive with Boolean c omplementand converse , from which it can be recovered by way ofthedefinition: (23) and An alternative way to view a Meyer monoid is takebothBoolean to De Morgancomplementsas primitiveandthendefine:
(24) DEFINITION 2.8 BooleanProtoDe MorganMonoid. A structure(L, /\, V, 0, -+, +-, -, -I, e) is a Boolean proto De Morgan monoid justwhen
(L, /\ , V,
-)
is a Booleanalgebra,
(25)
(L, /\, V, 0, -+, +-, r- , e) is a De Morganmonoid,
(26)
- "-'x = "-' -x.
(27)
IOThis see ms not quite an embedd ing result , except atthe level of thefree algebras,the y elements. issue being preservationof the identit 11 It isfitt ing that Meyer have a class of alge brasnamed after him , since he has populated theliteraturewithChurchmonoids , Dunnmonoids , etc. Cf. Anderson .and Belnap et al. 1975.
84
J. MICHAEL DUNN
THEOREM 2.9. A Meyer monoid becomes a Boolean proto De Morgan monoid given the definition (23). Proof. -
rv
X
= _[_(X-I)] = -[( -X)-I] =
rv
- x.
X = -([-(X- I )J-I) = - - X- I- I = x. a 0 b S c iff b S a -+ c iff b S -[( -c- I 0 a)-I J iff (_c- I oa)-I S -b iff _c- I oa S _b- I iff rvcoa S rvb. rv rv
THEOREM 2.10. A Boolean proto De Morgan monoid becomes a Meyer monoid given the definition (24). Proof. That-1 is an automorphismfollows from t hefactthateach of - and
are dualautomorphisms . That-1 is of period two follows from (27) and thefactthateach of - and rv are ofperiodtwo. We leave to thereaderthe proofof a -+ b = rv(rvboa) and itsdual,thehint being to use (6), (7), and (5) . ~ rv
DEFINITION 2.11 Pre RelationAlgebra . A pre relation algebra is a structure(L, /\, V, 0 , -1, e), where the structure (L, /\, V, 0, e) is a distributivelattice orderedmonoid and x-I is an automorphismof periodtwosatisfying e- I = e (xoy)-I =y-I OX-I .
(28)
(29)
The following definitioncomes ultimately from Tarski,thoughit is essentiallyin a form due to Birkhoff (1967), which emphasizes residuatton.P DEFINITION 2.12 Relationalgebra.A structure (L, /\, V, 0, -+, +-, -, -1, e) is a relation algebra justwhen (L ,/\,V,-) is a Boolean algebra, (L, 1\, V, 0 , -
1, e)
is a pre relation algebra,
(L ,/\, V,o,-+,+-,e) is a Peirce monoid, a -+ b = _(a- I 0 -b) , b +- a = -( -b 0 a-I).
(30) (31) (32) (33)
REMARK. Notethata prerelation algebradoesnotbecome arelation algebra simply by addingthe requirementt hatL be a Boolean algebra. We must also addthattheadditionof Booleancomplementto relativeproductand converse allows us to define residuals as in (33). above Notetheintriguing similarityof theearlier(22) with (33) above. To eyes trainedin relevance logic,thelatter seems a "typo" version of the former . We show nextthat these are in fact thesame. This is thekey link between relation algebrasa nd relevance logic. 12Cf. thedefinitionof Jonsson and Tarski and otherdefinitionsdiscussed by Maddux (1991) . The definitionbelow does not make it obvious thatrelation algebrashave an equational definition,b utall ofthe algebrass tudiedhere do.
A REPRESENTATION OF RELATION ALGEBRAS
85
THEOREM 2.13. A structure (L,I\,V,o,-+,i-,-,-\e) is a relation algebra if and only if (i) it is a Meyer monoid, and (ii) (L , 1\, V, 0, - 1 , e) is a pre relation algebra. , it is clearthatit suffices to show Proof. Upon comparingthedefinitions thatthetwo "definitions"of thearrowsin (22) and (33) areequivalent in a pre relation algebra,i.e., _[(_(b- 1 ) 0 a)-I] = _(a-I 0 -b), i.e., (34)
(_(b-
I ) 0 a)-I
=
a-I 0
-b.
But(35) follows from (15), (18) , and (29).
(35) -J
3. THE FRAMES
We nextdefinethestructures outof whichtherepresentations will be constructed . DEFINITION 3.1. By a frame we shallmean a pair (U,R), whereU is a nonempty setand R is a ternary r elation on U, i.e., R ~ U 3 . We augmenttheseframes withan explicitpartialo rder.P DEFINITION 3.2. By an articulated frame (forshort,a-frame), we shallmean a triple(U,~, R), where (U,~) is a non-emptyposet and R is a ternary relation on U, subjectto thefollowing conditions : Raf3, &, ~,' implies Raf3,' ,
(36)
Raf3, & a' ~ a implies Ra'f3"
(37)
Raf3, & f3' ~ f3 implies Raf3',.
(38)
We shallrefer totheelementsof U as "information s tates"(or sometimes "points") and we shall t hinkof ~ as the "informationorder,"interpreting a ~ f3 as saying that"f3 containsall oft heinformation in a" . Note thatany framecan be viewed as aspecialcase of ana-framewhere ~ is justtheidentityrelation r estricted to U. Adaptingthenotionof Meyer andRoutley1973, 1974, we shall call such a-framesclassical. In thispaper we shalloftentacitlyassume when wetalkof a frame(simpliciter) thatit has been madeaugmentedin thistrivialway ascontextrequires. Everya-framegives rise to aPeircegroupoidwhich weshallcall aframed Peirce groupoid. Thus letP(U) T be theset ofcones of U, where acone A is a member of thepowersetP(U) such thatVa, b(a E A and a ~ b implies bE A) . For A, BE p(U)T, we defineoperationsas follows : A 0 B = {X : 3a E A , f3 E B : Raf3x}
(39)
A=> B = {X: Raxf3 & a E A implies f3 E B}
(40)
B {::: A = {X: Rxaf3 & a E A implies f3 E B}.
(41)
13For Routleyand Meyer thiswas implicit, definedas ROOt{3.
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J. MICHAEL DUNN
'*,
'*,
DEFINITION 3.3. We shallcall(P(U) 1, n, U, 0 , ~) , with0 , ~ defined as in (39)-(41), the full framed Peirce groupoid on the a-frame (U,~, R). Any collection of members of P(U) T thatis closedundertheseoperations ones) will be calledframed a Peirce groupoid. (includingthelattice The name "framed Peirce groupoid" was chosen inanticipationof the factthatis easy to verifyt hatthis is in fact adistributivel attice-ordered residuatedg roupoid(with ~ as thelatticeo rder, and nandU as themeet andjoin), andso we havethe following. PROPOSITION 3.4. A "fram ed Peirce groupoid" is a Peirce groupoid. In orderto get aframed Peirce monoid we need bothassociativityand a two-sidedidentity , and we must impose more conditionson an a-frame. We developtheseby stagesso thatif anyonedesirestheycan abstract o uta numberof intermediate structures. We leave tothereadertheverificationthatone gets associativity(and hence a semi-group)by making thefollowing r equirement(as in Routley and Meyer 1973) on theaccessibilityrelation: 3x(Ra{3x & RX,~) iff 3x(Rax~ & R{3,X),
(42)
or inthenotation i ntroducedby Routleyand Meyer, R2(o{3h~ iff R20({3,)~ (R 2-associativity) .
(43)
We shall call any frame satisfying(42) associative. PROPOSITION 3.5. A framed Peirce groupoid defined on an associative aframe is a Peirce semi-group. In orderto get anidentityelement(andso a monoid), we need asetZ so that
Z 0A=A, A 0Z=A.
~
U,
(44) (45)
Given anon-emptycone Z of U, an a-framewill be called ordered in its 1st position by Z when 3( E Z(R(a{3) iff
~
(3,
(46)
0 ~
{3.
(47)
0
and ordered in its 2nd position by Z when 3( E Z(Ra({3) iff
DEFINITION 3.6. An assertional a-frame is a structure ( U,~, R , Z) where Z ~ U satisfiesboth(46) and (47), i.e., 3( E Z(R(o{3) iff
0
~
(3 iff 3( E Z(Ro({3)
(ordered).
(48)
A REPRESENTATION OF RELATION ALGEBRAS
87
We shallalso say thatan a-frame (U,~, R) is assertional if 3Z ~ U such that(U,~ , R, Z) is an assertional a-frame. Withrespectto a subset Z , an a-framewill becalled1st position weakly reflexive 14 if (49) "10: E U,3( E Z, R(o:o:, and 2nd position weakly reflexive if "10:
E
(50)
U,3( E Z , Ro:(o:.
LEMMA 3.7. An a-frame ordered in its 1st position by Z is 1st position weakly reflexive, and similarly, for the second position. 15
-t
Proof. Substitute 0: for j3 in (46) and (47).
REMARK. Note thatgiven our conventions , a frame simpliciter (U, R) is justwhen 3Z ~ U suchthat: assertional 3( E Z(R(o:j3) iff
0:
= {3 iff 3( E Z(Ro:({3)
(equality - ordered).
(51)
PROPOSITION 3.8 . A framed Peirce groupoid defined on an asseriumal aframe has both left and right identity elements.
Proof. We firstshow (44) . Suppose thatX E Z 8 A , i.e., 3( E Z, 0: E A , R(o:X. We know from orderingin thefirstposition (48) that0: ~ X. But since A is hereditary, then X E A. For theotherdirection, let us suppose thatX E A. For x E Z 8 A , we need that3( E Z , 30: E A, R(o:X. Butby Ist-positionreflexivi ty (49) , we have3( E Z , R(XX. The argumentfor (45) is similar,using thecorrespondingpropertiesfor the secondposition. -t The following follows from thepreviouspropositions. COROLLARY 3.9 . The fram ed Peirce groupoid defined on an asseriional associative a-frame is a Peirce monoid. 4. INVOLUTED FRAMES
We nextput conditionson a frameso as toaccommodateconverse . Let us suppose thereis a map U on U intoitselfs uch thatfor X E U, X U U = X (period two).
(52)
14Note the orderof the quantifi ers. If they were revers ed we would talkof "strong e Morgan m onoid, indeedin any reflexivity" in eachposition . They can be so reversed in a D Peir cemonoid satisfy ing squa reinc reas ingness(14). Moreoverin De Morgan monoid s, with theircommutativity (13), ther e is no need to distingui sh reflexivity in the two position s. 15Indeed, 1st position weak reflexi vity(49) can replacethe condition alfrom rightto left in the defin it ionof 1st p osi t ion orde ring(46), andsimilarl y for the 2nd position. Cf. Dunn 1993.
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J. MICHAEL DUNN
Such amap is commonlycalled aninvolution. 16 We alsoadd (writingX in place ofXU): (53) Rail, only if R/3iFt (tagging) . Let us define for A E P(U) r,
A-I={a :aEA}.
(54)
By virtueof (52), thisis equivalent to:
A- l = { a :aEA}.
(55)
DEFINITION 4.1. A structure (U,~, R, u, Z) is a pre-relation-algebra a-frame justwhen (U,~, R, Z) is an associativeassertional a-frame, U is an involutionsatisfyingtagging,and Z is converse closed, i.e., if ( E Z then (E Z . We shallcall(p(U)T ,n,U,0, -I,Z) , with 0 , -1 defined as in (39), (54), the full framed pre relation algebra on thepre-relation-algebra a-frame (U,~,R, U,Z). Any collection of members ofp(U)T thatis closedunder pre relatheseoperations(includingthelattice ones) will be calledframed a tion algebra. PROPOSITION 4.2. A "framed pre relation algebra" is a pre relation algebra. Proof. It is practically carriedby thenotation t hat(A U B)-1 = A-I U B- 1 (and similarlyfor n and complementrelativeto U). And clearlyZ -1 = Z (because oftherequirementst hatZ be closedunderu, and U be of period two). So we only need verify:
(A
° B)-1
= (B-
1
° A-I) ,
(56)
using taggingand period two (whichlastallows us to t reatany arbitrary pointas having a breve over . it) Thus:
X E (A0B)- 1
iff X E (A0B) iff 3a E A,{3 E B ,Ra{3x iff
3{3 E B- l,a E A-I,R/3ax iff
XE
B- 10 A-I.
-1
To obtaina-framessuitableforprotoDe Morganmonoids we add to an involution (52), insteadof tagging :
Ra{3,,! iff R1a/3 (antilogism).
(57)
REMARK. Routleyand Meyer (1973) can state(57) simply as (57') Ra{3,,! only if Ra1/3, becausein a De Morgan monoid0 is commutative,and this t hatthe first two positionsof R permute. But is reflected by p ostulating withoutthis postulate we must be moresubtle . Using thefactthatU is of periodtwo, (57) can beanalyzed as twoconditionals (thetwo"correct"forms 16FollowingRoutleyand Meyer, thisinvolutionis customarily enotedby d • in the relevance logicliteratur e. Here we mostlyuse U becausein the literatur e on rel a tions• is customarilyused forthe"ancestral ".
A REPRESENTATION OF RELATION ALGEBRAS
89
of (57') thatarisewhen oneis carefulnot to build in permutation) :
Ra/3-y only if R1'ajj,
(57a)
Ra/3"1 only if R/31'a.
(57b)
DEFINITION 4.3. A structure (U, ~ ,R, u, Z) is a proto De Morgan monoid aframe justwhen (U,~, R, Z) is an associativeassertional a-frameand U is an involution satisfyingantilogism.(p(U)T, n, U, 0 , =>, <=, r-, Z), with0, =>, <= defined as in(39), (40), (41), and rvA defined as{a : a ¢ A}, 17 is thefull a-frame framed proto De Morgan monoid on theproto-De-Morgan-monoid (U, ~,R, u, Z) . Any collection of members of P(U) T thatis closedunder theseoperations(includingthelattice ones) will be calledframed a proto De Morgan mono id. PROPOSITION 4.4. A ''framed proto De Morgan monoid" is a proto De Morgan mono id.
Proof. We showthatA
A0 B
~
0
B
~
C iff
rv
C
0
A
~
rv
B.
C means thatforarbitrary a, /3, 'Y=
a E A & /3 E B & Ra/3"1 implies "I E C. rvC 0 A
~
(58)
'" B means thatforarbitrary "I', a , /3' :
"I' E
rv
1"
C & a E A & R"I'a/3' implies f3' E
¢ C & a E A & R"I'af3' implies
/3'
rv
B, i.e.,
¢ B.
(59)
(60)
Substituting l' for"I', jj for/3' andusing periodtwo, (60) isequivalent to "I
¢ C & a E A & R1'a/3 implies /3 ¢ B.
(61)
It is easy to seet hat(61) and (58) areequivalent by usingantilogismand thelogicalprincipleof contraposition . ., The readercan easily verify thatone gets a DeMorganmonoid by adding thefollowing requirernents .l"
Ra/3"1 implies R/3a"l (R -commutativity) ,
(62)
Raaa (R-reftexivity).
(63)
To obtaina-framessuitablefor Meyermonoids we in effect a ddtherequirementthattheybe classical.T he reasonis that,in general , thecomplementof a cone is not a cone . Butby requiringthat~ be simply =u (theidentityrela17This definitionof e- was firstdue to A. Bialynicki-Birula and H. Rasiowa (1957) and was studiedin my dissertation ( d. Dunn 1966) as partof thealgbra ic semantics of relevance ced into the philosophicalsemantics of relevan celogic byR. logic. It was explicitl yintrodu Routl ey and V . Routl ey, and made a partof the semanticsfor R by R. Routleyand R . K . Meyer. For history , see Dunn 1986, or Anderson, Belnap, and Dunn et al. 1992. 18 S ee Routley and Meyer 1973.
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J. MICHAEL DUNN
tionrestricted to U), everysubset(and itscomplement)becomes a cone since everysubsetis closedundertheidentityrelation.Then P(U)t = P(U). We getthesame effect of working withclassical a-frames by simply not requiring thattheframes bearticulated . Finally,Z satisfies(51) . DEFINITION 4.5. A structure (U, R, u, Z) is a Meyer-monoid frame exactly when (U, , <=, - , -1, Z), with0, =>, <= defined as in (39), (40), (41) and -1 defined as in(54). A framed Meyer monoid is any collection ofsubsetsof U closedundertheabove operations . PROPOSITION 4.6. A "framed Meyer monoid" is a Meyer monoid.
Finallywe get therequiredframes forrepresenting relation algebrasby simply addingtaggingto therequirements for Meyer-monoid a-frame. DEFINITION 4.7. A structure (U, R, u, Z) is a relation-algebm frame just when (U, R, u, Z) is a Meyer-monoid frame with U satisfying tagging,i.e., the structure (U, =u , R, u, Z) is an associativeassertional classicala-frameand U is an involution satisfyingbothantilogismandtagging . A framed full relationalgebm on arelation-algebra frame (U, =u, R , Z) is a structure (P(U), n, u, 0, =>, <=, - , -1, Z), with0 , =>, <= defined as in (39),(40), (41) and -1 defined as in (54) . A framed relation-algebm is any collection subsetsof of U closed undertheseoperations . PROPOSITION 4.8. A ''framed relation algebm" is a relation-algebm.
Proof. We must verify
A=>B=-(A- 10-B), B<=A=-(-B0A- 1 ) .
(64)
We show the first identity , the second following insimilarfashion a . It clearly suffices to show that -(A => B)
= (A- 1 0 -B).
(65)
We notethat X E -(A => B) iff 3a ,,8 Rax,8 & a E A &,8 ¢ B, and X E (A-l 0 -B) iff 31i,,8 RIi,8x & Ii E A-I &,8 ¢ B.
Since a E A iff Ii E A-I, theproofboils down to showing that Rax,8 iff RIi,8X·
(66)
This follows by first applyingantilogismand thentagging(with some periodtwothrownin): Rax,8 iff R(3ax iff RIi,8X·
(67) 4
A REPRESENTATION OF RELATION ALGEBRAS
9]
5. REPRESENTATION RESULTS USING ROUTLEY-MEYER FRAMES
It turnsout thatnot only can one produceframed relation algebrasgiven but up to isomorphism, every an associativeassertional classical -frame, a relation a lgebracan beidentifiedwith such a framed relation a lgebra . We define thecanonical a-frame as follows.T he pointsof thea-frameare of canonical accessibility (proper)prime filters . The following definitionsthe relation R can all be shown equivalent usingresiduaeion.'?
R o o:{3"Y iff Va ,b, if a E 0:& bE {3 then a o b E "Y, K ... o:{3"Y iff Va, b, if a E 0: & a
R..-o:/h iff Va, b, if b +-
b E {3 then bE "Y, a E 0: & a E {3 then b E "Y. -+
(68) (69) (70)
We definethe canonical information order 0: ~ {3 iff 0: ~ {3. It is easy to verify the"tonicityconditions"(36)-(38) Beforeproceeding,it isconvenient to pause andstatea couple oflemmas.P THEOREM 5.1 ExtensionLemma. Let F be a filter, I be an ideal, with
FnG = 0. Then there exists a prime filter F ' 2 F and a prime ideal I' 2 I , so that F ' n I' = 0.
THEOREM 5.2 SqueezeLemma. Let "Y be a prime filter, and let 0:0 and f30 be filters. Suppose R o O:o{3o'Y . Then there exist prime filters 0: and {3 such that 0:0 ~ 0:, {30 ~ {3 and R o o:{3o"Y , R o O:o{3'Y· THEOREM 5.3 ExclusionLemma. Let X be a prime filter, and let 0:0 and {30 be filters . Suppose R o Xo:o/3o, and suppose that b I/. /30 , Then there exists a prime filter {3 such that /30 ~ {3 (and so R o Xo:o(3 ) with b I/. /3, and similarly,
when R o o: ox {3o.
Proofsof thesemay in effect be found Dunn in 1986. The first ofthese derives fromBelnap. The second derives fromRoutley and Meyer 1972, Lemma 4, where it is called the "Primeness Lemma. " The name "Squeeze The thirdis an easyconsequenceof Lemma" is due to Routley, I believe. theExtensionLemma, lettingF = {3 and I = (b J = {x : x S b}. The proofs deal withtheories(sets of es ntences) presentedin thereferencedliterature rather t hanwithfilters,b utit is easy totranslate betweenthetwo. We are now in apositionto prove the following . THEOREM 5.4. Every Peirce groupoid is isomorphic to a framed Peirce group-
oid on an a-frame . Proof. Fortherepresentation, wetakethecanonical a-frame(U, R,~) (whose pointsareprime filters) and define the canonical isomorphism, h( x)
= [o E U
: x EO:} .
19Cf. Dunn 1993a. terminologyfrom latti ce theorymay be foundin Birkhoff 1967.
20 Basic
(71)
J . MICHAEL DUNN
92
It is well-known from Stone(1937) thatthisis a 1-1 map preserving/\ and V, carryingthemintointersection and union . It is also known, as a special case, fromJonssonandTarski(1951-52) thath preserves0, and Dunn(1991) has provideda generalization of thetheirwork whichcontainsas a special case thath preserves--> and.- as well. Butit is difficult to "see thetreesfortheforest" inthisearlierwork, and so we prove (72) h(a 0 b) = h(a) 0 h(b), h(a
-->
b) = h(a)
=}
h(b),
h(a .- b) = h(a) ~ h(b).
(73) (74)
Thuswe showthataob E X iff 30, {3 R oo{3x&a E o&b E (3. Right-to-left is immediate. For left-to-right, let ussuppose thata 0 bE X. The principal filterdeterminedby x, [x) = {y : x S y}. Let 00 = [a), (30 = [b). Because o is isotone,R oo o{3oX . ApplyingtheSqueezeLemma twice gives us prime filters0 , {3 such thatRo{3X. We nextshow a --> b E X iff "la, {3, R_ox{3 & a E a only if b E {3. Again, half isimmediate, thistime fromleft-to-right . We prove theotherhalf by : a --> b ¢ X only if 30, {3, R oox{3 & a E 0& b ¢ (3. provingthecontrapositive Let usassume thenthata --> b ¢ X. Set 00 = [a), {30 = {b' : 3x E X, a ox::; b'}. It is easy to seet hatthis is the filter generated from [a) 0 X. It is clearthatRoxoo{3oj for ifx E X and a S a', thenx 0 a ::; x 0 a' . Further it is clearthatb ¢ {3j forotherwise(residuation)for some x E X, x S a --> b, so a --> b EX, contrary to ourassumption. We nowapplytheExclusionLemma and thentheSqueezeLemma to obtainfirst{3 2 {30 and thena 2 00, so that R oox{3. The argumentthath preserves.- is similar. --/ 5.5. Every Peirce semi-grou.p is isomorphic to a framed Peirce semi-grou.p on an associative a-frame.
COROLLARY
Proof. We need only verify thatthecanonicalrelation R ; is associative,i.e.,
(75)
We show only the directionfrom left-to-right, the otherdirectionbeing analogous.Suppose R oo{3X & RoX'o. Set ()o =
{x : 3b E {3, dE,, bod S x}.
(76)
This is a filter, and it almostimmediate is thatR o{3,()o. We must show thatRoo(}oo. Suppose a E 0, and x E ()o. Then 3b E {3, c E " b 0 c S x. By virtueof R oo{3x, aob E X, and usingRoX,o, weobtainthat(aob)oc E" i.e, ao(boc) E ,. Butsince 0 is isotone, we have ao(boc) S aox, by associativity, and soa 0 x E , as needed. We can nowapply theSqueeze Lemma toobtaina prime filter() with R oo()8. Moreover, sincet heSqueezeLemma assuresus that() 2 (}o, Rofh(} as well, and we are through . --/
A REPRESENTATION OF RELATION ALGEBRAS
93
We nextgo aboutthebusiness ofaccommodatingan identityelement . We actually analyzet heproofforrequirements weakerthana full left and right identityelement,b utwe statethetheoremonly fort hisstrongest case. 5.6. Every Peirce monoid is isomorphic to a fmmed Peirce monoid on an associative assertional a-frame.
THEOREM
Proof. Canonically , we defineZ = {( : e E (}. We show thatif 3( E Z, R(o.{3, thena. ~ {3. Suppose 3( E Z, R(o.{3 and a E o. Then since e E (, eo a = a E {3. If a. ~ {3, thenclearly R o[e)o.{3; for if e:::; x anda E o, thena = eoa:::; xoa andso x 0 a E {3. Thenby theSqueezeLemma, 3( E Z, R o(o.{3. A symmetricala rgument,usingaoe = a, showsthat3 ( E Z, R oo.({3 iff a. ~ ~
~
Let us nowaddressthequestionof whathappens if a Peircemonoid has to converse . a unaryoperation-Ion it corresponding 5.7. Every pre relation alqebra is isomorphic to a framed pre relation alqebra on an associative assertional a-frame on which is defined an involution satisfying tagging.
THEOREM
Proof. Given aprime filtero, define:
a = {a : a-I
Ea'}.
(77)
ClearlyU is of period two, since-1 is of period two. Z ={( : e E (}. It is easy to seethenthatZ is closedunderu, for if eE (, thene- 1 = e E ( . Allthatremainsis to demonstrate t hattaggingholds inthecanonicalaframe. Let us suppose thatR oo.{3-y . In orderto showR o /Ja7, assume x E /J v ' and y E o, i.e., x -1 E {3 , y -1 E a.. S'mce R 00. {3 -y, Y- 1 0 x -1 = ( x 0 y) -1 E -y, i.e., x 0 y E 7 as needed. ~ We nextshow: 5.8. Every proto De Moryan monoid is isomorphic to a fmmed proto De Moryan monoid on an associative assertional a-fmme on which is defined an involution satisfying antilogism.
THEOREM
Proof. The onlytrickis how to define t heinvolution given thatwe no longer have anoperationcorrespondingto inverse.T he idea comes rightoutof the canonicaldefinitionof theRoutley-Meyer (1973) "staroperator" ,21 and for reasonsof historyas well as for making a contrast w iththedefinition(77), in thisproof(againremindingthereaderthat we shalluse thestarnotation it hasnothingto do withtransitive closure): a.* 21 It
= {a : 'Va rt o}.
comes even earlierfrom A. Bialynicki-Birula and H. Rasiowa(1957) .
(78)
94
J. MICHAEL DUNN
It is straightforward thattheoperationso-defined is ofperiod two, since ...... is. We must showantilogismholds onthecanonical frame. Let ussuppose thatR oO:P'Y, andshow thatRo'Y·O:p· . To thisend, let ussuppose thatd E 'Y., a E 0:. We need thatc' 0 a E p., i.e., ......(c' 0 a) f/. p. Let us suppose the opposite: rv(c' 0 a) E p. Since ...... (c' 0 a) = a ---. ...... c', we havea ---. rvc' E p. Since Ro = R-..... we havervc' E 'Y. Butthiscontradicts d E 'Y", The other halfof antilogismfollows analogously. --I
We nextshow how torepresentMeyer monoids. THEOREM 5.9. Every Meyer monoid is isomorphic to a framed Meyer monoid on an associative assertional classical frame satisfying antilogism.
Proof. The two newitems we must addressare classicality and antilogism. The first oft hesecomes fromthefactthata relation a lgebrais Boolean,a nd in thecanonical r epresentation, as in Stone 1936, thepointsbecome maximal filters, since in Boolean a algebra(proper) prime filters and m aximal filters coincide. Since in the canonical representation of conversealgebras,[;;; is just ~, clearly t herelation must reduce to identitysince onemaximalfilterc annot extendanother . t hatthecanonicalframe satisfiesantilogism . We We can now also show shalldepend heavily uponthe"presence of absence" which Bool ean compley f/. X with -y E X. The point ment gives us, allowing us to interchange is thatit is well -known thatin a Booleanalgebraa prime (proper) filterX (being maximal) must containexactlyone of an elem ent y or its Boolean complement. Thus assume RoO:P'Y. We must show R o1o:i!J. Let usassume thatx E 1 , a E 0:. Then x- i E 'Y. In orderto showx 0 a E i!J, we must show (x 0 a)-i E P, which we do byassumingtheopposite in orderto derive a contradiction . From (x 0 a)-i f/. P, it follow s ( "presenceof absence") that - [(x 0 a)-IJ E P, since -[(x 0 a)-I J = -[a-lox-I] = _[a- i 0 - - (X-i)] = a ---. _ (X- i ). (79) So a E 0:, a ---. _(X -I) E p. Butsince R; = R_, we have R _o:P'Y, and so we inferthat_(X-i) E 'Y, i.e., ("presenceof absence") x- i f/. 'Y. But this contradicts ourassumptionthatx E 1, l.e., x - i E 'Y. We must also showtheotherform ofantilogism,b utthisfollowsimilarly using f - . --I REMARK. An enlightening way to seetheproofabove isthatwe hadalready previouslyshown intheproofof Theorem5.9 thatthecanonicalframe for a protoDe Morganmonoid satisfiesantilogism . Since a Meyer monoid is a protoDe Morganmonoid, wemightexpectthisto immediatelytransfer over frame for a M eyermonoid. Butwe aremomentarily s topped to thecanonical by thefactthatwe had done this for 0:. , defined by (78),ratherthana, defined by (77). Butfor a Meyermonoid, 0:. = a. Becauseof the" presence of absence," 0:. = {a: ......a f/. o:} can be exchang ed witha = {a : a- i E o} .
A REPRESENTATION OF RELATION ALGEBRAS
95
Let us now pulloutallthestops and dealwiththefullset of postulates forrelation a lgebras.
5.10. Every relation algebra is isom orphic to a framed relation algebra on an associative assertional frame on which is defined an involution satisfying tagging and antilogism.
THEOREM
Proof. This canbe constructed by combiningtheproofsabove representing pre-relation algebrasa nd Meyermonoids. The chiefpointsarethatwe can now show thatthecanonicalframe satisfiesbothtagging (as we did for a pre relation a lgebra)a nd antilogism(as we did for aMeyermonoid), since a -1 relation a lgebrais both. 6. PHILOSOPHICAL INTERPRETATION OF THE REPRESENTATION
We nowdiscussa possibleinterpretation of theaboverepresentation for relain termsof ternary f rames tionalalgebras . At firstblushtherepresentation must be admittedto be more abstract t hanone wouldwant. An idealrepresentation of relation a lgebraswouldsend theelementsintorelations,but Lyndon (1950) has shownthatsuch arepresentation is impossible. It turns outthatwe arein onesense close tosuchan idealrepresentation, butwe are off atype level.Elementsarenotcarriedinto relations , butinto sets of relations. 22 In thepresentsectionwe shallexplainthephilosophicalintuitions behindthisstatement , and in thenextsectionwe shallmake propermathematicsof itand alsorelat e it tothecomputersciencenotionof a "relat ional database ". PeterWoodruffsuggestedearlyon thattheternaryrelationRaf3'Y that arises in theRoutley -Meyersemanticsfor relevancelogicshouldbe viewed as an "indexed" binaryaccessibilityrelationR{3a'Y. 23 An equivalentidea, withslightly differentm etaphysicalo vertones,is to viewf3 as itselfa binary relation.More accurately , we view f3 as somethinglike the Fregean"objectidentifyingf3 witha setof correlate" of abinaryrelation.By thusnotliterally orderedpairs, butrather t hinkingof it asdetermininga setof orderedpairs, we avoid unnecessaryset-theoretical problems aboutwhathappens when as withe.g., Rf3f3f3. We thusthink one has an otherwise"untyped"relation, of theternaryrelationRaf3'Y as sayingthatthe(binary)relationf3 relates (is exemplifiedby) the pair of terms a and 'Y. This has definiteBradleian overtoneswheretherelationst hemselvesare objectsneedingto be related to theirterms,24 but we stop theBradleianregress(as did G. Bergmann, 22 It is importa nt ot noti ce as wellthatthis difference of type-l evel distinguish es our approachfrom ot herso-called"relat iona lsema nt ics"such asthatof, say, Orlowska (1992) . 23 Actually Woodruffthought of the first position 0 as the index, but for re asons of notational convenience w e take the second position 13. The paper of Dunn and Meyer 1997 takes thefirstposition. 24Cf. Bradley 1897.
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J. MICHAEL DUNN
1967 25 ) atthesecond level,notrequiringthattherebe a yet higher-level relation r elat ing the"exemplification relation " R to itsterms. Thereis another , more "technological" interpretation of essentially t he same ideas. Barwise (1993) distinguishesbetween"sites" and "channels" ; roughly , thefirst beingsituationsbetweenwhichinformationis to be communicatedand thesecondbeing whatenablesthecommunication.Barwise uses thenotation S I ~ S2 to indicatethe3-placedrelation t hatexistswhen thetwo sitesSI and S2 areconnectedby thechannelc. In thenotationof Routleyand Meyer thiscould beexpressedas RS I CS2 . Channelshence determinebinaryrelations betweensites, butthey need not be identifiedwith thoserelations . Barwiseleavesexplicitlyo pen theidea thatchannelsmight be sitesand vice versa, which is ofcoursethesituationwiththeframes we have beenexamininghere.26 Formnemonic reasonsweshalloftenuse p anda forthesecond(relational) termof theternary relation, andsometimes suppress R, e.g., writingapfJ for RapfJ·
Before going on it is good to explicitlyintroducemodificationsof the notational conventions of Routleyand Meyerintroducedabove in (43): R(afJho
=def
3x(RafJx & RXy8),
(80)
Ra(fJ-y)o
=de/
3X( RaX 8 & RfJ-yX)·
(81)
RoutleyandMeyeractually write"R 2 " where wewritesimply "R" above, butfor reasons of bothsuggestivenessa nd simplicitywe shalloftensimply letthenumberof termsaffixed tell thatthereis a power. One of therequirementsimposed by Routley-Meyer is the requirement (see (43)) thatthesetwocompositionsareequivalent : R(afJho iff Ra(fJ-y)o (R-associativity) .
(82)
Let us definet hecompositionof two"binaryrelations" p and a; DEFINITION
6.1. a(Rp ® R(7)fJ iff 3x(Rapx & RxafJ).
Clearlyusingthisdefinitionand R-associativity, we obtain a(Rp ® R(7)fJ iff R(ap)afJ iff Ra(pa)fJ,
(83)
wheretheright-hand sides of thelasttwoconditionsare understoodusing thecompositionalnotations of Routley-Meyer. One couldwrite(83) as 250ur R servesthesa me role as his "nexus". 261 was working on the present representationof relationalgebras roughlyduringthe same p eriod thatJon Barwise was workingon his "sites and chann els" . Publicationof ed (see footnote*) . My work wasnot motivatedby his, but my paper had been delay rath er from the Routley -M eyer semantics for relevan ce logicand from my "gaggl e theory ", althoughJon Barwise and I have been discussing the connection betweenhis ternary relation a nd theternary relationof Routl e y-Meyer. More recentlyGregRestallh as noticed thisrelationship and has been workingoutsome of its details .
A REPRESENTATION OF RELATION ALGEBRAS
97
anddoes notdenote but one shouldbe clearthat "(pa)" is a mere notation an actualp oint. The above expresseswhatone might labela "notionalh omomorphism" . Butit canbe made intoan actual h omomorphism if we "refine"theRoutleyMeyerrelationintosmaller ibts, in effectinterpreting Ra/3, as 0./3 !;:; ,. Computerscientistsmight thinkof thisas a "cur ried"version. This refinementis implicitin thedefinitionof thecanonicalf rame in the completenesstheoremsof Routley-Meyer and explicitin K. Fine's independentsemantics.F Our presentation is stronglyinfluencedby Fine, but we adapthis ideas tomake themfit thepresentcontext. DEFINITION 6.2. A refined relation algebm frame is a structure (U, M, o,!;:;, u, Z), withM, Z ~ U,. : U 2 -+ U, !;:; a partialo rderon U , U : U -+ U, all subjectto thefollowingconditions:
1.
0 is requiredto be associative and isotonicwithrespectto thepartial order,
2. Va, /3 E U : :I( E Z
( "ordered" ),
«( • a !;:; /3)
3. Va,/3" E U : a • /3 !;:;, only if 4. "10,/3" EM: a./3 !;:; , iff 5. V( E Z : ( E Z
iff a !;:; /3 iff :I( E Z (a • ( !;:; /3)
(J. Q !;:; l'
1'. a!;:; (J
( "tagging" ), ( "ant ilogism ") ,
( "convers e closure "),
6. Va E M, /3 E U : a !;:; /3 only if a = /3
( "m axim ality" ).
We put quotesaroundthelabelsbecausetheydo notliterally labelthe same conditionswe earlierimposed on Routley -Meyer frames, buttheyare closeenough,giventhereadingof a • /3 !;:; , as Ra/3, . In canonicalrefinedframes, U is theset of all(proper) filters,a nd M is theset of (proper) prime filters , and in relationalgebrastheselastareof . This is themeaningof (6). For two filters a coursejustthemaximal filters and /3 E U, we interpret a • /3 as thefiltergeneratedby theset {a 0 b : a E a and b E /3} . Immediatelythereis a wrinkle , becausether e is no reasonthat 0./3 shouldbe a prime filter , even whena and/3 arebothprime filters . This is thereasonthatRoutleyand Meyer wentto a ternar y relation,Roa/3" because,can be a prime filterextendinga • /3. So in a canonical refined frame, U must be takenas thesetof all filters , andthenM is interpreted as thesetof prime filters . Z and U aredefinedas before,withZ beingthesetof 2 7 A. Urq uhar sthouldalso be m ention ed. For furth er historyand det ail s, cf.D unn 1986, and Anderson, Belnap, and Dunn et al. 1992.
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J. MICHAEL DUNN
filterscontaining e, and a being defined as in(77).28 The proofs of (1), (2), (3), and (5) arebasicallythesame as forRoutley-Meyer frames. Regarding (4), thereadercansimilarlycheck the proof of Theorem5.9 and seethatthe "trick"used depends upon a, {J, 'Y being maximal. Given a refinedrelationalgebraframe, we can define relationalgebras on it by modifyingthe definitions (39), (40), (41) bysimply substituting a • {J !; 'Y for Ra{J'Y. We leave to the interested readerthetaskof redoing our representation of relationalgebras (and the otherrelatedalgebraswe introduced in section2) in termsof refined frames. propositions,which We do howeverw antto stateand provethefollowing aresomewhatpointlessin termsof theRoutley-Meyer styleclassical frames which we used in our first representation of relation algebras(forthesimple reasonthatp!; a neverhappensexcept in thedegenerate case whenp = a). PROPOSITION
6.3. For a refined relation algebra frame, for p, a E U: p !;
a iff p !; a.
(85)
Proof Suppose thatp !; a. Then by assertion3( E Z such thatp. ( !; a. Butthenby tagging,( • p !; a,and ( E Z since Z is closedunderu. So by assertion,p !; a. The converse follows since U is of periodtwo. --j
REMARK. The readercan verifythatin thedefinitionof a refined frame "tagging"can be replaced equivalently by (86) and (a 0 {J)U = /J 0 Q. PROPOSITION
6.4. For a refined relation algebra frame, for p, a E U: p!; a iff
n, "2 e;
(86)
Proof Assume thatp!; a andthat(XI,X2) E R u , i.e., RXlaX2' It follows using (38) thatRXIPX2, i.e., (Xl,X2) E R p • For theconverse assumethat R p "2 R u . By weak reflexivity (50), we have 3( E Z, R(aa. ButthenR(pa, and so by (48),p!; a. --j
REMARK. Notethatthedirectionof inclusion on t heright-hand side of (86) is thereverse ofw hatthereadermight haveexpected. The necessity of this is "information"orderbetweenbinary clear intheproof. Butwe can define an relations Rand S : R !; S iffS ~ R. This again reverses thedirection,so the right-hand side ends up R p !; R a as desired.P? Upon reflection, this is more thanmere notation since onerelation is "stronger"(carriesmore information thananother)justwhenthefirst is included theother.Mother-of in carries more informationthandoes parent-of. 28This definitionis thereasonthatwe can haveU definedupon all ofU, and not just on M as does Fine. The pointis thatourdefinition(78) using a-l carriesfiltersto filters, whereasthedefinition(78) using ~, which Fine uses, does not (althoughit does carry prime filterst o prime filters).Butour definitionis availableo nlyfor thosealgebrast hat have a-l as an operator(hereprimitivefor relationalgebras , or definedas _(a-l) as in a Meyermonoid) . s emanticsfor thecombinatorylogic of 29This same phenomenonoccursin theternary Dunnand Meyer (1997) and is discussedthere .
A REPRESENTATION OF RELATION ALGEBRAS
99
6.5. For a refined relation algebra frame , R (defined as a • (3 ~ ,) carries the composition of relations to the composition of their correlates, i.e.,
THEOREM
(R p ~ R a) = R(pea), i.e.,
(87)
a(Rp ~ R a)(3 iff aR(pea)(3.
(88)
-Meyer compositionalnotationdescribed above, Proof. Given the Routley a(Rp ~ R a)(3 can beexpressedas: R(ap)a(3.
(89)
Whatwe aretryingto establishis that R(ap)a(3 iff Roip» a)(3
(90)
or, usingourconvention of "R suppression", simply as (ap)a(3 iff a(pa)(3.
(91)
In otherbastardnotation, thismight beexpressedas (ap)a(3 = a(pa)(3.
(92)
Puttingit thiswaysuggeststhatassociativity o f. will figure in theformal proof, andthisis indeedthecase, once wecurryR, obtaining
3x(a. p ~ X & X. a
~
(3) iff a. (p. a) ~ X·
(93)
Right-to-left is easy. Set X = a • p, so a • p ~ X follows from reflexivity , andthenuse associativity toobtainX. a ~ (3. Left-to-right is barelyharder . Let usassume
(94)
Thenby isotonicity , (a.p).a~(3,
(95)
andso by associativity we obtain, a.(p.a)~(3,
as desired.
(96) -j
Given theinvolution U withthepostulates we requiredof it, we can now show thefollowing. 6.6. R carries the converse of a relation to the involute of its correlate, i. e., (97)
THEOREM
Proof. This is similarto thatfor (66) abovebutwithrespectto thesecond positioninsteadof thefirst. Thus using antilogism,t aggingand periodtwo, we obtain: a(Rp)-l(3 iff es,« iff R(3pa iff Rp6:/3 iff RajJ(3 iff aR(i5)(3.-j
J . MICHAEL DUNN
100
thatthefunctionwhichinterprets p as therelation We havedemonstrated
R p is a homomorphism from a refined frame (with . ,!;;;, U) , carryingany
point p of theframe to arelation R p (a set oforderedpairs), • to relative . This allows us to t hinkof thepointsof a frame product,and U to converse as "concept-correlates" of relations . Note thatif concept-correlates were intensional, theremight be two different concept-correlates thatdetermine thesame relation . Butit would be nice to be abletreat to t hemextensionally so thattwo conceptcorrelates are identicalif (and of course only they if) determinethesame set oforderedpairs. In theRoutley-Meyer semanticsthereis a favored point O. Theyintroduce thedefinitiona !;;; {3 iff ROa{3 andshow from various postulates that!;;;has thepropertiesof a pre-order(reflexive , trans itivity). In ourrepresentation of relation algebraswe were not able to utilize a single such favored point, butinstead,introduced a set Z and therequirement(51) thatthe frame be (51) into twoparts: assertional. Let us break down
=
{3,
(98)
3( E Z(Ra({3) iff a = {3.
(99)
3( E Z(R(a{3) iff a
The meaningof (99) is reasonably clear: each Rc, is an identityrelation on some subsetof U , and takentogether ( taketheirunion)theyconstitute the identityrelation on all of U . In symbols: THEOREM
6.7.
U Rc, =
Iu ·
(100)
C,EZ
But how are we tointerpret(98)? In all honesty, its sense is less than straightforward. Butlet ustryto thinkof it as follows . Let ussuppose thatwe could add some new points( as "identityindicators". The job ofthepoints ( is to tell us when two relations a and {3 are identical. We might thenpostulate(98) as above and, in effect, extendthemeaning ofR (and indirectly t hatof 0 , thoughtof as arelationcorrelate).The puzzle is : how cantheset Z , which by (99)containsthe concept-correlates of the various identityrelations,also atthesame time play thisotherrole of as the set "identity of indicators"?But (98) says it does, so let us justacceptit and follow the consequences. Using (98) we can now show: 6.8 Extensionality. Two relation correlates are identical just when they determine the same sets of ordered pairs, i.e.,
THEOREM
(101)
Proof. Left-to-right is justlogic(substitution of identicals).For right-to -left we assume R p = R,71 i.e., 'tIa, {3(Rap{3 iff Raa{3). Now for some( E Z ,
A REPRESENTATION OF RELATION ALGEBRAS
101
=
R(pp holds, sincep p. Instantiating both Q and {3 to that( we obtain R(pp iff R(ap, and hence R(ap. Butthenby (98), we obtaina = p. -l
We concludethis sectionby summarizingthe aboveobservationsin the following , appropriately using thefamous lambdanotation of Church. 6.9. The junction >.p(Rp ) is an isomorphism , carryin9. into relative product, U into converse, and Z into a set of partial identity relations whose union is the identity relation on U. In the refined semantics • is an actual operation, whereas in the Routley-Meyer version it is notional.
THEOREM
Thereis anotherway to make sense of (51) thatis in fact more useful in interpreting the framesemanticsin termsofrelations . We keep theidea that eachstatep can betradedoff for abinaryrelation R p = {( Q, {3) : RQp{3}, so our "concrete"frames will have relations as theirpoints. Butto saywhat the pointsare is not yet to say how the ernary t accessibilityrelationis to be interpreted. We shallgive up the"untyped"interpretation of theternary accessibilityrelation RQp{3 as " p relates Q to {3" , and adoptinsteadthemore mundane"typed"interpretation : "thecompositionof Ra and Rp is included in Rr/'. Then (99) and (98) make equal sense with Z as a set ofpartial identityrelations.We shallhave more to say a boutthis interpretation ofthe accessibilityrelation in thenextsection. 7 . REPRESENTATION RESULTS USING RELATIONS
The intuitiveidea of arepresentation is thatwe take somestructures viewed abstractly, and show howtheyare eachisomorphic to structures which are 3D A paradigmof this isStone's(1936) showing how (relatively) concrete. a Booleanalgebracan be regardedas a collection subsetsof of some given set, with the Boolean operationsinterpreted as intersection , union, and relativecomplement. Compared to this paradigm, the representation results of section5 must be acknowledged as disappointing. We haverepresented abstract . This relation algebrasusing frames, which themselves aboutas are logic, where one can is much worsethanwhathappensin, say, intuitionistic give arepresentation ofHeytingalgebrasin termsofbinaryframes. Thereat leastone can give somemeaningto the frames in termsof information s tates (Kripke talkedof "evidential s tates")and theinformation o rder("evidential statesbeing possiblerelative to oneanother") . But "attenyards," it is not ternary frames have to do with any intuitiveinterpretation so clear what our of relation algebras. But this is onlyat firstblush, because thereaderknows thatwe have alreadymade some headwayin section6 regardinga philosophicalinterpret hefollowing definition: tationof ourternary frames. We make this precise by 30Concr et enessis in the eye of the beholder (and not just formotes). I am rem indedof Bob Meyer's joke on this subject. A mathem atician says, "Ta ke a concreteexample, say an infinite-dimen sionalHilbertspace."
102
J. MICHAEL DUNN
DEFINITION 7.1. Bya concrete relation algebra frame we shallmean a structure(U,R, u, Z) whereU is a setof relations on some set X, Ro:j3-y is defined as -y ~ 0: 0 {3, U is justconverse,a nd Z ~ U is subjectto theconditionthat U Z = Ix (theidentityrelationon X) . PROPOSITION 7.2. Every "concrete relation algebra frame" is indeed a relation algebra frame . Proof. The needed factsc an be easily found in s tandardsourcesregarding thealgebraof relations''!and/oreasily verified by t hereader .
THEOREM 7.3. Every relation algebra is isomorphic to a framed relation algebra on a concrete relation algebra frame . The theoremis a directconsequenceof thefollowing: LEMMA 7.4. Every relation algebra frame (U, R, u, Z) is isomorphic to some concrete relation algebra frame. Proof. We set U = {R p : p E U} and Z = {RC; : ( E Z}. NotethatX = U . The desiredisomorphism shouldbe >.p(Rp ) , whichmaps each p E U to R p E U. We know from (6.9)thatthismap is 1-1, onto, and preservesu. We must accessibilityrelation R, i.e., also showthatthemap preservestheternary
Ro:{3-y iff RRoRf3R-y , i.e.,
(102)
Ro:j3-y iff
(103)
n; ~ s; 0
Rf3 .
For left-to-right, assume thatRo:{3-y andthat(Xl,Xz) E R-y, i.e., Rxnxz . By (42) we thenhavethatfor some c5,RXIO:c5 and Rc5j3Xz, i.e., (XI,c5) E R o and (c5,Xz) E Rf3, which is just the meaning of (XI ,XZ) E R o 0 Rf3 . For theconverseassume thatR: ~ Ro 0 Rf3 . By weakreflexivity (49) we have for some ( E Z thatR(-y-y, i.e., ((, -y) E R-y. But thenusing theassumed inclusion, we get ( (, -y) E Ro 0 Rf3, i.e., for some 15, R(o:c5 and Rc5j3-y. The first meansthat0: = 15 andso by substitution of identicalsin thesecondwe obtainRo:{3-y as needed. Finallywe mustshow thattheunion of Z ist heidentityrelation r estricted to X = U. This is just(100) . -J REMARK. In termsof anintuitive i nterpretation, wemightview theelements of U as relations in some relational database.T hesemightbe parent-of , sisterof, etc., and it is natural thattheyshouldbe closedunderrelative p roduct (so we shouldalso havegrandparent-of , aunt-of, etc .), as well as converse (so we would have child-of, grandchild -of, etc.). Becauseof Proposition(6.4) (and theRemark followingit), the elementsof M (being maximal in the informationorder)would beminimal in theinclusionorder,i.e., maximally strongrelations.Why shouldwe requireA to be closedunder ~? Let us view 31Cf. Tarski and Givant 1985, and Maddux 1991.
A REPRESENTATION OF RELATION ALGEBRAS
103
A as a "query". If we wantinformationaboutwho is aparentof whom, this querywould beatleastpartially answeredby thestronger i nformation a bout whom is a motherof whom. This accountis notentirely s atisfying, butthere it is.32 8. CORRESPONDING SUBSTRUCTURAL LOGICS
Peircemonoids correspondto the(associative)L ambek Calculuspermitting empty lefthandsides, supplementedwithconjunction a nddisjunction(which distributeover eachother). ProtoDe Morganmonoids ariseby theaddition arisingfrom of De Morgannegation , and Meyer monoids can be viewed as the additionof Booleannegationin additionto De Morgannegation(cf. Theorems 2.9 and 2.10) . We have shown inTheorem 2.13 thatrelation algebrascan be viewed as differing from Meyer monoids only by theaddition of (29). The questionoccursas towhetherthereare natural ways torestrictt he relations in aconcreter elation a lgebraframe so as tosatisfylawscorrespond ing to various structural rules ofG entzen , in particular permutation, contraction, and thinning . This would give ust heabilityto model"subst ruct ural relation algebras ," or maybe theyshouldbe calledsimply "structural relationalgebras,"since theyariseby addinglawscorresponding to theGentzen structural rules. Thecorresponding framepropertiesarerespectively R-commutativity (62), R-reftexivity (63), and 00
{3 [;;;
0
(R -lower-bound).
(104)
(Of coursein theabsenceof commutativity oneshouldalsoconsiderits commutedform.) Let us first a ddress thequestionofsquare-increasingness , wheretheanswer is straightforwardly "yes". Correspondingto R p • R p [;;; R p is thecondition thatn, ~ n, I8l R p, i.e., if (0, (3) E R p, then3X[(o, X), (X, (3) E RpJ. This is a standardpropertyof relations sometimes referredto as"pseudo-denseness ." Further,if a framesatisfiessquare-increasingness, it is easy to seethatthe corresponding c oncrete frame will be pseudo-dense.Thussuppose Rop{3, i.e., o. p [;;; {3. Set X =
0 •
p. Clearly 0
Rxp{3, i.e., (0. p). p
•
(105)
p [;;; X since [;;; is reflexive. We now m ust show that
= o. (p. p) [;;;
{3. Butthis follows from ourhypothesis
(105) using isotonicity .f'' 32Another possible interpr etationof a concret e relat ion algebra frame is to treat its pointsas relat ionson state s, i.e., as sets of transit ions, .e., i as possible acti ons on st ates. Butwe shallnotpursuethisher e. 33We carriedoutthis reasoningwithrespectto a refined frame, butit worksjust as well withan unrefinedframe. From Rppp and the hypothesis Rcrp{3 and we obtainR2 cr(pp){3, i.e., by R-associativity , R 2(crp)p{3, i.e., RcrPX and Rxp{3 for some X.
104
J. MICHAEL DUNN
propertyof TUrning now tocommutativity,I do notknow of anatural relationst hatmakes relativep roductcommutative. (Symmetry does not work.) I willpoint outthatour relationsR p correspondingto points on an R-commutativeframe do in fact c ommutewith eachotherusing relative product . Thus (a,{J) E Rp®R" iffRapx and RXCT{J iffR(ap)CT{J iffRa(pCT){J iff Ra(CTp) (J iff R(aCT)p{J iff (a, (J) E R" ® R p. So a crudeway toassure requirement t hatrelative p roductis commutativity is justto make the global commutativeon the collectionrelations of takenas thepointsin the frame. This works, but it is unsatisfying.A similarpoint seems to apply to the t hatthe collection relations of satisfiesR-Iower-bound . requirement It mustbe pointedoutthatit seems unlikely thataddingone or more of the above "structural postulates" will give a class of frames which characteristic is s ubstructural logic. It willratherbe characteristic for forthecorresponding the"relation algebra"version.The reasonis thatrelations alwayssatisfy T ~ R ® S only if
T ~ S® R
(tagging) ,
and soconcreteframes always validate (56). This observation can bemade to prove forexamplethatR-commutativeconcreteframes do notcharacterize commutativeMeyer monoids, orthatR-commutativeR-reflexiveconcrete frames donot characterize commutativesquare -increasingMeyer monoids ("BooleanDe Morganmonoids"). Unfortunately this does not decisively solve the problem for thecorre spondingsubstructural logics(distributive linearlogicwithoute xponent ials, R), since substructural logics arenotusuallyformulated withBooleannegation and converse (Meyer actually used "star")as primitive. It was R. K. R can be soformulated withBoolean Meyer'sinsightthatthe relevance logic , and thatthenthe usual De negationand staras conservativea dditions Morgannegationof R can be viewed asarisingby way of thedefinition rvcp = _(cp-l j). But (56) [(A 0 B)-1 = B-1 0 A-I] does notcorrespondto any formula in theprimitivevocabulary of R, nor do I know of any such formula that it implies whichis not also atheoremof R . So we areleft with n aagging question. 9. FRAME MODELS OF THE A-CALCULUS
Reflection on the above interpretation of therepresentation of relation algeatrelational frames asintuitive brasgives a perhapshelpful way of looking models ofcombinatorylogic and hence, by the well -knowntranslations betweenthetwo, asintuitivemodels oftheA-calculus . This sectionwill be programmaticin character and not focus on the detailsof development , which can be found in a recent paper I havewrittenwith R. K. Meyer.34 3
4
T he paper referenced in Dunn and Meyer 1997.
A REPRESENTATION OF RELATION ALGEBRAS
105
The A-calculus of Churchcan be viewed as atheoryof functionality. In its originalversions, it isuntyped. Its termsstoodfor func t ions,and given two terms M , N one canalways"apply" one tothe other to obtain M(N). Also given aterm M , one canalwaysform thefunctionAxM. The problemin interpret ing the(untyped)A-calculus has always beenhow to interpr et an expressionsuchas M(M), whichtreatsM as simultaneously standingforbotha functionand an argument . In theearlysevent ies,Scott (1976) gave twodifferent modelsof theA-calculus : thelimit modelD oo and thegraphmodel Pw . The lastof theseis most rel evantto ourpresentconcerns, and uses thewell-known correspondence of Cantorbetweena natural numberand a pair of natural numbers. Thus a set X ofnatural numbers can besimultaneously regardedas a relation betweennatural numbers,and so we can make sense of X(Y) as the X -irnageof Y. R. K. Meyer has a similar model, constructed by takinga set U, and closing itunderpairing so as toobtaintheset U· = U U U 2 U U 3 .• •. Then for X, Y ~ U·, X(Y) = {X: 31' E Y, < X, l' > E X}.35 The discussionof theprevioussectionshowed how to achieve theeffect of a similartype-defyinginterpretation: a propositionA can besimultaneously thought of asbotha setof statesanda set ofrelations betweenstates(note wellthatwe are up atype-level from ScottandMeyer). Wherep is a relation , and A is a set,thep-image of A = {X : 3a: E A , a:PX)} . Going up atype-level , where A is a set ofrelations , it isnatural to define a corr espondingimage BA = {X : 3p E B,3a: E A, a:PX} and to thinkof thisas a kind of application. a relationand a stat e, we stick in the Now regardingp as simultaneously ternary relation R of exemplification , rewritingthisas: BA = {X : 3p E B , 3a: E A, Ra:PX}
=A0
B.
It is clearthattheapplicationof onepropositionto anotheris justour old friend fusion (butwiththequantifiers in reverseorder,an awkwardness which isrepairedsimply by takingtherelation to be det e rmined by thefirst interpret A A = A 0 A. The key difference position: R a ) . We nownaturally betweentheinterpretation of 0 applicationand as erlativeproducthas to do withassociativity. This is aboutenoughforthepresentoccasion,although weshouldsay that rathert handiscuss thecare and feeding oftheA-operatordirectly , Meyer andI insteadexploitthetranslation of Curryof A-calculus intocombinatory logic,and show howthecombinatorscan be represe ntedas certainsets of states. This representation does notdepend upon theparticular choice of combinators(as long as th ey are"proper" ),andso the erpresentation canbe appliedto combinatorial bases appropriateto variouss ubstructural logics. 35A lt ho ugh M eyer ,Bunder , and Power s' (1991) "fool'smodel" wasnot publish ed until 1991, I rem ember discussing it in detailwithMeyer about1970. He was influe nce din this ththe so-called"P-W Probl em" from m od elby som e ideas L. Powers had in con nec t ion wi relevan celogic.
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J. MICHAEL DUNN
Thereis the well-known "Curry-Howard Isomorphism" thatestablishesa connectionbetweencombinatorsand provableimplicational formulas . The standard t ranslation of the >.I-calculus into combinatorylogic uses the combinatorsI , B , C , and S, and these are well-known correspond to to the implicational axioms of the relevance logic R by way ofCurry the -Howard Isomorphism. R. K. Meyer has saidthatit cannotbe a mere accident t hat Church was the creatorof boththe >.I-calculus and theWeak Calculus of Implication(theimplicational fragment of which agrees with the relevance logic R). I agree. REFERENCES
Allwein, G.,a nd J . M. Dunn 1993 A Kripkesemanticsforlinearlogic, The Journal of Symbolic Logic, vol. 58, pp. 514-545. Anderson, A. R., and N. D. Belnapet al. 1975 Entailment: The logic of relevance and necessity, vol. I, PrincetonUniversityPress, Princeton,New Jersey. Anderson, A. R, N. D. Belnap,and J. M. Dunn et al. 1992 Entailment: The logic of relevance and necessity, vol. II, PrincetonUniversity Press, Princeton,New Jersey. Barwise, J. 1993 Constraints , channels , and theflow ofinformation,Situation theory and , editors), vol. 3, CSLI Lecture its applications (S. Petersand Dlsrael Notes, Universityof ChicagoPress, Chicago, pp. 3-27. Bergmann,G. 1967 Realism, Universityof MilwaukeePress. Bialynicki-Birula , A., and H. Rasiowa ofquasi-Boolean algebras,Bulletin de l' Academie 1957 On therepresentation Polonaise des Sciences , vol. 5, pp. 259-261. Birkhoff , G. 1967 Lattice theory, AmericanMathematical Society,Providence . Bradley,F . H. 1897 Appearance and reality, OxfordUniversityPress, Oxford. Brown, C., and D. Gurr 1995 Relationsa ndnon-commutative linearlogic,Journal of Pure and Applied Algebra, vol. 105, pp. 117-136. Church,A . 1951 The weak theoryof implication,Kontrolliertes Denken, Untersuchgen zum Logikkalkul und zur Logik der Einzelwissenschaften, (A. Menne, A. Wilhelmy , and H. Angsil, editors),Kommissions- VerlagKarlAlber, Munich, pp . 22-37. Dosen, K. 1992 A briefsurveyof frames for t hepeirce calculus , Zeitschrift fur Mathema tische Logik und Grundlagen der Mathematik, vol. 38, pp. 179-187.
A REPRESENTATION OF RELATION ALGEBRAS
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Dunn,J. M. 1966 The algebraof intensionallogics, Doctoral Dissertation, Universityof Pittsburgh,UniversityMicrofilms, Ann Arbor; portionsrelevant tothis paper arereprintedin Anderson, Belnap et al. 1975 as §8 and§28.2. 1982 A Relational Representation ofquasi-Boolean algebras, Notre Dame Journal of Formal Logic, vol. 23,pp . 353-357. 1986 Relevancelogic andentailment , Handbook of philosophical logic (D. Gabbay and F. Guenthner , editors),vol. 3, Reidel, pp. 117-224. 1991 Gaggletheory:An abstraction of Galoisconnections andresiduation w ith applicationstonegationandvariouslogicalo perations,Logics in AI, Proceedings European workshop JELIA 1990 (J . van Eijck,editor),Lecture Notesin ComputerScience478, Springer-Verlag, Berlin. 1993a Partial-gaggles applied to substructural logics, Substructuml logic (P. Schroder-Heister and K Dosen, editors),OxfordPress, pp. 63-108. 1993b Perp and star:Two treatments of negation,Philosophical perspectives: Philosophy of language and logic (J . Tomberlin,editor),vol. 7, pp. 331357. Dunn,J. M., and R. K Meyer 1997 Combinatorsandstructurally free logic, Logic Journal of the IGPL, vol. 5, pp . 50&-537. Girard,J .-Y. 1987 Linearlogic, Theoretical Computer Science, vol. 50, pp. 1-102. Gyssens, M., L. V. Saxton,and D. van Gucht 1993 Taggingas analternative toobjectcreation , Proceedings of the conference on query processing in object-oriented, complex-object and nested relation databases. Jonsson, B., and A. Tarski 1951-52 Booleanalgebraswithoperators,A merican Journal of Mathematics, vols.73-74, pp. 891 -939, 127-162. Kraegeloh,K-D., and P. C. Lockemann 1976 Hierarchiesof databaselanguages:An example, Information Systems, vol. 1, pp. 79-90. Lambek, J. 1958 The mathematicsof sentences tructure, American Mathematical Monthly, vol. 65, pp. 153-172. Lyndon,R. C. 1950 The representation of relationalgebras,A nnals of Mathematics, ser. 2, vol. 51, pp. 707-729. Maddux,R . 1991 The originof relation a lgebrasin thedevelopmentof theaxiomatization of relations,Studio Logica, vol. 91, pp. 421-455. Meyer, R. K 1979 A Booleanvaluedsemanticsfor R, ResearchPaper no. 4, Logicgroup, ResearchSchool of Social Sciences, Australian N ationalUniversity , Canberra
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Meyer, R. K., M. Bunder, and L. Powers 1991 Implementingthe'fool's model' forcombinatorylogic, Journal of Automated Reasoning, vol. 7, pp. 597-630. Meyer, R. K., and M. A. McRobbie 1982 Multisetsa nd relevant implication , Austmlasian Journal of Philosophy, vol. 60, pp. 265-281. Meyer, R. K., and R. Routley 1972 Algebraicanalysisof entailment,Logique et Analyse, n.s., vol. 15, pp. 407-428. 1973 Classicalrelevant logics (I), Studio Logica, vol. 32, pp . 51-66. 1974 Classicalrelevant logics(II), Studia Logica, vol. 33, pp . 183-194 . Orlowska, E. 1992 Relational p roofsystems for relevantlogics, The Journal of Symbolic Logic, vol. 57, pp. 1425-1440 . Pratt , V. 1991 Actionlogic andpure induction,Logics in AI, Proceedings of the European workshop JELIA 1990 (J . van Eijck, editor),LectureNotes in ComputerScience 478,Springer-Verlag, Berlin. Routley,R., and R. K. Meyer 1972 The semanticsof entailment, II-III, Journal of Philosophical Logic, vol. 1, pp . 53-73, 192-208 . , Truth, syntax and modality (H. Leblanc , 1973 The semanticsof entailment editor), North-Holland, Amsterdam, pp . 199-243 . Scott, D. S. 1976 Datatypesas lattices,SIAM Journal of Computing , vol. 5, pp. 522-587. SchroderHeister,P., and K. Dosen 1993 (editors),Substructumllogic, OxfordUniversityPress, Oxford. Stone,M. 1936 The theoryof representations for Booleanalgebras,Transactions of the American Mathematical Society, vol. 40, pp . 37-111 . Stone,M. 1937 Topologicalr epresentations ofdistributive lattices and Brouwerian logics, Casopsis pro Pestovan{ Matematiky a Fysiky, vol. 67, pp. 1-25. Tarski, A. of relations,The Journal of Symbolic Logic, vol. 6, pp. 1941 On thecalculus 73-89 . Tarski, A., andS. Givant 1985 A formalization of set theory without variables, AmericanMathematical SocietyColloquiumPublications,vol. 41, Providence. Van Benthem, J . 1991 Language in action : Categories, lambdas and dynamic logic, Studiesin LogicandtheFoundations ofMathematics,vol. 130,North-Holland, New York. Ward, M., and R. P. Dilworth 1939 Residuatedlattices,Transactions of the American Mathematical Society, vol. 45, pp. 335-354 .
THOMAS FORSTER
CHURCH'S SET THEORY WITH A UNIVERSAL SET* For Alonzo Church on the occasion of his ninetieth birthday.
Abstract. A detailedand fairlyelementaryintroduction is given to the techniquesused by Church to prove theconsisten cy of his set theory with a models of it from models of ZF. The construction universalset by constructing is expla inedandsom e generalfacts aboutit proved.
1. INTRODUCTION
In 1974 AlonzoChurchandDrs Oswaldsimultaneously andindependently lit upona refinementof Rieger-Bernayspermutation modelswhichenabledthem to giveelementary proofs ofconsistencyfor somesettheoriesw itha universal set. My owninterest b eganmuch later,when Imade theaquaintance ofFlash Sheridan,a formerstudentof Church'swho was thenwritingan Oxford D.Phii. thesis onChurch'swork inthisarea.I am greatly endebted toFlash for kindling myinterestin thisengagingbywater,a nd forshowingme some relevant and (even now)u npublishedmaterial. Churchwas notgenerally known for having an interestin set theory, and one mightwonderwhatmoved him towritethislateand isolatedpiece. An interestin set theorywitha universalset was even less mainstreamtwenty years agothanit is now, but theconsistencyquestionfor suchtheoriesis somet hingthatthosewhose interestin set theorywas purelyphilosophical might well havemused aboutatany stagein its history,a nd I suspectthat Church'smotive was to mak e a smallpolemicalpointto theeffectthatthere are con s istentset theorieswitha universalset. This pointis worthmaking to studentseven now. Againstthisview is thatfactthattherewerealready twopapers making thesame point, and we knowthatone ofthem, Jensen 1967, was known toChurch. (The manuscriptof Ward Henson'sreview of Jensen 1967 in The Journal of Symbolic Logic archiveshas commentson it in Church'shand.) However it is possiblet hatChurchfeltthatthis paper did notmake thepointconvincingly,since thesystem whose consistencyis proved thereinis nota pureset theory,butadmits urelemente. The other, Grishin 1969, (which provestheconsistencyof NF3' a pureset theorywith a universal set and no urelemente) was apparently not known to Church.1
* There is considerable overlapbetweenthis paper and the final chapterof my book on s Forster 1995. set theory witha univer sal et, 1 I am endebtedto Herb Endert on for historicaldet ailsaboutTh e Journal of Symbolic 109 C. Anthony Anderson and M. Zeleny (eds.), Logic, Meaning and Compu tation , 109- 138. © 2001 Kluwer Academic Publishers. Printed in the Netherla nds .
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THOMAS FORSTER
The appropriateplace tostartis witha high-speedtreatment of RiegerBernayspermutation models. 2. RIEGER-BERNAYS PERMUTATION MODELS
We willstartby swiftlyrecapitulating and updatinga treatment of Riegerforthelanguage Bernaysmodels from Forster 1995. If (V, R) is a structure of settheory,a nd 1r is any (possiblyexternal) permutation ofV , thenwe say x R; y iffx R 1r'y.2 (V, R7r ) is a permutation model of (V, R). We call itV 7r • Alternatively we could define 'f>.x.(f"x). The map j commutes withgroup-theoretic operations : j'(1ru) = (j'1r)(j'u) . A n, j":« is (possibly external) permutationo of a set X is set/ike if, for all definedand is a permutationof pn ,x (if thislastthingexists). ZF proves thateverydefinablepermutation is setlikc,This is actually a formulation of theaxiom scheme of replacement! We shallstartw itha lemma and adefinition,b othdue to Henson (1973). The definitionarises fromtheneed totidyup
CHURCH'S SET THEORY WITH A UNIVERSAL SET 7, SO
111
that,for eachn, x E 7'Y iff 7n 'x E 7n+l 'yo
Thenby replacing'x E 7'y , by , 7n 'x E 7n+! 'y , whenever'x' has been assigned thesubscriptn, everyoccurrence of 'x' in ' T , will havet hesame prefix. Next we will want to know that7 n is a permutation,so thatin any wff in which ' x' occursbound-(Vx)(.. . 7n'x . . . )-it can be re-lettered (Vx)( .. . x ... ) so that'7' has beeneliminatedfrom theboundvariables . It is nothardto checkthatthedefinitionwe need to make this work is as follows DEFINITION 2.1.
70
= identity,7n+l = (jn'7)7n .
This definition issatisfactory as long asr '7 is always apermutationof V whenever7 is, for eachn . For this we need7 to be setlike . The trickof re-lettering variablest hatthis facilitates is of crucialimportanceandwill be used often. This gives usimmediatelya proof ofthefollowing result. LEMMA 2.2 (Henson, 1973). Let be stratified with free variables 'Xl" •••, 'X n " where 'Xi' has been assigned an integer k; in some stratification. Let 7 be a setlike permutation and V any model of NF. Then (VX)V
F ((xV
+---4
(7kt'Xl••• 7kn 'x n )
.
In thecase where is closedandstratified , we inferthatif 7 is a setlike permutation, then V F +---4 T • This was proved byScottin 1962. We can actually prove something withthe flavor ofcompleteness a theoremaboutthesethings: a formula is theclass of its models is closed under equivalent to stratified a formula iff permutation models using setlike permutations . The proof of this is too long to be done here, butsee Forster 1995, where it isTheorem3.0.4. REMARK 2.3. If (V, E) that(V, ET ) F ZF.
F
ZF and 7- 1 is a setlikepermutation of V, check
axioms are noproblem. The onlyunstratified axiom Proof The stratified scheme is replacement . Easy enough to check for any ¢ thatif Vx3!y¢, then VX3!y¢T , so thatfor any set Xtheimage of X in¢T is also a set. Call it Y. Butthen7- 1 'y is theimage-of-X-under-¢(in the sense of VT ). -l We will also need thefollowing observation: LEMMA 2.4. If f is a stratification of thought of as the partial function IN---+ IN that sends the variable's subscript(rather than the variable itself) to the type then <1>( Xl>' " , Xk )
+---4
). a Xk <1>« J'1' 1,a )' Xl,.. ·, (J'f'k')'
112
THOMAS FORSTER
Proof It might be a good idea to haveillustration an before a full proof. For example, thelemma tells ust hat x EY
~
a 'x E a"y.
The theoremis simply a moregeneralassertion,t rueforthesame reasons. Now for a full proof. By definitionof i we havex E y iffr'x E (j'r)'y for if 'x ' has been assignedtypenand'y' thetypen + 1, we any r . In particular r is j n'a to getx E y ~ (jn'a)'x E (jn+l'a)'y. By invoke the case where substitutivity ofthebiconditional we can do this simultaneously for all atomic subformulae in (XI , . " ,Xk) . Variables'y' thatwereboundin '(XI ... Xk)' now have prefixes like 'jn'a' in front of them but,since '(XI, ' " ,Xk)' was stratified , theywill beconstantfor each such variable'y' . We thenuse the of V so thatany formula ( Qy)( . . . (jn 'a )'y .. .) factthat[ ":« is a permutation (Q a quantifier) is equivalent to (Qy)( . .. y . . .). -1 DEFINITION 2.5. A P-embedding from A into B is a map i : A --; B for which the power set operationis absolute."No new members or subsetsof old sets."If i is theidentitywe say B is a P-extensionof A. THEOREM 2.6. Let M = (M, E) be a wellfounded model of ZF, and let a be a setlike permutation of M . Let i : M '-+ M" be recursively defined by i'x = : a-l(i"x). Then (i) i is a P-embedding and (ii) i is elementary for stmtified formula: Proof
(i) If x E" i'y, thenx is a value ofi, so i is an end-extension . Suppose (x ~ i'y)" : we wantx to be a value of i . (x ~ i'y)" is justa 's: ~ a(i'y) = i"y so a's: is a set of values of i , so x is a value of i. (ii)3 Let ¢(x) be a stratified formula whose free variablesare precisely t he X, a tupleof lengthk , Assume
A.( Z" XI,Z" X2, . . ·Z" Xk) . M " Lr- '+' By Lemma 2.2 this isthesame as
where, for eachj , thevariableX j has been assignedthetypenj in some fixed stratification of ¢. By therecursive definition i,ofi 'XI is equalto a- 1(i "x) or toa- 1 0 (j'a)- 1 0 (P'i)'XI andso on, byunravelling stepsin therecursive definitionof i'x ad libitum up tonl,which gives us(a- 1 )n, '((jn, 'i)'xI). So 3R.andallH olmes has foundan error intheproofof Theorem2.6, part(ii) , whichmust be withdrawnpro tern. A discussion will bepublis hed elsewherein due course.
CHURCH'S SET THEORY WITH A UNIVERSAL SET
113
applyingun! tothisgives us (jn! 'i)'x and similarly for theotherx variables , so the formula becomes:
M
F
COROLLARY 2.7. The axiom of foundation
is independent of ZF.
(A, {A}). (A is the empty set.) In thenew Proof. Use the transposition model the old empty set has become anobjectequal to its own singleton.-l The completenesst heoremfor setlike p ermutations andstratified formulre is a powerful and satisfyingpiece of 1950's-style model theory, not unlike Birkhoff 's theoremin flavor,butit is actually a nuisance . It's all very well if you have a model ofstratified a theory, and want generate to lots more models of it,but it does meanthatif youstartwith a model of taheory thatdoes not have u a niversalset, thenthe result will not have a universal set isstratified , set either.(After all , theassertionthatthereis no universal and if itstartsoff false, will remainfalse.) It gives us theindependenceof the axiom offoundation,butnottherelative consistencyof a universalset. If we want auniversal set we have to generalize construction the slightly . 3. CHURCH-OSWALD MODELS
Let (V, E) be a model of ZFand letk (for'kode') be a bijectionbetweenV and V x {0,1}. Next we define xEcoY
'ff { 1. k'y = (y', 0) and x E y', or 2. k'y=(y',1) and x fl-y' .
I
This is thesimplestversion oftheconstruction, and istheone in Oswald 1976-thoughOswaldconsideredspecifically the case where V was Vw and presentedit very differently . (Oswald'smodel is presentedmore fully in section3.1.1.) The firstthingto notice is thatevery set has complementin a thesense of E co · (Perhapsthefirstthingof all to noticethat is E c o really is extensional!) In facttheresultingmodel is a model of atheoryknown asNF2 , which Oswald wass tudyingat thetime. The axioms of NF2 are (i)Extensionality (ii) Complementation ( - x is a set, always) (iii) x n y and (iv) existence of {x} . This axiomatization is finitebutit isperhapsmoreconvenient to replace (iv) by the scheme giving the existence, for neach E IN andfor all -tuples n X, of the set{Xl,X2 ,··. x n } .
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THOMAS FORSTER
We tookk to be abijectionbetweenV and V x {O, I}, butit could have been Vx anythingK we like('K' for'Kode'), as long as 0E K andwe have suitableclauses for Ec o x wherethesecondcomponentof k'x is not O. The idea is thatin generalwe definex E co Y if either(i) snd(k'y) = 0 and x E fst(k'y) or4 (ii) variousotherclauses concerning (new) membership in sets y suchthatsnd(k'y) =I- O. The idea isthatotherentriesin K willcorrespond to otheroperationson sets (in the case we have justseen 1correspondsto complementation).In generalt hereis a problem ofextensionality. Thereis of course no difficulty in showing thatx and x' satisfyextensionality as long as snd(k'x) = snd(k'x') = 0, butthereareothercases to consider,The task of verifyingextensionality in the newstructure will be much easier xifand y such thatsnd(k'x) =I- snd(k'y) are so different thatthepossibilityof them having the same members in the new sense simply never arises. For example, in thecase we havejustseen, ifsnd(k'x)is 0, thenx has only a set (in the old sense) ofmembers-in-the-new-sense, whereas ifsnd(k'x) is 1, it has aproper class (in the old sense) members-in-the-new-sense. of This tells ust hatif the theoryfor which we are tryingto obtaina model bythisconstruction is T, thentheextensionality problemfor the model is deeply relatedto theword if we have a good notionof normalforms forT problem forT . In particular words over a set generators, of thenwe will be able to take K to be (roughly) a set of suchnormalwords. What this reveals is that this technique is not a great deal of use for constructing models of a theory T unless T has an easy word problem. Settheoriesw ithan easily solvable word problemare unlikely to be ofinterest. Forthemoment wewill-inthename ofgenerality -ignorethequestionof assumethatK is some whattheremainingcomponentsof k'x can be. We will arbitrary collection such that0 E K and there are rulesensurethat to E co is extensional and thatwhen snd'k'y = 0, thenx E co Y ~ x E fst(k'y) . We will try to prove some general resultsa boutthisconstruction . Let us call ("Church-Oswald") . structures built inthisway CO structures. To summarize,thereare threepartsto aCO construction over a model (V, E). Thereis a collection K of objectsavailable to be used as second componentsof orderedpairs; thereis a bijectionk betweenV and V x K, andfinallyt hereis a family of rules telling us how possible membership (in thenew sense) ofx in y depends on the secondcomponentof k'y. RiegerBernayspermutation model constructions can be seen asthosespecial cases of CO-constructions whereK is a singleton. Accordingly theonlythingone is free to play with thebijection is k. Since we areassuming thatone oftheclauses inthedefinitionof E co is alwayssnd(k'x) = 0 -+ (Vy)(y E co x ~ Y E fst(k'x)), we can make the following definition. DEFINITION
3.1. A lowsetis a set x suchthatsnd(k'x) = O.
4fst(x)and snd(x) arethefirstand secondcomponentsof theorderedpair x.
CHURCH'S SET THEORY WITH A UNIVERSAL SET
115
This is at variancewith otherdefinitionsin theliterature, but it is the thenotionof a hereditarily most useful.T his is notto be confused with lowset, for this will turnout to beimportantas well . DEFINITION 3.2. Hl ow is thegreatest fixedpointforthefunctionthatsends an argumentx to theset of lowsubsetsof x. Thatis to say, H 10 w is the collection of sets x such thateverything in thetransitive closure of x is low. (Noticethattheleastfixedpointmust consistentirely of wellfounded sets.) DEFINITION 3.3. The Axiom scheme ofLow Comprehensionstatesthat, for any formula ¢>(x , y), and for ally, if thecollection of allx such that (V, Ec o ) F ¢>(x, y) is a set of the original model, then (V, Ec o ) F "{x: ¢>(x, y)} is a (low) set". Thus it is by no meansobviousthatlowcomprehensionis axiomatizable. approachto thistopic is to thinkof the For thisreasonthemost profitable CO constructions as thingsthatgive us models,rathert hanto attemptto be specific insayingwhattheaxioms are ofthetheorywhoseconsistencywe have proved . The following t riviality is central to whatis to come. THEOREM 3.4. All CO structures satisfy low comprehension.
Proof. Let ¢>(x, y) satisfytheantecedent, and considertheclass of all x such that(V, Eco ) F ¢>(x, y) which is a set of the orig inalmodel, X , say. Then {x : ¢>(x, y)} in thesense ofthenew model is simplyk(-I )'(X,O) , andis of --j course low. In particular we have anaxiom of pairing. In a typicalCO construction therewill beplentyof newsets containing all low sets : V for one. However, PROPOSITION 3.5. No new set containing all low sets can be low.
Proof. Suppose therewas a low setcontainingall low sets.T hen, by low of all low sets is low . Thatis to say, thereis an comprehension , thecollection x such that(Vy)(y E c o x +--+ snd(k'y) = 0). Butthereis certainly a proper class ofy such thatsnd'(k1y) = 0, so this x has a properclass ofmembers notlow. -I and istherefore Attemptsto reconstruct theusualparadoxesin thisnew contextgive rise thatcertainsets are not low. Take Russell 's paradoxfor to demonstrations thingsthatare not example. The new modelcannotcontainthe set of all x such that-,(x Ec o x) were a members of themselves.If thecollection of set of theoriginalmodel, thentheRussell class would beset a of the new model. Sothereis a properclass ofx such that-,(x Ec o x).
116
THOMAS FORSTER
3.6. Every surjective image of a low set is low and every subset of a low set is low.
COROLLARY
Proof. The firstpartfollows fromreplacement in theoriginalmodelandthe secondfrom comprehension. -j The first follows from thesecondby AC buta proofwithoutAC is preferablesincewe willnototherwisebe makinganyuse of AC inthisdevelopment. COROLLARY
3.7. Every low set has a power set, which is also low.
Proof. When x is low, P'x (in thesense of Ec o ) must be k(-l)'(k(-l)"(P'(fst(k'x)) x {o}),0). COROLLARY
3.8. If x is a low set of low sets, then it has a sumset, which is
low. Proof. The objectwe need to playtherole ofU x is
k-1(U(fst0 k)"(fst 0 k)'x,O). This dependsonlyon theavailability of theaxiom of sumsetin themodelwe startwith, and does nottell usa boutstronger,less restricted forms ofthe axiom. -j This seems to be an argumentfor settingup this theorywithlow comprehensionin theway I have done it ratherthanwithlow replacementin theway Churchoriginally did. Low replacementin theform "T he image of a wellfounded setin a functionis a set" certainly i mplies existenceof power set for lowsets and sumset, butwe seem to need ACto deducesumset for lowsetsof lowsets. This is very messy. 3.1. Applications of the technique 3.1.1 Models of NF2
Oswalddid notset up his firstillustration withtheapparatusof K and k as here. Insteadhe defineda binaryrelation on IN as follows :
n Em iffeither 1. m is even and thenthbit of thebinaryexpansionof m/2 is 1; or 2. m is odd and thenthbit of thebinaryexpansionof (m - 1)/2 is 0.
This obviouslyderives from theold trick(due to Ackermann)of defining n E m (n, m E IN) iffthenthbit of thebinaryexpansionof m is 1. It mightbe a good ideatoconcentrate briefly onthethreesalientfeatures of thesimplestpossiblecase.
CHURCH'S SET THEORY WITH A UNIVERSAL SET
117
K = {O,l}, and theotherclause gives us complements; The E relation of the original modelwellfounded; is The rankof fst(k'x)is no greater t hantherankof x.
When theseconditionsare met we can say a lot. DEFINITION 3.9. An antimorphismis a permutation 7r of theuniversesatisfying(V'xy)(x E y +--> 7r'X (j. 7r'Y) . REMARK 3.10. Underthethreeassumptionsabove the new model admitsan antimorphism.
Proof Declare the following recursive definition a'x = : k-1«a"(fst(k'x)), (1 - snd'(k'x))} .
By consideringputativecounterexamples of minimal rankwe can show thatthis is everywhere defined . Suppose a'y E co a'x. This is a'y Eco k-1'(a"(fst(k'x)), (1 - snd'(k'x))} .
Now eithersnd'(k'x) = 0 or snd'(k'x)
= 1.
O. If snd'(k'x) = 0, the displayed formula becomes a'y Eco k-l'(a"(fst(k'x)), I},
which is a'y
which isy
(j.
(j. a"(fst(k'x)),
(fst(k'x)). Butif snd'(k'x) = 0, thisbecomes y
(j.co
x.
1. On theotherhandif snd'(k'x) = 1, then
a'y Eco k-l'(a"(fst(k'x)), (1 - snd'(k'x))}
is a'y Eco k-I'(a"(fst(k'x)),O}.
This is a'y E a"(fst(k'x)), which of course simplifies to y E fst(k'x), and (sincesnd'(k'x) = 1) this becomesy (j.co x. So eitherway we havea'y E c o a'x
+-->
y
(j.co
x,
-j
Thereis another nice resultwe get as a reward for making these assumptions. Considerthe game G x , played as follows, by two players, I and II. I picks Xl E X, II picks X2 E Xl, I picks X3 E X2 ..• , with the first playerunableto move (youcannotpick a member of theempty set!) losing.
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THOMAS FORSTER
Clearly t heaxiom of foundation implies thatG x is alwaysdeterminate in the sense ofadmittinga winningstrategyfor oneplayeror theother . This is E-determinacy . The axiom of foundation is obviouslyimplicatedbecause if x = {x}, G x is clearly notdeterminate.The converseis nottrue, however, andwe canprove thefollowing. 3.11. Under the threeassumptionsabove, the new model obeys E-determinacy .
REMARK
er I if thereis ayE x such thatG y is a win for Proof. G x is a win for play playerII. SimilarlyG x is a win forplayerII iff for every y E x , G y is a win forplayerI. First we prove thatthereare x of arbitrarily high ranksuch thatx is hereditarily lowand G x is a win forplayerII. (And playerI, but we do notneed that .) This is because, undertheassumptionsgiven, H 10w is an isomorphiccopy oftheoriginalmodel and thereare clearly x of arbitrarily high rankwiththoseproperties. Next we prove byinductionon rankof x that(in thesense of thenew model) G x is determinate ( admitsa winningstrategy for I orII). Suppose this is not true,and let x be a counterexample of minimal rank. Then k'x = (y,l) or k'x = (y,O), for some y. If k'x = (y,O) for some y, thenby inductionhypothesisG, is determinatefor every z cEo x and G x must be determinate as well . If k'x = (y,l) , thenthereis some ordinalQ (namely rankof y + 1) such that , for all z, if z is an elementof theoriginalmodel of rankatleastQ, thenz rt y, so (since k'x = (y, 1) z Ec o x. Now we havejust provedthatatleastsome of thesez give rise toG, thatare wins forplayer II. SO thereis atleastone z Ec o x suchthatII has awinningstrategy for G z • ButthenI has awinningstrategy forG x which helaunchesby picking
z.
~
3.1.2 An elementary example We are going toconsidertwo models, morecomplicatedthantheoriginal Oswaldmodel, in which inadditionto complementsfor allx, we also have B'x for allx. B'x is {y : x E y}.5 The theorythathas booleanaxioms (U, nand- ), singletonand B is NFO. (The '0' is intendedto suggestOpen. This theoryis more naturally axiomatizedas Extensionality plus existence of {x : ¢(x,y)} where ¢ is stratified a nd quantifier-free. It is a relatively thattheseaxiomatizations areequivalent.) straightforward exercise to check The secondmodel willsatisfyallthebooleanaxioms (and so is a model of NFO) , butwe startwiththefirst, which does not . 51 use this notationbecause it was Boffa who fir s t im p ressed on me the importan ce of this operation. In fact the first use of this op erationwas by Quine, and Whitehead suggestedtohim that{y : x E y} should becalledthe essence of z:
CHURCH'S SET THEORY WITH A UNIVERSAL SET
119
ity with ConsiderS = theset ofreducedwords inthesemigroup-with-un twogenerators e and b, and theequatione2 = Is. ' e' is intendedto recall "Complement"and'b' to recall "B'x". We now letk be a bijectionbetween V andV x S. We define Eco by recursionby cases: 3.12. 1. If snd(k'x) = ew for some wordw E S, then
DEFINITION
yEcox iff yrt.cok(-l)(fst(k'x),w) . 2. If snd(k'x) = bw for some wordw E S then y Eco x iff k(-l)'(fst(k'x) ,w) Eco y.
3. If snd(k'x) is Is , theny Eco x iff y E fst(k'x) . PROPOSITION
3.13. The E co of Definition 3.12 is extensional.
Proof. The proofis an inductiveproof(by cases) onthe lengthof words in S. The situationwe arecontemplating is two distinctXl and X2 which we will beinterested in thesecond have thesame Eco-members. Naturally componentsof k'Xl and k'X2. Thereareseveralcases toconsider: 1. The secondcomponentsarebothIs, theunitof thesemigroup. In this case thefirstcomponentsmust be thesame, and Xl = X2 follows.
2. One of thesecond componentsis Is and theotheris not. Suppose snd(k'xt} = Is and snd(k'x2) is bx or ebw. (We cannothave two adjacente's since the words arereducedand e2 = Is .) Then the collection of y such thaty Eco Xl is fst(k'XI), which is a set inthe sense of (V, E). In contrast , thecollection of y such thaty Eco X2 is y such thaty rt.co k(-l)(fst(k'X2) ,b(x»either(i) thecollection of which is to say, by an a pplication of part2 of Definition3.12-thesame of y such thatk(-I)(fst(k'X2),X) rt.coy, or (ii) is the as thecollection collection of y such thatk(-I)'(fst(k'X2), w) Eco y. The collection in case (i) cannotbe a set becausefor anyobjecta we can easily find proper-elass-many u norderedpairs which do not have a as a member. The collection in case (ii) cannotbe a setbecausefor anyobjecta we can easily find proper-elass-many u norderedpairs which do have a as a member. of both 3. snd(k'xt} = e(xI) andsnd(k'x2) = e(x2) wherethefirstletter Xl and X2 is b. Assume (Vy)(y Eco Xl +---4 Y Eco X2) with a view to deducingXl = X2 . We can expand thisin accordancewith part2 of definition
120
THOMAS FORSTER which is to say
Thatis to say, if we have d istinctXl andXz withthesame membersg,, wherethefirstletter of snd(k'xt}is thesame as In-the-sense-of-s thefirstletter of snd(k'xz),namelyc, thenthereare distinctYI and yz (namely k(- 1) (fst(k'XI),XI) and k(- 1) (fst(k'xz),xz))satisfying (V'z)(z E co YI +---+ Z E c o yz) and snd(k'YI) and snd(k'yz)areshorter thansnd(k'xt}and snd(k'xz)-indeedthey areterminalsegments. in are Xl This clearly reducesto thecase where the two wordsquestion and Xz bothbeginningwithb, which we nowtreat. 4. The second componentsof k'XI and k'xz are words b(XI)a nd b(xz). We have(V'y)(y E c o Xl +---+ Y E c o xz) . Expandingthisby clause (2) in Definition .12 3 we obtain
Now as long as
we can falsify this biconditional by takingY to bethesingletonof one of these two.Singletonsexist by lowcomprehension : thesingletonof X (in the sense of cEo ) is justk(-I),({X},O).
5. k'XI has secondcomponentCbWI andk'xz has secondcomponentb(xz). and This is really the case where one word begins with a the c other reduced, we also knowthat begins with a b,butsince thewords are all theletter following t hec in theword beginningwitha c must be a b. If we have this becomes
This is thesame as
iff (by clause (2) in Definition3.12) Now Y Ec o k(-1)(fst(k'xt},b(xt}) k(-I)(fst(k'xt},XI) Eco Y
CHURCH'S SET THEORY WITH A UNIVERSAL SET
121
andsubstituting thisfor'y E co k(-l)(fst(k'xd,b(xd}' in
we obtain
Now anythinglike (Vy}(a E y t - - 4 b 'f- y) must always be false because -1 of theexistenceof theempty set. The restis easy:
3.14. With Ec o as in Definition 3.12, (V, Ec o ) exists} and (Vx}(B'x exists) .
PROPOSITION
F
(Vx) (- x
Proof. The complementof x will be k( -l)(fst(k'x),csnd(k'x)} , for, by clause (1) in Definition3.12, we have Y E co k(-l)(fst(k'x),csnd(k'x)} iff y 'f-co k(-l)(fst(k'x),snd(k ' x) iff y 'f-co x .
B 'x will be (k-l)(fst(k'x) ,bsnd(k'x)}, for, by clause (2) in Definition3.12, y E c o k(- l)(fst(k ' x),bsnd(k'x)} iff k(-l)(fst(k'x), snd(k'x)} E co Y iff x E c o y.
-1
3.1.3 P-extend ing model s of Zermelo to models of NFO
Before wecontemplate t hesecondconstruction (which gives us a model containingx n y andxU y for allx and y) we had betterask ourselves why we did not get it lasttime. After all, theseaxioms hold wh en K = {O, 1}. The point is thatin thatcase everythingis thesame size as an old set or the complement of an oldet.s Also ifx and yareboththesame size as an old set orthecomplementof an old et, s thenso are x n y and xU y, andso n and U do notconstruct a nythingthatis not alreadythere. Once we have B 'x this breaksdown, and if we wantx U y and x n y in generalwe have to construct themspe cially.T his makes theconstruction of models of NFO altog e thermore daunt ing. THEOREM
3.15. Ev ery model of ZF+foundation has a P -extension that is a
model of NFO
+
low com prehensi on.
122
THOMAS FORSTER
Proof. (We certainly need notrestrict ourselves to model s of ZF, for this is certainly t ruefor well-founded modelsZof and presumablyweakertheories as well .) The idea oftheconstruction was originally thatk'x is to be apair (y, w) wherey is a set andw is a reducedword in somealgebrawithoperations thatcorrespondto theoperationsu nderwhich wewantthe universe of the new model to be closed. Although this can be made to work, theapproachit gives is very much less smooththanan approachthatcreatesa giganticfree NFO model over aproperclass ofgenerators wherether e is one generator for eachsetof theold model.Unfortunately this second , smootherapproach is not really CO a construction , and sostrictly does not belong here as an illustration . The excuse forputtingit in here ist hattheconstruction is, in spirit, very close to the CO constructions thatprecede it and will succeed it, and its presence here will help. In the last case the only operationsin which we were interested wereunary (complementation and B) so the case had spurioussimplicity a . Recallthat the axioms ofNFO are (apartfrom extensionality) existence of{x} and au pied de la lettre closureunderB, U and n. If this were aCO construction we would get closure undersingletonfreebecauseof lowcomprehension , so we couldexplicitly forgetaboutit. Here too we can forget aboutit, andwill indeed doso-forthemoment. Laterwe will see why this is all right after all. We should be able to makewithonly do one of nandU, butconjunctive and disjunctivenormalforms for boolean words are so useful thatwe will retainboth. Our languageof terms has a constantt erm gx for each old set x. The constants willeventually correspondto low sets of thenew model. We also havefunctionletters U, n - and B . We have to do a bit of work to find thecorrectnotionof reduced word for thisalgebra.Thereis theirritating t hatwe do not want to have bothanb and bna butwe can getround feature thatby wellordering the alphabetand extendingtheorderlexicographically turnoutthatwe will want to augmentthelanguage by to thewords. It will addingA (symmetricdifference) . DEFINITION 3.16. A restricted wordis eithera constant, or is WAg where W is a booleancombinationof Bs of restricted words, and9 is a constant . If 9 is missing, we canspeak of apurerestricted word.
Now we have to show t hateverythingt hatwe wish toconstruct can be denotedby a restricted word. We can thinkof a wordw as a booleancombinationas a union ofinter sections, where thethingsbeing intersected areconstants or Bs and complements ofeither.Consider anintersection like anbncnd ....
CHURCH'S SET THEORY WITH A UNIVERSAL SET
123
If even one oftheseis a constant(i.e., willcorrespondto a low set) , then thewholeintersection can be representedby justone (new)constant . Intersections of anynumberof complementsof low sets can be represented as one complementof a low set . Thus theintersections areeitherconstants, or intersections of Bs and - Bs withthecomplementof a lowset, which si to say, anintersection of Bs and - Bs minus some low set. Thus w can berewritten in theform (1)
wheretheWi areintersections of values of B or complementsof values of B, and the9i are low sets. We will work throughthis inthecase wheren = 3, is so thatthe readerthansee how to dothegeneral case. (This probably more helpful t hana rigorousproofof thegeneral case would be!) For the moment we areinterested only intheexpression insidethesquarebracketof formula (1): (WI
n 9I) U (W2 n 92) U (W3 n 93)'
This expandsto aninterse ctionof 23 (WI U W2 U W3)
= 8 subformulre as follows :
n sevenotherterms.
A typicalexample oftheseothertermsis WI U W2 U 93' It is typicalin that it containsat least one entryof the kind9i . Now WI U W2 U 93 is thesame as
This is the complementof a low set , so we canthinkof this as 9novell for some novel low set 9novell' This can be done to all the remainingterms, six so theexpression-inside-the-square-bracket in formula (1) now looks like
Subtracting finitely many low sets thesame is as subtracting one, so the expression-inside-the-square-bracket in formula (1), reduces to
reducedw to So, in thegeneral case, we have
(U i~n
This is actually
W n - 9novel) U 9n+!'
THOMAS FORSTER
124
where G is i~ n
i~n
which is a low set. Notice thatwe startedwithW as a booleancombinationof B's and constantsand have ended up with somethingrathersimpler: U i
CHURCH'S SET THEORY WITH A UNIVERSAL SET
125
We'd betterverifythatthisdoes actually define aP-extension.Suppose y E co i' x . Then y E c o 9 (k (- l)oi) "x which is to sayk- 1'y E (k(- 1) 0 i )"x , . Next whencey E i "x and y is a value ofi. So therange of i is transitive suppose i'y E c o i'x. This is i'y E c o 9 (k (- l)o i ) "x, which inturnis k- 1 'i'y E (M-1 ) 0 i) "x , which isjusty Ex. So i is an isomorphism. Finallylety be a low set of values iof . Then i' y will have to be 9 (k l - l)o i ) "y' if thislastthing is defined. But (k( - l) 0 i)"y is certainly a set, so 9 (k (-I )oi ) " y is definedand is available to bei'y. -j One corollary of thisis thatany fragmentof Z strongenoughto execute this construction proves every theoremof NFO . See Forster-Kaye 1991 for anexplanation of Df expressions.
Ilr
3.1.4 Church's Model
in which everyhereditarily Church'smodels are all roughCO constructions low set has an n-cardinaJ.W hatis an n-cardinal?T he l-cardinal of x is the cardinal ofx in theusualsense. And forlarger n? Actually it does notmatter a greatdealwhathappens forlargern, since whatev er we decide to mean t hat by it it can bemade to work. Sheridanis developinga construction accomodatesa versiondifferentfrom theone here(withtheeffectthatthe singletonfunction , consideredas a set ofWiener-Kuratowski orderedpairs, is a union of finitely many of thesecardinals) . One natural version ofthe idea ofn-cardinal is quite well-developed in NF studies. Recallthatj is the operationon permutationsdefined by(j '1I") 'x = 1l" "x. Then we say x is nequivalent to y if thereis some permutation 11" of V so that (j n'1l")'x = y. See Forster 1995 for more onthissortof n-cardinaJ.These ideas areanticipated in the eminentlyreadableTarski 1986. , two sets have t hesame n+l-cardinal iffthereis In theversion used here a bijectionbetweenthem such thattheelementspaired by thebijectionhave thesame n-cardinaJ.We startoff withthecase n = 1 forsimplicity'ssake. As withtheexampleof thelastsection, thisis not astrictCO construction. We need abijection(k, as ever) betweenV and a set of codes for objects. We will usethenotation' x = Y , to mean thatx andyarethesame as size. We will see very soon how thesefaketerms(' x ' etc.) canbe treated genuinedenotingterms. DEFINITION
3.17. The thingsthatare are values kofareeither:
1. orderedpairs (x , i), where x is an arbitrary setand i is 0 or 1 (thiswill providelowsetsand complementsof lowsetsas usual), or
2. orderedpairs (i , K), whereK is a cardinal(otherthan0) andi is either I or II. I and II are two unspecified d istinctobjects thatwere notin
THOMAS FORSTER
126
theoriginalgroundmodel. The idea is thatthese objectsare to be cardinals(andcomplementsof cardinals)in thenew model. Now we sayy E co x iff 1. snd(k'x) = 0 and y E fst(k'x), or
2. snd(k'x) = 1 and y
~
fst(k'x) , or
3. fst(k'x) = I and snd(k'y) = 0 (so y is low)and fst(k'y)= snd(k'x) , or 4. fst(k'x)= II and snd(k'y) f= 0 (so y is low) orfst(k'y)f= snd(k'x). Clauses1 and 2 make surethatevery low set has complement. a Notice thatnothinghas beensaid aboutwhatcardinalnumbers are. Notice also thatthisdoes notmatter!All we need ist hatthereshouldbe a definable class Canda definablerelationbelongs-tobetweensetsand members of C satisfying (\lx)(3!y E C)(x belongs-to y), (\lx\ly)( x = 11 <----> (\lz E C)(x belongs-to z
<---->
y belongs-to z)).
The termx canthen be taken todenotetheappropriatem ember ofC. We do not needtheaxiom of choice to define cardinaln umberssince as long as we havefoundation (which we areassuminghere) we canuse Scott cardinals. The Scottcardinalof x is theset of all thingsthesame size as x that are of minimal rankwiththisproperty. Clause3 willensurethatevery lowset has a cardinalin thenew model y low setx, thecollection of all setst hat (in thestrongsense thatfor ever have thesame cardinalas x is a set ofthenew model). We stipulatet hat cardinalsused do notinclud e O. We do this for tworeasons: (i) to keep 0 free tosignallow sets as usual , also(ii) becausetheexte nsion of the cardinal number0 (theset of all empty sets) is a set by low comprehensionanyway, and we do not wishit to make difficulties ours for elveswithextensionality by manufacturing it twice.Clause 4 ensuresthatthecomplementof every such cardinalis a set. The usualapparatusof lowcomprehensioncan now betakenforgranted. It shouldby now be clear t hatthis model is a model of complementation . It is theexistenceof cardinals thatwe hadbetterspend a bit oftime verifying . PROPOSITION 3.18. The clauses of Definition 3.17 give a model in which every low set x has a cardinal: {y : y = x }.
Proof. Notice thatthe cardinalst hatwe havecreatedby this means are t hecomplementsof lowsets, which makes demonstrably n eitherlow nor are
CHURCH'S SET THEORY WITH A UNIVERSAL SET
127
set. Considertheorderedpair(I, fst(k'x» . life mucheasier.Let x be any low We will check t hatk(-l)'(I , fst(k'x» is thecardinalof x (in thesense that it is theset of all t hingsthesame size as x) in thenew model. Byclause 3 we have y E co k(-l)'(I , fst(k'x» iff y is low and fst(k'y)=fst(k'x). Since x and yareboth low, this is the same as saying thatthe set (in theold sense) ofthings E co Y is thesame size (in the old sense)as the set ofthings Ec o x , so thereis a bijectionbetweenthesetwo (old)sets. This bijectionis an (old)set of (old)orderedpairs. By lowcomprehension (Theorem3.4) thecorresponding(new) set of (new) orderedpairs is also a set, so x and yareof thesame size in the new senseas well . The other directionis easy. Thereforek (-l) '(I , fst(k'x») is indeedthecardinalof x in thenew model. Correspondingly k (-l) '(II, fst(k'x» is thecomplementof thatcardinal , which we have to havecomplementation if is to betrue. --l We have to do al ittle b it of work to see how to generalize t hiscorrectly setof lowsetshas a2-cardinal. tothecase n = 2, themodel where every low WhatChurchactually claims isthatfor eachn his construction gives us a model where every well-founded set has ann-cardinal.I preferthestatement in termsof lowsets, lowsetsof . .. n lowsets. For thecase n = 2 we have toadd two moreclauses5 and 6 to Definition 3.17 in thesame style . We will need two more novel constantsin thestyle of I and II, which we mayas wellwrite'III' and 'IV'. Objectsx such that fst(k'x) = III will be2-cardinalsa nd objectsx such thatfst(k'x) = IV will becomplementsof 2-cardinals . We will needthenotation'2-card'x' for the2-cardinalof x , and we will use lower case Greek lettersto range over 2-cardinalsas overcardinals.Then thereareto be twofurtherkinds of orderedpairs in therangeof k: pairs whose firstcomponentsare III and pairs whose firstcomponentsare IV. In bothcases thesecond components are2-cardinals . We will needthetwo following new clausesthedefinition in of y E co z: DEFINITION
3.19.
5. fst(k'x)
=
III and y is a low set of low setsand
2-card'{fst(k'z) : z E fst(k'y)} 6. fst(k'x)
=
= snd(k'x) .
IV and (y is not a low setof lowsetsor
2-card'{fst(k'z) : Z E fst(k'y)} =I- snd(k'x» . Clause6, of course, ensuresthat2-cardinals , too, have complements. The detailswill beomitted. 3.20. The membership relation of Definition 3.19 gives a model in which each low set of low sets has a 2-cardinal.
PROPOSITION
128
THOMAS FORSTER
Proof. Let x be a low set of low sets. Then the 2-cardinal(in the sense of E co ) of x will bek(-l)'(III, 2-card'{fst(k'z) : z E fst(k'x)}) . We had
bettercheck this. Suppos e y is a low set of low sets. Then y E co k(-l)'(III, 2-card'{fst(k'z) : z E fst(k'x)})
iff
= 2-card'{fst(k' z)
2-card'{fst(k'z) : z E fst(k'x)}
: z E fst(k'y)} .
Whatwe actually want is forx and y to havethesame 2-cardinalin thenew sense. As before, if thereis an (old)bijectionbetweenfst(k'x) and fst(k 'y), therewill be a new bijection between x and y by lowcomprehension. And thesame goes not only for x and y, butfor eachz' E co x and y' E co y thatare paired by the bijection: ifthereis an (old) bijection between fst(k 'x') and fst(k'y') , therewill be a new bijection between z' and y' by lowcomprehensionas desired. --I It should now be clear howtinker to with thisconstruction to add simultaneously for all n E IN, theassertionthatevery (low set of')" low sets has an n-cardinal.It is perhapsworthnotingthatit does not seem to be necessary to arguethat,for eachn E IN, we can do this for all< mn and thenuse compactness. 3.2. Wellfounded sets in CO-structures The rootsthatthistechniquehas inRieger-Bernays permutation models still have fruit to bear, as witness the following theorems. two 3.21. For a given choice of K and V and rules, all CO structures are permutation models of each other.
THEOREM
over a model(V, E) with the Proof. Suppose we have twoCO structures same K butdifferent coding functions k and k' respectively. We wish to find a E Symm(V) such thatthefirst modelthinksthatx E y iff the second --I thinksx E u'y. o must be (k')- 1 0 k. We also havet hefollowing: 3.22. Every permutation model (V, E CO ) 17 of (V, E co ) is obtained from it by replacing k by some k' with a corresponding new membership relation Eco'. If the permutation is a , then the new k' is o : 1 k . PROPOSITION
Proof.
(V, E C O ) 17
FxEy
iff {
(V, E co ) (V, E) (V, E)
F x E u'y,
Fx
E co u 'y ,
F x Eco' (V, Eco') F x E y.
y,
--I
CHURCH'S SET THEORY WITH A UNIVERSAL SET THEOREM
129
3.23.
(1) Hlow is always isomorphic to a permutation model of the original universe.
(2) Whatever K we started with, for any permutation a of the old universe we can find a coding function k so that (Hlow , Eco ) ~ (V, E,,). Proof.
(1) Therewill be abijection7f : V (Vxy)(x E a'y.
f--->
f--->
H 1ow ' We seek aa so that .7f'X E c o 7f'Y).
Whatis a'y? Clearlyit has to be{x : 7f'X Eco 7f'y}, This is a set, since7f'y is low. We must check t hatthis definitiongives us aa thatis 1-1 andonto. 1-1 by extensionality of Eco . Is it onto?Given z we must find It is certainly a y so thatz = {x: 7f'X Eco 7f'y}, This y must be 7f- 1'k- 1 '(7f"Z,0). (2) We know-howeverwe choosek-thatH 10w is a properclass whose complementis a properclass, so let7f be a bijectionbetweenV and such a class, and let us fasten that on class to beH10w and resolve to cook up k so thatit actually is the H 10w of the new model.DugaldMacpherson has used the word "moiety" for thingsthatare bothinfinite and coinfinite: we will borrow it here describe to proper classes whosecomplementsare of V. We want to cook up k so that properclasses. Leta be a permutation (Hlow, Eco ) ~ (V, E,,). We want (Vxy)(x E a'y.
f--->
.7f' X E co 7f'Y).
°
The righthandside is 7f'X E fst(k'(7f'y» (and snd(k'7f'y)= since 7f'y is a lowset). Now fst(k'7f'y)= 7f"a'y, so we wantk'7f'y = (7f"a'y,O). It is true thatthisonly tells us whatk shoulddo to values of n, but sincetherangeof 7f is a moiety andtherangeof k 0 7f is also a moiety,therewill be noproblem orderedpairswe need. -j extendingthisto abijectionbetweenV and all the Analogously withtheembeddingi : V '-+ V" defined inthetheoremmodel intothenew model(V, Eco ) defined byrecursionon E: DEFINITION
3.24. i'x = : k(-l)'(i"x,O).
Like thei of Theorem2.6, thisembeddingis a P-embedding. 3.25. If (V, E) is well-founded, then i is defined and is a P-embedding from (V, E) into (V, E co ) .
THEOREM
Proof. Firstwe prove by E-inductionthati is defined on all sets. We must nextcheckthati is an isomorphism, so we wanti'x Eco i'y f---> X E y. Since thesecond componentof k '(i'y) is 0, i'x Eco i'y iff i'x E fst(k'(i'y» = i"y iffx E y.
130
THOMAS FORSTER
Next we showthatthe range of i is transitive' Eco). Suppose y is in the range ofi, and y = i'z. So x E co i'z = k(-l)'(i"z, 0) iffx E i"z, so x would also be intherange ofi . Finallywe must checkthatany subsetof somethingin therangeof i is likewise in the rangeof i . Suppose x is in therange ofi, so thatk'x = (i"z, 0) for some z. Suppose alsothat(Vw)(w E co Y . . . . wEco x) . Considerthe set t heoriginal model) of thosethingsthatare E c o y. This is (in the sense of indeed a set of the originalmodel, since it is a subsetof i"z. If it is i"u, then k'y must be (i"u,O), so y is in therangeofi. --i (Essentially, this proofis in Church 1974 thoughhe prefers to allege that thewell-founded sets form a model of ZF and does nottheconceptof have a P-embedding.) The difference here from Theorem2.6 is thatthis time wecannotexpecti to beelementary forstratified forrnulee.(It's a goodquestionwhichforrnulee it iselementary for,actually!) However we can at leastderive the following: COROLLARY 3.26. The P-embedding i : (V, E) ....... (Htow,E co ) is elementary for stratified [ormulai.
Proof. Since Hl ow is isomorphic to a permutationmodel ofV (by Theo--i rem 3.23) we can useTheorem2.6.
Does this meanthatwe can takethewell-founded sets thenew of model to be the range of i? No, because we arrange can for the range iofto be a set ofthenew model,and it will be in some sense well-founded . PROPOSITION
3.27. The range of the canonical embedding can be a set of the
new model. Proof. We use a slightmodificationof Oswald'soriginalconstruction from section3.1.1: if x is a finitesubsetof H, thenE n Ex2n +l E H. Then say x E co Y iff either: y is 2n and thexthbit ofn is 1, or y is 2n + 3 and the xthbit ofn is 0, or y = 1 and x E H. To completetheproofwe thinkof IN as a copy ofVw by associatingwith it theAckermannrelation alluded to above. The canonicalembedding i is thendefined oneverything in Vw which is a set of thenew model. --i
However,although it is possible for the rangei of to be a set, it is not always a set. PROPOSITION
3.28. If the three conditions of section 3.1.1 are met, the range
of i is not a set . Proof. We wantto showthatif X satisfies(Vy E co X)(y is in therange of i), thenX is in therange ofi too. The case snd(k'X) = 0 we havealready considered.Thereremainsthe casesnd(k'X = 1).
CHURCH'S SET THEORY WITH A UNIVERSAL SET
131
We will showthatthis case cannotoccur. If it did, x Eco X iff x f/. fst(k'X). So X would be so large thatthereis only aset of thingsthat are not members(Eco) of it, andallthethingsthatare would be values of i. Butthecollection of values of i is not so bigthatits complementis a set, since ifx E rangei, snd(k'x) = 0, and thereis a properclass ofx such that snd(k'x) = 1. Therefores nd(k'X) i= 1. -l The possibilityof addingnew well-founded setsin thenew model inthis ad hoc wayrestricts thethingswe can say ingeneral a boutwell-founded sets, at leastif we takewell-founded sets to be thosedefined inthenatural way s ets (Definition3.29). by means of theinductivedefinitionor asregular The inductivedefinitionis theobviousone: if wethinkofwell-founded sets as thoseover which we can do E-inductionthenwe areled totheinductive definition:
WF(x)
+----t
(Vy)«Vz)(z ~ y
-->
z E y)
-->
x E y).
(2)
The trouble w iththisdefinitionis thatif we have very l ittle comprehension theremay be so fewy such that(Vz)(z ~ y --> Z E y) thatlots ofs etsmight turnoutto bewell-founded thatshouldnot. In fact, in ZF thereare no such sets atalland thisdefinitionis completelyuseless! Wetalkof regular s ets instead. DEFINITION 3.29. x is regular iff (Vy)(x E y
-->
(3z E y)(z n y = 0)).
If we have anaxiom of complementation , we can prove:
PROPOSITION 3.30. A set is well-founded in the inductive sense iff it is reg-
ular. Proof. Suppose (Vy)«Vz)(z ~ y --> z E y) --> x E y). Substitute- y for y, getting(Vy)«Vz)(z n y = 0 --> z f/. y) --> x f/. y). Contrapose,g etting (Vy)(x E y --> -.(Vz )(z n y = 0 --> z f/. y)). This is (Vy)(x E Y --> (3z)(z n y = 0/\ z E y)), which saysthatx is regular. -l The inductivedefinitionenablesus to provethateverywell-founded set has ¢ as long astheextensionof ¢ is a set and any set ofthings-whichare-¢is itself¢ . This is a lot less applicablethanit mightseem becauseof theextremeunlikelihood t hattheextensionof ¢ shouldbe a set. We can prove thatif x E Xl E X2 . • 'X n E X then-{X,XI" . x n } is a set y such that (Vz)(z ~ y --> Z E y), andso no well-founded set ismember" a of itself,b ut one wouldexpectto be able to prove a lot more. Thereis also aprincipleof inductionforregular s ets. This is standardin moderntreatments of settheory,a nd we do it as follows . Suppose we know (Vx)«Vy E x)(¢(y)) --> ¢(x)), andsuppose thereis a regular set z such that -.¢(z) . Let Y be a transitive s et containingz: Let X = {y E Y : -.¢(y)} . Thenz E X andz is regular, so X must be disjointfrom one of itsmembers. Butthiscontradicts theinductionhypothesis,so ¢(z).
132
THOMAS FORSTER
How does thisfare intheChurch-Oswald c ontext?If we knowthatany set ofthings -which-are-eis itself, and thatfor anyregularz thereis a transitive s et Y containingz such that{y E Y : -,(y)} exists,thenwe can prove thateveryregularset is . Can we do this for allr egularz and all ? If we could showthatfor everyregular z thereis a lowtransitive Y with z E Y, thenwe cando therestby lowcomprehension . We could dothisif we in i, knewthatevery well-founded set istherangeof H 1ow , thehereditarily low sets, mightc apturebettertheidea ofsetswellfounded inthenew model. Here we have to rememberthatfoundation fails andthattherefore thereare several n otionsof hereditarily low. Inparticular theremay be hereditarily low setsthatare notwell-founded, for example Quine atoms, which aresets equalto theirown singletons . Thoughthis is easy toarrange,it is also easy to avoid. We can arrange t hatevery setthatis hereditarily low (inthesense thateverything in its transitive closure is low) is also intherangeof i . Suppose (x n : n E IN) is a descendingw-sequence underEc o and thatXo is hereditarily low. Thenfor all n , snd(k'xn ) is 0, so x n +! E c o X n is justXn+l E fst(k'x n ) , so Xn+l is a set of lowerrankthan fst(k'x n ) . If we set upk so thatfor ally therankof fst(k'y)is no greater thantherankof y, thentherecan be nosuchinfinitedescendingsequencein thenew model andallhereditarily lowsets are well-founded . In particular thiswillhappenwheneverthethreeconditionsof section3.1.1 are met. Whatthissays isthatwe canarranget hattheonlysetsthatareillfounded are large, or atleasthavesomethinglarge intheirtransitive closure . There is a similarproblemin NF: must self -membered setsbe large?WhenI asked thisquestionin thefirsteditionof Forster 1995, MauriceBoffa came upwith a permutationmodel whichpartlyansweredmy question . The permutation he used (and which heand Petryreportedin 1993) uses a device abit like a rankfunctionand it admits a greatdeal ofrefinement . It is now possible using resultsof Korner(1994) to showthatit is consistentrelativeto NF thatE restricted to finite sets is well-founded . This is done byRieger-Bernays permutation models. It soundsa weakerresultt hanarranging forE restricted to lowsetsto be well -founded, butbearin mind thatin thepresentcase all we have to do ispreventtheappearanceof newill-founded smallsetsin the new model,whereasin theNF case wenotonly have to do thatbutwe have to kill off any smallill-founded sets inthemodel westarted w ith, and-being a model ofNF-itmay have had lots . fixed pointfor "set of all low Nice thoughit is to knowthatthegreatest subsetsof' andtheleastfixedpointcanbe thesame, whatreallymattersis thatit istheleastfixedpointwe wantforourconceptof well-founded the in new model. Atreatment now follows . DEFINITION
X)
-+
3.31. Let us sayx is well-founded" (V'X)«V'y)((low(y) if /\ y -+ x EX).
Y E X)
~
CHURCH'S SET THEORY WITH A UNIVERSAL SET
133
In otherwordsthecollection of well-founded" sets istheleastfixed point corresponding to thegreatest fixed point Blow' We willjustifya principleof unrestricted E-inductionfor well-founded" sets. Firstwe check (i)thatthecollection of well-founded" sets istransitive and (ii) thateverythingin it is low. (i) If x is well-founded" , and(Vy)(low(y)l\y ~ X) -+ Y E X), thenx EX . It will suffice to show thatx ~ X as well . Suppose x ~ X. We will showthat(Vy)(low(y) 1\ (y ~ (X - {x})) -+ Y E (X - {x})) , whencex E (X - {x}) (since x is well-founded") . This is impossible. Suppose y ~ (X - {x}) and is low. Then y ~ X and y EX. To deduce y E (X - {x}) it will suffice to show y :f x, which would follow from x ~ (X - {x}) . Butwe haveassumed thatx ~ X , so a fortiori x ~ (X - {x}) . (ii) Suppose x is well-founded" butnot low , and that(Vy)«low(y) 1\ y ~ X) -+ Y E X) . Let y be a lowsubsetof (X - {x}). Then y is a lowsubsetof X and is therefore a member of X . We wantit to be amember of X - {x} so we wanty :f x. Buty is lowand x is not. Therefore(Vy)«low(y) 1\ y ~ X) -+ Y E (X - {x})) and x E (X - {x}), since x is well-founded" . Nextwe need to know thatTC(x) existsif x is well-founded" . We prove by inductionon IN thatif x is well-founded" , thenUn X existsand is low. Low comprehensiontells ust hata low set of low sets has a sumset. low The inductionis good becauseIN is a low setandso thereis enoughcomprehension for full inductionover it. So thecollection { unx : n E IN} is a lowsetof low setsand its sumset-TC(x)-is a lowset. Nextby relativising theproofof proposition3.30, we showthatx is wellfounded"iff (Vy)(x E y -+ (3z E y)(x n y = A 1\ low(z))) . (A is theempty set.) Finally, we prove E-inductionas follows.Suppose we know (Vx)«Vy E x)(¢(y)) -+ ¢(x)), and suppose thereis a well-founded" set z such that .¢(z). LetX = {y E TC({z}): .¢(y)} whichexistsby lowcomprehension. z E X and z is well -founded",so X must be disjointfrom one of itsmembers. Butthiscontradicts theinductionhypothesis,so ¢(z). -j We can prove this unrestricted scheme of E-inductionscheme for wellfounded" setsd espite being unableto prove it for sets well-founded in the inductivesense webeganwithbecausetheremay well be large well-founded setsaboutwhich we knownothing . Armed withthisscheme ofE-inductionfor well-founded" setswe canprove thatevery well-founded" set istherangeof in i. Perhapsthemoral ofthisdiscussionis thatwe shouldrestrictourselves to codingfunctionsk whichadd no new well-founded sets. thosecircumIn thenewlanguage as "same size as awell-founded stances"low" is definable in set" asChurchoriginally intended,a ndthewholeenterprisebecomes axiom-
134
THOMAS FORSTER
atizable.It is truethatwe lose somegenerality by making this restriction, butwe sacrificedgenerality attheoutsetof thissectionby notconsidering modelsof ZF withoutfoundation.Neitherof these seems aharshsacrifice . 4. OPEN PROBLEMS
4.1 . The Axioms of Sumset and Power set At presentthe only form of theaxiom of sumset we can proveforthisconstruction is thata lowsetof lowsetshas a sumset (which will be low) . Can we tweakthis construction to get a lessrestricted axiom of sumset, dropping thesecondor perhaps even thefirstoccurenceof 'low' intheabove? Similarlytheonly form oft heaxiom of powerset thatthisconstruction apparently gives us is powersetsof lowsets. Withneitherof thesetwoaxioms is therea standard p aradoxobviouslyskulkingin thewings waitingto cause forms ofthem. It's natural to see if we trouble s houldwe adoptunrestricted cando betterthanthis. Mitchell 's settheoryallows poweret, s buthas other disadvantages.For example, x U y and x n y do notexistin general. The questionis alsodiscussedin Sheridan'sthesis.
4.2. Natural strengthenings of Theorem 3.15 This is themost generala ndthemost importantof theopen problemsin this area. Such extensionswould be theoremsof the form "Ev ery well-founded modelof T 1 is thewell -foundedpartof a model ofT2 " . It is-perhapsin this connectionthatChurch's remarkin 1974 seems most apposite: "One source of added axions to bestudiedfor theirconsistencywith the basic axioms is ... Quine set theory . . .. An inter e stingpossibility . .. is a synthesisor partialsynthesisof ZF and Quine set theory. " Let KF be the subsystem of Zermelosettheoryobtainedby droppingtheaxiom ofinfinit y andrestricting theassonderung (comprehension)scheme tostratified ~o formulre.A natural conjecture forNF to makein thiscontextwould bethat: 4.1. If NF is consistent, then every (well-founded) model of KF is the well-founded part of a model of NF.
CONJECTURE
Anotherpossibilityis: CONJECTURE 4.2. Every model of KF has a P- extension that is a model of NF + low replacement .
Thereis a basic difficulty thepathof in anyonetryingto proveanything likeConjectures 4.1 and 4.2. The CO construction is a methodwhich-on being presentedwith(i) a robustmethodof constructing termmodels for a trivialsubsystemT of NF, and (ii) a model M F= ZF-outputsa model of
CHURCH'S SET THEORY WITH A UNIVERSAL SET
135
T which hasM as an initialsegmentof itswell-founded part.To make this themethodof generating t erm models for T in the work, we have to have hand, as it were . It is notenoughto knowthatT is consistent . (The system NW of Forster(1987) has a canonicalt ermmodel so wemight be able to use Church-Oswald constructions to showthatevery (well-founded) model of KF is thewell-founded partof a model ofN W .) A fortiori we do not have a methodof provingconditionals of thekind: if T ~ NF is consistent,and M is a model ofZF, thenthereis M' F T of whichM is thewell -founded part.Thereare plentyof interesting assertionsof thiskind, andit would be very nice to know which themweretrue of .
4.3. Axiomatizability We shouldaxiomatizethetheoriest hatCO constructions give us; low comprehensionis obviouslyimportantb utnotobviouslyaxiomatizable . We could of E co and = a predicatewhichmeant do thisif wecouldfind inthelanguage "low". We can dothisif themodel westartedwithsatisifesthethreeconditionof section3.1.1, but it justmay be possible to saysomethingeven withouttheseextraassumptions.
4.4. Constructive CO constructions Anotherquestionis: is therea sensibleconstructive treatment of CO constructions?The problemis thevery classicalnatureof CO constructions : snd(k'x) is alwaysequalto 0, or tosomethingelse. This seems to meanthat if we want to executeintuitionistic CO constructions, we will have to assume tertium non datur foratomics. It would bequiteilluminating to developCO constructions insideintuitionistic ZF andsee whathappenstothetransfinite recursiveversionsof thenegativeinterpretation. It is probablyquitehard. Indeed,as unpublishedwork of Dzierzgowski has shown,it isextraordinarily diffcult to get nontrivial models forintuitionistic versionseven ofsystems as straightforward as NF2 •
4.5. Schroder-Bernstein The point has been made thatin Church'stheoryand its kin we have no comprehensionto speak of for big sets." This means thateven apparent banalitiescan turnout to be hardto prove. For example can we prove Schroder-Bernstein for big sets inChurch'sset theory? Thereare various proofs oftheSchroder-Bernstein theorem . Thereis a slick onesuitablefor andotherswho have been exposed to the use withcomputersciencestudents Tarski-Knaster fixpointtheoremforcompletelattices . Suppose f : A "--t B 7 Big sets arethingslike complement s of single tons. They arenot large in the way that , say, measurablecardinal s arelarge .
136
THOMAS FORSTER
and9 : B <--+ A. Then AA'.(A - 9 "( B - f "(A')) is a continuous functionon the 'P'A and must have a fixpoint . Howeverthisproofdepends completelattice on thepower setaxiom, which is not available Church in 's theory . There theoutlookis is a much more low-tech proofnot using power set for which slightly more hopeful. It goes like this. : Considerthetwosequencesdefined by amutualrecursion
b« = B - J"A;
bn + 1 = f"a n ,
ao = A - g "B ;
an+l
= g"b n .
Set The bijectionwe want isflA' U g-ll(A - A'). Thereseem to be twomajorhurdles . (i) Are all the b.; andthean sets? If so then{an: n E IN} is a set by low comprehension . (ii) If {an: n E IN} is a , all set, is its sumseta set? If A and B arelow,then, by lowcomprehension thebn and thean arelowsets,so {an : n E IN} is a low set of low setsand its sumset is a low set , and thereseems no reason to e xpectthatthebijection flA' U g-ll(A - A') will not also be a low set. However A and if Bare merelysets, not low sets, theredoes not seem to be any reasontoexpectthis proofto work . Theremay be some otherway of provingSchroder-Bernstein butthereis no reasonto expectthat . One specialcase can bedisposed of easily . In modelssatisfyingthethree conditionsofsection3.1.1, Schroder-Bernstein will betrue,forthefollowing veryunsatisfactory reason. In such models every set either is low or co-low . Schroder -Bernsteinwillcertainly hold for the low sets, and no lowset will . If A and B are two co -lowsets thenthe be thesame size as a co-low set model cannotcontainany injectionsfrom A into B or B into A since any moiety) and must such injectionwould beneitherlow nor co-low (it will be a will bevacuouslytruefor be absentfrom themodel. SoSchroder-Bernstein non-low sets . (Indeed, becauseoftheabsence of moieties, no non-low set can even be the same size as itself!) This argumentcannotbe used in the case . of Church'smodel, becausethereareplentyof moietiesthere
4.6. Mitchell's set theory Emerson Mitchell'sP h.D. Thesis (1976) containsa CO-like construction . His system has power set for all sets (notjustlowsets) butdoes not have closureunder(binary)U and n.
4.7. Extensional quotients? I close by raising an obvious but completelyunexploredpossibility . The enduringdifficulty with theseconstructions is extensionality.It's easy to
CHURCH'S SET THEORY WITH A UNIVERSAL SET
137
show thatx and y havethesame members (in thesense ofE co ) as long as snd(k'x) = snd(k'y) , butif snd(k'x) # snd(k'y), we have our work cut out, and ittendsto bedoableonly fortheoriest hathave an easily solvable word problem. One thingone couldnaturally do is startby being much more reckless in one's choice K of and membership conditionsfor non-low sets , quotientof theresult . and pick upthepieces laterby takingan extensional It's hard to see w hatmight bepreservedin a developmentlike this,b utthat means thattheremay be many consistencyproofswaitingto be revealed by such anextensionof themethod. My presentfeeling isthatthepossibility of developingCO techniquesalong these linestheirmost is excitingfeature. Thereis no obvious reason why this should not thekey hold thatwill one day unlock the consistencyquestionof NF. REFERENCES
Boffa, M., and Petry,A. 1993 On self-memberedsetsin Quine'sset theoryNF, Logique et Analyse, vol. 141-142, pp. 59-60. Church,A . 1974 Set theorywith a universalset, Proceedings of the Tarski symposium, Proceedingsof Symposia in PureMathematics(L. Henkin,editor),vol. XXV, Providence,pp. 297-308; also inInternational Logic Review, vol. 15, pp . 11-23. Forster,T. E. 1987 Term models for weakset theorieswitha universalset, The Journal of Symbolic Logic, vol. 52, pp . 374-387. 1995 Set theory with a universal set: Exploring an untyped universe, second edition, OxfordLogic Guides, no. 31, ClarendonPress, Oxford. Forster,T . E., and R. Kaye 1991 End-extensionsp reservingpowerset, The Journal of Symbolic Logic, vol. 56, pp . 323-328.8 Grishin, V. N. 1969 Consistencyof a fragmentof Quine's NF system, Soviet Mathematics Doklady, vol. 10, pp. 1387-1390. Henson, C. W. 1973b Permutation m ethodsappliedto NF, The Journal of Symbolic Logic, vol. 38, pp . 69-76. Jensen, R. B. 1969 On theconsistencyof a slight(?)modificationof Quine's NF, Synthese, vol. 19, pp . 250--263. 8Errata , p . 327: Line 11 shouldread'a nd a E M such thatM ex pression following 'M
1= ' shouldbe
==
1=
7l"'a=P'a' . Line 13: the
'7l"'a=P'a '. Line 26: ' (notjust7l"'a = P 'a) ' should
read' (notjust 7l"'a =P'a)'. Line 28: '7l"'a' shouldread'P'7l"'a' .
138
THOMAS FORSTER
Korner,F. 1994 Cofinalindiscerniblesand someapplicationsto NewFoundations,Mathematical Logic Quarterly, vol. 40, pp. 347-356 . Mitchell, .E 1976 A model of settheorywith auniversalset, Ph.D. thesis, Madison, Wisconsin. Oswald, U. Ph.D. thesis, ETH , 1976 Fragmentevon "New Foundations " undtypentheorie, Ziirich, 46 pp. Scott,D. S. 1962 Quine's individuals , Logic, Methodology and Philosophy of Science (E . Nagel, editor),S tanfordUniversityPress, pp. 111-115. Tarski, A. notions?,History and Philosophy of Logic, vol. 7, pp . 1986 Whatare logical 143-154.
ROBIN O. GANDY
AXIOMS OF INFINITY IN CHURCH'S TYPE THEORY
Abstract.In his 1940 paper Church gav e an elega nt for mulation of the c is represent ed ni it si m ple theoryof functi on-types. Higher order arithmeti almostwithoutartifi ce; the only artificialdevice used is a particular formulat ion of the axiom of infinity. We discu ss the significanc e of thisaxiom, and possible alt e rnat ives.
1.
Church'ssystem has two basictypes: the type 0 of propositionsand the type £ of individuals . If 0 and {3 aretype symbols then(0{3) is thesymbol for thetype ({3 -> 0) of functionsfrom type {3 into type o. Terms (with types indicatedby subscripts)are formed fromvariablesa nd constantsby applicationand A-abstraction.Bracketsmay be omittedfrom both type symbols and terms withassociationto theleft. Weadopttheconvention thattypesubscriptsmay be omittedfrom constants introduc ed by definition or by hypothesis,when this can be donewithoutambiguity,and from all occurrences of a boundvariablee xceptits bindingoccurence. A Church'srulesI-VI and axioms (1)-(6) covertherules of -conversion, classicalpropositional calculus andquantification theoryat all types. Equality, atalltypes, is defined by:
A"
=
B"
=df
(/0")(/A" :J / B,,).
This system, together withChurch'saxiom:
(7) (3x., y.)(x =I- y) we shallcall (C). Furtherpossible axioms aretheaxioms of descriptions: andof extensionality
10"13 (/"13,9"13)(X,,)(fx = gx) :J / = g); we shall also count 10 0 (Po, qo)(p == q :J P = q)
as anaxiomofextensionality.We use (C), (CD), (CDE) forthecorresponding systems. 139
C. Anth ony Anderson and M. Zeleny (eds.}; Logic, Meaning and Computation, 139-147. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.
ROBIN O. GANDY
140
2.
Churchintroducest henatural numbers asiterators over thetype. Let L' standforu(u) . Thenwe define: 0" S"" No,'
=df =df =df
>'f" x, . x , >.n" f"x,. f(nfx), >.n" . (Jo,' )[fO & (m, )(Jm :J f(Sm)) . :J fn] .
These definitionsmay also be made over an a rbitrary t ype a (with a' for aa(aa)). We shall omitthetype subscriptsfrom 0", S" " , No,' and adopt the conventionthatthe lettersm, n, p, q (withoutsubscripts)standfor variables of type L' restrictedto range only over No," A consequence of axiom (7) is
(m)(O 1= Sm). Church 's axiom (8) is
(m, n)(Sm = Sn :J m = n) . Thus (in (C)) thenumbersin type L' satisfy all of Peano'saxioms, and hence alltheoremsof classical higher-order arithmetic can be proved in(CDE)+(8). Unlikemost recursive functions , + and x can be defined withoutusing descriptions:
+n m xn
m
=df =df
>.f"x, . mf(nfx) ; >'f"x, . m(>.y,.nfy)x. 3.
We willwrite(P3) 01 for the result replacing of L by a in (8); it is Peano's thirdpostulate for thenumbersin typea'. In Newman and Turing 1942 it was shownthatif L occurs ina then(P3)01 is provable in(CDE)+(8). They introduced t henotion'n belongs to its own posterity'defined by BOP o, '
= df
>.n. (3m)(n
and they proved , in (C); ""(8)
=
S(m + n))
== 3n.BOPn
(3.1) (3.2)
(and thecorresponding theoremwith L replaced bya). And, assuming (3n)BOPn, they prove,in effect,
(3!p)(3!q)["" BOPp & S(Sp + q) = Sp].
(3.3)
Plainlyif p and q satisfytheformula within the [ ] for .3),(3thenBOP(Sp) and in anintuitive sense (which will be formalized thenext in section) Sp+q is
INFINITY IN CHURCH'S TYPE THEORY
141
the'largest' possible number. We need aname for it.Whenwe areassuming ""(8) (or (3n)BOPn) we assume thatM satisfies NOL,M & (3p)(""BOPp & SM = Sp).
(3.4)
In fact,M is unique;eitherit is its ownsuccessoror itssuccessor'precedes' it. 4.
We are going to prove that(8) is equivalent to 'thereare notfinitelymany individuals'.We need adefinitionof finite which can be appliedeven when we assume (8) is false. Toconstruct thiswe need to work in (CD) . Church showed thatin thissystem, definitionby primitiverecursionis permissible (as, also, is theuse ofconditional definitions).Of course if we are assuming thatM satisfies(3.4), thentherecursionequationcannotbe used to define thevalue of afunctionatSM, but it will work at otherarguments . Also when usinginduction,t heinductivestep is (n)(n
f=
M & A(n). ::> A(Sn)).
4·1. In (CD) we define afunctionD.,.,., by recursionas follows:
(ii) DmO = O. (iii) Dm(Sn) = Dmn if n f= M andm f= n, = 1 if n f= M andm = n . And wewritem < n forDmn = 1. Then < is a linearorderingof thenumbers,withM as thegreatest.Specifically in (CD)thefollowing c an be (successively) inferred from (3.4) using inductionon n wherenecessary .
4·2. (1) n
f= M ::> n <
Sn.
(2) n f= M::>.m < Sn == m ~ n. (3) p < m ::> • m < n ::> p < n.
(4) n.,:n. (5) n
f=
M ::> n
f= Sn.
142
ROBIN O. GANDY
(6) m =1= M :::> • Sm < n == (m < n & n (7) m < n V m = n V n < m . (8) n::; M.
=1=
8m).
(9) 8M::; M.
4.3. Remarks (1) Ratherthanusing descriptionsit would be possible simply to postulate theexistenceof afunctionsatisfying4.1. (2) If one defines m::; n by (3p)(n = m + p) , thenBOPm & BOPn. :::> m ::; n, so thisdefinitionis notsuitable . (3) If we assume (8) insteadof "'(8), then4.1 and 4.2 are to be modified by M. omittingall clauses which involve
4·4· Now wewrite
8egn =dfAx ., . N X & x ::; n. We shallnot beconcernedwiththeempty set, so we use::; rathert han< ; 8eg n has, intuitively , n + 1 members, and is a paradigmfor se tsof cardinal n + 1. Since we have not a ssumed axiom 100 ouruse oftheterm'set' here, and in whatfollows shouldbe treated as a convenienta buseof language.
4·5. Now wewrite
This says that!is an injectionfrom thedomain given by X O{3 intothetype o . SimilarlywecandefineSURa(fa{3 : X o/3 ) to saythat! a{3 maps thedomain X o /3 ontotype 0, and BIJ a(fa{3 : X O (3 ) forbijection , and we define
to expressthat!a{3 is an injectionfrom X o/3 which is strictly intotype o. 4·6.
Finallywe write
m any individual s. It is astrongassertion , to express thatthereare finitely as it impliesthatthereare many functionsin type u:
143
INFINITY IN CHURCH'S TYPE THEORY 5.
Now we prove THEOREM 5.1. ""FIN J (8) is provable in (CD). We take>- FIN as a hypothesis, H say. LEMMA 5.2. Let A(n) be (3/LL' )(INTL(f : Seg n ferred from H in (CD).
». Then (n)A(n) can be in-
Proof. A(O) is an immediateconsequenceof axiom (7). And
H we have is easily proved; hence, using A(n) ::> A(Sn).
'»' does not
Note thatthis proof, unlike aproofof (3/,,' )(INJL(f : N o L requirean axiom of choice.
Now to provethetheoremwe assume ""(8) and thatM satisfies(3.4) and seek acontradiction. By theLemma we may suppose thatF" , satisfies
and thereis a ZL not intherangeof F. DefineGLL , by
GXL = F(S(J1.m)(m < M & x = Fm» if (3m)(m < M & x = Fm), =zow. (In (CD) theJ1.-operator isdefinable.)Then, since (3p)(p < M & M follows from .4), (3 by inductionwe prove
mG(FO)
= Gm
= Sp)
for m :::; M.
ButthenSMG(FO) = G(FM) = z . ThereforeSMG(FO) =I- mG(FO) , and SM =I- m, form :::; M . Thus SM 1:. M whichcontradicts 4.2(1). So, on eliminatingthevarioushypotheseswe have aproofof "" FIN::> (8) in (CD). --j
144
ROBIN O. GANDY
extensionality. To provetheconverse ofTheorem5.1 we need an axiom of THEOREM
5.3. In (CDE) one can prove
FIN::) '" (8).
Proof. We take FIN and (8) as hypotheses , and seek acontradiction. Since allordinarymathematical argumentscan be formalized in (CDE)+(8) we argue informally (and write 1m in place ofmI). Suppose thereare N + 1 individuals . Thenfor anyz, and anyItt
cannotall bedistinct . Hence for some 0~ m
So if n I q then
~
N, 1 ~ n
~
N + 1.
INx = IN+Qx.
Let P be the LCM of1,2, .. . , N + 1. Thenfor allI, x we have
INx
= IN+Px.
Then, using the axiom of extensionality, we have
N=N+P. Thus BOP N, which by (3.2), contradicts (8).
(*) --I
5.1. Remarks (1) The use ofextensionality is unavoidable . One can make a model for (CD)+FIN in which all the terms N, SN, S(SN), . .. ,
are distinct , and soBOP(N) fails and (8) holds . (2) The numbers in(*) are the best possible. In (CD) one can prove thatif n < N or p < P then n =I n+p. (3) In particular, by considering a cyclic permutation of theN + 1 individuals one can proveN + 1 =I N and hencethatif M satisfies (3.4),then M>N.
INFINITY IN CHURCH'S TYPE THEORY
145
5.2. t Churchalsoconsidersthe proposition,called 30 , whichstatesthat the functionTt't/l defined onNot/l by
is injective : thereare not morenumbersin NOt /l thanthereare inNot" He proves in (C) (8) :J 30t • In a footnotehe creditsTuringwith aproofof theconverse in (CD) . We can useremark(3), lifted up to Not/l to get such a proof. Suppose "'(8) and hencethatthereis a largestnumber M as in sections 3a nd4. Define m~" = (JLnt /l )(Tn = m) . ThenT gives a bijectionbetweenSegM' and Not" Now we replacet hetype of individualsby NOt' and considernumbersin Not/l. Let Qt/l be thelargest number: i.e. it satisfies (3.4) with c replaced byi" , Liftingremark(3) to £" we seethat
Q>M'. ButthenT cannotbe aninjection ; thus
rv(8):J rv30t is a theoremof (CD). Whetherit is also atheoremof (C) remainsopen (cf. 5.4 below). 5.3.
It is nothardto seethatthe use of descriptionsin theproofofTheorem5.1 is essential. For one can construct a model for (CE) which has threeindividuals I\:O,I\:},1I:2 butin which the only functions in type u. are theidentityand the constantfunctions. Hence in this mod el Sl = 1. One can define< by Xmn, m = 0 & n = 1, and thentheresultsof §4 hold, with M = 1. Even if, as one may, oneaddsan injectionfrom Seq, intotype £, it is plainthatFIN and (8), and hence " 'FIN :J (8) are all false in this model.
In fact I have discovered a general methodforconstructing models (likethat in Theorem5.3) of (C). Let (C+) beobtainedfrom (C) byaddingconstants forindividuals , forequalitybetween individuals,and if requir ed, for(extensional) functions in various types, together with axioms governing the added constants.Thenone can showthatthe(normalforms of the) closed termsof
146
ROBIN O. GANDY
(C+) form a model for ( C+E). A proofof thistheoremwillappearin Gandy 1995. It can be used to c onstruct a model in which(P3)t' holdsbut(8) fails. I have tried(using a smallnumber of individuals,and selectedfunctions) to make models in which (30)t is truebut(8) is false. I had hoped to make good use ofSchwichtenberg 's characterization (in his 1976) of the A-definable functions from N Ot' to Not', butthepresenceof addedconstants complicates matters.' 6. C ONC LU DI NG REMARKS
(1) It is well known thatwithoutthe axiom of choice (for type L) one cannot infer from",FIN thattype Lis Dedekind infinite. Hence Theorem5.1 fails if one replacesFIN by 'Dedekindfinite'. (2) From a conceptual ( and from amodel-theoretic) pointof view,",FIN is axiom of infinityt han(8). ButTuring,who had done a lot of a morenatural formal work on Church's system, was surprisedwhen in1953, I showed him theinformalproofthatthey areequivalent. Theemphasis, bothatPrinceton and G6ttingen,on formal proofs and metamathematics tendedto discourage a bouttheintuitivemeaningof axioms. thought that (3) Church's papershows howarithmeticcan be founded using the idea the natural numbers areiterators, ratherthanfinitecardinalsor ordinals . If, likeChurch,one seeks to avoid the usedescriptions of and extensionality whenever possible , then(8) is thepreferredaxiom of infinity.B uttheproofs y modified to show th at (CDE) is consistent in Gandy 1956 can be easil details relative to (C). So, providedone is read y to put up with the formal of sections3 and4 (which ar e necessary for t hedefinitionof FIN), one can . take""FIN as theaxiom of infinity REFERENCES
Church,A. 1940 A formulation of thesimple theoryof types, The Journal of Symbolic Logic, vol. 5, pp. 56-68. Gandy, R. O. 1956 On theaxiom of extensionality , The Journal of Symbolic Logic, vol. 21, pp.36-48. 1995 A closedtermmodel forChurch's type theory,in preparation. Newman, M. H. A., andA. M. Turing 1942 A formaltheoremin Church's theoryof types, Th e Journal of Symbolic Logic, vol. 7, pp. 28-33. ISince writingthe above, I have managedto show that30' :J (8) is a theor em of (CE) , butis falsein a non-e xtension alm od el of (C ).
INFINITY IN CHURCH'S TYPE THEORY
147
Schwichtenberg , H. m it typen,Archiv fUr mathematische 1976 Definierbare f unktionen im A-kalciil Logik und Grundlagenforschung, vol. 17, pp . 113-114 .
EDWARD 1. KEENAN
LOGICAL OBJECTS
INTROD UCTION
This paper began,and begins, as a reflection on the following claim Putby nam (1981, page 217): " Theorem : Let I:- be a languagew ithpredicatesF I , F2 , • • • , Fk (not necessarilymonadic). Let I be an interpr etation,in the sense of anassignmentof anintensionto everypredicateof 1:-. Thenif I is non-trivial in thesense thatat leastone predicatehas anextensionwhich is neitherempty nor universalin atleastone possible world, thereexistsa second interpretation J which disagreeswithI , butwhich makesthesame sentencest ruein everypossibleworld asI does. . . . If atleastone predicate, say, F u , has anextensionR u j which isneitherempty nor all of U j , selecta P j of U j such thatP j (Ruj) i= R u j ' . . . " permutation Lewis (1984) acceptssome form ofPutnam's claim, and Lakoff(1987, y of ModelTheoretic chapter15: "P ut nam' sTheorem . . . The Inconsistenc Semantics")uses it toargueagainsttheapplicability of modeltheoretic semanticsto thestudyof natural language. But we show, in esction1, that Putnam's claim, as stated , is false. We theclaim, provingthatpermutation invariance (Plness)rather reformulate thannon-tr ivialityis whatbinds reference(extensions ofpredicates)totruth. In section2 we argue,contraEtchemendy(1990) , thatinvarianceunder automorphisms(AIness, a conceptual generalization of PIness) characterizes thedenotations of "logical"e xpressions ,includingbutnotlimitedto classical logicalconstants.In section3 we defendourmain thesis: thatthelogical objects of a giventype in a givenontologyare the AI ones.Our analysis derives withoutcircularity (Theorem22) thatbeing "logical " on ourthesis is itselfa logicalproperty.Section4 concludesw ithan open problem. Previouswork,originating w ithKlein (1872)andincludingWeyl1949, Mostowski 1957, Silva 1945, van Fraassen 1990, K eenan 1991, Keenan and Stabler 1991; Keenan and Stabler 1995 identifiesthestructural propertiesof objectsin a givendomain withthoseinvariantunder thestructure preserving permutations of thatdomain. Thatthelogical propertiesof objectsmay be identifiedwiththoseinvariant u nderallpermutationswas, to my knowledge, firstproposed in a lectur e by Tarskiin 1966, edited and publishedas Tarski 1986 by J. Corcoran.Tarski thereexplicitlybuildson Klein'sErlangerProgram(1872 ; see Klein 1892 and van Fraassen 1990, page 266) in 1 49 C. Anthony Anderson and M. Zeleny (eds.}, Logic, Meaning and Computation, 149-180. © 200 I Kluwer Academic Publishers. Printed in the Netherlands .
150
EDWARD L. KEENAN
whichgeometriesofvarioussortsaredistinguishedaccordingas theobjectsof studyareinvariant u nderone oranother class T of transformations (= permutations) . E.g., "Euclideanobjects"are onesinvariant u ndertranslations , rotations , etc. Then theweakertheinvarianceconditionsthetransformationsmust satisfythelargertheequivalence classes ofobjects. And Tarski makes thenatural extrapolation: themost general o bjects,the"logical" ones, are thoseinvariantu nderalltransformations.' It is alsothisassociationof generality with PIness undertheheading"symmetry" in thephysicalscienceliterature thatliesbehindvan Fraassen'suse oftherole ofs ymmetryin characterizing scientific laws . The idea that"PIness" = "logicality" developed is most extensively in van Benthem 1989a. Plotkin1980, Keenanand Stavi1986, van Benthem1991, Marshall andChuaqui1991, andSher 1991 alsosupportthisidea. We differ onlyslightlyfrom thesein emphasizingthe type and structure dependency of "logical"objects. This enablesus to explicatethesense in which , e.g., set membership bothis, and is not,a logicalrelation . We drawon natural language studiestoexhibita varietyof novel logical objectsin hightypes(van Benthem 1983, 1991 ; Westersttihl1985 ; Keenan and Moss 1985; Keenan and Stavi 1986).
Notation.[A ~ B] is thesetof functionsfrom A intoB . If 1 E [A ~ B] and K ~ A, I(K) is {f(x) I x E K}; for sEAn, I(s) is (f(sI) , .. . , I(sn))' The power set ofA is denotedPA or P(A). LanguagesL are assumed first orderrelational (withoutzero placerelation symbols) unlessindicatedotherwise. Models forsuch LS are pairs M = (EM,I) , EM a non-empty set, the universe of M (usuallydenotedE) , and I a functionmapping therelation(function)symbols of L to relations(functions)over E. As usual,I liftsrecursively to aninterpretation I M of L relative toassignmentsof values to thevariables . For ¢ a sentencewriteM F ¢ forI M (¢) = 1, thelatter meaning thatI M(¢)(9) = 1, for allassignmentsg. Th[M], the theoryof M, =df {¢ E Sent(L) 1M F ¢}. ModelsM,M' are elementarily equivalent, M ~ M', if Th[M] = Th[M']. For M = (E,I) and a a bijectionwithdomain E, o M =df (aE, a 0 I) is a model. For M , M' models, M' isisomorphic (~) to M iff M'= o M, some bijectiona : EM ~ EM'. And ofcourse (1) M ~ M' ::} M ~ M'
(so M ~ aM, allbijectionsa withdomain EM)'
IThe notionof invariancedeveloped here differs in tworespects from thatin Tarski T ruthand Falsity,or truthfunctions( AND, OR, . . .) built 1986: (1) Tarski does nottreat on them, as objectsto be preservedby permutations . He justconsiders a set theor eti c hierarchybuiltup from theuniverseand notesthatnone ofit s elementsare PI , onlytwo subsetsare, onlyfourbinaryrelation s are. At higher levelsrelations likesubset anddisjoint will be PI. (2)Tarski'spaper is informal . He does notexplicitly d efinea typehierarchy a nd identifythePI objectsas thosefixed by allp ermutations . He even remarksthatwhile the t oformulat e logicalpropertiesof classesarethenumericalones, theidea "is quitedifficult in a precise way" (page 151) .
151
LOGICAL OBJECTS 1. BINDING REFERENCE AND TRUTH
To evaluate P utnam's claim let£,p be ofsignature{ P}, P a Pi (one place predicatesymbol). Then, in a possibleworldw withuniverseU w , if I inter prets P as non-emptyand non-universal, so that0 C I(P) C U w, let7r be some x E I(P) withsome y (j. I(P) and a permutation of U w interchanging fixingeverything else. As per (1) then, (Uw, I) ~ (Uw, 7r 0 I) , butI =1= 7r 0 I since x E I(P) and x (j. (7r 0 I)(P) , in conformitywithPutnam'sclaim. ButPutnam's claim failsw henP is a P n , n 2: 2. LetU w be thedoubleton {a , b} and letI(P) = {(a , b}, (b, a)} . I(P) is not 0 noruniversal= {a , b} x {a, b}, butallmodels ({a, b} ,.1} with .1 =1= I disagreeon some sentenceof £,p.
Proof. (a), (b) and (c) below are false M in = (Uw,I) :
(a) 3x(xPx)
(c) 3x3y(xPy & -.yPx}
(b) VxVy-.(xPy)
Let ({a, b},.1) be a modelof E with.1(P) =1= I(P) . If (c, c} E .1(P) for some c E Uw then(a) above is truein.1. So we may assume thatfor all c E E, (c, c} (j. .1(P). If no pair (c, d} of distinctelementsof U w is in .1(P) then .1(P) is empty and (b) above is true. So at leastone of (a,b} , (b,a} is in .1(P). If bothare then.1 = I contrary to assumption. So justone is, and in each case (c) istrue . This exhaustst hecases. -J Lemma 2 reformulates Putnam'sclaim in terms of permutation invariance (PIness) notnon-triviality (thetwo notionscoincidingon PiS). And Theorem4 tells ust hatPIness is equivalent to covariation of truth a ndrefer ence (holdingt heuniverseconstant) . Theorems7 and12 partially generalize Theorem4 to showthatfor suffici entlyrich £S, PIness guarant ees a kind of expressivecompleteness:such £S describetheirmodels up toisomorphism. First, DEFINITION 1.
(i) An n-aryrelation R over auniverseE is permutation invariant (PI} iff 7rR = R, for all p ermutations 7r of E. (Recall:7r(R) = {7r(d) IdE R}) . For R E [Em --+ E] thisjustsays thatR(7rd) = 7r(R(d», alldEEm. (ii) A model (E,I) is PI iff for all P E Dom(I) , I(P) is PI. LEMMA 1. For R S;;; En , R is not PI iff 3s E En 37r E PERM(E), s E Rand (j. R.
7rS
Proof. {=: If s E R then7rS E 7rR =}
=1=
R since 7rS (j. R; so R is not PI.
Let R suchthatfor8 E PERM(E), 8R
=1=
R.
152
EDWARD 1. KEENAN Case 1: 3t E R-8R. Thentherighthand side above holds choosing
s
= t and 1r = 8- 1 •
Case 2: 3t E 8R - R. Then 8- l t E 8- 18R = Rand t 1 righth andside above holds choosing 1r = 8 and s = 8- t.
tf. R so the --J
LEMMA 2. Let M = (E, I) a model for I:- and P a relation symbol of I:- with I(P) not PI. Then I:- has a model M' = (E,.1) ~ M such that .1(P) =!= I(P). Proof Given M = (E,I) withI(P) not PI. ByLemma 2 thereis a tuple s and a permutation1r of E suchthats E I(P) and 1r(s) tf. I(P). Then 1rM = (E,1r 0 I) ~ M so Th[1rM] = Th[M]. ButI =!= 1r 0 I since 1r(s) tf. I(P) but 1r(s) E (noI)(P) = 1r(I(P)) since s EI(P) . --J
COROLLARY 3. Every non-PI model has an elementarily equivalent model with the same universe which disagrees with it on some predicate symbol.
THEOREM 4 Stability o f Reference.Let I:- have equality. models M for 1:- , M is PI {:} \1M' with EM'
= EM,
M' ~ M ...... M'
Then for all
= M.
Proof
=> Let M = (E ,I) be PI. Let M' = (E,.1). The +- directionof the consequentis trivial.Assume M' =!= M, so .1 =!= I . We show M' ~ M. Let P be of arityn with.1(P) =!= I(P). Let t E (I(P) - .1(P)) U (.1(P) - I(P)), let x be an n-tupleXl, •.• ,xn of distinctvariablesand let Dist(x) bethe formulaAND{(x; = Xj) I t; = tj, 1 :::; i =!= j :::; n} & AND{(x; =!= Xj) I t, =!= t j , l :::; i =!= j :::; n} , where for K a finite set of formulas, AND(K) = 'ix(x = x) if K = 0, and otherwiseAND(K) is the conjunction of theformulasin K. Consider¢ = 'ix(Dist(x) -> Px). If t E I(P) - .1(P), thenM F ¢ and M' ~ ¢; if t E .1(P) - I(P), thenM' F ¢ and M ~ ¢ . ¢:: Followsimmediatelyfrom Lemma 2. --J Theorem4 tells us thatit is preciselyPIness thatguarantees thatchanging extensionsof predicateschangesthe truthof sentences. Does Theorem 4 generalize,replacingidentityof universeswithsameness of cardinality and equalityof models withisomorphism? In fact only h alfthegeneralization obtains.We firstestablishtheuseful: LEMMA 5. A model M is PI iff for all
t:
E PERM(E),1rM = M.
Proof
=> Let M be PI and 1r E PERM(E). Then for allP E Dom(I) , I(P) = 1r(I(P)) = (1roI)(P). Since P was arbitrary I = 1roI, so 1rM = (1rE, 1roI) = (E,I) = M. ¢:: Let P E Dom(I). For 1r E PERM(E), 1r(I(P)) = (1r oI)(P) = I(P), so I(P) is PI. --J
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LOGICAL OBJECTS
LEMMA 6. Let M be a model for 1:-. Then [a] ::} [b] and if I:- has equality, [bJ ::} raj. [a] For all models M' , M ~ M' ::} for all bijections 1f M' =1fM,
EM
-+
EM',
[b] M is PI. Proof. [aJ ::} [b], Assume [a] and let M =(E,I), let8 E PERM(E). Show 8M = M, whence M is PI byLemma 5. Since M ~ 8M, thenM ~ 8M, so for allbijections1f: E -+ E6M = 8E = E, 8M = 1fM. Choosing1f= idE yields 8M=M. [bJ ::} [a]. Let I:- have=, let M =(E,I) be PI and let M' =(E',I') with M ~ M'. If lEI f= IE'I, then[a] holds vacuously. So let 1f be a bijection: E -+ E'. Show 1fM= M'. Assume, leadingto acontradiction, thatequality fails.Then1f 0 Y f= I', so for some P E Dom(I), Y'(P) f= (1f 0 I)(P). Now if in thereis atE(1foI)(P) -I'(P) , then,where Dist(x) is formed as Theorem 4, ¢ = V'x(Dist(x) -+ Px) is truein 1fM and false in M'. But 1fM ~ M M ~ M'. And if thereis a so 1fM ~ M so ¢ is truein M, contradicting t E I'(P) - (1foI)(P) , then¢ is false in1fM andtruein M'. Andagainsince Th[1fM] = Th[M], ¢ is false in M , contradicting thatTh[M] = Th[M']. This exhaustst hecases provingthelemma. --l
THEOREM 7. Let M be a model of a relational language I:- with equality . Then M is PI::} \1M' with IEM,I = IEMI, M' ~ M iff M' ~ M. Proof Given M PI and M' with IEM'I = IEMI . Let 1f : EM' -+ EM be a --l bijection.Assume M' ~ M. By Lemma 6, 1fM= M' and, (1), M ~ 1fM.
FACT 1. The converse toTheorem7 is false. Proof Let I:- have signature{P}, P of arity2. Let M = ({a,b} ,I) with I(P) = {(a, b)}. Thenfor all M' withIEM'I = 2, Th[M'J = Th[M] ::} M' ~ M. In general(Enderton 1972, page 96, example 17) finite models which are elementarily equivalentare isomorphic. Here is a demonstration for this case. NotethatTh[M] includes:
(a) ..., 3xPxx
(b) 3xyPxy
and
(c) V'x, y(Pxy
-+ ..., Pyx).
Let M' = ({a,13},I') with a f= (3. Assume Th[M'] = Th[M]. By (a) neither (a ,a) nor (13 ,(3) is in I'(P) . By (b) either(a ,(3) or ((3, a) is in I'(P) and, by (c), notboth. So I'(P) = {(a ,(3)} or I'(P) = {((3,a)} . In thefirst case the map sending a to a and b to (3 is an isomorphism; in thesecond the map sendinga to (3 and b to a is. So M' ~ M. But I(P) is not PI. The 1f interchanging a and b fails tofix {(a, b)}. --l
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EDWARD L. KEENAN
. FACT 2. In Theorem7 theconditionthatIEMI = IEM'I is necessary Proof. Let C= be thezero-signature firstorderC withequality . For any E, (E ,I) is a model ofC=, since I(=) is requiredto be {(a,a) I a E E}. Now numbers, and letM and M' be models ofC=, where EM= N, thenatural EM' = P(N). By Theorem 9, Th[M] = Th[M'] ; but M f:. M' since their universesdifferin cardinality . -1
LEMMA 8. Let M, M' be infinite models for C=. Then for all formulas ¢ in C= , for all assignments 9 : VAR -+ EM and all assignments g' : VAR -+ EM' if g(x) = g(y) iff g'(x) = g'(y) , for all x, y E FREEVAR(¢) , then M F ¢[g] iff M'
F ¢[g'].
Proof. By recursionon formulacomplexity(Appendix).
THEOREM 9. All infinite models of C= have the same theory. Proof. Let ¢ E Sent(C=), let M, M' infinitemodels ofC=. Clearlyall9 : VAR -+ EM andg' : VAR -+ EM' identifythesame variablesfree in¢. So for all o.s'. M F ¢[g] iff M' F ¢[g'l, so M F ¢ iff M' F ¢. Thus Th[M ] = Th[M']. -1
N .B.: It is only theinabilitytostatethecardinality of theuniversewhich preventst hemore general s tatement t hatfor PI models,samenessof theories and model isomorphism coincide. This latter statementis naturally understood as a kind ofexpressivecompletenessrequirement:languageswhose models satisfyit candescribethemselvesup to isomorphism (Theorem12).
DEFINITION 2 Cardinality Quantifiers.Let C be a firstorderlanguage. We expandC to C(CARD) by addingthecardinality quantifiers Q" foreach cardinal.x, as follows:
.x,
syntax: For eachcardinal Q" is a quantifier symbol (as areV and 3) . For allquantifier symbolsQ, allvariablesx , andallformulas¢, Qx¢ is a formula. semantics: M F Q"x¢[g] iff I{a E EM 1M F ¢[g[x/a]]} I
=.x.
In expandingC to C(CARD) we add new clauses to t hedefinitionofsatisfactionbut we do notadd new symbols to be interpreted . Hence themodels M = (E, I) forC(CARD) arc exactlythoseforC; thedefinitionof PIness for models isunchanged.We need: LEMMA 10. PIness is preserved by model isomorphism.
155
LOGICAL OBJECTS
Proof Let M = (E,I) be PI, let6: M ~ M' = (E',I') . Show M' is PI. Let rr E PERM(E'). Show rr(I'(P)) = I'(P) , for allP E Dom(I') . Note that I'(P) = (6oI)(P) = 6(I(P)). And since6-1 orro6 E PERM(E) and M isPI, (6-1 0 rr0 6)(I(P))= I(P), so 6«6- 1 0 rr0 6)(I(P))) = 6(I(P)) = I'(P) . But 6«6- 1 orro6)(I(P))) = (rro6)(I(P)) = rr(I'(p)) , whencerr(I'(P)) = I'(P), so M' is PI. --I
LEMMA 11. Let rr : M ~ M' . Show V¢ E .c(CARD), vs : VAR - EM, M
1= ¢[g]
iff M' 1= ¢[rr 0 g].
Proof Set S = {¢ E .c(CARD) I vs : VAR - EM,M 1= ¢[g] iff M' 1= ¢[rr 0 g]}. See Enderton 1972 (pages 91-92) for theproofthatS includes theatomicformulasandis closedunderconjunction, negationand universal quantification. We show thatS is closedundercardinalquantification.
g, allcardinalsA thatM Let ¢ E S. Show for all Q.xx¢[rr 0 g].
1=
Q.xx¢[g] iff M'
1=
ObservethatV9 : VAR - EM, Vx E VAR, Va E EM, rr0 g[x/a] = (rr0 9)[x/rra] : (a) (rr0 g[x/a])(y)
= {rr(g[x/a](y)) = 7I"(a) if y = X,
(b) (rr0 g)[x/rra](y)
={
And we infer :M
rr(g(y))
ify~x ,
rra (71" 0 g)(y)
1= ¢[g[x/alJ
= rr(g(y))
ify=x, if y ~ x.
iff M' 1= ¢[rr 0 g[x/alJ InductionHypothesis. iff M' 1= ¢[(rr 0 g)[x/rralJ theobservation .
Thus themap rrt{a E EM 1M 1= ¢[g[x/a]]} - {rra E rr(EM)= EM' 1M' ¢[(rr 0 g)[x/rra]]} is onto(andone-to-one, since rris one-to-one).T hus
I{a E EM 1M
1= ¢[g[x/a]]} I =
Whencefor all c ardinals A, M theproof.
I{rra E EM' 1M'
1= Q.xx¢[g]
1= ¢[(rr
0
1=
g)[x/rra]]} I.
iff M' 1= Q.xx¢[rrog], completing --I
THEOREM 12 ExpressiveCompleteness . For all E with =, for all models M for .c(CARD), M is PI ::::} for all models M' for.c( CARD) , M ~ M' iff M ~ M' . Proof Given.cwith= , let Mand M' be models of .c(CARD)w ithM PI. ::::} Let Th[M] = Th[M']. For A = IEMI, M' 1= Q.x(x = x) since M 1= Q.x(x = x) , so IEM'I = IEMI . Let rr:EM ~ EM" By Lemma 11, M' = rrM, so M' ~ M by Lemma 10.
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EDWARD L. KEENAN
<= For M,M' isomorphic models of .c(CARD) and ¢ E Sent(.c(CARD)), Lemma 11 implies thatM 1= ¢ iff M' 1= ¢, whenceTh[M ] = Th[M '] , completingtheproof. -1 Theorem 12 entailsthatelementaryequivalence of models of .c(C ARD) preservesPlness. But we have astrongerresult , whose proofincorporates some observationsused later : THEOREM 13. For.c with =, M and M' models of .c, if M is PI & M ~ M' then M' is PI. Proof sketch. As notedearlier , thePI n-aryrelations form acompleteatomic subalgebra of P(E n ) , denotedPI(E n). 13.1 For eachsEEn, set [s] =df {t E En I i, = tj iff s, = Sj , for all 1~ i,j ~ n} . Then [s] is non-empty (s E Is]) and PI, and no proper non-emptysubsetof [s] is PI, so [s] is an atomof PI(E n). Whencefor R an n-aryPI relation on E, R = UsE R[S]. 13.2 For s e E" lets= be thatequivalencerelation on {I, .. . , n} given by: i s= j iff s, = Sj. Let F(s) be theformula(AN D{(Xi = X j) Ii s= j} & AND {(X i =1= Xj) I--,i s= j}) , where AND(K) is as defined earlierfor K a finite set of formulas. Also we writeOR(K) for 3x(x =1= x) if K = 0 ; otherwis e oR(K) is thedisjunctionof theelementsof K. 13.3 For M = (E,I) a model of..c with= and P a P n , one proves that
I M(P) is PI iff M
1=
'lxl , . . .
, X n(PXl " ' "
X n ......
OR{F(s) Is E IM(P)}) ,
Write' P' is PI for theformulato therightof 'p ' above. It says that' P' denotes0 or thenon-trivial union ofthe [s] for thetupless in itsextension. Thatis, it says'" P' is PI". 13.4 Now, let .c have =, letM and M' be models withM ~ M' and M PI. Thenfor eachpredicatesymbol P (including' =' ), 'P' is PI is truein M and thusin M', so M' is PI. -1 2. LOGICAL OBJECTS AND LOGICAL CONSTANTS We have seenthatPI relations play aspecialrole withrespectto thetruth of sentences.Here weextendthisobservationto acharacterization of "logicalobjects", objectswe intendto includeas denotations of classicallogical languag e. constants as well as an infinitely enrichedclassdrawnfrom natural thelogicalobjectsto be thedenotations of a So we cannotad hocly define listedset ofconstants . Ratherwe need a cr iterionforincludingnew objects contraEtchemendy(1990) , we seek a property amongthe"logical" ones. So common to denotations of logicalexpressions. And we claimthata certain . generalization of PIness is thatproperty First, just as the "equals " predicateis a standardlogical on e and for each E, its denotation IDE = {(a, a) I a E E} is PI , so, when interpret ed
LOGICAL OBJECTS
157
directly,thedenotations of otherstandardlogicalexpressionswill be "logt hestandardquantifiersdirectly ical objects". For example,interpreting (Westersttihl 1985) we thinkof 3x¢> as 3Ax¢>, where AX¢> is interpreted as a set(or itscharacteristic function)a nd 3 is interpreted as a function from sets ., thatfunctionmapping a subset A of theuniverseto to truthvalues, viz True iff A is not empty. Similarly'V maps A to True iff A is theuniverse. (Note theuniversedependenceof 'V.) These functionsmust countas logical objects,thoughwe must saysomethingabouttheirPIness. Similarlywe want to say t hatthe truthfunctionsdenotedby classical and, or, and not, are logical objects, thoughnaive applicationof PIness thetablefor AND containsthe line here appears to yield a wrong result: «T,F),F) . Butundera permutation of {T,F} which interchanges T andF theresulting tablecontains« F, T), T) which is not a line in the tableforand, so thedenotation of and is (obviously)notinvariant u nderallpermutations of {T,F}. values is Now in distinctionto theuniverseof a model, the set truth of structured. It has two elements that so Ss (sentences)a ndtheirnegationscan structure is given by a~ relation. be interpreted differently and boolean its " in thatit bears the ~ relation to both One ofthetwoelementsis "least elements;it istheone assigned to disjunction a of Ss justin case eachdisjunct is assignedthatvalue.The otherelementis "greatest"in thatit bearsthe~ only to itself; itthevalue is assignedto aconjunction of Ss iff eachconjunct 'Tr of truth values we want are is assignedthatvalue.Thus thepermutations justthebooleanautomorphisms, thosethatsatisfyX ~ y iff 'TrX ~ 'Try (which t hatall two element boolean is to say7l"(~) =~, thatis, 7l" fixes ~). We recall lattices are isomorphic and thatthe onlyautomorphismof such alatticeis theidentityfunction . In generalthe automorphismsof a structure (A, R 1 , R 2 , •.• ) are theperRs , The automorphismsof mutationsofthedomain A whichfix therelations (liketheuniverse of a model) with relations no (functions)given a structure like are simply thepermutations ofthedomain. Notethatbeing structured, lanthe truthvalues, istypicalof theprimitiveswe need to modeln atural guage. Forexampleto representtensemarking(present,past, ...) we might equipped with includeas a primitivea set T of pointsof time (orintervals), a precedence(or overlap)relation.To accountforintensional operatorswe might include setW a of possible worlds, equipped with anaccessibilityrelation. Doubtlessmore is needed. And thepermutations we need arejustthe automorphisms.This guarantees thatthepropertiesof modelsdeterminedby theprimitiveshold as well their of images underan automorphism,and thus those images are also models for language the . And thenotionof PIness we under automorphisms, AIness. requireofobjectsto be "logical" invariance is Before formally defining AInessconsider we some ofthenovelexpressions whosedenotations we wish tocountamongthelogicalo bjects. Theseinclude type. So we ones which aren otfirstorderdefinableandones of high logical
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EDWARD L. KEENAN
must considerautomorphismsofstructures otherthantheprimitiveones we buildourmodels on. m anyexpressionsofthesame category -DetermiFirst,thereareinfinitely ner (Det) inlinguisticparlance-as some and all. (See Keenan and Westersttihl1994 and referencescitedtherefor an overview of recentwork inthis area; notein particular thecollections van Benthem and ter Meulen 1985, Giirdenfors 1987 and van der Does and van Eijck 1996 ; Keenan and Moss 1985 and Keenan and Stavi 1986 are probablythemost extensiveempirical studies.) (1) a. Not more than fifty studentsheremajorin math. b. Most undergraduates studysome Linguisticsbutless than two percent majorin it. c. All but two of the studentssigned up for advancedUlithianthis quarter . b ut neither got an A d. Both studentsansweredquestionsix correctly on theexam. Many oftheseDets are first order: no, both, neither and thoseof the form morelfewer than n , exactly n, between nand m, all but n, the n. Butmany are not:just finitely many, all but fin itely many, infinitely many, as well as (Barwise and Cooper 1981) theproperlyproportional Dets likemost, morel less than half, exactly ten percent, less than a third (even iftheuniverse si requiredto be finite) . Butallsharethelogicalcharacter of theclassical first orderquantifiers. Now, given anontology fl, namely a setEn ofentitiesa nda booleanset 2n of truth values(subscriptsusually o mitted), we treatPIS likemajor in math, , represented as get an A on the exam, etc. as denotingpropertiesof entities functionsfrom E into2. So PIS, likecommon nouns (student , underyraduate , etc.), denotein [E -> 2]. NPs (Noun Phrases)like some stud ent , John and some student, etc. combine withPIS to formSs (P os) anddenotein [[E -> 2] -> 2], theset ofgeneralized quantifiers over E. So wethinkof asentencelike Every student laughed as denotingthetruth valueassignedtothedenotation of laugh by thefunctiondenotedby every student. And as Dets likeevery combine withnounsto form NPs we cantreatthem as denotingin [[E -> 2] -> [[E -> 2] -> 2]]. More systematically now let usrepresentdenotation sets relativeto a choice ofontology(E ,2) in termsof types. The notionof type used in linguisticsemanticsderives fromChurch (1940) butreplaceshis "L,O" notation withMontague 's (1973) more mnemonic "e, t"- "e" forentit y, "t" fortruth value. We extendit toincludeproducttypes: DEFINITION 3. type is theleastset containing t heinitial types "e' and"t ' and satisfying(a) and (b) :
LOGICAL OBJECTS (a) 0:,13 E type :::} (0:,13) E type
159
(functional types),
(b) O:I , ... , O:k E type:::} (al • ...• ak) E type (producttypes).
Type indexesdenotation s etsrelative to anontology n = (E ,2) . An expresas an elementof sion is oftype r if, for eachontologyn, it is interpreted Dennr,defined by:
(2) a. Denn= E, b. Dennt = {02' 12}, c. Denn(o:,l3)= [Denn(o:) -> Denn(I3)], d. Denn(O:I•.. .• O:k) = DennO:lx . .. x DennO:k. DEFINITION 4. For eachontology n, then objects arejustthosein theunion oftheDennr.
Our concernhere is tocharacterize those n objectsappropriately called logical. Butwe anticipate . First,certainof the Dennrarealwaysboolean, as defined by : DEFINITION 5. baal-type
=df
theleastsubsetof type satisfying :
a. t E bool-tupe, b. If 0: E type and 13 E bool-type then(0:,13) E bool-type, and c. If 0:1, . .. , O:k E baal-type then(0:1 •. ..• O:k) bool-type. For r boolean,D ennris a completeatomicbooleanlattice , given by: (3) a. For x, y E Dennt, x ~ y iff x = 02 or Y = h, b. For I,s E Denn(o:,I3), f ~ 9 iff for all a E Denn(o:), f(a) ~ g(a), c. For f,9 E Denn(O:I • ...• O:k), f ~ 9 iff for each1 ~ i ~ k , fi ~ gi· And we are now in positionto a defineautomorphism invariance: DEFINITION 6. A map 11" withdomaintypeis anautomorphismof anontology (E,2) iff
n=
1. For allr E {e,t}, 1I"(r) is an automorphi sm of Dennr, 2. 11"(0:,13) is thatmap sendingeach f in Denn(a, 13) to {(1I"(0:)(a),11"(13)(b») I (a, b) E J)}, and 3. 11"(0:1 • . . .• O:k) sends eachk-tuplea in Denn(O:I• . . .• O:k) to (11"(0:1)(al), . .. , 11"( O:k)(ak))'
FACT 3. For allautomorphisms11" of n, alltypes r, 1I"(r) is a permutation of Denn(r). If r E bool-typethen1I"(r) is a booleanautomorphismof Denn(r). Moreover, every p ermutation of E extendsuniquely to an automorphismof n = (E,2).
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EDWARD L. KEENAN
DEFINITION
7.
(i) For all ontologies n andtypes r, ad E Dennris automorphism in variant (AI) iffn(r)(d) = d, allautomorphismsn of n . Write AInrfortheAI elementsof Dennr.
(ii) For d a closedexpression of .c, d is AI iff for all models M for , LM(d) L is AI.2
Notation: We usually write n(d) forn(r)(d) . And forn some n-automorph ism we writen- 1 forthatmap sendingeach rEtype to (n(r))-I, theinverse of the permutationn(r) . no n- 1 is understoodsimilarlyand is thusthe identityautomorphism. THEOREM 14. Let.chave '= ' of type ((e · e ), t ), '3'and 'V' of type ((e,t),t), and and or of type ((t · t) , t) and not of type (t , t), all interpreted as indicated earlier. Then each of these expressions is AI.
Below wepresentmany theoremsof thissortomittingproofs forreasons ofspace. We illustrate one case her e to givethereadera feel for t healgebraic character of theseproofs. Observefirstthatforn an automorphism,(1) a E A iff n(a) E n(A) , and (2) (nF)(na) = xb says thesame as (na ,nb) E n F: Now we showthat,given E, F : P(E) -+ 2 is AI, where F(A) = 1 iff A i- 0 . automorph ism. We show thatn F = F : Let n an arbitrary (nF)(A) = 1 iff (nF)(n(n - 1 A)) = 1 rr, n- I are inverses iffn(F(n- 1(A)) = 1 observat ion (2) iff F(n-1(A)) = 1 n is theidentityon truth values iff n-1(A) i- 0 def F observation(1); n- 1 an automorphism iff Ai- 0 iff F(A) = 1 def F
Of coursethetruth function s denotedby and, or, and not are nottheonly AI ones. Alltruth functionsare AI (van Benthem 1989a) . Let us define the pure t-types to be theclosure of{ t} undertheformationof functional and producttypes. Then, THEOREM
15. For all ontologies (E ,2) , all rEt-type, each d
E
Den(E,2)r is
AI.
And as well , the Dets discussed above, of type ((e, t) , ((e, t) , t)) , areAI. Here are somesampledefinitions , assumingan ontology w ithuniverseE and A, B arbitrary subsetsof E. 2We need not er quire thatd have no free variabl es; butthen the definit ionmust m en t ion IM(d )(g) , for alla ssignme nts g.
LOGICAL OBJECTS (4) a. b. c. d. e. f. g.
161
ALL(A)(B) = 1 iff A - B = 0. NO(A)(B) = 1 iff An B = 0. SOME(A)(B) = 1 iff An B =1= 0. MOST(A)(B) = 1 iff IA n B I > IA - BI. (ALL BUT TWO)(A)(B) = 1 iff IA - BI = 2. (EXACTLY TEN)(A)(B) = 1 iff IA n BI = 10. (THE TWO)(A)(B) = 1 iff IAI = 2 and A - B = 0 .
(4a) says in effect thatAll poets daydream is trueiffthepoets lessthedaydreamersis theempty set. (5) belowillustrates thetype structure and semanticinterpretation we assume: (5)
t
((e,t)~
»<>; I
I
(e,t)
((e ,t),((e,t),t)) (e,t)
I I
all
poets
daydream
ALL
POET
DAYDREAM
I
I
A¥~ ALL(POET)(DAYDREAM) THEOREM
16. The Dets given in (4) are AI.
expressionswhichshare the syntacticd istribution of Observethatnot all theDets above do pr esenta "logical " character. Contrast: (6) a. All / John's catsare black, b. Most / More male than female studentsr eadtheTimes. of John's depends bothon whichindividualJohn In (6a) theinterpretation denotesandon whatobjectshe possesses. Manynon-Al functionscan interpretJohn's (Keenan and Stavi 1986) . In (6b) more male than f emale admits of non-AIinterpretations, as itdependson whichobjectsare maleandwhich female,a contingent matter . language . They share We returnnow to"logical"expressionsin natural withtheDets in (4)thepropertyof being AI. Consider first the comparative quantifiers in (7) (Keenan and Moss 1985, B eghelli 1992). Their type stru ctureis representedin (8), using 'p' ( "property") for (e, t) :
EDWARD L. KEENAN
162
(7) a. b. c. d.
More/Fewer studentst han teachers signed thepetition, Exactly as many studentsas teachers signed thepetition, More than twice (n times) as many studentsas teacherssigned, The same number of studentsas teachers signed thepetition .
t
(8)
(P't)~
~pep
((pep), (p,t»)
I
p
A
I
p
I
p
I
more-than (student,teacher)smiled (more studentst hanteacherssmiled) The interpretation of thesecardinal comparativesis straightforward : MORE As THAN Bs have 0 iff IAn 01 > IB n 01. Quitegenerally cardinal comparativesare not first orderdefinable:more-than defines most (over a universe E) byMOST(A)(B) iff (MORE An B THAN A - B)(E) . Thatis, thanAs who are not Bs exist. So most As are Bs iff more As who are Bs if MORE- THAN were firstorderdefinable MOST would ,be butby Barwise andCooper (1981) it is not. The examples given so far suggestthatlogical expressions of high type depend on cardinalities . But this is not in general the case (See Keenan 1987, page 92; van Benthem 1989b): (9)
a. Different people likedifferent things, b. No two studentsr eadthe same papers, c. Each studentansweredthe same questions(on theexam).
Keenan (1992) shows thatwe do notobtaina correctsemanticanalysis of thesesentencesif forexample different things , . .. , the same questions are interpreted as generalized quantifiers.A semantically correctanalysis interpretsdifferent-different (no two-the same, etc.) as functions mapping pairs ofpropertiesto functionsfrom binaryrelations into 2. So writingr for (e, (e, t», different-different has type ((p e p), (r, t». As illustrated in (lOa) anddefined in (lOb) this functionis AI, and its valuesdepend primarilyon identityofobjects, notcardinality . (10) a. Different people like different things is trueiff at least two people likesomething,and for any two such people, thingsone the likes are notexactlythesame as thosethesecond likes. b. (DIFF-DIFF)(A, B)(R) = 1 iff IDom(R n (A x B»)I a' E Dom(R n (A x B)
{y E B
I (a,y)
E Rn (A x Bn =1= {y E B
I (a',y)
~
2 and Va =1=
E Rn (A x Bn·
LOGICAL OBJECTS
163
theDeterminerclass ofexpressions An examplewhich moves away from are reflexives, (11a), and reciprocals , (llb): (11) a. Everysenatora dmires himself, b. The candidatescriticizedeach other. w ithnon-AI Noun Phrases(NPs) like the himself and each other contrast President in Every senator admires the President and the reporters in The candidates criticized the reporters. We treath imself as denotingthatfunction SELF from binaryrelations (e.g., ADMIRE) topropertiesgiven by: SELF(R) = {a EEl (a,a) E R}. Clearly SELF is AI. And weinterpret each other as that AI map EO from binaryrelations tomaps from setsto truth values given (on firstapproximation)by: EO(R)(A) = 1 iff Va i= a' E A , aRa' . A last,non-quantificational, example: ArguablythePI is opened in (12b) is derivedby a logical o peration , Passive, from theP 2 opens mapping binary relations to sets: PAss(R) = {b EEl 3a E E aRb} = Ran(R) . Obviously PASS is AI. In (12c) therole oftheprepositionby is to saythatits object, John , is the"logicalsubject"of theP 2 opens. It combines withNPs to yield binaryrelation modifiers,andso has type «p, t), (r , r)). (12) a. Johnopens thedooreverymorningatnine, b. The dooris openedeverymorningatnine, c. The dooris opened by Johneverymorningatnine. Keenan(1980) defines: BY(z)(R)(y)(x) = 1 iff (x, y) E R and x = z . Analyzing(12c) as (the door)(Pass(byJohn)(open))) yields correcttruth conditions.And clearly B Y E Alo«p, t) , (r, r)) . 2.1. Logical constants We have found paroperty , thatof being AI, whichclassicallogicalconstants have incommon. And we have shownthata varietyof novelexpressions whicharepretheoretically "logical " also havethisproperty.This would seem torefuteE tchemendy(1990, chapter15 "TheMythoftheLogicalConstant ", page 128) whoclaims thatthere is no propertyof expressionswhose inter pretationswe hold fixedthataccountsfortheirrole inestablishinglogical truths andvalidinferences . ButdenotingAI elementsin theirtypeis a good candidate.In fact: THEOREM 17 (KeenanandStavi,1986). Constantly interpreted expressions denote AI objects.
LS, richenoughtoincludelogical We extendthetheoremtotypetheoretic expressionsof thesortdiscussedabove. And we define constantly interpreted fortheseLS. Syntax. Type theoretic LS withequalityhave, for each t ype T, countably many (possiblyzero)constantsCon, and denumerablymany variableswith
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EDWARD L. KEENAN
VarT n VarT' = 0 if r =I- r'. If d and d' areexpressionsofthesame type,then (d = d') is an expressionof type t. Expressionsof type (0 , {3) concatenate withones oftype a to form ones oft ype {3. If d1 , . • • ,d k are expressions , then (d1, . .. , dk) is an expressionof type of types r1 , . . . , rk respectively (r1 • ...• rk) . Booleancompounding,universal(existential) quantification applyjustto expressionsof booleantypes. Semantics. M = (EM,2M,I) is a model for atype theoreticI: iff (EM,2M) is an ontologyand I is a functionmapping each c E Con, into D(E,2)(r), allrEtype (subscriptsomitted). An assignment s maps each x EVarT into D(E,2)r. An interpretation IM of I: relative to M maps 1:expressionsto functionsfrom assignmentsintotheD(E,2)r suchthat : For c E ConT> IM(c)(s) = I(c). For v E VarT,IM(v)(s) = s(v) . For d of type (a,{3) and b of typea, IM(d ~ b)(s) =df IM(d)(s)(IM(b)(s» . IM(d 1, .. . , dk)(s) = (IM(d 1)(s), .. . ,IM(dk)(s». IM(d = d')(s) = 1 iff IM(d)(s) = IM(d')(s). Booleancompounds are interpreted pointwiseon theassignments,and for each x EVarT and eachexpression ¢ of booleantype, IM(3x¢)(s) = V{IM(¢)[sja] I a E D(E,2)(r)}. (Each booleanD(E,2)r is complete.) Note: 1:-models are closed u nderisomorphism: if M = (E, 2, I) is a model and 7r an automorphismof E andof 2, then7rM =df (7rE,7r2,7r 0 I) is an I:-model. For d an expressionof I: andM a model M, whenIM(d) is constant on the assignmentswe writeIM(d) forIM(d)(s) , s anyassignment. If d is closed (= no freevariables) , IM(d) is constant on theassignments.In whatfollow s we need Theorem20, theappropriategeneralization of M c:::: M' :::} M ~ M'; the proofuses thetechnical Lemmas 18 and 19. 'C rangesovertype theoretic languages. LEMMA 18. Given M = (E,1) , 7r a bijection with domain E, s an assignment, x EVarT>a E Dor and writing 7rS for 7r 0 s, we see that 7r(s[xja]) = (7rs)(xj7ra). Moreover, A = B, where
A = {I"M(d)«7rs)[xja])) I a E Dor} and B = {I"M(d)(7rs)(xj7ra» I a E Dor}. Proof. (1) A ~ B. Let 8 E A . Then for some a E Dor, 8 = I"M(d)(7rs) [x/a]. Since 7r t Dor is onto, a = na' for some a' E Dor, so 8 = I"M(d)(7rs) [xj7ra'], so 8 E B . (2) B ~ A. Let 8 E B. Then for some a E Dor, 8 = I"M(d)(7rs)[xj7ra]). Hence for somebE Dor, 8 = I"M(d)(7rs)[xjb]), so 8 EA. -1 LEMMA 19. For M = (E , I) a model for 1:, 7r a bijection with domain E, and s an assignment,
for all expressions d of E:
LOGICAL OBJECTS
165
We give theproofin full intheAppendix. THEOREM 20. For type theoretic Ls, Model Isomorphism => Elementary Equivalence.
Proof. Let 1r: M ~ M', l/J E Sent(£),s any M-assignment.Then, IWp = IMl/J(s) = 1r(I Ml/J(s», 1r fixestruth values,= IM'l/J(1rs), Lemma -j 19, = IM'l/J since l/J is closed. DEFINITION 8. An expressiond of £ is constant iff for all models M, IMd is constant on theassignmentsand for all models M', all bijectionsrr : EM-+ EM', 1r(IM(d» = IM ,(d). THEOREM 21. A closed expression d of type T is constant => 'v'M = (E,2,I),IM(d) E AI(E,2)T.
Proof. Given M = (E,2 ,I), let1rE PERM(EM). Show IM(d) = 1r(IM(d». Let S an arbitrary assignment. Then1r(IM(d» = 1r(I M(d)(s» = I 7rM(d)(1rs), Lemma 18, = I M(d)(1rs) since EM = E".M and d is locallyconstant, IM(d)(s) since IMd is constanton theassignments,= IM(d). -j THESIS 1. The logicalconstantsin a type theoretic £ are theconstantsas defined in (8). Logicalconstants like'= ', 'and ' ,'not', and 'V' are in factc onstant.Some be syntacti callysimple, though might wantto requirethatlogicalconstants this does not seemessentialand would havethe unpleasantconsequence thata definedsimple expressioncouldbe logically constant even thoughthe complex definingstatementwas not. (E.g., letTaut abbreviateVx, y(x = y or x =1= y». Also Westerstdhl (1985) , van Benthem(1989a), Keenanand Stavi(1986) andSher (1991)notethatDefinition 8 allows as logical constants expressionsdefined by AIconditions,likethehypotheticallottsa below, where for all models M = (E,I), ALL if E is finite, (13) I(lottsa) = { INFINITELY MANY if E is infinite. We stressthatTheorem21 andThesis 1 assumeclosureunderisomorphism fortheclass of models of the£s in question.Here forexampleis an attempt to construct a constant whosedenotation is not AI: Let blik be anexpression of type (e, t) andsuppose thatthedefinitionof interpretation requiresthat for M = (E,2,I),I(blik) = {3} if 3 E E, otherwiseZ(blik) = E. Blik is constantly interpreted b utI(blik) is not AI when E= {3,4}. But this conditionviolates closure underisomorphism. If 1ris a bijection:{3, 4} -+ {5,6} then1rM = ({5,6},1r2,1roI) is not a model of E since (1roI)(blik)= {1r3} is a unitset,butthedefinitionof interpretation requiresit to be {5 , 6}.
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EDWARD L. KEENAN
Note thatEs may present"logical"expressionswhich arenot constant . Candidatesare Dets likeseveral and a jew (# many and jew). Several seems vague. Just how many sparrowson a clotheline countas several? At leasttwo, and perhapson some occasionsthatsuffices. One accountof this vagueness(Keenan and Stavi 1986) wouldmerelyrequireof models of EnglishthatI(several)E {AT LEAST n I 2 :::; n} . Then we would have differently , models withthesame universein which several was interpreted andso notconstantly interpreted, butits denotation would always be AI in its type. Weturnnow tothecentral thesisof thisarticle : 3. THESIS 2
THESIS 2. For allontologies fl andalltypesT, thelogical elementsof DennT are theAI ones. 3.1 .
EmpiricalsupportforThesis 2 is given byTheorem21: given anontology fl, AInT providesthedenotations of logicalc onstantsof type T . And this observation generalizes . The constants consideredso far are ones that,given an ontologyfl, uniquelydeterminean element of someDennT. But many expressions,includingones alreadyconsidered , determineobjectsin multiple types, where, in eachtype, theobjectis AI. Herearesome examples: (14) and, or, neither.. .nor ... combine inEnglishwithexpressionsin most booleantypes to formexpressionsin thattype: VerbPhrases : John bothlaughedandcried; neitherlaughednor cried . Transitive VPs : John both hugged and kissed / neitherhugged nor kissed his cat. Adjectives: a smart and industrious/ neithersmart nor industrious student . Prepositions : Johnliveseitherin or near n / eitherin nornearNew York City.
FACT 4. For allfl, allbooleanT, ANDT of type ((T e T), T) which maps each (a , b) to aA T b, thegreatest lowerboundof {a, b} in DnT, is in AIn((TeT),T). And needless to say, all booleanly definableconstants, functions,andrelations determineobjectswhich are logical in the appropriatetype. Here are some additional AI functors:
167
LOGICAL OBJECTS (15) a. Typed Equality:For alln, allr , =r of type «r =r (a,b) = 12 iff a = b, is in AIn«rer),t)
e
r), t) given by:
b. Typed Membership: For alln, allr, E r, of type «r e (r, t )), t) whichsends each (a,g) in itsdomain to g(a) is in AIn«re (r, t)) , t) , as is f/.Tl defined intheobvious way. Note thatE r is thespecialcase ofAPPLY r ,u witha
= t:
c. Typed Application:For alln, alltypes r,tr, APPLYr,u is that elementof Dn«re (r ,a)) ,a) mapping each (a,g) in its domain to g(a) . It is in AIn«r e (r, a)) , a) . (15b,c) characterize thesense in which setm embership is a "logical"notion . Note thatin a I:- in which' E' is a P2 of type « e· e), t) interpreted as set membership, its denotationis not ingeneralAIn«e . e), t) . The only AI binaryrelations over E are0, Ex E, IDE = {(a , a) I a E E}, and --, IDE = {(a, b) I a, bEE & a i- b}; see Theorem28. Set membership fails to coincide with any of theserelations exceptwhen nox in E is a member of anyy in E. Other"transcendental " (van Benthem 1 gS9a) functorslike Composition andSubstitution , so useful inlinguisticdescription , arealsoeasily seen to be AI in theappropriatetype. d. COMPu,r,v in Dn«(a, r ) e (r,v)) , (a, v)) mapping (f,g) to gof , is in AIn(((a, r)(r, v ))(a, v )) e. SUBu,r,v, thatelement S of Dn(a,«a, (v ,r)) , (a,v ))) such that S(a)(J)(g) = f(a)(g(a)), is in AIn«a, «a,r) , (a , v )) ). Finally,observe thatthe propertyof being "logical" is its elf "logical" . Thatis, withoutcircularity, thepropertyof being AI si itselfAI: THEOREM 22 "Being logicalis logical".For all ontologies n = (E ,2) 4 and all types r, that element AIr of Denn(r,t) which maps each din Dennr to 12 iff dE AInr, is in AIn(r, t) .
Proof We sketchthedirectproofby cases to confirm t heabsenceofparadox. Let n be an arbitrary ontology,r an arbitrary t ype. Eitherr is 'e', 't', a functional type or a producttype. We show thatin each casethefunction AIr is AI (thatis, is an elementof AIn(r,t)) . In whatfollows we often write "d is AI in type r" for d E AInr. Also we use rep eatedly thatan objectis AI in a type iff its imageunderany automorphismis AI in thattype. r = t. Eachelement(omittingsubscripts)0,1 of Dennt is AI, so AIt maps each to1. We must show that7I"(AId = AIt, for eachn automorphism71". Now 7I"(AI t)(x) = 7I"(AI t(7I"-l x)) = 71"(1) = 1, so 7I"(AI t) = Alt. r = e. If Denne has just one el e ment, b, thenb is AI so Ale(b) = 1. And 7I"(AIe)(b) = 7I"(AI e(7I"-1(b)) = 7I"(AI e(b)) = 71"(1) = 1, so 7I"(AI e) = Ale. If
168
EDWARD L. KEENAN
Dennehas morethanone element,t henfor eachx E Denne x is not AI, so 1(x)) Ale(x) = 0 and7r(Ale)(x) = 7r(Al = 7r(0) = 0, so 7r(Ale) = Ale. e(7rT is a junctional type (a, b). We must show thatAI(a,b) = 7r(AI(a,b») ' Assume firstthat9 E Denna is AI, so AI (a,b)(g) = 1. Then 7r(AI(a ,b»)(g) = 7r(AI(a ,b»)(7rg) = 7r(AI(a,b) (g)) = 7r(I) = 1, so the two functions takethesame values at AIarguments.Now assume 9 is not AI. SoAI(a,b)(g) = O. And 7r(AI(a ,b»)(g) = 7r(AICa,b) (7r-lg)) = 7r(0) = O. ThusAI(a,b) and 7r(AI(a,b») take thesame valuesatall9 and so arethesame function. T is a product type a = al . . . .. an' Let x E Denna . Observe thatx is AI in its producttype iffeach Xi is AI in its type ai . If x is AI then l(x))) = 7r(Al Ala(x) = 1. And 7r(Ala )(x) = 7r(Al a (x)) = 7r(I) = 1, so a(7requalityholds inthiscase. If x is not AIthenAla(x) = 0 and 7r(Ala )(x) = 7r(Ala(7r-1(x))) = 7r(0) = 0, completingtheproof. --I 3.2. Plausibility
Grantedthatthetechnically definedAlnTprovidedenotations of logic alexpressions, whyshouldwe expect this be so? We offer answers from two perspectives. 1. Regardingour typetheoretical primitives,theelementsof 2 are given of sentenceshas iff each as logical: 21is uniquelythe value caonjunction conjuncthas thatvalue, andO2 is uniquelythe(distinct)value assigned to thenegationof asentencewith value h . The permutationswe studydo not changetheselogicalobjects. So to some extentwe are askingwhatother objectsmust not change as raesultof holdingthetruthvaluesconstant? of the These are ones whose "logicality" follows from the logicalitytruth values. E, however, is randomlychosen; itselementsdo notin generalhave any logicalcharacter.B ut the permutations of E do. They are, in effect,t he functionswhich fully respectthebooleanstructure of propertiesof elements of E, thatis, theyarethebooleanautomorphismsof P E . WritingATOM(P E ) forthesetof unitsetsin P E , observe:
5. (a) If 0 is a booleanautomorphismof P E , then0 i ATOM(P E ) is a permutationof ATOM(P E ) . So 0 determines0' E PERM(E), where o'(x) = y iff o{x} = {y}.
FACT
(b) Converselyeach permutation n of E uniquelydeterminesa boolean automorphism7r* of P E by: 7r*(K)= {7r(k) IkE K} . To say thata permutation 7r of E preservesthe boolean s tructure of PE is to sayinformally t hat7r preservesthe"and,or, and not"structure of PE . E.g., 7r(LAUGHOR CRY) = 7r(LAUGH)U 7r(CRY); 7r(NEITHER LAUGH
LOGICAL OBJECTS
169
NOR CRY) = -'(1f(LAUGH) U 1f(CRY)). 1f alsopreservestheuniversaland existential quantificational structure of PE. Note thattheset denotedby praise every boy is theintersection of thesetsdenotedby pmise x takenover all boysx. And 1f(PRAISE EVERY BOY) = 1f(nXEBOYPRAISE X) = nXEBOY 1f(PRAISE X). Similarly1f(PRAISE SOME BOY) = 1f(UXEBOY PRAISE X) = UXEBOY 1f(PRAISE X) . Finally , as a bijection,a permutation 1f alsorespectsdistinctnessof individuals: 1f does notmap distinctindividualsto thesame one, and1f does not that1f maps theidentityrelation to omit any individuals . This guarantees itself.Thus,
(16) AI objects are ones that remain invariant under substitutions which don't change truth, boolean structure, or identity. ThatAI objectshave a logical character thenis notsurprising. 2. Butthe intuitionbehindusing AIness tocharacterize logicality is not thatpermutations fix propertieswe pre-theoretically considerlogical -most of theseobservationsare theorems, i.e., derived. Rather,o urintuitionconcernsthegenemlity-specijicity dimension: purelylogicalrelat ionsc annotdisand thus criminateoneobjectfrom anotherb utmust treatt hemuniformally must remainunchangedundersystematicsubstitutions of individuals.One way this uniformtreatment shows up is indefinabilityproperties. For example, considerdefinability of n-aryrelations overtheuniverseof models for firstorderLS. Given L and a model M= (E, 2, I) forL , we saythatR ;2 E k is L-definable(in M) iff for someL-formula4> with exactlythe k distinct k Xll " " Xk free,R = {t E E I M F 4>[tJ}, where4>[tJ is 4>[gJ, 9 : VAR -> E such thatg(Xi) = ti , all1 :::; i :::; k. Then, recalling t hatL= is thezerosignature firstorderL withequality,
THEOREM 23. An n -ary relation Rover E is AI iff R is definable in L=.
Proof Appendix.
-j
Note thatin distinctionto definabilityin highertypes (Liiuchli 1970; Plotkin 1980; van Benthem 1989a, page 330) Theorem 23 does notrequire thatthe universe be finite. No matterhow large E is, theAI n-aryrelations 28) , so thenumberof suchrelations of over E are finite number(Theorem in allaritiesnever exceeds the numberofexpressions in L=. COROLLARY 24. For R s;;: En, R is AI iff for all first order L with =, R is definable in all L-models M with EM = E.
Proof The proofconsists in observingthatR is preservedunderlanguage and modelexpansions,and any first orderL with= is an expansionof L=. -j
170
EDWARD 1. KEENAN
Clearlyany relation R definable in1:.= is "logical", as all we can use to defineR is equality,booleanoperators,and universal and existential quantification.The factthatall AIn-aryrelations over anarbitrary universe are definable in 1:.= thensays thatwe cannotlimitthe"logical"relations to a propersubsetof the AI ones unless we are preparedto acceptthatnonlogicalobjectscan be defined interms of logical ones (equality)by logical operations . And a much more general resultgiven by vanBenthem(1989a) is: (17) All closed t ermsin thetheoryof typesdefine PIobjectsin theirtype. Further,in a type hierarchyb uiltfrom a finiteE, every PI item in any type is definable by some closed type-theoretic termof thattype. (Type-theoretic t ermshere differ from expressions in the type-theoretic Es discussed earlierin that(1) theylackconstantsof arbitrary type, and (2) theymay be built bylambdaabstraction.)Our lastresulthere concerns uniform definability. First, DEFINITION 9. Let t: = T1 •. . .• rk be a producttype and R be a functor of type (T, t), i.e., R maps each f2 = (E,2) into Deno(r,t) . (So R(f2) ~ DOT} x . . . x Dork .) Then for all languages 1:., R is uniformly definable in I:. iff for somev = (VI, .. . , Vk) of k distinctvariables of types T1, . . . , Tk respectively, for some I:.-formula¢ free forv and for all models M = (E ,2,I) for1:., M F ¢[sJ iff (SV1," " SVk) E R(f2) . THEOREM 25. VI:., Vfunctors R of type (r,t), R is uniformly definable in I:. ~ R(f2) is AI, all f2.
Proof. Let R be uniformally definable inI:. by ¢. Let f2 = (E,2) arbitrary, 1r an automorphismof 11. Show R(11) = 1r(R(11». ~ Let u E R(11). By uniformdefinability 3M , M F ¢[uJ . So 1r- 1M F 1uJ 1 ¢[1rby Lemma 19 since 1r- is an automorphism,so 1r- I U E R(11), whence u E 1r(R(11». {:= Let u E 1r(R(11». So u = ttt: for some t E R(11). Thus 3M' , M' F ¢[tJ. So 1rM' F ¢[1rtJ by Lemma 19, so by uniform definability nt = u E R(1r11) = R(11) since 1r is an automorphism . --/ These resultshold because Alness is preserved by operations the in thedefinitions: THEOREM 26. For 11 = (E,2) given, (1) The type forming operations preserve AIness: if A and B are sets of AI objects then all elements of [A -+ BJ, A x B and of course P(A) ~ [A -+ 2J are AI.
(2) For each boolean r , Alor is a (complete, atomic) boolean subalgebra of DenOT. In other words, boolean functions preserve AI ness.
171
LOGICAL OBJECTS
(3) Generalized converses: If R is a AI k-ary relation and"( is a permutation of {I, . . . ,k} , then "(R =df {(a-y(l)" " ,a-Y(k» I (al,." ,ak)R} is a AI k-ary relation. (4) Cylindrification (existential quantification): If R is a k+l-ary AI relation, then for all 1 $ i $ k, Ci(R) =df {(al,"" ai-I , ai+I, ' .. , ak+l) I 3a E Ai(al, ... , ai-I, a, ai+I, ... , ak+l) E R} is AI. (5) The value of a AI function at a AI argument is itself AI. As a special case, composition of AI relations (functions) yields a AI relation (Junction) . 3.3 Additionalc onditions?Grantingthelogicalobjectsare AI ones, are wereadytograntthatall AIo bjects(of a giventypc,in a givenontology) are "logical"?Ourremarksabove ondefinability suggestthatwe do, pending plausibleadditional conditions.Here are two suggestions,t hefirst of which is onlyapparent,t hesecondof which rules outtoo much. 3.3.1 The notionof Alnessgeneralizes naturally to invarianceu nderisomorphisms (ISOM) and trivially objectswhich areISOM are also AI. In fact equality holds, sotheapparently s tronger r equirement does notin fact eliminateany' candidates.We sketchtheresult. First, a map 7r with domain type is an isomorphism from an ontology n = (E,2) to anontologyn' = (E',2') iff for allinitialtypes T, 7r(T) is an isomorphism: DennT ~ Denn'T andthevalue of7r at functional andproduct types liftsjustas in thecase ofautomorphisms. An elementd of DennT is said to be isomorphism invariant iff for all ontologiesn', allisomorphisms 7r,0 from n to n', 7r(d)= o(d). And we prove:
THEOREM 27. For all dE AlnT.
n=
(E,2), all
T
E type, all d E DnT, d
s:
ISOMnT {:}
Proof Only ¢= is non-obvious . The proofis by recursionon type.
Set K
= {T E type I for alln = (E, 2),
alls e DnT,
dE AlnT ::::} dE ISOMnT}.
i. Obviouslyt E K since for anyn' = (E', 2') thereis only one isomorphism: 2 ---> 2'. And e E K since ifAlneis non-empty,thenE has justone element,whenceIE/I = 1 so againthereis only onebijectionfrom E to E/. ii. Let 0., {3 E K, show (0., {3) E K. Let d E Dn(o., {3), let7r,O be isomorphisms from n to n'. Then 7r(o,,B)(d)(7ro (a» = 7r,B(d(a», by def ISOM, = o,B(d(a» by theIH, = O(o,,B) (d)(8o (a» = 8(o,,B) (d)(7ro (a» by IH , so 7r(o,,B)(d) = 8(o,,B) (d), so (o. ,{3) E K. iii. Closureunderproductsis straightforward. --l 3.2 Universeindependence?One might have hoped for a more"absolute"notionof logicalo bject,one thatdid notdepend on thechoice of universe ortype. Note thaton theview presentedhere,
172
EDWARD L. KEENAN
FACT 6. For each ontology 0 andtype T, DenOT is in AIo(T, t) . It is in fact its boolean unit. So each element in thetypehierarchy b uilton an (E,2) is AI in a certaintype relative tothatontology.Butit doesn't make sense to claim that , e.g., {3,4} . is "absolutely" PI. It is inAI(E,2)(e,t) when E = {3,4} but not otherwise Similarly {(2 , 2), (3,3)} is AI over the universe =E {2,3} butfails to be AI over anyotheruniverse, say {I, 2, 3} or {I , 2}. More generally , for0 = (E, 2) and 0' = (E' , 2') with E, E' disjointand 2,2' disjoint,we havethatfor all T, DOT and DEfT are disjoint. Thus the attemptto characterize invariants independentof ontology seems bootless. Nor canomit we mentionof type. An objectd in some AIoT fails to be inAhde), where the universe 0' of includes{d, (d, 0), (d, I)}. And we havethusshown: FACT 7. For d in any AIoT thereis an ontology 0 ' and type T' such that d E DenOT - AIoT. 4. AN OPEN PROBLEM
Van Benthem(1989a) notes as opentheproblem ofcharacterizing the AI elements of the typehierarchy built from an a rbitrary ontology (E, 2), an obvious desiderat umgiventhecloseassociationbetween AInessa ndlogicality . We do not have a general answer, butwe shall answer two basic cases, and make a fewremarksconcerningsome others . (We tendto write PIinstead of AI when dealing with types not involving 't'.) THEOREM 28. The PI n -ary relations over a universe E with at least n elements correspond one-to-one to the sets of equivalence relations over an n-element set . . Directcalculation Proof sketch.P(E n ) is a complete,atomicbooleanalgebra shows thatPIness is preservedundercomplementsand arbitrary intersections, so the PIn -aryrelations form a completeand thusatomicsubalgebra relationover of peEn). To constructits atoms let T be any equivalence {I, . .. ,n}. Set R T = {s E En I VI:::; i,j:::; n'Si = Sj iff iTj} . Then R T is AI, non-empty(since lEI:::: n), and nonon-emptypropersubsetof R T is AI. So R T is an atom. Clearly different r'« give rise to different R'TS, andif R is AI and non-empty,t henfor eachs E R , R; =df {t E En I ti = tj iff s, = Sj, all 1:::; i, j :::; n} is an atom which is asubsetof R . So the R'TS are all t he atoms,and since each AIR is the union of its atoms,theAI R's correspond one-to-oneto the sets of atoms. Hence I{R E peEn) I R is AI}I = 2EQ(n), where EQ(n) isthenumberof equivalence relations over ann element set.--l Westerstahl (1985) also givesthefigure EQ(n) 2 and provides a recursive means ofcalculating EQ(n). Ourinterestin the figure is at ththenumberof AI n-aryrelations over E does not grow with the size of (sufficiently large) E
LOGICAL OBJECTS
173
butis boundedas a functionof thearityof therelations.The same property obtainsforthePI functionsfrom En intoE: THEOREM 29. If lEI 2: n + 2, then IU E [En -> E] I f is PI}I = n· rrl~k~n-1(n - k)C(n;k+IJ •
Proof sketch. The PI maps from En into E are the projectionfunctions whichcanchoosedifferent c oordinates atnon-isomorphictuples,e.g., (a, b, c) and (a, b, a). If f E [En -> E] is PI, then\/0" E En f(O") = a, for some 1 ::; i ::; n, otherwisepermutingf(O") withsome elementnot f(O") or any coordinate of 0" fails topreservef . So f restricted to thesetof n-tuplesw ith no coordinatesidentifiedis one ofthe n projectionfunctions . For k > 0, C[n; k + 1] =df n!/(k + I)! · (n - (k + I»! is thenumberof ways ofidentifying k+l coordinatesof an n-tuple . For eachsuch way thenumberof different coordinates of theresulting t upleand hencethenumberof possiblevaluesa PI functioncan take,is (n - k) . It takesits valuesi ndependently on tuples formed bydifferentways ofidentifyingk+ 1 coordinates. -1 So (forE sufficiently large)no x E E ~ [EO -> E] is PI (so therecanbe no logically c onstant p ropernouns);onlytheidentitymap in [E -> E] is PI, and onlythetwoprojectionfunctionsin [E x E -> E] are PI. Thereare3.2 3 = 24 PI maps in [E3 -> E] and 4 . 3 6 • 24 = 46, 656 PI maps in [E4 -> E]. Richer e types seem to behavesomewhaterratically. For lEI> 1, there are no PI maps in [[E -> E] -> E]: Let h be in thatset and let 1f move h(id) . Then (1fh)(id) = (1fh)(1fid) = 1f(h(id» i= h(id), so h is notPI. This observation is a specialcase of: THEOREM 30 (Van Benthem,1989a). For T in the closure of {e, t} under junctional types, Denf/T has no PI elements iJJT = (0"1,(0"2" . . (O"n,e) . . .») where each Denf/O"i has PI elements. By contrastin [[E -> E] -> [E -> Ellthe number of PI maps grows exponentially in lEI, whenceforinfiniteE thenumberoutstripst henumber of expressionsin thelanguage.N otethatdifferent s ubsets K of {O, ... , lEI} in (18) give rise todifferentiK's, where ( 18)
fK( )(a) = 9
df
{g(a) if I{~ I ga = a}1 E K, a otherwise .
Similarly,thegrowthof AI generalized quantifiers is exponentialin the size oftheuniverse: (19)
IAI[P(E)
->
211 = 2IE1+l.
Butwe stilllack ageneralmeans of computingcardinalities of AI sets in a type hierarchy.
EDWARD L. KEENAN
174
APPENDIX LEMMA 8. Let M, M' be infinite models for L=. Then for all formulas 4> in L=, all 9 : VAR ----t EM and all g' ; VAR ----t EM'
if g(x) = g(y) iff g'(x) = g'(y), for all x, y
E FREEVAR(4)),
then M
F 4>[g]
iff M'
F 4>[g'] .
Proof. By recursionon theformulas of L=. Let M,M' be infinite models for L=. Set S = {4> E Fmla(L=) I for all 9 : VAR ----t EM and allg' : VAR ----t EM' if g(x) = g(y) iff g'(x) = g'(y), for allx ,y E FREEVAR(4)) , thenM F 4>[g] iff M' F 4> ls'll . (i) Allatomic formulas4> are in S. 4> is (x = y) for some x, y E VAR. Let g,g' be as in thelemma withg(x) = g(y) iff g'(x) = g'(y) . Then M F (x = y)[gJ iffg(x) = g(y), iffg'(x) = g'(y), iffM' F (x = y)[g'J, so (x = y) E S. (ii) S is closedunderconjunction a nd negation. (a) Let 4>, 1/J E S. Show (4) & 1/J) E S. Let g, g' be such that g(x) = g(y) iffg'(x) = g'(y), for allx, y free in(4) & 1/J). Then for all x ,y free in4>, g(x) = g(y) iff g'(x) = g'(y) since if, e.g., 9 identified free variables in 4> thatg' distinguished , then9 would have identified themin (4) & 1/J) butg' woulddistinguishthem,contrary tohypothesis. Then M
F (4) & 1/J)[gJ
So (4) & 1/J) E S.
iff M F 4>[g] and M F 1/J[g] def interpretation, iff M' F 4>[g'J and M F 1/J[g'] IH + theobservation, def into iff M' F (4) & 1/I)[g'J
(b) Let 4> E S. Show that-, 4> E S. Note thatFREEVAR( -,4» = So
FREEVAR( 4».
So -,4> E S.
iff M ~ 4>[gJ iff M' ~ 4>[g'] iff M' F -, 4>[g'J
def int, IH + note, def into
(iii) S is closedunderexistential quantification . Let 4> E S. Show for all variables3z4> z, E S. Let z be arbitrary, letg,g' as in the lemmadistinguishthesame variables in 3z4>. Show M F 3z4>[g] iff M' F 3z4>[g'J.
LOGICAL OBJECTS
Left to right. Let M 1= 3z4>[g] . ThenM
of g.
1=
175
4>[gz], wheregz is a z-variant
case 1: gz(z) = g(x) for some x E FREEVAR(3z4» . Let g~ be thatz-variantof g' such thatg~(z) = g'(x). Then for allx ,y free in 4>, gz(x) = gz(y) iff g~(x) = g~(y) . So by IH, M' 1= 4>[g~], whence M' 1= 3z4>[g']. case 2: gz(z) =I- g(x) for anyx free in3z4>. Thenfind az-variant of g' such thatg~ (z) =I- g'(x), for anyx free in4>. Such a g~ exists since FREEVAR(4)) is finiteandEM' is infinite,so thereis an objectnot in therangeof g~ t FREEVAR(4)). And againfor all x, y E FREEVAR(4)), gz(x) = gz(y) iff g~(x) = g~(y). So by IH, M' 1= 4>[g~], whence M' 1= g~
3z4>[g'] .
This exhaustst hecases. The rightto leftd irectionis done analogously. -1 THEOREM 20. Model Isomorphism ==> Elementary Equivalence for Type The-
oretic Languages. For M = (E , I) a model for1:-, 11" a bijectionwith domain E, s any M-assignmentand d an expressionof anytype in 1:-,
Set S = {d E I:Then,
I for
allassignmentss, 1I"(IM(d)(s)) = I 1rM(d)(1I"s)} .
1. For allT , Con, ~ S. Let c ECon,; Then,
1I"(IM(C)(S)) = 1I"(I(c)), def int, andI 1rM(c)(1I"s) =
(11"
0
I)(c) , def int,= 1I"(I(c)).
2. For allT , VarT ~ S. Let x EVarT. Then,
1I"(IM(V)(S)) = 1I"(s(v)), def int, and I 1rM(V)(1I" 0 s) = (11" 0 s)(v) = 1I"(s(v)). 3. For d, d' E S, d, d' bothof type T . Show (d = d') E s. 1I"(IM(d = d')(s)) = 1 iffIM(d = d')(s) = 1 iffIM(d)(s) = IM(d')(s) iff1I"(IM(d)(s)) = 1I"(IM(d')(s)) iffI 1rM(d)(1I"s) = I 1rM(d')(1I"s) iffI 1rM(d = d')(1I"s) = 1
fixes eachx E 2, def interpretation , 11" is injective , InductionHypothesis, def interpretation .
11"
176
EDWARD L. KEENAN
7r(IM(d)(s) = = = =
7r((IM(di)(s), ,IM(dk)(s))) (7r(IM(dd(s)), , 7r(IM(dk)(S))) (I"M(di)(7rs)), ,I"M(dk)(7r 0 S))) I"M((di , . . . ,dk))(7r 0 S)
def interpretation, def 7r on tuples, InductionHypothesis, def interpretation .
5. For d, s « S, d of type (a, (3) and b of type (3, show d? bE S. 7r(IM(d~
b)(s))
= 7r(IM(d)(s)(IM(b)(s))
= 7r(IM(d)(s)(7r(IM(b)(s))) =
I"M(d)(7rs)(I"M(b)(7r 0 s))
= I"M(d~ b)(7r 0 s)
def interpretation, def 7r on functionspaces, InductionHypothesis, definterpretation .
6. For d, d' E S, d, d' of booleantype -r, show (d & d') E Sand
..,dE S.
7r(IM(d & d')(s)) = 7r(IM(d)(s) t\ IM (d')(s)) def interpretation, = 7r(IM(d)(s)) t\ 7r(IM(d')(s») 7r a booleanautomorphism, = I"M(d)(7r 0 s)) t\ I"M(d')(7r 0 s)) InductionHypothesis, = I"M(d & d')(7r 0 s) def interpretation. The proofthatS is closedundernegationfollowsthesame pattern. 7. For x E Varr and ¢ of booleantype a , assume ¢ E S. Show
3x¢ E S.
7r(IM(3x¢)(s)) = 7r(V{I M(¢)(s[xja] I a E Dnr}) = V{7r(I M(¢)(s[xja] I a E Dnr)}
def interpretation, 7r t Dnr a boolean automorphism, InductionHypothesis, = V{I"M(¢)(7r(s[xja])) I a E Dnr} = V{I"M(¢)(7r 0 s)[x/7ra])) I a E Dnr} Lemma 18, = V{I"M(¢)(7r 0 s)[xja])) I a E Dnr} Lemma 18, def interpretation. = I"M(3x¢)(7r 0 s) --I
23. :::} The £=-definable n ~ 1-ary relations over E are just the {t E E k 1M F ¢[t]}, any M with EM = E, v = (Vi,'" ,Vk). We show that each R,p ,v is AI, ¢ free for {Vi 11 ~ i ~ k} . Set S = {¢ E c; I Rq"v is AI, all V such that ¢ is free for v} .
THEOREM Rq"v =df
LOGICAL OBJECTS
177
Proof a. all atomic formulas ¢ arein S. case 1: ¢ is (x = y) for x, y distinct.Let v be a tupleof distinct t hatfor some i ,j Vi = x, Vj = y . R ,v = {s E E k 1M 1= variables such (x = y)[s]} = {s E Ek l s, = sj ], which is AI. case 2: ¢ is (x = x) . Let v be a tuple ofdistinct variableswith k k V j = x . Then Rc/>,v = {s E E I M 1= (x = xHSj]} = E , againAI. b. S is closedunderconjunction a ndnegation 1. Let ¢, '1/1 E S. Let (¢ & '1/1) be free inv, so ¢, '1/1 are also free in v. Then R&1/J ,v =
{s E E k 1M
1= (¢ & 1/J)[sj}
=
Ek 1M 1= (¢Hs]} n {s E E k I M 1= ('I/1H sj}, = R,v n R,p,v, which is AI sinceintersection preserves AIness. 2. Let ¢ E S. Let --, ¢, and hence also¢, be free inv.
{s
E
R~ ,v = {s E E k I M ~ ¢[sj} = -R,v is again AI sincecomplement preserves AIness .
c. S is closedunderexistential quantification. Let ¢ E S, let3z¢ be free inv. We may assume z is free in¢, otherwise k R3z,v = R,v, which is AI. ThenR3z,v = {s E E 1M 1= 3z¢[sj} = k {s E E 13bM 1= ¢[vi/si,z/bj}. If R3z ,v = 0 , thenit is AI. Setv' = (VI, .. . , Vk, z). Let U E R3z c/>,v, so 3b, M 1= ¢[s/b] and (u, b) E R ,v Let 1f E PERM(E). So (1fu,1fb) E R,v ' by the IR, so 1fU E R3 z,v, whenceR3z,v is AI. l •
{:: Let R
~
En be AI. ShowthatR is definable inL=.
= 0, thenR = R ,v where¢ is (x =I- x) and V = (x) . Let R =I- 0. Let S E R . RecallR, = {t E En I ti = tj iff s, = Sj,
i. If R ii.
1
s i ,j $
all
n} .
Then fixing the sequence v = (Xl,. . . , x n ) of distinctvariables,R , is defined bytheconjunction ofAND{ (Xi = Xj) I s, = Sj} withAND{ (Xi =IXj) I s, =I- Sj} . Since R = U SER R s' R is defined bythedisjunction of the finitely many sentencesused to define each Rs . -j ACKNOWLEDGEMENTS For constructive discussion of specific issues raised inisthpaper I would like tothankJohn Corcoran,David Kaplan, Jim Lambek, MichaelMakkai, TonyMartin , Uwe Moennich,YiannisMoschovakis, EdStabler,and Gonzalo
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Reyes. Specificthanksto: MichaelBarrforsubstantive help with issues concerningdefinability in languages withoutequality;to Gila Sher for constructivediscussionof theearlytheorems,most notablya significantsimplication drawingmy of my originalproofof Theorem4; and to Michael Zeleny for attention to Putnam's work andremindingme of Lakoff's discussion . REFERENCE S
Barwise, J ., and R . Cooper 1981 Generalized quantifiers andnatural language , Linguistics and Philosophy, vol. 4, pp . 159-219 .
Beghelli, .F 1992 Comparativequantifiers,Proceedings of the VIII Amsterdam colloquium on logic, language, and information (P. Dekker et al., editors),ITLC, Amsterdam. Church , A. 1940 A formulat ion of thesimple theoryof types, The Journal of Symbolic Logic, vol. 5, pp . 56-68. Enderton , H. B. 1972 A mathematical introduction to logic, Academic Press, New York. Etchemendy,J . 1990 The concept of logical consequence, HarvardUniversityPress, Cambridge, Massachusetts. Giirdenfor s, P. 1987 Generalized quantifiers . Linguistic and logical approaches, Reidel, Dordrecht . Keenan, E. L. 1980 Passive is phrasal(notsentential or lexical) , Lexical grammar (T. Hoekstra , H. van derHulst,and M. Moortgat , editors),Foris. 1987 Unreduciblen-ary quantifier s in naturallanguage,in Gardenfors 1987. 1991 Anaphorainvariantsa nd languageuniversals , Proceedings of the West Coast Conference of Formal Linguistics (D. Bates,editor),vol. X,StanfordLinguisticsAssociation,S tanfordUniversity. 1992 BeyondtheFregeboundary , Linguistics and Philosophy, vol. 15, pp . 199221. Keenan, E. L., and L. Faltz 1985 Boolean semantics for natural language, Reidel, Dordrecht . Keenan, E. L., and L. Moss 1985 Generalized q uantifiers and the expressivepower ofnatural language , in van Benthem and ter Meulen 1985.
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Keenan,E. L., and E. P. Stabler 1991 Languageinvariants, Proceedings of the eighth Amsterdam colloquium (P. , editors),I nstitute for Logic,Language , and ComDekkerandM. Stokhof putation , Universityof Amsterdam,Rodopi, Atlanta , pp, 309--329. 1995 Thereis more thanone language,Proceedings of langues et grammaire I (G. Tsoulasand L. Nash, editors). Keenan,E. L., and J . Stavi 1986 A semanticcharacterization of natural language determiners,Linguistics and Philosophy, vol.9, pp . 253-326. Keenan,E. L., and D. Westerstahl 1994 Generalized quantifiers in linguisticsa ndlogic, The handbook of logic and , editors),Elsevier, ch . X, linguistics (J . vanBenthemand A. ter Meulen pp. 837-893. Klein, F. A. 1872 Comparativereview ofrecentresearchesin geometry(M. W. Haskell, translator), Bulletin of the New York Mathematical Society, vol.II, 189293, pp. 215-249. Lakoff, G. 1987 Women, fire and dangerous things, Universityof Chicago Press, Ch icago. Liiuchli, H. 1970 An abstract notionof realizability for whichintuitionistic predicatecalculus iscomplete, Intuitionism and proof theory, North-Holland, Amsterdam. Lewis, D. 1984 Putnam'sparadox,Australas ian Journal of Philosophy, vol.62, no. 3. Marshall, M. V. and R. Chuaqui 1991 Sentencesof type theory : The onlysentencespreservedunderisomorphisms, The Journal of Symbolic Logic, vol.56, no. 3, pp, 932-948. Montague,R. 1973 The propertreatment of quantification in ordinaryEnglish, Approaches to natural language (J. Hintikka et al., editors), Reidel; reprintedin Formal philosophy (R. H. Thomason, editor),YaleUniversityPress, New Haven. Mostowski, A. 1957 A generalization of quantifiers , Fundamenta Mathematicae, vol. 44, pp . 12-36. Plotkin,G . 1980 Lambda-definability in thefulltype hierarchy,To: H. B. Curry : Essays on combinatory logic, lambda calculus and formalism, Academic Press, New York. Putnam, H. 1981 Reason, truth and history, CambridgeUniversityPress, Cambridge, England. Sher, G. 1991 The bounds of logic, MIT Press, Cambridge,Massachusett .
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Silva, J. S. 1945 On automorphismsof arbitrary m athematical systems (A. J . FrancoDe Oliveira , translator), History and Philosophy of Logic, vol. 6, 1985, pp . 91-116. Tarski, A. 1986 Whatare logicalnotions?(J. Corcoran,e ditor), from a 1966 lecture, History and Philosophy of Logic, vol. 7, pp. 143-154. Van Benthem,J. 1983 Determinersand logic , Linguistics and Philosophy, vol. 6.4, pp . 447-478. 1989a Logicalconstantsacross varyingtypes, Notre Dame Journal of Formal Logic, vol. 30, no. 3, pp . 315-342. 1989b Polyadicquantifiers,Linguistics and Philosophy, vol. 12, pp. 437-464. 1991 Language in action, North-Holland, Amsterdam. Van Benthem, J. , and A. terMeulen 1985 (editors), Generalized quantifiers, Foris, Dordrecht . Van der Does,J., and J. van Eijck 1996 Quantifiers, logic, and language, CSLI LectureNotes54, CSLI Publications,Stanford . Van Fraassen, B. 1989 Laws and symmetry, OxfordUniversityPress, Oxford; reprintedin 1990. Westerstahl, D. 1985 Logicalconstants in quantifier languages , Linguistics and Philosophy, vol. 8.4, pp . 387-413. Weyl,H. 1949 Philosophy of mathematics and natural science, Princeton University Press, Princeton, New Jersey; reprintedby Athenaum, 1963.
SAUNDERS MAC LANE
THE LAMBDA CALCULUS AND ADJOINT FUNCTORS
Abstract . The wellknownlambdaca lculus was first formulated by Alonzo Church(1932) . It was originally i ntendedas a newfoundationof mathematics ; but soon theremarkabl e connectionbetweenlambdadefinabilitya nd recursive o ped. Onlymuch laterwas the connection withrewrit e sysfunctionswas devel tems noted, whiletheremarkabl e influenc eof theca lculus in c omputerlanguag es er. This note is to pointoutthattheGaloisconnectionsand was alsonotedlat thislambdacalculusa re p erhapsthe firstappearan ces of an explicitpair of adjointfunctors.T hese functorsin generalwere notfound untilthework ofDaniel Kan in 1958.1 1. LAMBDA CALCULUS AND ADJOINTS
In thetyped lambdacalculusconsiderfunctionsf(x, y) where x and yare variablesof types X and Y, andwithvalues of type W . Then Axf(x,y) defunctionsof x given by(g(y))(x) = notesthefunction9 of y whose values are f(x, y). This defineslambda.Or, considera category withobjectsX, Y, and W withproductobjectsX x Y. Thentheexponential set of all functionson X to W is written W x. It is determinedby the fact t hatthereis a "natural" bijection(one-to-onecorrespondence) betweenfunctions
f : (X
x Y)
andfunctions g: Y
---t
---t
W
WX•
One saysthatthe operation"productwith X" is leftadjointto theopera"functors" This . tion, "raise tothepower ofX "; such operationsare called bijectionis oftenexpressedby thenatural isomorphism Hom(X x Y, W) = Hom(Y, W X ) ,
with theleftadjointon theleft, and where Hom(Y, W) is the conventional from Y notation for the set of all morphisms, i.e., all functions thecategory in to W . A categorywithproductsa ndsuch exponentials is called aCartesian closed category. IThis and otherrefere nces may be foundin Mac Lane and Moerdijk 1993. Also many fine furth er commentson the lambdacalculusare presentedin an articl e by Henk Barendregt,"T he impact of the lambdaca lculus in logic and computer science" , Bulletin of Symbolic Logic, vol. 3, no. 2 (1997) , pp. 155-215. 181
C. Anthony Anderson and M. Zeleny (eds.), Logic, Meaning and Computation, 181-184 . © 2001 Kluwer Academ ic Publishers . Printed in the Netherlands.
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SAUNDERS MAC LANE
thetwonotionsoflambdaandexponential adjunctions In thispresentation are almostidentical.This can bestatedmore precisely , as in thebook by Lambek and Scott,as a suitableequivalencebetweenthecategoryof typed lambdacalculii and t hatof Cartesianclosedcategorieswith anatural numthisconnectionto beclearly formulated . bers object. It tooka long time for The notionof anadjointcovers many more cases thanthatofproductand exponent.Given categoriesB and A , a functorF : B -+ A is said to be left adjointto afunctorG : A -+ B whenthereis a bijection Hom(Fb, a)
= Hom(b, Ga)
whichis "natural " for all b in B anda in A. Forexample,letB be thecategory of sets and A thatof groups, whileG sends each group to itsunderlying groupgeneratedby thatset. The set and F turnseach set into the free bijectionthenexpressesthefactthata given map ofthegenerators b into this underlying set ofa uniquelydeterminesa group homomorphism from the freeg roupto thegroupa.2 2. THE CATEGORY OF SETS
In his initialwork onthelambdacalculus,Churchconsideredthiscalculus as a formulation of basicpropertiesoffunctions,so his considerations appliedto thecategory ofsets. The presencethereofproducts whatwould now be called and exponentialsis an essentialpartof the Lawvereproposalto use the of mathematics-a elementary t heoryof thecategoryofsetsas a foundation beautifulidea which isneglectedby many peopleenmeshed in therococo the astounding propertiesof thehigherset theory. However the1930's, in connections oflambdasto recursive functionsandcomputability, as developed by Churchandhis studentsKleene, Rosser, and Turing,wereproperlyin the centerof interest. A functorial point of view became possible onlyafter the developmentof algebraictopologyand itsrepresentation of geometrical maps by algebraic ones, thatis, withfunctors-just as the real understanding of forcingdepends on thegeometricalnotionof sheaf(see Mac Lane and Moerdijk). The discovery ofadjointfunctorsalso came instages: in theadjunction bijection above, seta = Fb to get the Hom(Fb, Fb)
= Hom(b, GFb).
Then theidentitymap on theleft (every c ategoryhas identities,in keeping withancientwisdom) must correspondto somemap n : b -+ GFbj in the case to itself , ofgroupsthis istheevidentmap which sends eachgroupgenerator 20ne might recallthephilosophicalorigins of theterminology : "Category " was purloined fromAristotle or Kant, at your choice , while "functo r " was takenfrom a book y b RudolfCarnap.
THE LAMBDA CALCULUS AND ADJOINT FUNCTORS
183
t hentakestheform thateach as seen in theunderlying set. The adjunction map 9 of setsdeterminesa uniquemap 1 of groupsso thatthe triangular diagramd below"commutes". T/
b ····· .. CFb
~
II
Ca
Fb
11
t
a
One commonly saysthatthis is the universal wayembedding of the setb in a group a. If follows from this thatthe freegroup is determinedonly up to anisomorphism. This reveals a basic a spectof mathematics:its well knownobjectsare not unique, butonly so up toisomorphism. Despite von Neuman, he did not have the ordinalnumbers,butonly one of many possible isomorphicmodels-andthe back of myhandtothefamiliar definition the of orderedpair. 3. QUANTIFIERS AS AGENTS
Thereareotherstrikingconnections with logic.Theconnective"And" isjust a product,so is properlydefined by anadjunction,as is "Or" and "Not". Moresurprisingis thefactthatquantifiers are really adjoints.For consider a propositionalfunctionH(x, y) as if it were asubsetof theproductset X x Y, so that(Vy)H(x, y) and (3y)H(x, y) aresubsetsof X. Writep for theprojection p:X xY->X. For anysubsetSeX x Y, writep" S for its "inverse image" in X x Y; that is, fortheset of all pairs (x, y) withxES. Thenfor any set T eX, one has theinclusions p*T c H(x, y) iffT c (Vy)X(x, y),
H(x, y) C p*T iff(3y)H(x, y) cT. Now read inclusion as an arrow ; these equivalences thenstatethattheexistential quantifier is leftadjointand theuniversal one rightadjointto inverse image. This strikingidea, due to Lawvere, finally replaces those previously clumsy attemptsto make quantifiersalgebraic,as in cylindricalgebraor polyadic logic. These examplesgo to exemplifythemain thesis: Logic is abranchof mathematics,a ndnot vice versa." 30ftenit is notnotedthattherecognition thatlogiccouldbe practicedin departments of mathematicswas possibleonlybecauseof Hilbert,GOdel, and in thiscountryAlonzo Church. BeforeChurch, therewas nohope. In 1928, as anundergraduate atYale, Iasked my pro-
184
SAUNDERS MAC LANE REFERENCES
Church, A. 1936 Mathematical logic, Departmentof Mathematics , PrincetonUniversity , mimeographednotes(113 pages); reviewedin The Journal of Symbolic Logic, vol. 2, p . 39. 1941 The calculi of lambda conversion, PrincetonUniversityPress, Princeton, New Jersey. Lambek, J. , and P. J. Scott 1986 Introduction to higher order categorical logic, CambridgeUniversityP ress, Cambridge, England. Lawvere, F. W. 1970 Equalityin Hyperdoctrinesand comprehensionscheme as an adjoint functor , Applications of categorical algebm, AmericanMathematical Society,Providence . Mac Lane, S. 1938 Review of "Logicalsyntaxof language"by RudolfCarnap, Bulletin of the American Mathematical Society, vol. 44, pp . 171-176. Mac Lane, S.,and 1. Moerdijk 1993 Sheaves in geometry and logic, A first introduction to topos theory, Springer- Verlag , New York.
fessor of mathematics to direct my readingof Principia Math em a ti ca .He responded, "Oh no, readHausdorffinstead." In 1930 , as a gradua te studentof mathem atics at Chicago, I found no one to direct my intended thesis in logic , In 1933 , when I presented my (G ot ting en ) thesisin logic to the annua l m eetingof the Am er ican Ma thematical Society, the leadingmathematicianat Yal e , Oystein Ore, followed my 10minute talkon logicwitha e did he know the futur e. ten minutedenunciation: Algebra, not logic! Littl American helpwas on th e way. OswaldVeblen, of PrincetonUniversitymathematic s, in his retiringaddress as Presidentof the Am ericanMathematic al Society, said: "T he conclus ion seems inescapable thatformallogic has to be tak en over by themathematic ians. The factis thattheredoes notexist an adeq ua te log ic at thepresenttim e, and unless the mathematician s create one , noone else si likely to do so." Unlikemany words in a retiringaddress , this statem ent led to result s. Vebl en him self had (at elast)two PhD. students in logi c: A. A. Bennett , who was su bse que nt ly influ ential in forming the Asso cia ti onfor Symbo lic Logic, and C hu rch, in 1927. The striking and p enetrat ing res u lts ofC hur chhimself and his earl y Am erican st udents Kleen e and Rosser together made logic an appropriate subject for mathematical dep artments.
GERALD J. MASSEY
ATOMIC BOOLEAN ALGEBRAS AND CLASSICAL PROPOSITIONAL LOGIC
1. BOOLEAN ALGEBRA AND CLASSICAL PROPOSITIONAL LOGIC
From thetime of Boole and De Morgan, algebraicm ethodshave been used thecustomaryvehicle for the application to studylogics. In this century, of algebrato logic has been t heso-callednatural or Lindenbaum algebra of thelogicalsystem being investigated . 1 In thecase of a formal axiomatizaPL), the tion (logistic system) S of classicalpropositionallogic(hereafter LindenbaumalgebraA happens to be aBoolean Algebra. (The elements of the LindenbaumalgebraA are theequivalence classes IAI of the wffs of S with respectto therelationA +-+ B , where A +-+ B if and only if thereis an S-proofof B from A and ofA from B , and thecomplement,meet, and join operationsof A are definedthus: -IAI = I rv AI, IAI /\ IBI = IA & BI , and IAI VIBI = IA V BI, wherethetilde,ampersand,and wedge areeither primitiveor defined connectives S.) of As an illustration of theuse and power ofalgebraicmethods, consider thederivationof thecompletenesst heoremfor a (formal) axiomatization of PL from Stone's Representation Theorem, whichstatesthatevery Boolean algebrais isomorphicto the set algebraof a field of sets. (In a set algebra,the theunion of the elements algebraic operationsare setcomplement[relative to of the algebra], set intersection, and set union .) This derivation employs two additional and easily proved results, namely : 1.1. If tI> is the set algebra of a field of sets and x is an element of tI> other than the unit, then there is a 2-valued homomorphism h of tI> such that h(x) = O. (A 2-valued homomorphism of a Boolean algebra is a structurepreserving mapping of the algebra into the 2-element Boolean algebra.)
LEMMA
1.2. Every 2-valued homomorphism h of the Lindenbaum algebra of an axiomatization S of PL determines a classical valuation V on the wffs of S, where V(A) = t or f according as h(IAI) = 1 or O.
LEMMA
(By a classical valuation on the wffs of S is meanta mapping from the wffs of S ontothetruthvalues (t, f)thatconforms to the classical truthtable s for theconnectivesof S.) 1 For the algebr a ic terminologyused in this paper and for a lu cid presentation of the applicationof algebratologic, see Rasiowa and Sikorski 1970.
185 C. Anthony Anderson and M. Zeleny (eds.), Logic, Meaning and Computation, 185-189 . © 2001 Kluwer Academ ic Publishers. Printed in the Netherlands.
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GERALD J. MASSEY
The derivation ofthecompletenesstheoremfor anaxiomatization S ofPL from Stone'sRepresentation Theoremgoes thus: Proof Let A be any non-theoremof S. Then, by theconstruction of the LindenbaumalgebraforS, IAI =I- 1. So, by Stone'sRepresentation Theorem together with Lemma 1.1,thereis a 2-valuedhomomorphism h of the Lindenbaum algebrasuchthath(lAI) = O. Hence by Lemma 1.2, thereis a classical valuation V such thatV(A) = f. So, A is not atautology.Therefore,by contra position,if A is a tautology , thenA is a theoremof S. 2 -j 2. THE ALLEGED IRRELEVANCE OF ATOMIC BOOLEAN ALGEBRAS
Stone'sRepresentation Theoremis difficult to prove , a claimsupportedby the factthatit was not proved until 1936 . In brief, whatmakes Stone's Representation Theoremhardto prove is thenon-existencein an arbitrary Booleanalgebraof a classX of elementst hatobey thefollowing threelaws (wherex ~ y, which isreadas x is included in y, is defined asx /\ y = x): (2.1) If x E X, then(y)(x /\ y
= 0 or x /\ y = x) .
(2.2) (y)[if y =I- 0, then(3x)(x E X and x (2.3) If x E X, then(y)(z)[x
~
(y
Vz)
~
iffx
y)]. ~
y or x ~ z].
If a Booleanalgebra containsa set X of elements thatsatisfythesethreelaws, one canreadilyconstruct an isomorphic setalgebrathus: For any element z of the givenalgebra,let f(z) = {x : x E X and x ~ z}. Then, let the elements of the set algebrabe the f-images of the elementsof thegiven algebra . Finally, let f map thecomplement,meet, andjoinoperationsof the givenalgebrawithset complementation (relative to thef-image of theunit and union respectively . It is easy toestablish of thealgebra),intersection, thatf is an isomorphism from the given algebraontotheset algebra. Certainsimple butnevertheless importantBooleanalgebras -theatomic thatsatisfythelaws (2.1), Boolean algebras-boast a class X of elements (2.2), (2.3), namely, theset ofatoms of thealgebra . (An atom a is an elementthatstrictly includes only the zero element , i.e., if y < a, theny = O. A Booleanalgebrais atomic if and only if every non-zero element includes an atom.") It is easy to show , by theline of reasoning sketchedin the preceding paragraph,thateveryatomicBooleanalgebrais isomorphicto asub-algebra of the powersetalgebraof its atoms(Representation Theorem for Atomic Boolean Algebras) . One simply establishest hatthefunctionthatmaps each element oft hegiven Booleanalgebratheset oftheelement'satoms (the 2This derivationof thecompletenesstheorem from Stone's RepresentationTheorem follows closely thatof Belland Siomson (1971) . t heor ems for bothatomicand finite 3See Mendelson 1970 forproofsof representation BooleanAlgebras .
ATOMIC BOOLEAN ALGEBRAS
187
atoms included inthe element)is an isomorphism from thegiven atomic algebrato asub-algebraof thepowersetalgebraof the set of atoms of the givenalgebra . (Such asub-algebra is, of course, itself a fieldsets.) of If theLindenbaumalgebraof anaxiomatization of PL wereatomic, one could derive the completenesst heoremforit from theRepresentation Theorem for AtomicBooleanAlgebras -in conjunctionwith Lemmas 1.1and 1.2-by the samereasoningused to derive it from Stone'sRepresentation Theoremin section1 above. Butnone oftheLindenbaumalgebrasof axiomatizations of PL are atomic; rather , thesealgebrasare altogether atomless. This elementary observation has causedmathematicians and logicians to con cludethattheRepresentation Theoremfor Atomic Boolean Algebras is not germaneto theinvestigation of classicalpropositionallogic. Theprincipal burdenof thispaperis to debunkthis widely -sharedbelief. 3. CLASSICAL PROPOSITIONAL LOGIC AS AN ENSEMBLE OF FRAGMENTS
An axiomatization S of PL will be said to be regular if andonly if , for each varitheoremT of S, thereis a proofof T in S thatcontainsno propositional ablesotherthanthosethatoccur inT. (This terminologyis motivatedby the conviction thatthereis somethingintuitively irregular aboutaxiomatizations ofPL thatare not regularthesensejustdefined.) in It is importantto notethataxiomatizations of PL thatemploy axiom schemes andstructural inference rules (rules given by means of rule schemes , e.g., modus ponens) are regular. To see thatthis is so,notethatsimultaneoussubstitution of variablesnot in thepropositional variablesof a theoremT forpropositional T preservesthepropertyof being a proof in any axiomatization of PL that uses axiom schemesandstructural inference rules . Hence, we do not forfeit any generality when intheremainderof thispaperwe confine our a ttention to regular axiomatizations of classical propositional logic. By a finite -variable fragment of aregular axiomatization S of PL is meant the axiomsystem thatresultswhen onerestricts theprimitivebasis of S to a finitesubsetof itspropositional variables . Noticethatnothinggoes on, so its fragments, to speak, in S unless it also goes on in some offinite-variable and conversely. In particular , a finite sequence of wffs qualifies proof as a of a wffA in S if and only if it also qualifies as proofof a A in any or all of thefinite-variable fragmentsof S, thevocabularies of whichcontainevery propositional variablet hatoccursanywherein the given sequence of wffs S.of Hence, A is a theoremof S if andonly ifA is a theoremofthe finite-variable fragmentdeterminedby thepropositional variables of A. Note thata finite-variable fragmentF of S containsonly finitely many equivalence classes IAI of wffs. The Lindenbaumalgebraof F, therefore,will be a finite Boolean algebra.All finite Boolean algebrasareatomic,so the Lindenbaumalgebraof F will be anatomicBooleanalgebra,(One can prove a Representation TheoremforFiniteBooleanalgebrast hatis slightly s tronger
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GERALD J. MASSEY
thantheone foratomicBooleanalgebras,to wit: every finite Booleanalgebra is isomorphic to thepowersetalgebraof theset of itsatoms.) The derivaof PL from tionin section1 ofthecompletenesst heoremforaxiomatizations Stone'sRepresentation T heorem made use ofthelemmas (1.1) and (1.2). Lemma 1.2 needs to beamendedslightly before it can be appliedto the Linfragmentsof aregular axiomatization denbaumalgebrasofthefinite-variable S of PL because the 2-valued homomorphisms of thesealgebrasd etermine classicalvaluations only on the wffs theirrespectivefinite-variable of fragments, and noton allthe wffs ofS. Here, then,is the requiredamended version ofLemma 1.2: 1.2*. Every 2-valued homomorphism h of the Lindenbaum algebra of a finite-variable fragment F of a regular axiomatization of PL determines a classical valuation V on the wfJs of F, where V(A) = t or f according as h(IAI) = 1 or O.
LEMMA
Bya tautology of aregular axiomatization S of PL is meanta wffthathas thevaluetruth in all classical valuations on the wffs of S. Note, then,thata if andonly ifA has the value truth in all classical wffA of S is a tautology valuations on the wffs of any finite-variable fragmentof S thatcontainsA. Therefore,by making use of Lemmas 1.1 and 1.2*, for each finite-variable completenesst heoremfrom the fragmentF of S we can derive the following Representation Theoremfor AtomicBooleanAlgebras: If a wfJ A of F is a tautology, then A is a theorem of F. The completenesstheoremfor S itself follows as caorollary from thesecompletenesstheoremsfor itsfinite-variable fragments. S of PL to the Why should one b otherto relatea regular a xiomatization ensemble of itsfinite-variable fragmentsrathert handeal withS as a single unifiedsystem? Thereis a significantalgebraicadvantagein so doing, for the Lindenbaum algebras of the finite-variable fragments are atomic Boolean algebras; whereas, theLindenbaumalgebraof S is atomless. (It is theinfinity of thepropositional variablesof S thatmakes its Lindenbaumalgebranont heRepresentation atomic,indeedatomless.)Consequently, by invoking only Theoremfor AtomicBooleanAlgebras, we can derivecompleteness a t heorem for eachfinite-variable fragmentof S from Lemmas 1.1 and 1.2*. Then, S totheensemble of itsfinite-variable fragments,weobtain simply byrelating of theRepresentation Theorem thecompletenesst heoremforS as a corollary for AtomicBooleanAlgebras. The methodused inthis paper to bringthe Representation Theoremfor l ogic-namely, Atomic BooleanAlgebrasto bearupon classical propositional to relateor reduce aregular a xiomatization of the logic to theensemble of -canbe used tostudymany otherpropositional its finite-variable fragments logics. Forexample,one caninvestigate t hemodal propositional logic S5 by relating or reducingsome regular a xiomatization of it totheensembleof its finite-variable fragments . Similarly, one can studymodalproposition allogic
ATOMIC BOOLEAN ALGEBRAS
189
84, or intuitionistic propositionallogic, byrelating r egulara xiomatizations fragments . of theselogics totheensemblesof theirrespectivefinite-variable S of propositional logic will be said to be infinitistic if An axiomatization wffs and only ifS containsan infiniteset A of pairwisenon-interderivable (i.e., no two wffs inA can be proved from each other).S will be said to be intrinsically infinitistic if and only if itcontainsan infinite set of pairwise non-interderivable wffsA suchthatonly finitely many propositional variables occurin themembers of A. Finally, an infinitistica xiomatization of propositionallogic will besaid to be extrinsically infinitistic if andonly if it is not intrinsically infinitistic.For example, axiomatizations of classical propositionallogic and of S5m odalpropositional logic areextrinsically infinitistic; onlythepresenceof infinitelymany propositionalvariablesin theirrespectivevocabularies makes thesesystemsinfinitistic. On t heotherhand,regular axiomatizations of 84 modalpropositional logic areintrinsically infinitistic ; are(intrinsically) even thefinite-variable fragmentsof theseaxiomatizations infinitistic . The advantage of investigating an extrinsically infinitisticregular axiomatization of some propositional logic byrelating or reducingit totheensemtheparentsystem, ble of itsfinite-variable fragmentsis obvious; for, unlike none of itsfinite-variable fragmentsis infinitistic . Butthe methodof relating orreducinga regular a xiomaticpropositional s ystem to theensemble of its finite-variable fragmentsis advantageous even whenapplied to intrinsicallyinfinitisticpropositional systems such asregular a xiomatizations of 84 modal propositionallogic. Eventhoughthe finite-variable fragmentsof a regular a xiomatization of 84 will themselves intrinsically be infinitistic,they thantheparentsystem itself. arenonetheless more tractable REFERENCES
Bell,J ., and A. Slomson 1971 Models and uliraproducts: An introduction, North-Holland, Amsterdam, pp.42 -43. Mendelson , E. 1970 Boolean algebms and switching circuits, Schaum's OutlineSeries, MeGraw-Hill, New York , pp. 135-137. Rasiowa, H., and R. Sikorski 1970 The mathematics of metamathematics, thirdedition,Polish Scientific Publishers,Warsaw.
ROBERT K. MEYER
IMPROVED DECISION PROCEDURES FOR PURE RELEVANT LOGIC Dedicated to Professor Alonzo Church, in his 91st year, and to Professor Saul Kripke , who provided the impetus for these results .
Abstract.We study Church's "weak implicat io nalcalculu s" of 1951 , which is thepure implica ti on alpartR _ of the relevantlogic R . We investigate and ow ing Church, the effect ofaddingprop ositionalquantifiersto develop, againfoll R _ ; thisallow s also the specificationof a minimal, a De Morganand finallya . The bulkof the paper dealswith finite model properties for Boolean negation va rio usfragments of R , includingR_ . A modificationof an argumentdue to Saul Kripke is thechief to olin this project, yielding anInfinit e D ivision Prinicple (IDP ) ates ourfinitizat ion. Thus system s inte rm edi a te for "C hurch monoids" thatfacilit ive R) have the finitemodel propert y and betweenR_ and LR (non-distribut are hen ce m od el-theoreticall ydecid abl e. A concludingsecti ons hows how to define various logicalparticles using C hurch's propositionalquantifi ers , which turn s what are conservat ive exte nsion esult r s atthequantifi er-fr eelevel into axiomat ic extensions when quantifiers are present.
1. INTRODUCTION TO RELEVANT IMPLICATION
Among themany contributions of AlonzoChurchto Logic, his inventionof relevant implicationis by no means the least.' His principalpaper on the subjectis Church 1951a; so far as I know, he has never returned to it. As Churchdeals with relevantimplication,it has two levels : a fundamental one, justbased on --+ and propositionalvariables ; and an advancedone, in which propositional quantifiers(based on V) are alsointroduced.Following Church'snotational conventions.?we axiomatizeR_ , choosingthe following 1 In fact, Church(195I a) ca lledit th e "weak implication al ca lculus ", while in Church 1951 he againsp eaks of "weak" implication. Also he used ~, which I reser vehere for a material implication . C hurch' s inventionof thispure --+ sys te m (afterwards calledR_ by Anderson and Belnap,and BCIW by combinatory logi c fans) wasmildlyant icipated by Moh Shaw-K wei(1950) , and more significant ly a nticipatedin Orlov 1928; also see Doser: 1999. Adaptinga terminological ugges s t ion ofB acon, R_was exte nde d tothe system R of relevantimplica ti on. See Anderson and B elnap 1975. 2T hat is, we associa te equa l o c nnectives to the left, usin g dots (s pa ringly)as p arentheses to reverse the directionof associ at ion. W e exte ndChurch 's conventions by ranking binar y connectives, includin g additionalprimitives or defin ed ones , in orderof increasing scope: &,0, v , +, $ , ~ , --+, +-+ . Of these , 0, $ , and +-+ arethe "re leva nt"a nalogu es of 'and' , 'o r', and 'co-imp lies' intro duced in Church 1951a ; + (fission) is in Anderson and Belnap 1975 and is a DeMorgandual of 0 (fusion), whil e &, v, and ~ are "t rut h-fu nct ional" particles for R inthe sense of Anderson and Belnap 1975 .
19[ C. Anth ony Anderson and M. Zeleny (eds.), Logic, Meaning and Computation, 19[-217 .
© 2001 Kluwer Academic Publishers . Printed in the Netherlands.
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axiom schemes and rule:3 Axl. AxB . AxC . AxW.
A -> A B ->C ->. A ->B->. A-> C (A-> . B -> C )-> . B ->. A ->C (A ->. A -> B ) ->. A -> B
Rule ->E. A
->
B=> . A => B
Churchviewed R_ as theresultof weaken ingthe int uit ionistmplica i tiona l calculus J _ by tradingin the (paradoxical) ax iom AxK. A->. B ->A forAxl, Thus R_ bearsthesame sort of elation r to J _ as the .xlcalculi of Church (1941) bears to thepreferred.xK calculus of Curry and Feys (1958).4 2. P ROPOS IT IO NAL QUANTIFIERS
We ext endR_ to R':1> with the following definitionsa nd additionalaxiom schemes and rul e:" Do. A 0 B =df Vq«A -+ . B -+ q) -> q), q does notoccurin A or B D....... A ...... B =d f (A -+ B ) 0 (B -+ A ) Dffi. A ffi B = df (B -+ A -+ . A -+ B -+ B) D3 . 3pA(p) =d f Vq(Vp(A (p) -+ q) -> q), q does not occurin A(p) DF . F = d f Vpp DT. T = d f F -+F Dt. t = d f Vp(p -+ p) AxVE. VpA(p) -+ A(B) Ax-+V. Vp(A -+ B (p)) -> Rule VI. A => VpA
.
A -+ VpB(p) , p not fre e in A
The definitionsabove are larg elytaken from Church, who has h imself adaptedthemfrom Russell.Among them, Church 'sanalogu e0 ofconjunction 3C hurch(1951a) stated part iculara xioms, building subs t it ut ionnti ohis not ion of deducibility . We disp ense withan explicit subs t it ut ion rul e by stati ngaxioms schematic a lly, following on v Neumann. Rules are formula ted with =} as a m et alogicalf'.'i Thus, read -+ E below as I"f A -+ B is a theore m then if A is a theo re m then B is a theorem ." 40r, as Curryonce p utit, "C hurch neverdid mu ch like the comb i natorK ." Inciden tall y, C hu rch(1951a ) notes thatothers have previo usly sugges ted doing witho utAxK. 5 R~ is what Church (1951a) calls th e "weak theo ryof imp lication", adapti ng Russell' s use of the "theory of impli ca t ion" tocharacte rize the class ical theory based on material l pro posit ion al ua q ntificatio n (near enough). In stating quant ifiim plication and universa ca t iona l axiom sche mes for R~ , we adoptthe convention thatA(p) stands for a for mul a in which p may occur free; where B is any for m ula, A( B) s ha llthen denotethe result fo proper substitution of B for p in A(p) , rewr iting bound variables as necessary to avoid confusion of bound variables.
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has played acentral role insubsequentdevelopments , largely onccount a of Dunn's algebraicworkcontributed to Anderson and Belnap 1975. Similarly, Church'sco-implication~ comes to the samethingin R as the~ defined in Anderson and Belnap 1975. ButChurch'sdisjunction,for which I useEB, has (more or less) vanished withoutt race . Churchalso has various thoughts aboutnegation,none of which have really caughton (yet, anyway) ; we look atthese below . Whathas caughton, however, isChurch'sform ofthedeductiontheorem. Whatit amountsto, inoutline , is thatif ther e is a deductionof B from AI, ... ,An in R_ , theneitherthereis alreadya deductionof B from AI, . .. , An-I, or elsethereis a deductionof An -+ B from A I, ... ,An-I. The usualformulation of thedeductiontheoreminvolves only t hesecond clause ; butit makesessentialuse ofAxK to provide a "fictitious" useAn of in case therewere no "real" ones . Church'sdeductiontheoremforR_is more vertebrate ; onlythoseA i,j actually used (by thelights ofR _) show up in the theoremAi ,1 -+ •.. , Ai ,m -+ B thatrecords thededuction.In thelimitingcase, whereB itself is atheorem,theremay (or may not) be a way to use one or more of theAi in thedeductionof B ; still , the first clauseChurch's of version of thedeductiontheoremlets us off the hook here , since thetheoremhood (whatAndersonandBelnap, 1975, calls) its Official deof B is sufficient for ducibility . Church(1951a) shows how toextendChurch'sdeductiontheorem to all of R'j>. Among thepropertiesof ourjustdefined connectives 0 and ~, are the ~ elimination theorems(A ~ B) -+. A -+ B, (A ~ B) -+. B -+ A, and the introduction theoremA-+B-+. B-+A-+. A~B. Even nicer ist hefollowing: ResiduationTheorem. (A
-+.
B
-+
C)
~
A
0
B
-+
C.
3. NEGATION
Church(1951a) triesoutvarious definitions negation of . The most straightforward is am inimal negation,in the style of Johansson(1936) , obtainedby simply adding aconstant f tothelanguage of R_, together with D",. ",A
=df
A
-+
f.
We call this system MR!:... . Since f, in this case, is simply playing the role of apropositional variable, it lacks those propertiesof classical negation which intuitionist logic lacks (like full double negationand contraposition principles).But,to getthoseproperties,it is sufficient to formulate a system R!:... with fprimitive, D"" andthesingleadditional axiom scheme Ax""",E. """,A
-+
A.
WhileAx"''''E would induce collapse to classical logic if applied to intuitionist(or even minimal) logic, producesa it conservative extensionof R_ .
ROBERT K. MEYER
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And itpermits another definitionof therelevant analogue of conjunction in R!:... , namelythefusion 0, as follows: " Do. A 0 B
=df '"
(A -+ "'B) .
Given Do, we can even define an equivalenceconnectiveby D...... above. And we also may define Dt. t =df
",r.
As we have seen above, 0 and...... were, to all intentsa ndpurposes, already presentin Church 1951a. We have alsothepromised DcMorgandual+ of 0 in R!:... .
D+. A + B
=df
"'A -+ B .
R!:... is an interesting alternative, notconsideredby ChurchbutbuiltintoR in Anderson and Belnap 1975 as anadaptation of Ackermann 1956. Making properties doublenegationexplicitsuffices togive", its usualt ruth-functional (more or less).Thereis moreovera simple Gentzensystem, due toKripke, which yields(prettytrivially) thatR!:... is a conservative extensionof R_ . Thus, unlikeintuitionist logic, R_ livesquitecomfortably w itha standard truth-functional view ofnegation ; on theotherhand,it resist s nonsenselike A -r-e , ",A -+ B .7 4. LATTICE EXTENSIONS OF R_
We may also, followingA ndersonand Belnap(1975, and event ually Ackermann), introducelattice connectives & and v, subject to the following additional axiom schemes and rule: Ax&E. Ax-+&1. Ax-+vE. AxVI.
A & B -+ A and A & B -+ B (A -+ B) & (A -+ C ) -r-e , A -+ B & C (A -+ C) & (B -+ C) -r-e , A V B -+ C A -+ A V Band B -+ A V B
Rule&1. A and B => A & B 51n fact,ourtwo ver s ions of Do become equival e ntin the presen ce of A x~~ E above. Otherwise, it seems reasonabl e to take the previous Do (from Church 19S1a) as themore fundamental.Another app roac h si s imp ly to take fusion 0 as an additional primitive, subjectto -+ axioms cor respond ing to t he two hal ves of the residuati on theorem schem e above. On allsuchschem es , residuati on stillholds. So do the elim inatio nand introdu ction prop erties of +-+ . 7C hurc h(l9S1a) does conside r f -+ ~A as a possibl e additi on alaxiom scheme, which em s. I thinkthat he got carrie daway C hurch sued to prove vario us Glivenko-s tyle theor withtrying to make MR_ f as much like Johansson's minimal logic as possibl e. Not e that f -+ ~A does become a theorem sc he me if one defines f as VpVq(p-+ . q -+ p) , which Church (l9S1a ) alsoconsiders. But, while AxK may be bad, it hardl y seems sufficiently bad to qualify as das Falsche.
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In thesequel,theextensionof R-. obtainedby addingjust& and its governingaxioms and rules (Ax&E, Ax- &1, and Rule &1) will have a distinguishedplace. We call t his system R-.&, and we notethatit is a conservative extensionof R-..8 We may alsoconsiderit asystem MR~&, since anypropositional variableirrelevant to acontextmay playtherole of a minimal f. But thesystems splitif we add theAx",,,,E form ofdouble negation,n aturally enough. In thepresenceof doublenegation,V becomes definable intheusualDeMorganway from & and"', and its latticeaxioms Ax-VE andAxVI become provable;b utonlytheAxvI axioms areprovable inMR~&. Instead, let ustakeV as an additionalprimitive, subjectto thelattice axioms andruleabove. This producesvariantsofthesystemLR (for"lattice R").9 We thenhavesystems as follows : MLR-.&v = R-.& + v-axioms!" MLR~&v = MLR-.&v + D", (minimal negation) LR+ = MLR-.&v LR = MLR~&v + Ax",,,,,E = LR+ + D", + Ax""""E Churchin stickingto inferential definitions Notethat(hereanyway)we follow of negation . Finally, to get thefullsystem R, Andersonand Belnap(1975) add the distribution principle AxDist. A& (B V C) - A& B V A& C. ButR , being undecidable,will only conc ernus peripherally in thepresent essay. 5. THE COMBINATORICS OF RELEVANT DECIDABILITY
Kripke (1959) notes (by aGentzenargument)t hatR-. is decidable . Kripke's argumentwas adaptedto all of LR in Meyer 1966. 11 Unfortunately, the most thatone getsoutof Kripke is raw decidability;thesense thatone has obtainedthesortof control over asystem thatdecidabilityproofsoftenyield is sadlymissing. 12 8Moreover , R is a conservative extension ofR_&. 9Mechanizedin Thistlewaite, McRobbie, and Meyer 1988. lOWe may call th is system simply MLR+. 1l See Thistlewaite, McRobbie , and Meyer 1988 forthesm ooth est Gentz en-type formulationof LR. 120neoughtnot,however,readtoomu ch intothislack ofc ontrol.E ven truth-function al ststearing their logic is no be ttert hanNP-complet e,whichtendstoleavecomplexi tytheori s areluckyto haveany hairleft to ar te. In thefirstplace, Urquhart hair. Butrelevantist (1984) showedthatR (and it s close cousins, like E) are undecidable. More distressing(for R_ fans) si the factthatUrquhart(1990) placedfright ening boundson thecomplexity of any decision procedur e for R_ and relevantdecidable extensions (like LR) .
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Still, the decision questioneven forR_ remainedopen for some years. insight.P We shallexpress Kripke solved itwitha very nice combinatorial thisinsight, in thefirst plac e, in termsof thedivisibility orderingI on theset N+ of positiveintegers . 14 We may writethenon-zeromembers s of N+ in the form S
= 2/(1 ) . . . . . pl(i ) . pl(n (s » , •. . n (s ) ,
wherePi is the i t h prime; f : N+ -+ N+ is a functiongiving theexponent on Pi in prime decompositionof s , and n : N+ -+ N+ yieldsthe last relevant prime in thedecompositionof s.1 5 Let now 8 be anysubsetof N+ . We shall callfinitely 8 grounded iffthere is some prime p suchthat,for all s in 8, everyprime Pi withpositive exponent in theabove prime decompositionis such thatPi ~ p. This means, in our notation above, thatP 2: n(s) forall s E 8 . Alternatively, whereNk is theset of allpositiveintegersbuiltoutof thefirstk primes, 8 is finitelygrounded iffthereexists ak such that8 l:;Nk . I shallshortlyprove my favoriteequivalent of Kripke's lemma, expressed as a fact of numbertheory . Again, let 8 be any set positiveintegers of . We calls E 8 an infinite divisor for 8 iffthereexists aninfinitesubset8' of 8 , for alls' in 8', we havesis'. Thatis, an infinitedivisorfor 8 such that divides infinitely many othermembers of 8. And now we showthefollowing : THEOREM 5.1 Infinit e Division Principle(lOP). Let 8 be any infinite, finit ely grounded subset of N+. Then there exists an infinite divisor s for 8 .
Proof. We prove lOP for allNk' by inductionon k. The firstapplicablecase members of 8 aremultiplesof a singleprime p is k = 1, in which case all (which, as we have set thingsup, is 2). Let s be the least member of 8. It will, evidently , divide alltheothers . So 8' = 8 will do , withs as theinfinite divisor. The inductivecase, which is more fun , we leave tot hereaderto look up or workout, endingtheproofof thelOP. -1 DEFINITION 5.2. Let us call saequence(nih
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We thenhave,adaptedto thepresentcontext, LEMMA 5.3 Kripke's Lemma. Every finitely grounded irredundant sequence of positive integers is finite . 17
Proof. Let (ni) be a finitely g roundedsequenceof positiveintegers.To prove thelemma, assume (ni) infinite ; we showthat,for someh < i. nh Inj; whence, contraposing , if (ni) is irredundant, it must also be finite. Let S now be the setof allt heni. Thereis a k such thatS ~ Nk, since (ni) is finitelygrounded . If S is finite,thensome nh is repeatedinfinitely often inthesequence,which is thensurelyredundant . So we may assume S infinite . Butthen,by the IDP, thereis a member nh of S which is aninfinitedivisorfor S, for h a positiveinteger.I.e., thereis an infinite subsetS' of S such thatnh divides every member of S'. Evidently,infinitelym any of theseproperlysucceed nh in thesequenceordering;l etting j be theleastpositiveinteger>h such that nj E S', we havenh Inj, endingtheproofof (thepresentversionof) Kripke's -1 lemma. Thereare a number of interesting principleswhich eitherimply or are equivalent to Kripke's lemma. IS A final onethatI shallconsiderhere goes back to Dickson(1913) . Let S ~ N+, andlet Smin~ S be thosemembers of S thatare minimal on thedivisibilityordering . Dickson showedthat : LEMMA 5.4 Dickson'sLemma. If S
~
N k , then Sm in is finite.
Proof. Again weapply the IDP. Suppose thatSmin is infinite,for reductio. Then, by theIDP, it has aninfinitedivisor. Butthe minimal members of a set, by definition,a retotally unordered , which yields acontradiction. This ends theproofof Dickson'slemma. -1 This is enoughcombinatorics . Now we need somemodel theory. 6. OPERATIONAL SEMANTICS FOR R_& Therehave been anumberofsemantical analyses,datingfrom theearly'70 's, of relevant logics. I shallconcentrate here onthosethathavetargeted R and its subsystemsandshallstickto astyleofsemanticalanalysist hat,in a wider perspective,I more or less donotprefer. Still, it works presentpurposes for . DEFINITION 6.1. A commutativemonoid (henceforth,c-rnonoid)is a structure M = (M,o, 1)
-17Forsomething - - - - -clos -e-r to-Kripke's - - originallemm a, towards whichwe are workinghere,
fromKripketoAnderson seeAnderson and Belnap 1975, page 139. This is based ona letter andBelnapin 1959. 18Fora surv eyofsome of these pr inciples, see Riche'sANU thesis(1991) .
ROBERT K. MEYER
198
whereM is a set, 1 is an (identity)e lementin M, and 0 is a commutative, associativebinaryoperationon M. We shalloftenuse simple juxtaposition for '0'. DEFINITION 6.2. A partially ordered c-monoid (henceforth , c-po-monoid) is a structure (M, $) , whereM is a c-monoid and $ partially orders M; i.e., $ is a reflexive , transitive, antisymmetricrelation on M; moreover, it hasthemonotonicityproperty;i.e., for all a, b, e in M, (m) If a $ b thenea $ eb. (The othermonotonicityproperty,"a $ b implies ae $ be," is immediate by commutativity of 0.) The idea of using c-monoids to give asemanticsfor R ..... & goes back to Urquhartand to Routley,independently ; also independently,but a little later , the ideas wereextendedto all of R (and otherrelevantlogics) by Fine.19 DEFINITION 6.3. Let L bethesetof formulasof R..... &, let M be ac-monoid, and let2 be theset {T,F} of truth-values. By an interpretation we mean a functionI : L x M -+ 2 subjectto thefollowing c onditions,for allA, BEL, xEM: (T&) (T-»
I(A & B , x) = T iff I(A , x) = 1:(B, x) = T; I(A -> B ,x) = Tiff, for all a 1'E.1 if I(A, a) = T, thenI(B, xa) = T.
DEFINITION 6.4. A formulaA is verified on I justin caseI(A, 1) = T . And theidea of themodellingis thatA shouldbe a theoremof R..... & justin case A is verified on all interpretations in all cmonoids. Unfortunately, this idea does not worko utfor R.....&. Since it does work outfor thecorrespondingfragmentof Girard'slinearlogic,theproblemis withAxW , whichseparatest hesystems. To dodgethisproblem,we define: DEFINITION 6.5. A Church monoid is a c-po-monoid, subject for all a in 1'.1 to Dunn'ssquare-increasing postulate (Sq) a $ aoa. 19Urquhart (1972) provided a semilattice semanticsfor R_& . This was Routl e y's orig, though hedid drop hintstowardthe relational semantics thatRoutley inal idea as well and Meyer (1973) event ua llyevelop d ed. F ine's 1974 sets out his operotional-relational ideas-which,withextra(andsom ewhatcumbersome) machinery, provides a completesemantics for R and otherrelevantlogics. See Routley, Plumwood , Meyer, and Brody 1982 (if you can find it) for a er gen al es tt ingoutof Routley' s andmy ideas; whatI think(which differs from just abouteveryone onsome salientpoints) is covered inthepapers (m any withRoutl ey) listedin Wolf'sbibliography(Anderson, Belnap, and Dunn 1992).
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PURE RELEVANT LOGIC And:
DEFINITION 6.6. A hereditary interpretation I (henceforth, h-interpretation)in a Churchmonoid is requiredto satisfy (H) If a $ b, thenif I(A, b) = T, thenI(A, a) = T,
A E L, a, b E M. for all
A simple inductionsuffices to show thatit is enough to impose (H) on propoL. of sitionalvariablesto get it for all formulas DEFINITION 6.7. A formulaA is h-validin (M, $) justin case A is verified on allh-interpretations therein.And A is h-valid,simpliciter, justin case A is h-valid in all Churchmonoids. For asnappy completenessp roofforR_&, we appealfirst to a canonical model in theLindenbaumalgebraof (all of)R, identifying provably equiva20 lent formulas of R and defining 0 and t by Do and Dt section3. of Taking R as synonymouswithits Lindenbaumalgebra(which will do for thepresent) we define: DEFINITION 6.8. C is thecanonicalinterpretation of R _& in R iff C(B, A)
= T iff A --+
B is a theoremof R,
for all formulas B in thelanguage of R_& and A in thelanguage of R. So parsed,R is a DeMorganmonoid (intheterminology ofAndersonand Belnap, 1975), anda fortiori C a hurchmonoid (inpresentterms). Moreover, it is trivialt hatC satisfies the h ereditary condition(H) andthetruth-conditions (T --+) and (T&) above, by deductivepropertiesof R. Accordingly we have : THEOREM 6.9 CanonicalCompletenessTheorem forR_&. A is a theorem of R_& iff A is verified in R on the canonical h-interpretation C. Proof. Littleneeds to be said . In view oftheR-theoremA +-+ t --+ A, exactly thetheoremsof R_& will betrueat1 on C. We havealreadynotedthatC is an h-interpretation . -1
THEOREM 6.10 FirstCompletenessTheoremforR_&. R . . . & iff A is h-valid.
A is a theorem of
Proof. For "onlyif', argue bydeductiveinductionthatI(A , 1) = T , for an arbitrary h -interpretation I in an arbitrary Churchmonoid M. If A is a -1 non-theorem,it is canonically refutedas above. 20T his is as in Dunn's (1966) thesis. The monoid 1 is identified with(theset of formul as provablyequival e ntto) t, while$ is theprovabl e entailment relation in R . We dep end on thefactthatR is a conservat ive extensionof R_&, which I have provedelsewhere .
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ROBERT K. MEYER
6.11. A is a theorem of R-.& iff A is h-valid in all countable Church monoids.
COROLLARY
Proof. Observethat , formulated in a countable vocabulary , theLindenbaum many elements. --I algebraR has countably
THEOREM 6.12 SecondCompletenessTheoremforR-.& . A is a theorem of R_ & iff A is h-valid in all finitelygeneratedChurch monoids. Proof. Only right-to-left requiresthought . So assume thatA is a nongenerated Churchmonoid (M, S:) and an theoremof R-. &. We find a finitely I thereinsuch thatI(A, 1) = F. Let Sub(A) be thecollection interpretation of subformulasof A , and letSub(R) be thesub-monoidof theLindenbaum algebraR generatedby (equivalence classes of)members of Sub(A) and t, i nterpretation C to impose closingunderfusion 0. 21 We thenuse thecanonical an interpretation I in Sub(R), by settingI(p, x) = C(p, x) for allpropositionalvariablesp and elementsx of Sub(R). Imposing T --+ and T&, I is thendefined on all formulasR-.& of . Moreover, since(H) continuesto hold on propositional variables(since C is anh-interpretation), it will hold on all theproofby showingthat , for allsubformulas formulas. We can conclude B of A and for all elementsx of Sub(R), T(B ,x) = C(B,x). We show this case occurs when B is of the by structural induction . The onlyinteresting form D --+ E, when C(B ,x) = F. This means that,in R , x does notentail D --+ E. Butthen, byresiduation,x 0 D does notentailE ; i.e., in bastard symbols, C(E, xD) = F, while of course C(D, D) = T. Butthen, since D is a subformulaof A (becauseB is), we have oninductivehypothesisthe same . Othercases counterexample underT ; whence, byT--+, I(B, x) = F as well being trivial , we have inparticular thatI(A, 1) = F . So eachnon-theorem of R-.& is refutedin some finitelyg enerated Sub(R). --I
It is notimmediatelyobvious how toextendthetheoremjustproved to theUrquhart-Routley semilattice semanticsset out inAnderson, Belnap, and Dunn 1992. (If it were, we shouldalreadyhave a finite model propertyfor R_& , since finitely g eneratedsemilattic es are finite .) This aspect will be consideredbelow. 7. RELEVANT DIVISION AND FREE COMMUTATIVE MONOIDS
We shall not, for the time being, pursuesemilattices . Our emphasis will be on Churchmonoids and whatit takesto finitizethem. We pause for thefollowing disconcerting observation.L ittlechildren,we haveheard,are indoctrinated with the ideat hat2 divides 12. While this is OK for some 21 We alsostipul a te that$ shallagree in Sub(R) and R. The trick of clo sing und er 0 butnotunderother connectives of R wassuggestedto me by Segerberg, when I discus sed these problems with him in 1972.
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purposes, like beingpromotedto thirdgrade, it runsimmediatelyinto a violationof theChurchuse criterion.For 12 has afactor , namely3, which is not used in performingthe division. But we could no t object ,on such grounds, to "6 divides 12" . For every prime factor of 12 (specifically,2 and 3) is also aprime factorof 6. Let usinventsome notation forthissituation. We use, as before,min for ordinary divisibility. Butwe shallwritem / n to mean thatm relevantly divides n. Specifically , for allpositiveintegersm and n, we have (Df) m / n iff (a) min and (b) a(m) = a(n) , wherea : N+ ~ 2Primes is a functionthatassigns to eachpositive integerthe , satisfyingly , we have not only 6/12 but set of itsprime factors.Notethat also111/999. But,despite (a), both2/12 and 37/999 fail. We nextobservethatall oftherefutation builtintoourfirstsemantical completenesstheoremcan alreadybe done inN+, orderedby [, Indeed, it can be done inthe Nk. Let us call a formula A numerically valid iff A is verified on every h ereditary i nterpretation I in every (Nk , /}.22 We then have: THEOREM 7.1 NumericalCompletenessT heoremforR--.&. A is a theorem of R--.& iff A is numerically valid. ce of ourfirstcomplete Proof From left to r ight, thetheorem is aconsequen ness theorem . For therest , assume that A is not atheorem. We shallfind an Nk and ahereditary i nterpretation T thereinsuch thatT(A , 1) = F. By thesecond completenesstheor em, ther e is a finitelygenerat edChurch monoid (M,:S) and a hereditaryinterpretation 3 therein such that 3(A,I) = Fin M. Every finitelygeneratedc-rnonoid si an image ofthe free c-monoid N k in k generators , for some k in N+. Moreover , relevant divisibility/ is thesmallest relationunderwhichNk becomes a Church monoid. So, where G is a set of k generators for M, leth : N k ~ M be thehomomorphism determinedby anysurjection fromthefirstk primes ontoG. Note, in particular, thatif m / n, thenh(m) :S h(n) . And let sunow define3 by setting,for all p ropositionalvariablesp and numbersm in Nk , 3(p, m) = T iffT(p,h(m)) = T . Note thatthis satisfies (H) on propositionalvariables; for if m /n, thencertainly h (m) :S h(n) . We completethespecificationof J by imposing thetruth-conditions T~ and T&. This makes 3, certainly , an h-interpretation in theChurchmonoid Nk, orderedby [ , We concludetheproofby showing,for all m in Nk andformulasD of R--.&, 3(D ,m) = T(D , h(m)). We showthisby structural inductionon D. It is true by fiat ifD is a propositional variable , andit follows quicklyDifis B & C , on 22We m ake / exp licit h er e onlyto m ake the point that the I\Ik are C hurch mon oids, by an easy verific ation. But, since / is defin abl ein term s of mon oid operations, we essentially get it for free.
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inductivehypothesis. So suppose D is B -+ C. Then J(D, m) = F iffthere exists ann in Nk such thatJ(B, n) = T and J(C, mn) = F , iff (oninductive hypothesis)I(B, h(n)) = T and I(C, h(mn)) = F = I(B, h(m) 0 h(n)). Since h is an epimorphism, thisargumentgoes smoothlyin bothdirections; its upshotis thatJ(A,l) = I(A, h(l)) = I(A, 1) = F, transmuting our g enerated M into itsrefutation in one oftheNk. -f refutation of A in finitely 8 . FINITIZING FREE COMMUTATIVE MONOIDS
Primitivepeoples, nothavingbeen indoctrinated in thesophistrythatthe natural numbers fail to runout, are said to have counted1,2, . . . , HEAP, where HEAP was (from theirviewpoint)the greatestn atural number. It is time toresuscitate t hisprimordialintuitionof ourspecles.F' Ouraim is to refuteeverynon-theoremof R--+& in a finite model.B uteven theN'k , our preferredmodels tothis point, do not run out. We wish to use Kripke's s hrinkthem until combinatorial insights,jammed in a few pages back, to theyare finite . And forthiswe shallrequirea value for HEAP. At anyrate, we alreadyhave a value for k. Our strategy in thesecond semanticalcompletenessp roofabove was to buildrefuting a model for a nontheoremA out ofSub(A), throwingin 1 and closingundero. Thus k, the ourgenerating set.24 cardinality of Sub(A), is an upper boundfor the size of Note toothatour manipulations of the last section, tradingin M for one of theN'k, did notincreasethevalue ofk . Still , unless we can use A to induce a value for HEAP, afterwhichiterated multiplication willjustproduceHEAP again, we haven aughtbutcold comdo In fort. ButKripke's lemma and its cousins give us a way to justthis. thefirst place, recall ourthreeprinciplesfrom section5-theIDP, Kripke's lemma, andDickson'slemma. We noteimmediatelythatallthreeprinciples continueto gothroughif alltalkof ordinary divisibility, I, is replacedwith talkof relevant divisibility, / .25 For example,let us call : DEFINITION 8.1. s E S ~ N k is a relevant i nfinitedivisorfor S justin case thereis an infinitesubsetS' of S such thats / s' for every s' in S'.
Thenwe have: 8.2 RelevantIDP. Let S be an infinite subset of an N k • Then there exists a relevant infinite divisor for S.
THEOREM
23Being a Sixist, I have alwaysthoughtt hatHEAP = 7; i.e., therearejustsix objects in theworld,withan extraone if one throwsin themetaphysicals ubject. ("Monism is right in whatit intends," as Wittgenst ein taught ; butit is off by 5,a tleast .) ButSlaney,a uthor of theMost MagnificentResult(theMMR) ever achieved in logic,showedin Slaney 1985, thatthereare exactly 3088 Ackermannconstants in R . 24Alas, it tooofteninducesan enonnous upper bound. u nfairto Kripke-since, to allintentsa nd purposes, 25So puttingthematteris a little his lemma dealtwithrelevant divisibilityfrom thestart .
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Proof. All ofthemembers of N k, and hence ofS, are builtoutof the same finite set Pk = PI, . .. , Pk of primes. Thereare only finitely many subsets of Pk • Accordingly, since S is infinite,thereis an infinitesubset S* of S and aparticular subsetP of Pk suchthatallmembers of S* are built out of members of P. By theordinarylOP, S* has an infinite divisor s, which will be a relevant infinite divisor for S*. Accordingly,s will also be a relevant infinite divisor for S. --I
COROLLARY 8.3. Dickson's and Kripke's lemmas hold in relevant form. Proof. From therelevant lOP, as above(mutatis mutandis).
--I
Now we can get back to the business of finding HEAP. DEFINITION 8.4. Let A be a non-theoremof R_&. Fix k and .:J by the numericalcompletenesstheoremso that.:J(A, 1) = F in Nk. Call an element n in Nk A-critical j ustin case, for somesubformulaB of A, we haveboth (i) .:J(B,n) = F, and
(ii) For all m such t hatm / n, .:J(B,m)
= T.
condition(H) assuresthatTruthis propagateddownSince thehereditary wards underrelevant division, it similarly assuresthatFalsehood is propaof are numbersat which some gatedupwards . So theA-critical elements Nk subformulaof A (including,perhaps, A itself) isminimally false. This sets thingsup for a relevant applicationof Dickson's lemma. FACT 8.5 Dickson's Fact. Let.:J be an h-interpretation in Nk such that .:J(A, 1) = F. Thereare finitely many A-criticalelements ofNk . . Suppose thereare infinitely many A-critical n in Proof. By contradiction N k • Since thenumberof subformulasof A is finite,thereis some particular subformulaB of A which hasproperties(i) and (ii) immediatelyabove, for infinitely manyn . But all thosen are minimal, under/, with respectto .:J(B, n) = F. By (relevant) Dickson, this cannotbe; any totally unordered subsetof N k must be finite. --I We now use Dickson's fact to find HEAP. Let Nk, .:J, and A be as in the fact,andletS be theset of A-critical elementsNkof . Since S is finite,there is some highestexponentm - 1 which occursa mongtheprime powers which characterize its members. We define a new c-monoidNk' whose elements shall bejust those members ofNk which, inprime power decomposition, 26 shall have no exponentgreaterthanm on any prime factor. Thus, for example, N~ = {I, 2, 3, 4, 6, 8, 9,12,18,24,27,36,54,72,108, 216}. 26 A concernfor efficiencyherewouldsuggestthatwe keep trackof exponentson particular prime powers individually . Afterall, if noA-criticale lementhas exponentgreater than2 on theprime 3, it is otioseto allow theseexponentsto rise to 10 j ustbecausesome otherA-criticale lementhas divisor512. Butwe arechasinga finitemodel propertyhere; it is notationally simpler, and no impediment to thedevelopment , if we allowexponents on any prime to rise tothemaximal one.
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Nk. Having setHEAP = m, we define / onNJ:' so thatit agrees with / on But,since some productsof members of NJ:' threaten to "go off the edge", we redefine 0 on prime powers sothatpi 0 ph = pi+h if j + h ::; m, and pi 0 ph = p": otherwise; on the obvious epimorphism from Nk to NJ:', 0 is thendefined inthelatter for all elements . We setout,justfor fun, the 0 and / tablesforN~ . 1 2 3 1 1 2 3 2 2 4 6 3 3 6 9 4 4 4 12 6 6 12 18 9 9 18 9 12 12 12 36 18 18 36 18 36 36 36 36 0
4 4 4 12 4 12 36 12 36 36
6 6 12 18 12 36 18 36 36 36
9 9 18 9 36 18 9 36 18 36
12 12 12 36 12 36 36 36 36 36
18 18 36 18 36 36 18 36 36 36
36 36 36 36 36 36 36 36 36 36
/
1 2 3 4 6 9 12 18 36
1 T F F F F F F F F
2 F T F F F F F F F
3 F F T F F F F F F
4 6 9 12 18 36 F F F F F F T F F F F F F F T F F F T F F F F F F T F T T T F F T F F F F F F T F T F F F F T T F F F F F T
t hemingle axiomp ~. p ~ p While it is overkill, let usaboutrefuting set in N~. (This is also arefutation o fAxK.) Set T(p ,2) = T , T(p ,3) = T, I(p,6) = F. Then, by T~, Ttp ~ p,2) = F, since forj = 3 we have I(p,j) = T and Tip, 2j) = F. But thenI(p -r-e , p ~ p, 1) = F , since it has a trueantecedent and a false cons equenton T at2. Whatwe have done for mingle we canquitegenerally do . For we havethe: LEMMA 8.6 FinitizingLemma forR_ &. Let Nk be a finitely genemted submonoid of N+, and let .:J be an h-interpretation (under /) and A be a formula ofR_ such that .:J(A, 1) = F. Then there exists a natural numberm and an h-interpretation T such that I(A , 1) = F in NJ:'.
Proof. We fix m - 1 by Dickson's fact , and defineNJ:' accordingly . For each propositional variablep and element x ofNJ:', setTip, x) = .:J(p, x). Complete T~ andT&. Clearly T the definition Iofby imposing thetruth-conditions is hereditary , because it agrees with the h-interpretation .:J on propositional variables, while T~ and T& preserve (H) . We finish the proof by showing , for all subformulasB of A and for all x in NJ:', thatI(B , x) = .:J(B, x). We proceed bystructural induction.T he conditionis imposed on propositionalvariables.I t is evidenton inductivehypothesiswhen B is of the form C & D. So suppose B is C --> D. Assume firstthatT(B,x) = F. Then thereis a c in NJ:' such thatT(C , c) = T and T(D ,x 0 c) = F, where0 is defined in the NJ:' way. But then, on inductivehypothesis,.:J(C, c) = T and .:J(D, xoc) = F. Of coursexoc may not be thetrue productxc. Never mind. Falsehoodis propagatedupward under[, and x 0 c / xc. So .:J (C ~ D, x) = F as well.
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We finishtheargumentby assuming .J (C -+ D, x) = F andshowingthe same for I , for x in Nk' and C -+ D a subformulaof A. Thereis now a c in N k such thatI(C,c) = T and I(D, xc) = F in Nk. If xc = xoc, we have oninductivehypothesisnothingto prove; for we shall have , on inductive hypothesis,t hesame counterexample in Nk'. Otherwise,wehadbetterreduce exponentsuntilwe put bothc and x 0 c in Nk'. We can reduceexponents freely on c; for, since Truthpropagatesdoum underI on J in Nk , we shall have oninductivehypothesisI(C,c') = T for any c' ENk' such thatc' I c. And we can reduce exponentsa little bit on x 0 Cj for we chose m sothatAcritical elementshave inNk exponentsa tmost m - 1 on any prime. SinceD is a subformula of A, which is false on J at xc, thereis a y suchthat,in prime decomposition,everyexponenti on anyprime p satisfiesi < rn; moreover, y I xc, and J(D, y) = F. Fix c' by reducinganyexponentsg reater t henm to m . Thenc' E Nk' and I( C, c') = T. Moreovery I x 0 c' in Nk', whence, since Falsehoodpropagatesup, J(D, y) = I(D, y) = I(D, x 0 c') = F . So we still have acounterexample to C-+ D atx underI . This completestheinductive argument,whenceI(A, 1) = .J(A, 1) = F. So A is refutedin a finite model. -l
We sum allthisup in thefollowing: THEOREM 8.7 FiniteModelTheoremfor R-+&. are equivalent:
The following conditions
(1) A is a theorem ofR_&. (2) A is h-valid. (3) A is h-valid in all the N k • (4) A is h-valid in all finite Church monoids. (5) A is h-valid in all the Nk'. Proof. by previoustheorems,lemmas, and facts.
COROLLARY 8.8. R_& is decidable. We knewthecorollary anyway, of course, by Kripke. Indeed, so far weare behindKripke (thoughwe shallrectifythatshortly),since hismethodsyield decidabilityfor all of LR. Meanwhile,t hedecidabilitythatwe have is less thancompletely satisfying.Usually, finite model propertiesplace somesortof bound(usually horrible,butthat'sa nother s tory)on atestfortheoremhood. Here, however, we have naked decidability . We set someone, perhapsBelnap, thetaskof enumerating proofs inR--+ &. We set someone else , maybe Dunn, thetaskof enumerating r efutations in theNk'. Dunn's job is eased because . He may he canbound k, for a givenA, justby countingup its subformulas curse usnonetheless, since we have given him no recipe bound to m . All
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he knows (or we know) that, is if A is refutable , thereexistsan m thatwill do thejob. In brief, this is the "Don' t call us, we'llcall you " method; if A is provable,Dunn must stillwait for Belnap(or Thistlewaite's KRIPKE program,or its refinementby Riche) to pick upthephone, beforehe knows thatthejob is done. 9. EXTENDING THE FINITE MODEL PROPERTY TO LR
The result s of the previoussectionapply immediatelyto R_ (since it is an exactsubsystem of R_&) and to MR~& (since it is just K ....&, with f identifiedsyntactically with a variableirrelevant to context) .27 A little trickieris R~, for which",,""E is explicit. But it's notmuch trickier,s ince R~ exactlytranslates intoMR~ withthetranslation of A beingPI --+ f --+ f --+ PI --+ ••• • --+. Pn --+ f --+ f --+ Pn --+ A, wherePI, .. . , Pn is some selection v ariablesof A,28 So ourfinitemodels for R~ from among thepropositional will go onrefuting(translations of) non-theo rems of R~. Whenit comes toaddingV, thesituation becomes trickier . For thetranslationthatembeds R~ in MR~ relieson thefactthat",,,,,B --+ B --+. """" (A --+ B) --+. A --+ B is alreadya theoremof MR~. A similartrickworksin the presenceof &; butthe inductionbreaksdown on V. Butif one trick won't workanother will- aGlivenko-style translation, whichwe borrowin principle from Church(l951a) . DEFINITION 9.1. For eachformulaA of LR, we defineits *-translation A* recursively as follows : (i) f* = f by D"" (ii) p* = """"p, whererv is definedinferentially
(iii) (B
--+
C)* = B*
(iv) (B & C)*
--+
C*
= B* & C*
(v) (B V C)* = "" (rvB* & rvC*) Evidently,A* is alwaysin thevocabulary of MR~& . LEMMA 9.2 """,E Lemma. For all formulas A ofLR, ",rvA* --+A* is a theorem of MR~& . 27T he R_ resultwas proved in Meyer and Ona 1994, together with a sim ila rone for BCK. Indeed, the BCK result wasthe excuse for Meyer and Ona 1994 . For I realized , ow) wouldapply in thinking aboutit, that the presentm ethods (which had long lainfall to BCK as well. 281f A containsno p roposi t iona lari v abl es ,it transl ates sim ply as itself. As anoth er triumphforSixism, ther e areexac tl y 6such non-equivalentformulasin MR~ .
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Proof. By structural inductionon A. The most interesting case (since we were quickaboutit before) is when A is of the formB & C. On inductive hypothesis, rvrvB* --> B* and rvrvC* --> C* are theoremsof MR~& . Elementarypropertiesof & and--> yieldrvrvB* & rvrvC* --> B* & C* as another theorem . But, suffixing twice in Ax&E, we have rvrv(B* & C*) --> rvrvB* on minimal propertiesof r-«, The same holds for C*, whenceadjoiningand applyingAx-->&I, we haverv",(B* & C*) --> rvrvB* & rvrvC* . Transitivity of provable--> in MR~& thenyieldsrvrvA* --> A* in this case . -l
THEOREM 9.3 LR Containment Theorem. MR~& under * translation.
LR is an exact subsystem of
Proof. We must show thatA is a theoremof LR iff A * is a theoremof MR~&. From rightto left this is easy, sinceA*ifis provablewithout rvrvE it iscertainly provablewith it; moreoverA * +-+ A is a theoremscheme ofLR. Fortheconverse, proceed bydeductiveinduction . Sincetherules of LR and MR~& arethesame, it suffices to show thatwhenA is an axiom ofLR, then A* is a theorem of MR~&. Even the axiomsdon'trequire much checking; theonlyadditional one isrvrvE , which ourrvrvElemma suffices to handleon * translation . Butwe hadbetterlook inparticular attheV axioms, since this is theparticleleftoutof theMR~& vocabulary. As for AxVI, half of it will suffice ; anyway, by Ax&E we havervB* & rvC* --> rvB* . Butthen, minimally, we getrvrvB* --> rv (rvB* & rvC*) by suffixing . ButB* --> rvrvB* is minimally OK. (There is nothingwrong withrvrvIeven at the MR~ level.) So, by transitivity , B* --> (B V C)*. Finally, we look at Ax-->vE. We need to show (B* --> D*) & (C* --> D*) -r-e , rv (",B* & rvC*) --> D* in MR~& . At anyrate, by Ax&I we have(rvD* --> rvB*) & (",D* --> rvC*) -->. D*) & (C* --> D*) -->. rv(rvB* & rvC*) --> rvrvD* . The rvrvE lemma andreplacement theoremsallow us to remove thervrv on theultimate consequent, endingtheverification of the lastpotentially troubling axiom on * translation. -l
One is disturbedoverthefailureo fAxDisttoenterinto thesepleasantre29 flections. As an axiom relating & and V,thereis much to be said for AxDist. Butas a principlerelating & and minimalrv,it isn't veryappetizing. THEOREM 9.4 FiniteModelTheoremforLR. The following conditions are equivalent: (i) A is a theorem of LR.
(ii) A * is h-valid. (iii) A * is h-valid in all the Nk'. 29Much ofthis is said by Belnap(1993).
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ROBERT K. MEYER
Proof By theLR containmentt heoremjustproved, and thefinite model theoremforR-+& above. -l Well, thatwasn't much of a finite model theorem .P" Still, it suffices for yet anotherproofof thedecidabilityof LR. 10. THE FINITE MODEL PROPERTY FOR THE SEMILATIICE SEMANTICS
It might beexpected,from ourresultsso far, thatit is easy to get a finite
model propertyfor thesemilatticesemanticsthatUrquhart c ontributed to Anderson, Belnap, and Dunn 1992, and whichRoutleydiscoveredindependently.i'! For we quickly found above, for a non-theorem given A, a finitely generated commutativemonoid N k thatrefutesA. If we could pull thesame trickwithsemilattices,thefinite modelpropertywould be a piece of cake. For finitely g enerated s emilatticesare, afterall, finite. As thereaderprobablyexpectsby now,thingsare notnecessarily so simoriginalformalintuitions,thesemilatticeelementsare ple. On Urquhart's (or might as well be) sets32 of relevance numerals, corresponding to thetags on formulast hatappearin theadaptation in Anderson and Belnap 1975 of Fitch-stylen atural d eductionto R ..... and otherrelevantlogics. Sothinking does yield acompletenessprooffor thesemilatticesemanticsfor R-+, which Urquhart c ontributed to Anderson, Belnap , and Dunn 1992. In our terminology so far: DEFINITION 10.1. An Urquhart semilattice may be parsedas a c-rnonoid M = (M,o,l),where M is asetsofsetsclosedunderfinite union ,Ufusion0 is U, and theunit1 is thenull set0 , which belongs to M. We may atthe same time consideran Urquhart semilatticeto be aChurchmonoid (M, =), since idempotenceof U assuresthatthesquare-increasing conditionwill be met.
Similarly, onthis construal , every interpretation satisfyingT ~ (and T &, if present)will be anh-interpretation . Accordingly , in discussingUrquhart semilattices we candrop alltalkof heredity,takingthenotionsof h-interpretation and interpretation to besynonymous. Because ofthissynonymy,we callvalid the h-validformulasin an Urquhart semilatticeM. And a formula valid in all Urquhart semilatticesis justUrquhart valid. 3° 1 had,I think,a bett er way ofdealingwithLR, once upon a time. I even gavea seminar to theANU logicgroupon thetopic, after Kripke had suggeste d to em in correspondence thatof coursethemethodsof the presentpaper worked for all LR. of 31Conscie ncerequir es the admission of my profounddebtto Urquhart , in allthatI have thought s emantically aboutrelevantlogics. It washardl y appropriat e to say so in myjoint workwithRoutley,since he was notso indebted. ButI never thoughtindep endentlyabout anythinglike T-+ above. And since that,t hroughmany a metamorphosis, is for my money s emanticalins ight as th ere has been, it is pleasantto recordmy the key to such relevant gratitude to thefellow whoslipped me thekey. 32Urquha rtm otivat es these as pieces of information.
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I think,as I reportedwithMcRobbie (Meyer and McRobbie 1982) , that thedata type most naturally associatedwithsystemsofrelevant implicationis thatof multisets. All ofo urtalkofl~+ , theproduct" anddivisibilityrelations I and /, is justa way ofrecallingw hatwe have knownaboutmultisetsfor a longtime. Butit is possible (even if lesst hancompletelynatural) to represent multisetsas sets. Both the success and (again, as I see it) the ultimate failure of boththeFitch-style n atural deductionsystems and thesemilattice semanticsforR and its fragmentsrest on suchrepresentation. At anyrate, whatwe shall do here will pressthefore to aset-theoretic representation of multisetsas sets, transforming t hec-rnonoidsemanticsintoan Urquhart semantics,which we canthenfinitize along lines laid down above . It is trivialthatalltheoremsof R_& are Urquhart valid, byourfirst semanticalcompletenesst heorem . And we need not redo Urquhart by proving the converse; ourinterest,rather,is in jinitizing him. To thisend we concentrate on theN k . Witheachprime pin N k , we associatem copies PI,' .. ,Pm' We letthe Urquhart universe Uk consistof allthesecopies of primes; and we letP k be thepower set of Uk' We define arepresentation function r from P k to N k as follows:
(i) r(0) = 1. (ii) WherePi is a copy oftheprime P, r(Pi) (iii) Suppose S ~ in N k .
Uk' T
c:;;;
Uk' S n T =
= p.
0. Then r(S
U
T) = r (S)
0
r(T)
Since everymember S of P k is expressibleas a disjointunion of(unit setsof) copies ofprimes E Nk ' andsince thec-monoid propertiesof Nk make as a function it indifferent whichrepresentation we choose, r is well-defined from P k ontoN k . Moreover,since we makeexactlym cop ies of eachprime in Uk' images underr stay in a natural way within Nk . Now suppose thatwe havebuiltan N k for therefutation of a particular 33 lemma that non-theoremA of R_ &. We shall now prove a Xerox shows that we can do thesame work inP k . LEMMA 10 .2 XeroxLemma. Let I be an h-interpretation of R_ & in N k , for some positive integers k, m . And let A be a formula of R_& such that I(A, 1) = F , where m-l is the highest exponent that occurs on any A-critical elements of Nk under I. There is then an interpretation I' in the associated Urquhart semilattice such that I'(A , 1) = F .
Proof. Thatthereare k , m, and I thatm eet the conditions oft helemma for anynon-theoremA is partof thecontentof ourfinite modeltheoremfor R_ & in section8. Where r is therepresentation functiondefined by (i)-(iii) 33La ngfordsuggest s thatuse of this trad e nam e witn esseshow far beh ind the times I am . Maybe so. He wouldsuggest, I guess, thatI call it th e "p hotoco py lemm a" .
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justabove, we defineI' by setting , for eachpropositional variableq in A and subset S of UI:, I'(q, S) = I(q , r(S» . Imposing T -+ and T& to complete PI: , we show for each s ubformulaB of A andfor the definitionofI' on all of allS in PI: thatI'(B,S) = I(B ,r(S», by structural inductionon B. This is laid down by fiat for propositional variables ,and it is readilyapparenton inductivehypothesiswhen B is C & D, justapplyingT& in bothcases. So theinteresting case is B = C -+ D . Suppose firstthatI'(B , S) = F . Then thereis a counterexample-s-a V ~ UI: such thatI'(C , V) = T and I'(D,SU V) = F . On inductivehypothesis,I (C ,r(V» = T and I(D,r(SU V» = F as well. We wish now to massage this counterexample so thatit is also one onthec-monoidsemanticsforNI: as well. It couldfail, How could itfail, one wonders, to remaina counterexample? alas, if S and V are notdisjoint,whence reS U V) is not the reS) 0 reV) thatwe requirefor abonafidecounterexample underT -+ in WI: on I . But this is no realproblem; if S and V have copies of some ap rticular prime p in common, allthismeans is thatreS U V) / reS) 0 r( V). Since Falsehoodis propagatedup underrelevant divisibilitywe shall still have I(D, r(S)or(V», falsifyingC -+ D atreS) on I as required. t urnto thecase whereI(C -+ D , reS»~ = F. We need to show Finally, we thatI'(C -+ D , S) = F as well , on inductivehypothesis . At anyrate,there is acE NI: such thatI(C,c) = T and I(D,r(S) oc) = F, by T-+ . We need to convertthis to arefutation in PI:. Thereis atleast some setV ~ UI: (and probablymany) such thatreV) = c. We wantto choose V so thatit is as disjointfrom S as possible. If thereis a V such thatS n V = 0 and r( V) = c, we shallhave oninductivehypothesisthesame counterexample in P I: thatwe alreadyhave inNI: , and we are done. Butwe may not be done . For it is conceivable thatthereis no V disjoint from S such thatreV) = c. This couldhappen, forexample,if all m copies of some prime p alreadyoccurin S, andpic. However ,we chose m sothat all A -criticalelementsof NI: would havee xponentat most m - 1 in prime decomposition,on everyprime p. So letS' be likeS , except fortheremoval of one copy ofeveryprime p which divides c, when m cop ies occuralreadyin S. And let c' bet heproduct(withoutiteration) of theprimes thatdivide c. Since Truthpropagatesdown under/, andsince c' / c, we haveI (C,c') = T as well. And since r(S')oc' = r(S)oc in NI:, we still have I(D , r(S')oc') = F as well.T his suffices torefuteC -+ D atr(S') in NI: . (Well, we knewthat C-+D couldnotbe minimally false atr(S).) And choosing nowV' so thatit containsone copy of eachprime divisorof c', disjointfromthecopies in S' , we have oninductivehypothesisthatI'(D, S' U V') = F. But S' U V' = Su V' , whencerefutation of C -+ D on I' at S is completedwith theobservation, on inductivehypothesis, thatI'(C, V') = T . --j We can nowstat e thefollowing :
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THEOREM 10.3 FiniteModelTheoremfortheSemilattice Semantics. The following are equivalent : (1) A is a theorem ofR_&.
(2) A is Urquhart valid. (3) A is valid in all finite semilattices. Proof By thefinite modeltheoremforR_& and theXeroxlemma.
-j
We caneven useprevioustricksto getLR underthetent. COROLLARY 10.4 FiniteSemilatticeC orollary forLR. Let A be a formula of LR, and let A * be the result of tmnslating A into R -.& as above. Then the following conditions are equivalent: (1) A is a theorem of LR.
(2) A* is a theorem ofR_&. (3) A* is valid in all finite semilattices. Proof Allimmediatefrom previoustheorems. 11. EXTENDING THE THEORY OF PROPOSITIONAL QUANTIFICATION We leftpropositionalquantifiersback insection2, withthe observation that Churchhad providedthem as originalequipmentin his theoryof relevant implication . In theclimate of thetime, when any use of the wordproposition tendedto expose one to t hewrath of Quine , thiswas adaringthingto do.3 4 It istime to delve al ittle more intowhatpropositional quantifiers cando for us, in a theoryof relevant implication. A conspicuoususe ofpropositional quantifiers, already p lumbedby Church (l951a) , lies intheirutilityfor definingthings.We have seen how alreadyin R~ thatwe can define fusion 0, equivalence <-+, existential quantification 3, andso forth . Moreover, in so defining them, we givethemthepropertiest hat theyought to have. Clearlythis kind ofthingcan be carriedconsiderably further.How much further is a questionthatwe shall look at now , borrowing resultsfrom thedraftpaper Meyer 1978.35 Churchcontented himselfwithanalogues of thetruth -functional connectives. But could we , followingAckermann,Anderson, and Belnap,define somethingcloser totherealthings?I do notsee how, although I havemade . First, thereis negation.I am contentwiththeinferential some attempts f. ButI am not content w iththe definitionofnegationvia Dr-anda constant 34 1 used to avoid propositions my self,untilBelnapconvin ced m e at la stthat 1 was ebing silly . 35 Ander son , Belnap, and Dunn 1992 pass on some of my observationson page 68f.
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ROBERT K. MEYER
failureof ".""E and associatedprinciples.Solution(atleastfor my money): Justadd ","'E as a newaxiom scheme. The result,w ithoutquantifiers,is R!..... Let us look now at this system with quantifiers . Call itR~P, and subjectit toboththequantificational principlesand definitionsof section2 andtheconstant f andthe",,,,E axiom scheme ofsection3. We noteimmediatelythatR~P has aconspicuoustheoremthatR:1> lacks. andthenexist entally gen eralizing on the",,,,E axiom, we get For universally
(1) 3qVp(p-;. q -;. q -;. pl . The closestthatwe cangetto (1) in R:1> itselfis
(2) Vp3q(p -;. q -;. q -;. p), which is a long way from thesame thing. So, contradicting the expectations of conservative e xtensionthatwe have come toexpectin systems based on R---+, we have a failure here . Perhaps, though , we can dosomethingwith thatfailure. It occursto one thatwe can introducesomethinglike aHilbert(indefinitedescription) E-operatoraccordingto a rubriclike DE. E q(A(q ))
= df
3q(q 0 A(q)).
Let us see howtheDE trickworks on f. Define, in R~P, Df. f = E qVp(p -tq -tq -tp), thenwe can prove the equivalence of the f just defined tothe primitive constant of R~p. So we do not need t hatconstant a tall.(Beforecontinuing, we do need the",,,,E axiom; withoutit, back in R~P, thedefinitionis st ill legal , but",,,,E remains, unfortunately , a non-theorem.) It is time also, in R~P , to clearup ouralternative definitionsof 0 (and hence of...... ). We need toshow, inthatsystem, that>- (A -t"'B) andVq((A-;. . B -;. q) -;. q) are in factequivalent.F rom rightto elft ,simply instantiate q to '" (A -t"'B) ; after this, right-to-left goes throughin R!.... . From left to andconfinementpropertiesof right,it will suffic e to show, bygeneralization V, that", (A -;. "'B) entailsin R!.... the formula(A -t. B -tq) -tq. Minor exercise,left toreader. In additionto definingconstants, it is also int erestingto experimentwith the directdefinitionof connectives , via a rubriclikeourE operator . Here is '" in R~p. D",. ",A
=df
3q«A -tq) 0 Vp(p -tq -tq -;. p)).
Againthedefinitioncorrespondsto a provabl e equival ence . From lefttoright, ",A does imply A -tf, to which we may fuse theR!.... theoremVp(p -tf-tf-tp) in theconsequent;existent ially generalizing on f completesthe leftto right
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argument . In theotherdirection , it suffices (bypropertiesof 3) simply to show that(A -+ q) 0 "i/p(p -+ q -+ q -+ p) entailse -A in R~p . The trickhereis to instantiate thesecond "fusejunct"to rvq; in view oftheR~ theoremhood of rvq -+ q -+ q; this "fusejunct " willentailrvq, afterwhicha step of "modus tollens" getsus rvA. Thatwill do. Whathappensif onethrowsin thelattice c onnectives?Still morei nteresting things. For example, form R~& by addingto R_ & thequantificational machineryof section2. It thenturnsoutthatlattice V is alreadydefinable, by DV. A V B
=df
"i/q((A
-+
q) & (B
-+
q)
-+
q).
It is thenan easy exerciseto useAx&E and Ax -+ &1 to provethe dual principlesAxVI and Ax -+ vE in R~& . So, while V isindependentof the otherpositiveparticlesa tthesentential level,t hisis no longerthecase inthe presenceof v, ButAxDist, afterall,remains independent . Not alllattices aredistributive. This merelyscratchest hesurfaceof whatcan be done. Especiallyinterestingis whathappenswhenwe pass to all of R, addingnot onlyAxDistbut, atthequantifier level , theaccompanying"infinitedistribution " principles Ax"i/v. "i/p(A V B(p))
-+
A V "i/pB(p) , if p is notfree inA ,
Ax & 3. A & 3pB(p)
-+
3p(A & B(p)), if p is notfree inA .
The resultingsystem is the R'VP of Anderson,Belnap,and Dunn (1992). Indeed,one cango further . One getsa conservative e xtensionCR of R by addinga new,explicitly B ooleannegation-, to theDeMorganrv which comes in R as original equipment.P" Sufficient to g overn-, are the(paradoxical) axioms: AxAbsurd. A & -,A
B, AxTrivial.A -+ B V -,B. -+
Ourrubricabovethatproducedsomethinglike aHilbertE-operator a ppealed crucially to o. We cando somethingsimilarin CR'VP using &, via
D-, . -,A
=df
3p(p & (A & p ---+ F) & (T
-+
A V p)), wherep does notoccurin A.
First, theleft side ofD-, implies therightside in CR when wedrop the initial3p, replacingallthesubsequento ccurrences of p with-'A j existentially 3 6 W hilethe extensi on is conservativein theoriginalvocabul a ryof Anderson-BelnapR , builtup from variable s under&, V, "', and _ , it is not conservat ive as we are formulating thingsher e, withprimitive Ackermannconsta nt f. To ethcontrary , CR (of which my most recentstudyis Meyer 1979) counte rac ts ethSlaney explosion of Ackermannconst ants by reducingthenon-equivalent ones to 8- almost, butnotquite, enough togladden the Sixist heart .
ROBERT K. MEYER
214
generalizing completestheargumentin CR\fP. To go fromrightto left, it thatB = p & (A & p ---. F) & (T ---. A V p) entails- ,A in CR. suffices to show Well, certainly - ,A & B entails-,A, by &E. And A & B has (nearenough) A&p&(A&p---.F) as a conjunct , whichentailsF, whichentailsa nything -in particular , -,A. So, by Ax---. vE andAxDist,we get(A V-,A) & B ---. -,A. But, appealingto AxTrivial,t hisis equivalent to B itself. SoB does entailA by thelights of CR, whence(slappinganexistential quantifier on theantecedent, t hat which is OKbecausep does notoccurin A) wecompletetheverification -,A is CR\fp equivalent to itssuggesteddefiniensY Note againthatwhat is atthequantifier-free level alinguistic extensionbecomes, withquantifiers, . For D-, is, of course,a lreadyavailablea tthe R vp an axiomatic extension levelas a definition. But, like ourdefinitionof f inR~, ourdefinedBoolean -,A onlyhas Booleanpropertieswhen we makethemexplicitthrough specific axioms. 12. CONCLUSION
I beganthispaperwitha noteofappreciationfor Alonzo C hurch(who, alas, t hesame note. Ideas has died since itwas written).Let me conclude on thathe threwoff more or less in passing have become, formany of us, the ", work of years. He b uiltbetterthanhe knew. The Church"use criterion in particular, is susceptibleof almostindefinitevariation . It need not imply AxW, if one is fussyaboutreusing premisses; thatway lieslinearlogic, and much else. Nor need the use criterion even implythatfusion iscommutative, and othersubstructural logics, or associative ; thatway lieminimal relevant in the style,.g., e of Lambek. In another direction , one mayexpandtheR_ insightsso thatfusion becomesidempotent , as in theDunn-McCall s ystem RM of Andersonand Belnap(1975). But,however modified,m utilated,or extended,R _ and R~ lieatthe verycenterof agreatdeal of mostvaluable researchin logic.Thanks, ProfessorChurch. ACKNOWLEDGEMENTS
In additionto Church,I am also seriously in debt to a number of other scholars. I have mentionedKripke, who firstthought up thetricksthathere producefinite modelproperties.Indeed,I would havepreferreda jointpaper with Kripke, which proposed I to him on several occasions. His consistent responsewas thathe wouldthinkaboutit. I am also muchindebtedto Urquhart,who was my colleaguea tthe Universityof Torontowhen I proconversations with ducedthefirstdraftofthispaper; I had manystimulating 37W hile we usedAxTrivialin theverification thatD. defines Boolean negationin CR , we may chop its stand-in T -+ A V P from thesuggesteddefiniens. I.e., the follow ing also worksin CRIIp : D~. ~A =df
3p(p& (A&p-+F)) .
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him, especiallyaboutthesemilatticesemantics, atthattime. I am always non-theoremA of R--. , a fiindebtedto Dunn;thejob of finding, for each niteDeMorganmonoid, a la Dunn, thatwouldrefuteA was my first aim. (Thereis always one;j usttweaktheNk' as in Routley and Meyer 1973 to find it.) This materialis presentedatsome lengthin Anderson, Belnap, and Dunn 1992 and in Routley, Plumwood, Meyer, and Brady 1982. Pastand presentmembers of theLogic Groupof theAustralian National U niversitythelateRichardSylvan, Slaney, Martin,McRobbie, Meglicki,Restall,a nd many others-have been a constantsourceof fruitful ideas.(This group, whichbeganlife inthePhilosophydepartmentof theResearchSchool of Social Science, has now mainly migratedto theAutomatedReasoningProject in the ResearchSchool ofInformationScience andEngineering .) Jacques Riche, aPh.D. graduate of theARP, related t helOP and its cousins to many well-known m athematical principlesin Riche 1991. I projectwithhim (and withKripke as well,t histime, I hope) a retrospective "39 years ofKripke's lemma" tostudytheseprinciples.ThoughI was not soexplicitlyinfluenced recognize by him, Kit Fine-alsoa sometime visitortotheLogicgroup-will thecoincidencebetweensemanticalideas setouthereandones thathe had quiteindependently. Nearenough,h-validityin c-monoids is Fine's criterion fortheoremhood in R--.&. Naturally I am alsoindebtedtoAndersonandparticularly to Belnap,who not only i ntroducedme to R (startingfrom R--.) butwho suggestedthedecision questionforR (towardwhich I oncehoped thatthispaperwouldcontribute) as a goodproblem. Finally , I am indebted to ProfessorHiroakiraOno, whobuggedme intoturningt hisargumenton its headto producea finite model propertyforBCK. In Meyer and Ono 1994, R--. was almostan afterthought. So it ispleasantto return theargumentto its feet,e xtendingit asindicated.T hanksto all.
REFERENCES
Ackermann,W. 1956
BegriindungeinerstrengenImplikation,The Journal of Symbolic Logic,
vol.21, pp. 113-128 . Anderson, A. R., and N. D. Belnap,Jr.
1975
Entailment, vol. I, PrincetonUniversityPress, Princeton , New Jersey.
Anderson,A. R., N. D. Belnap, Jr., and J . M. Dunn
1992
Entailment, vol. II, PrincetonUniversityPress, Princeton , New Jersey.
Belnap,N. D. 1993
middle, Substructural logics (P. Schroeder-He isLife intheundistributed terand K. Dosen, editors),OxfordUniversityPress, Oxford, pp . 31-41.
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Church,A. 1941 The calculi of lambda-conversion, PrincetonUniversityPress, Princeton, New Jersey. 1951 The weak positive implicationalp ropositionalcalculus(abstract),Th e Journal oj Symbolic Logic, vol. 16, p. 238. 1951a The weak theoryof implication,Kontrolliertes Denken, Untersuchungen zum Logikkalkiil und zurLogik der Einzelunss enschaften, (A. Menne, A. Wilhelmy,and H. Angsil, editors), Kommissions-VerlagKarlAlber, Munich, pp. 22-37. Curry , H. B., and R. Feys 1958 Combinatory logic, vol. 1, North-Holland, Amsterdam. Dickson, L. E. n umbers withn distinct 1913 Finitenessof theodd perfectprimitiveabundant prime factors,American Journal of Mathematics, vol. 35, pp. 413-422. Dosen, K i ntroduction to substructural logics, Substructuml logics (P. 1993 A historical Schroeder-H e isterand K Dosen, editors),OxfordUniversity Press, Oxford, pp. 1-30. Dunn, J . M. 1966 The algebraof intensional logics, Doctoral diss e~ation, Universityof Pittsburgh , UniversityMicrofilms, Ann Arbor; see Anderson and Belby Dunn. nap 1975, pp . 352-369, contributed Fine, K . , The Journal of Philosophical Logic, vol. 3, pp. 1974 Models forentailment 347-372; an updatedversion appears in Anderson, Belnap , and Dunn 1992, pp. 20 8-231. Johansson,I. 1936 Der Minimalkalkiil, einer re duzierter intuitioni stischerFormalismus, Compositio Math emat ica, vol. 4, pp. 119--136. Kripke, S. 1959 The problem of ent ailment(abst ract ),Th e Journal oj Symbolic Logic, vol. 24, p. 324. Meyer ,R. K 1966 Topics in modalandmany-valuedlogic, Doctoral dissertation, University of Pittsburgh , UniversityMicrofilms, AnnArbor. 1978 The relevant t heoryof propositionsis undecidable,unpublishedmanuscript. 1979 A Boolean-valued semanticsforR , researchpaperno. 4, Research School of SocialSciences, Logic Group, Australian N ationalUniversity. Meyer ,R. K, and M. A. McRobbie implicat ion,Austmlasian Journal oj Philosophy, 1982 Multisetsa nd relevant vol. 60, pp. 265-281. Meyer ,R. K , and H. Ono 1994 The finitemodel propertyfor BCK and BCIW, Studio. Logica, vol. 53, pp . 107-118.
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Orlov,I. E. 1928 The calculus of compatibilityof propositions(Russian), Matematicheskii Sbornik, vol. 35, pp. 263-286 . Riche, J. 1991 Decidability,complexityand automatedreasoningin relevantlogic, Doctoral dissertation, Australian N ationalUniversity,C anberra . Routley,R., and R. K. Meyer 1973 The semanticsof entailment(I), Truth, syntax, modality (H. Leblanc, -Holland,Amsterdam,pp. 199-243. editor),North Routley,R., V. Plumwood,R. K. Meyer, andR. T . Brady 1982 Releuani logics and their rivals, PartI, Ridgeview, Atascadero,California. Shaw-Kwei, M. 1950 The deductiontheoremsand two new logical systems, Methodos, vol. 2, pp .56--75. Slaney,J. K. 1985 3088 varieties.A solutionto theAckermannconstant problem, The Journal of Symbolic Logic, vol. 52, pp. 487-501. Thistlewaite, P. B., M. A. McRobbie, and R. K. Meyer 1988 Automated theorem-proving in non-classical logics, Wiley, New York. Urquhart, A. 1972 Semanticsforrelevant logics, The Journal of Symbolic Logic, vol. 37. pp . 159-169; an updatedversion appears in Anderson, Belnap, and Dunn 1992, pp . 142-155. 1984 The undecidabilty ofentailment andrelevant implication,The Journal of material Symbolic Logic, vol. 49, pp. 1059-1073; reprintedw ithadditional in Anderson, Belnap, and Dunn 1992, pp. 348--375. 1990 The complexityof decisionproceduresin relevancelogic, Truth or consequences: Essays in honor of Nuel Belnap (J. M. Dunnand A. Gupta, editors), Kluwer, London,pp . 61-76.
STEWART SHAPIRO
THE "TRIUMPH" OF FIRST-ORDER LANGUAGES*
If "all" and "there exists" are applied to variable propositional func-
tions, the question arises : what is the totality of all propositional functions? Skolem 1928
It is truethatthe. " notion of consequence . . . presupposes a certain absolute notion of ALL propositional functions . . . But th is is presupposed also in classical mathematics, especially classicalanalysis .
Church 1956, page 326n 1. INTRODUCTION
Is second-orderlogic the answer? As thesaying goes, it depends on the question . Thereareseveralimportantmetaphysicalandepistemic issues in thevicinity: thenatureand statusof classes,and theirplace in logic,and therelationship betweenlogicand mathematics. We also have to a ddress significantdetailsof logic : variables,connectives,andquantifiers. A fewahistorical remarkson "thequestion " mightbe in order.Here si its form: Whatis thecorrector best [language or logic] in which to [
? I
The most importantitem is theblank,whichconcernsthegoals orpurposes for which alanguage/logic is chosen. Withoutattemptingto beexhaustive, thepossibilitieshave included: (1) to develop calculi that(more or less)accurately describe(correct)
inferencepatterns of mathematics-t-each calculus to describeinference in one or more a reas;
* Originally published in Foundations without Foundationalism: A Case for Secondorder Logic by Stewart Shapiro (1991) . Reproduced in this new revised form by perm ission of Oxford University Press . lowea larg e debt to GregoryMoor e,bothfor suggest ionsa nd encourag em entto pursue this project . I would also lik e to thankthe participants of theOhio UniversityConferenc e e Logic Colloquium(January1987) , the Center on Inferenc e (October 1986) , the Ohio Stat gh (April 1987) , and the Tel Aviv for Philosophyof Scien ce at theUniversityof Pittsbur University PhilosophyColloquium(December 1987) . This pap er also benefit edfrom John Corcoran , MichaelDetl efsen ,MatthewForeman, Harvey Friedman,HaroldHodes , Ulrich Majer , David McCarty , BarbaraScholz,RobertTurnbull , and twoanonymousrefer ees. 219 C. Anthony Anderson and M. Zeleny (eds.), Logic, Meaning and Computation, 219·259. © 1991 Stewart Shapiro . Printed by Kluwer Academic Publishers, The Netherlands.
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(2) to codifytheunderlying logic ofall rational or scientific discourse-stocapturethemost generalfeatures ofreasoningcompetence(logicism) ; (3) to provide aframeworkin which all (or most) of mathematicsc an be (re)formulated -afoundation ; (4) to formalize particular a branchofmathematics , such asarithmetic orgeometry,whichcaninvolve (4a) codifying eth trueor knowable propositionsof thatbranch(Hilbert),or (4b) describingthestructure(or structures) studiedby thatbranch,or (4c) both; (5) to enhancehumanreason,by providingclear models ; (6) to establisha canonicallanguage of science (logical positivism). Some of thesegoals areassociatedwithdifferentphilosophicalp rograms. Thesecondpresupposes,or postulates, thatthere is an underlying logic for all rational or scientific discourse,p resumablya single logic . Thethirdpostulates thatthereis (or can be) afoundationfor all of mathematics. The fourth seems to involve v aarietyof realismtowardthebranchesin quest ion,in that it ass umes thatthereis a collection of truthsor knowablepropositionsto codifyand/orsome structures to describe. On theotherhand,a substantial anti-metaphysical agendaunderliest hepositivistprogram, item (6). goalssuggestdifferent kinds of logical s ystems, and Clearly,t hedifferent theyinvolve different criteriaof success. Forexample, if thedevelopmentof , as in (1), thenan eff ectively a calculus is takento be partof thegoal of logic enumerable(deductive)consequencerelation is requisite. Indeed,it seems to be partof thenotionof "calculus"t hatthecorrec tness of deductionsshould be mechanically checkable. Lesstrivially, if one isconcern ed with codif ying a reasat once (2), correct inference, eitherin a givenarea(4a) or in all thenone wouldrequiretheconsequencerelation to matchhumaninference competencein appropriateways. If one furthertakesinference competence to be effective in some sense, thenthedeductiveconsequencerelation s hould correspondand be effective.' If, however, one holds a mor e grandioseview of humaninferencecompetence,thenan effectively enumerableconsequence relation is not requisite,a nd may not be desired . As we shall see, Zermelo held a view like this. G6del did as well , butapparently he did notaccept thegoal ofdescribinginferencecompetence. In cases wherethegoal isthe description of amathematical stru cture(4b), one would also not insiston an effect ive(semantic)consequencerelation .? 1 For a Fregeanaccount of th e relevance of effect ivenessorf goa l (2), see Wagner 1987. In 1983, I argue thatproblem s like these wer e central factors in the mathem atical cha ra tion ofthenot ion of effecti veness m or e thanthree decad es later. See also Gandy acteriz 1988. 2Vebl en (1904) seem ed aware of the factthatdifferentgoals suggest differentlogical system s . The purpose of his axiom atization of geometry was to describe Euclidean space.
THE "TRIUMPH" OF FIRST-ORDER LANGUAGES
221
The otheritem in thequestionform, "languageor logic" is only mata terof terminology,butclaritycan forestall confusion. Thereare, I believe, " to be a deeperissues thatlie nearby. It is now common to define a "logic logicalvocabulary , a set ofoperatorsand variablesorts. Such vocabulary is combined with non-logical terminologyto determinea languag e. Although this is a usefulconceptionfor manypurposes (see the papers in Barwise and Feferman 1985), it is unnatural for historicalanalysis . The problem " and is thatthedefinitionpresupposesthattheboundarybetween "logical "non-logical" terminology has alreadybeen drawn.T hereis a sense in which thedisputesbetweenadvocatesof first-order logicand advocatesof higher. In particular, thestatusof certain orderlogicconcernjustthis boundary , or membership, relation is in question . The quantifiers and thepredication higher-order camp takestheseitems to be logical, while the first-order camp thusbe imprudentto decidethisissue takesthemto be non-logical. It would in advance. Withsome hindsight,let us definelogic a to consist of a formal language , or class of formal languages,together with adeductivesystemand/ora semantics. First, a language is a collection of formulas . By itself, it is neitherfirstordernor higher-order.As Gilmore (1957) haspointedout, languageswith "predicatevariables"can beinterpreted as many-sortedfirst-order systems. theinsightbehind the so-called "non-st andard" semantics In effect, this is forhigher-order languagesdeveloped by Henkin(1950) (for whichcommon deductivesystems are complete; see Shapiro 1991, chapters3-4). Thatis to say, afirst-order model-theoretic semanticsis available forvirtually all of the(finitary)languages underconsideration here, andthisfirst-order semantics issound and completefor common deductivesystems. In such cases , the"predication"relation (or themembership relation) betweenobjectsand properties(or sets) is takento be non-logical . The terms "second-order" or "higher-order" refer tointerpreted , or partiallyinterpreted, formallanguagest hatcontainvariablesrangingover all collections, or all properties,or allrelations, or allpropositional functions,of whateveris in therangeof theordinary,or first -order,variables.At leastin part,this is amatterof the"meaning" of sometermsand,thus, "semantics" . broadlyconstrued.For ourpurposes, it is amatterof interpretation If a languagewith predicate(or relation)variablesis presentedwithan An explicitaim was "to show thatthereis essentially only one classof which . . . the axioms are valid . . . " He added thatonce such a "categorical " characterization is accomplis hed, er axiom wouldhave to beconsideredred undant " , towhicha footnot e is added "any furth ". . . even were it notdeducibl efrom the[other]axioms by a finitenumber of sy llog isms ." a xiom" is redundantonlyto thedescriptionof the structure , not to the Such a "further codificationof thetru e principl es thereof. Compare Veblen's orientationto thatof Hilbert (1900a, problem 2) : "When we are investigatingthe foundation s of a scienc e,we must set up a syst em of axioms which cont a ins anexact and com plete des criptionof the . . . ideas of thatscien ce . . . no stat em entwithinthe realmof the scienc e .. . is held to be tru e unless it can be deducedfrom theaxioms by m eansof a finitenumberof logicalsteps."
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explicitlyarticulated model-theoretic semantics,thesituationis, or can be, straightforward. We only need to see the if stated rangeof the predicate variables(in each model) ist heentirepowersetof thedomain, or includesall of itsproperties(or relations , etc.). If so, thelogic ishigher-order. Of course, we can still wonderwhethert heuse oftheword "all " in themeta-languag e meta-meta-theory (if there really meansall, and forthiswe can look to the is one). Butperhapsthisis perverse. More substantially , one can wonder whethert heauthorhas thesame notionof "property"thatwe do. It may be thatnot everysubclass(in thepresentsense) ofthedomain is theextension of a "property".Some authors,forexample,are notdisposed to countenance propertiesthatare notdefinable,or aredefinableonlyimpredicatively . Of course,model theoryis not much help in historical analysis,being a relatively latedevelopment(see Etchemendy 1988). Typically,we are often leftwithoutan articulated semanticsatthevery beginning. In such cases, if predicate(or relation)variablesare introduced,we must figureoutwhat they are"intended"to rangeover. Onepossibility,of course,is thatthey aren'tintendedto rangeover anything.The authormight be developinga "meaningless"formallanguage,p erhaps with adeductivesystem. In this case, thelogic isneitherfirst-order nor higher-order . Anotherpossibilityis thatthelanguageis to be regardedas atleastpartially interpreted, butthe interpretation is leftinformal,or intuitive. Then it is a matterof exegesis. Some authorsclearly i ntendtheirpredicatevariablesto rangeover "all" properties,b ut,as noted,theymay not mean by"property"whatwe mean by "property " . As shall be shown, in almostallthecases understudyhere before, say, 1930, even if standard a higher-order interpretation distortst he author'sintentions,a Henkininterpretation is a much greater d istortion. r ootscan be traced , I must make a numberof disBeforethehistorical tinctions,all of which concernthedevelopmentof logic sincethelatter half thescope of logic , of thelastcentury.We must attendto differing views on andvariousconceptionsof class, variable , and quantifier. Threetraditions can bedistinguishedin thehistoryof modernlogic (provided one does not i nsiston sharpboundaries).One ofthemoriginateswith Booleandincludes,a mongothers,Peirce, Venn,a ndSchroder.T his algebraic oranalogues,betweencanonsof inference a nd school focused onrelationships, algebraico perationslikeadditionand multiplication. A primaryaim was to develop calculi common to thereasoningin differenta reasof mathematics. Boole'sseminalwork (1847,1854), forexample,was a formalcalculus that could beinterpreted a mong propositions,classesand probabilities . As the name suggests,theorientation is thatof abstract a lgebra,alongthelines of grouptheoryor fieldtheory.One beginswithone or moresystemsof related operationsa ndabstracts a common structure . One set of axioms ist henformulatedwhich issatisfiedby eachsystem and whichilluminates all oft hem. In some ofthecases athand,theaxioms are dubbed "laws ofthought" . The second tradition d atesback atleastto Euclid,and, forpresentpur-
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poses, includesDedekind, Peano, Hilbert,and the postulatet heoristslike Veblenand Huntington.The aim of this mathematical school is theaxiomatization of particular branchesof mathematics.Euclid,Hilbert(1899), and Veblen(1904) developedgeometry ; Dedekind(1888) andPeano(1889) devel(1902, 1905) developed oped arithmetic ; andHilbert(1900) andHuntington analysis.An axiomatization producedby thealgebraicschool isintendedto school are applyto severalsystems atonce, whilethoseof themathematical primarilyintendedto apply to a singlesystem. It shouldbe notedthatin at least some cases, thisis a difference of emphasis rathert hana difference of principle.The mathematicians occasionally consideredalternate interpretationsof axiom systems, typicallywhenestablishingindependenceresults. Some members of themathematical school didnotindicateinterestin the studyof logicper se. They are includedhere becauseof theirrole inthe historyof logic. Ourthirdtradition, calledthelogicist school, includesFrege, Russell, and, . Theiraim was to codifytheunderlying logic perhaps,theearlyWittgenstein of all rational, scientificdiscourse. Forthem,logic isnot theresult ofabstracin Logic tionsfrom thereasoningin various disciplines, or varioussystems. concernsthemost generalfeaturesof actual(precise) discourse, featuresindependentofsubjectmatter.Frege'swritingsc ontainnumerouscomparisons betweenhis workandthatof thealgebraists.In 1883, forexample, he wrote : I did notwish torepresentan abstract logic byformulas,b uttoexpress a content.. . in more exactand clearfashion . .
The difference seems to be t hatthe algebraists(and themathematicians) wereconcernedwithcertainsystems,or algebras,andwithreasoningrelative to thesesystems, while the logicists were attemptingto codifyunrelativized reasoning,reasoningthatis applicablein everycontext.i' Goldfarb(1979) and van Heijenoort(1967a) arguethatthethesisthat logic does not involve (a process abstraction of) a nd theidea of a universal languageapplicablein allcontextsseparatest helogicistsnot only from t he mathematiciansand algebraists,butalso fromcontemporarymodel theory. To use presentterminology, logicistsystems have nonon-logical terminology . Today,typicaltreatments do not refer toparticular a subjectmatter.R ather, schematicletters are used for nonlogical terminology, suchas predicates,relations, and individualconstants.These letters get different i nterpretations in thevariousmodelsof thelanguage . In thisrespect,theorientation resembles 3Using contemporary techniques, relativiz ed reasoningcan be accommodatedin a logicistsystem. One simply restrict s thequantifiers in some formulasto particular structures . When an algebraistor mathematiciansays "for allx . . . " in referen ce to a domain d, a logicist couldwrite"for allx, if x is in d, then. . . " From the other p erspective, the notion of unrelativized reasoningwas for eignto thealgebrai c approach.Som ething similar m ightbe obtainedif, concei vab ly, on e could invoke anotionof "allsystems whatsoev er " and abstract"laws ofthought " from it, or iftheentireuniver secouldbe takenas a single system. Butthisis hindsight.
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thatof thealgebraists.Instead of schematicletters,logicistsystems have variables,which haveassociatedquantifiers.T he wholenotionof "model" or "interpret at ion" is foreignthe to logicistproject. There is nothingto "interpret" in variousdomains." One importantdistinctionconcernstheintendedranges of variablesa nd quantifiers.Thereare threecases. The algebraicschool of logicians took variablesto rangeover anunspec ified , butfixed domain. As in algebra,this domain is the resultof an abstraction from more specificcontexts .P For the mathematical school, thevariables of each system are intendedto range over afixed and specified domain, such asthenatural numbersor Euclidean took their space. The logicistsFrege, Russell,and theearlyWittgenstein, systems to representt heunderlying logic of all rational discourse.For them, (first-order)variablesr angeover all objects whatsoever . All oftheschoolsdevelopedquantifiers,buttheywereunderstood differa nd Peano, forexample, tookfree va riablesand quantifiers in ently. Frege thecontemporary sense except, perhaps,thattheyallowed(atleast)secondorderterminology . The algebraicschool, on theotherhand, tookquantified propositionsto be ext e nded conjunctions and disjunctions . This led tothe consideration of infinit elylongexpressions, discussion of "convergence" , etc. Noticethata domain is needed toexpandquantifiers intoconjunctions and disjunctions,and, thus,once againthe algebraicschool is rel ying on a fixed domain. The finaldistinction(for now)concernsthe notionof class andits role in logic,togetherw iththestatu s of theconcomitantm embership relation. There is a traditiondatingat least to Boole thattakes classes (or intensionalcounterparts like propertiesor propositionalfunctions) to be under thepurviewof logic. God elacknowledged t hisin his paperon Russell(Godel 1944), writingthatmathematical logic dealsw ith"classes [and]relations ... insteadof numbers, geometricforms, etc." As noted, the algebraicschool tookthesubsetsof a fixeddomain to beunderstudyby logic- thiswas one of thestatedinterpretations of theirsystems. Thereis some hindsighthere, however, sincetheydid notalwaysdistinguishbetweenthe membership relationand thesubsetrelation , and sometimes theydid notdistinguishthe singletonfrom itselement.Frege alsotookthestudyof conceptsand their extensionsto be withinlogic (see forexample, Frege 1884), as did Russell before his no-classeriod. p Whatis crucialhere isthatthe focus is on the subsetsof a fixed universe, or domain. Thatis, thecontextof thetheory 4 As noted , it is notclaimed thatthis taxonomyis exclus iveor com prehe ns ive. Indeed , m embers of the differ en t choo s lsdrew from eac h other. Mor eover,the division is limited e re. Later logicians cannotbe classified along these lines , and to the periodunder st udy h contemp orary logic has eleme ntsof allthree. Ther e is, of course, no need to forc e ea rlier logicians , such as Aristotl e,the Stoics, and the Scholastics, into this classifi cati on. 5In the axioms forgro up theor y, for exa m ple, the (object- la ng uage) variab les ra nge over a fixed, butunsp ecified group . The purpose is to make st a te me ntsap plicab le to allgroups .
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determines,or presupposes, a universe ofd iscourse-arangeof the (first order)variables.A setis a subdomainof thisuniverse. Call this the logical conception of set. Thus, in arithmetic , a logical s et is a collection of natural numbers;in geometry,a logical set iscollection a of points,etc. Toreiterate, s ets simpliciter, only logical sets withina giventheory , thereareno logical although, of course, thec ontextcan be left unspecified (see Shapiro 1991, section1.3). On theotherhand, currentaxiomaticset theory , as it developed from the work ofC antor,Zermelo, Skolem , Fraenkel,etc., fits the mold of the mathematical school. The axiomatization refers to abinaryrelationon a domainconsistingof a (possiblyempty) collection ofurelements, sets of those , sets ofsets of those, etc., with the"set of" relationpossibly iteratedinto iterative conception of set. Aniterative set is the transfinite . Call this the sometimes called a "well-founded .set" In short,t hen,an iterative set is amember of theset-theoretic hierarchy, . The corresponding memthedomainof a single theory, axiomaticsettheory bership relation is non-logical (like thepredicationrelation in non-standard semantics). In contrast , the logical conceptionof set always refers to do-a main fixed bycontext . A logical set issubclass a oftheuniverseof discourse. systemsof thealgebraicand Its membership relation is a partof the logical . Partof the logicist schools, as wellas second-orderlogicand type theory are issue concerninghigher-order systems is the extentto which thel atter . "legitimate"p artsof logic Confusion between the two conceptsof set can arise in set theory, since therethe"fixeddomain" (for the logical conception)is itselftheset-theoretic hierarchy. An i terative set is amember ofthathierarchy and, in thiscontext , of iterative s ets-whatis todaycalled a "class ". a logical set is collection a entirehierarchy,t hecollection of alliterative Thus, withinset theory, the sets, is a subdomainof itself. So the entirehierarchyis a logical set . With 's paradoxshows thatin set this terminology , theargumentyielding Russell thatare notiterative sets, or inother theory , thereare logical sets (classes) words, thereare collections of iterat ive sets thatare not members of the set-theoretic hierarchy . In particular , theentirehierarchyis not aniterative , theword "set" is set. The customtodayis that,whendiscussingsettheory used only for "iterative set" and, in thisc ontext,theword "class" is used for setthatis not aniterative set. "logicalset". A "proper class" is a logical This suggeststhatRussell'sparadoxshouldtroubleonly thosetheorists notionof setand an all-inclusive domain, onethatmight usingbotha logical include all of its subdomainsas members. To some extent,historyconfirms this "expectation" . For the mostpart,onlythelogicists were affected, for they tookvariablestorangeover all objectswhatsoever , which at least prima facie includes all collections objects of . AlthoughFrege clearly separated conceptsfrom objects,he came toassociateeachconceptwith anobject- its extension . The assertionof the existence extensions, of and the axiomthat
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twoconceptshavethesame extensiononlyif theyareco-extensiveled tothe paradox(Prege 1893) . Russellhimselfavoidedthis (and other)antinomies by separating t heallinclusivedomain intoindividuals,classes (or properties) of individuals,classes ofclassesof individuals , etc. For presentpurposes, theinteresting cases arethoseauthorswho did not feelthreatened by theparadox. Zermeloindependently discoveredRussell 's paradoxa fully earbeforeRussell(see Rang and Thomas 1981) , butdid not publishit, andcontinuedhis work insettheory.W hetheror notZermeloheld the"iterat ive conception"of set atthistime, he apparently did notbelieve thatset theory,as heconstruedit, wasundermined.fIn theaforementioned paper on Russell , Codel (1944) also saysthatthe iterativec onceptof set was neveraffected bytheparadoxes.For themost part,thealgebraicschool was alsounaffected, since theyonlyconsideredthesubclassesof a fixed domain, and did notenvisiona universald omain-onethatcontainsall of its subdomainsas members. In 1890 (page 245), Schroderremarkedthatone cannot , withoutcontradiction , letthe fixed domain consist of all o bjects. The contrast w iththelogicistsis striking." It shouldbe mentionedthatthisrelatively neatdivision of treatments of setandclassrequiressome hindsight . Aroundtheturnof thecentury , Hilbert was more or less in themathematical school(withsom e influence from t he algebraists), andhis correspondence withFregeindicatesj usthow far he was from thelogicists (s ee Resnik 1980, chapter3). As far as I know , he never considereda universal d omain. Thus, accordingtothethesisathand, Hilbert shouldnothave beenbotheredby Russell 's paradox. Yet, afterlearningof theparadox, he tooka cautiousview ofsets. In 1905 (page 176), he wrote: . .. theconception s . .. of logic, conceivedin thetraditional sense, do notmeasureup totherigorousdemands thatsettheorymakes . . . Yet treatment if we observeattentively, we realizethatin thetraditional of . . . logic, certainfundamental notionsare used, such as thenotion of set.
The proposedsolution,of course, was todeveloplogicand mathematics,includingsettheory,simultaneously, andthento providea finitaryconsistency proofof thewholesystem. This was notto be. 6See Moore 1978. Zermelo (1908a) does m entionRussell'sparadox in the op eningparagraph, notingthat"the very existe nce of et [s theory]seem s to be thr e aten ed". However , by carefulanalysisof the histor ical co n te xt,Moore shows thatthe ope ra t iveword hereis "seems", forZermelotookthe thre atto be more apparentthan real(see also Zermelo 1908, sect ion 20). For example, the stated planis to "st a rtwithset theor y as it is historically given ". This programwouldbe a rather po or one if settheor y "as it is historically g iven" is not fundamentall y sound. In Zermelo'smind, the culprit s i an unrestrict ed principle that every "collecti on.. . of .. . well-distinguished obj ect s" is a set. This seems close to the the parad ox above-notevery logical set is a member of the hierar chy. way I formulated ggestion by Boole thatone 7In the relevantpassage, Schrod er criticizes an ea rlier su inte r pret at ion "class1" of is the univer salclass, the classwhich cont a insevery t hingconceivable. Schroder's argume nt si ra th erobscure, and includ es the conflat ion of the eleme nt and subset relation s. Frege (1895 ) had littl e troubl e disposing of Schr oder 's argument.
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2. NARRATIVE
In themore directsketchesof therelevant histories , thereare threemajor items, two of whichoverlapconsiderably.The first istheemergenceof firstorderlanguages (andsemantics)as thestandard in formal logic . The second is thedevelopmentof set theory and theemergenceof first-order ZFC. 8 The thirdis thecontemporary controversy overthestatusof second-orderlogic. The first two are presentedhere, thethirdis deferredto thenextsection. Much of thehistorical materialin thissectionis based onthework ofGregory Moore (1980, 1988), andon severalsuggestionshe made to me. Of course, I maintainresponsibilityfor anyerrors . Once disagreementsovertheseissues beganin earnest , Skolem andGodel werethemain proponentsoffirst-order languages.Fordifferent reasons,each heldthatonlyfirst-order a xiomatizations arelegitimate.The higher-order language"opposition" was championedby Zermelo,Hilbert,a nd Bernays.? Thereis an interesting absenceof correlation betweentheline-upacrossthis battlelineand thatover moretraditional philosophicalm atters.Godeland BernayswerePlatonists (each in his own way), whereasSkolem andHilbert embracedvariousversionsof formalism,relativism , andfinitism. 2.1. The development of first-order logic
Priorto Lowenheirn's theorem(1915) , no one haddiscussedfirst-order logic . In various ways, authors from all schools as even adistinguishedpartof logic introducedvariablesrangingover both objectsand properties(or propoof the sitionalfunctions, or classes). As noted,one statedinterpretation systems of Peirceand Schroderwas thesubclassesof a fixeddomain. Each proposition(with a freevariable)d eterminesa class. They alsointroduced butdid notcarry second-order variablesrangingoverproposition-subclasses, this very far. By contrast, Frege (1879, 1884) introduceda nd brilliantly exploitedsecond-ordervariablesrangingoverconcepts(but, again,extensions of conceptswereregardedas objectsand thusin therangeof thefirst-order and theancestral were all variables).The notionsof identity,equipollence, defined insecond-order language,thelatter playing acentral role inthefornumbers. Frege did notseparatethefirst-order part mulationof thenatural ofthesystem,andin a sense hecouldn't, sincethelanguage c ontainsno predsymbols otherthanhigher-order variables . A variation on icateand relation this theme is Russell, who went on to develop ramified type theory,with different variablesortsforindividuals , propositional functions,p ropositional 8The words "emergence" and "standard"in this paragraph(as well as th e word "triumph" in thetitle)are not m eantto endorse a th esis thatthe first-order theoriesought to be standard . As notedin the final ectionof s thispaper, my own sympathiesareon the higher-orderside. See Shapiro 1991 for afurther a rticulation of higher-order logic . 9 John Corcoranin for m s me thatTarski eventuall y rejectedhigher-ord er logic, even thoughmuch of his earlywork concernssuch syst ems.
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functions of propositional functions , etc. For Russell,a tleast,thepropertiesor propositional functionsa reintent urnson this,however , since anotionof "identitybetween sional. Not much properties"plays nosignificantrole. Indeed, such notionwas a not formuonlyappliesto objects,the items lated . In Frege's case, theidentityrelation in therangeof thefirst-order variables . The closestthingto "identity"that Frege applies to conceptsis coextensiveness , which, of course, is the criterion of identityamong extensional items.10 It does not take major a exegeticalstudyto showthatthe logicist systems areatleastclose tocontemporary ( standard)higher-order logics, and it tothinkof logicistsystemsas first-order (e.g., to would be amajordistortion impose a non-standard semantics). First,as notedabove, thesystems were notintendedto beuninterpreted formal calculi. Second, and more important , in contemporary t erms,thesystemsdo not have any non-logical terminology, much less a non-logical predicationrelation.E veryitem has a fixed meaning, everyvariablea fixed range . It is truethatRussell(at least)may not mnge ofpropertyvariables in contemporary higher-order have envisioned the systems, becausehis propositional functionswerepredicative,b uteven this difference is not clear in light of the axiom reducibility of . In any case, the difference is due to Russell'sviews on thenatureof propositional functions, In contrast with non-standard senot ontheextentof thevariable-ranges. e-thevariablesrange mantics, he did notregardthe ranges as indeterminat functionsof appropriatetypeand level. over all propositional Peano also introducedvariablesrangingover "classes" ofn atural numbers, and similarpracticeswereadoptedby othermembers of themathematof geometryby Hilbert(1902) and Veblen ical school.T he axiomatizations (1904), and theaxiomatizations of analysisby Huntington (1902, 1905), all containsecond-ord er terminology , or somethingequivalent, essentially . Veblen andHuntington employ the more usual second-orderaxioms of conti nuity,essentially due toDedekind. For Hilbert , it wastheextremalaxiom thatthereare to be noproperextensionsof thesystem.II It must benoted, however , thattheconsequencerelation was notdiscussed. Some of theseauButthesecond-order variables thorshad verylittle to sayaboutlogic at all. were clearly intendedto range over all classespropertiesof or the relevant thesesystems to firstdomain. If, throughan actof hindsight,one restricts orderlanguages,Hilbert'sextremalaxiom is notsatisfiableand, of course, a refalse.12 thecategoricity theoremsproved by Veblen and H untington JOlt is consistentwithFrege's view thatcoextensi veness is acongruence relat ion among concepts. Thatis, if \lx(Yx == Xx) and 4>(X), then 4>(Y) (provided thatY is free for X in 4>(X» . I am gratefulto HaroldHodes for pointingthis out. 11 As Moore (1988) notes, this "complet eness axiom" did not appear in the original ons . Germanedition(1899), butwas addedin the French (1902) and Englishtranslati 121t follows fromtheso-calledupward and downwardLowenheirn-Skol em theor e ms that no first-ordert heorywith an infinitemod el is categori cal. Indeed, for any such theory T and any infinite cardinalK , ther e is a model of T whose domain has cardi na lityK .
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Our attemptto characterize theemergenceof first-order languagesa nd logic iscomplicatedby the use or discussionof infinitaryformulas. This of thequantifi ers in thealgebrai c datesback (atleast)to theinterpretation school, and continuedthroughtheperiod understudyhere. For example, Hilbertadoptedsuch aninterpretation in 1905 (butlaterc ame to rejectinPrincipia Mathematica, finitarysystems), and in thediscussionsurrounding Lewis (1918) and Ramsey (1925) bothintroduced infinitaryformulas . With hindsight,suchsystems providea hierarchy of logicst hatare "intermediate " in some sensebetweenfirst-order and higher-order. The algebraicschool, as well as Whiteheadand Russell(1910), oftendiscussedthelogic ofpropositions, the logic of classes, and thelogic ofrelations, but thesedid notcorrespondto thepresentdistinctionbetweenfirst-order logic andhigher-order logics. Concerningthealgebraists,thereis thecomplicationof infinitaryformulas,and theconflation of syntaxand semantics left anunclarity concerningwhattherestricted s ystems were like. Theseparation of first-order systems,or sub-systems, is due to Lowenheim and Hilbert.I'llbegin with thelatter. In a series oflectures in 1917, Hilbertappliedtheaxiomaticmethodto geometryand logic (see Moore 1988). This is easily recognized as an extension ofthemethodology of themathematical school. Heexplicitly developed primitive symbolsandaxioms forfirst-ord er logic, which he n amed "t hefunctional calculus " . Hilbertnoted thatthissystem is sufficient to codify edductions within variousbranch es of mathematics ; buthe added thatit is notadequate of mathematicaltheoriesthemselves become] [ an obwhenthe"foundations ject of investigation " . For "foundational study", an "extendedcalculus ", containingvariablesr angingover properties,was developed. This extended calculus is, in effect, a version of ramifi ed typetheory(withtheaxiom of reducibility).Hilbert'slogicalsystems did notappearin print,however ,until Principles of mathematical logic, by Hilbertand Ackermann, was published (1928) . Eventually , Hilbertcame to acceptthe simple theoryof types, an w-orderlogic. The importantpointis thatin bothplacesHilbertregarded first-order logic as adistinctsub-system of all of logic. It was dubbed the "restricted functional calculus".He suggestedthatset theory - and even -is incapableof anadequatetreatment in therestricted system. arithmetic Hilbert'spoint seems to be thatthe first-order s ystems cannotadequately formulate many propositionsandconceptsnecessaryformeta-mathematical Thus, if the categoricity state me ntsproved by Vebl enand Huntingt on are applied to fir s tom s tates thatther e are orde r th eori es, theyare false. As noted , Hilbert'sext rema l axi to be no proper extension s of the syste ms. But if the theor y is first-orde rand if it has an infinitemodel (which it does ), then ther e is no maximal m od el. Ever y infinit e m od el o put this histori calp oint in Quinean of any first-order theory has a prop er exte nsio n. T terms, suppose we were engaged in "tra ns la ti ng" ethsystem s of these au t ho rs in t o cur rent m a thematicaldialect. If it is insistedtha tthe "out put " of the tra ns la t ion be formulat ed in the correspo nd ing conte mpo ra ry fir st-order systems , then we mu st at t ri butesome rath er . elementary mistakes to the auth ors
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studyofvarioustheories . Examplesincludetheprincipleofinductionfor formulas,themembership relation, number,andcardinality (see Shapiro 1991, chapter5). He showed how all of these notionshavestraightforward formulationsin thefull,higher-order system. Hilbert'sco-workers , such as Bernays (1918, 1928), Schonfinkel(1924) (and Bernays and SchOnfinkel1928), Ackermann(1924), andvon Neumann (1927) , all used essent iallythesame system. In 1918, Bernaysformulated and proved thecompletenessof propositional logic.WithintheHilbertprogram, this resultwas thefirst precisesolutionto acompletenessproblem for a proper part of logic. Similarproblems forothersub-systemsof logic became crucialcomponentsof theprogram,and representedstagestoward whatwouldultimately be a consistencyprooffor all of mathematics.In this spirit,theproblemconcerningthecompletenessof first-order logic was proposed in Hilbert 1929 (and Hilbert and Ackermann 1928) .13 This, of course, was establishedby Godel (1930) (who alsoadoptedthephrase "restricted functional calculus"to denotefirst-orderlogic). t healgebraic In contrastw ithHilbert,Lowenheim was firmly placed in school. Henames Schroderas thesourceof his interestin logic,and all of Lowenheim'sworkconcernsvariantsof Schroder'ssystems (see Thiel 1977, page 237, and Moore 1988) . The celebrated t heorem,appearingin 1915, was noexception. Lowenheirnfirstmade a distinctionbetween a"relational had to be of finite expression" and an "individualexpression". The latter lengthand couldonlycontainvariablesover individuals . This is close to formulas,but in Hilbert 's distinctionbetween first -orderand higher-order keepingwiththealgebraic school,Lowenheimthought oftheseas expandable intoinfinitelylongstringsof connectivesa nd quantifiers.On this matter, however,thecrucialdifferencebetweenLowenheim and his predecessorsis thattheonlyinfinitary formulasenvisionedhere weret hoseequivalent (in the system) tofinitaryformulas.This variation on thealgebraict hemepermitted ofthefirst-order partof his Lowenheimtocarefully distinguishtheequivalent system. His version oftheLowenheim-Skolemtheoremconcernedthatpart, buthis proof relied heavily on thefullsystem. Thatis, theproofinvolved bothhigher-order and infinitary formulas. Lowenheirnalso showedthatthe resultdoes not hold for thefull(higher-order) system. It was Skolem who showed t hatLowenheim's theoremcould be proved in a first-order ( meta-)language(by using Skolem functions) . He extended thetheoremsomewhat,applyingit tocountably infinite sets of formulasa nd to some infinitarylanguages(Skolem 1920) . By 1922, however, he ceased mentionof the extensionsof the theor em to infinitarylanguages , and he 13By this time, the distinctionbetw eenproof theory and sema nt ics wasgettingclear . Some of theproblem s posed in Hilbert 1929 concern th e deductivecompleteness of (secondorder)arithmetic . After notingthatthePeano postul a tesare categorical , Hilbertasked sistentwitharithmetic, thenit s negation for aproofthatif a sente nce isshown to be con cannotbe shownconsistentwitharithmetic.
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logic istheproper basis forset-theory a nd, began to urgethatfirst-order indeed, for all of mathematics. I say "urge" here , not "argue". As far as I can determine,neitherSkolem nor Godel gave detailedreasons for their insistenceon first-order logic. Nevertheless , theirassertionsare notmerely logic isthe most interesting or fruitful a reafor directivesthatfirst-order research . They areclaimsthathigher-order variablesdo not belong in logic . It was atthistime thatSkolem began idscussing the"Skolem paradox", whichconcernstheinterpretation of axiomaticsystems. It is totheaxiomatizationof settheorythatI now turn . 2.2. Axiomatization of set theory
In thispartof ourstory,thecrucialitem is theaxiom of separation: Vx3yVz(z E Y == z E x & P z).
The issue concernsthestatusof thevariableP . In first-order ZFC, the axiom is a scheme, eachinstanceof which isobtainedby replacingt helett er P with afirst-order formula(notcontaining y free). Insecond-order t reatments , thelett er P is a predicatevariable,regardedas eitherfree or asboundby a prenexuniversalquantifier (see Shapiro 1991, chapter5). Here we beginwithZermelo. Earlierset theorists,likeCantor , were not particularly concernedwith axiomatization a nd underlyinglogic. Late in 1904, Zermelo provedthewell-ordering theorem(Zermelo 1904) and, almost immediately , theproofdrew intenseresistance(sec Moore 1982). Virtually was attacked . During1907, in response everyaspectof Zermelo's treatment to hiscritics,Zermelocomposed his firstaxiomatization of set theory(Zermelo 1908 and 1908a). Withinthetradition ofHilbert 's methods,andwhatI have calledthe"mat hemat ical school" , the goal was to elucidate (anddefend) theassumptionsbehindtheproofof thewell-ordering theorem . The version oftheaxiom of separationthatappearedin Zermelo'sfirst formulation a ssertedthatfor everypropositionalfunctionP(z) , if P(z) is definit for a setS , thenthereexistsa setcontaininge xactlythoseelements of S for whichP(z) is true. Presentconcern,of course, is withtheopening universalquantifier a nd withtheword "definit". At thetime, Zermelo only on the notedthata given P(z) is definit for S if themembership relation domain and the"universally valid laws of logic" determinewhetherP holds aws of logic"thatare even hintedat here for eachelementof S . The only "l are theprinciplesof bivalenceand excludedmiddle. Perhapsther e is little point in speculatingon this, but as Moore (1980, page 109) notes, it is reasonably clearthatZermelo did not i ntenda restriction to thoseproperties definable infirst-orderlogic. First, the axiomatization o ccurredalmosta decadebeforeLowenheimand Hilbertseparatedfirst-order logic as a subsystem of logic;second, theaxiom of separationwas presentedas a single
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sentencewith avariablerangingover propositionalfunctions;and, third, Zermelolatera ttacked proposalsto limit settheoryto afirst-order language . Like thepreviousproofof the well-ordering theorem,this axiomatization attracted criticism, but now thecriticshad somethingmore tangible(or, directedattheaxiom shallwe say, moredefinit) to attack.Much of this was example, Russell,writingto of choice,butseparationwas not ignored. For Jourdainin 1908, called it "so vague as to be useless"Grattan-Guinness (see 1977, page 109); andPoincare(1909) objectedthata propertythatis definit might not be well-defined . 14 In attemptingto improvetheaxiom ofseparation,HermannWeyl(1910) proposedthata propertyis definit iff it iseitherof the formx E y or x = y, or is obtainedfrom suchpropositionsby finitely many uses negation, of conjunction,disjunction,existential quantification, andsubstitution of aconstant for a variable . He did notindicatewhatsortsofexistential quantification are perthatfirst-order languages had yet topublished be mittedhere. Again, recall as separatesystems. Weyl 1917 containsa similarproposal,but therehe s tatedthatvariablesrangingoverpropertiesa renotcountenanced. explicitly variables asu nsuitable for logic, He went on to explicitlyreject higher-order t hat probablythe first personto do so inprint. However, oneshouldnot infer Weyl (1917)explicitlyaccepted classicalfirst-order logic;sometime afterthis, perspectivein mathematics(see Moore 1988) . Weyladoptedtheintuitionist Weyl(1910) notedthathis formulation of definit seems to presupposethe notionof natural number, as evidencedby the phrase "finitely many uses of". This raises afoundational issue concerningw hethersets ornumbersare more fundamental, thequestionbeing which of these are toformulated be in for all of mathetermsof theother.If set theoryis to serve as afoundation it matics, it would seemthatone oughtto be able to developindependently thenatural numberswithinset theory(as, of arithmetic,and then formulate forexample, Dedekind 1888).15 Afterseveralattemptsto explicatedefinit withoutusing thenotionof number, Weyl came to reject Zermelo's(and 14See also Schoenflies 1911 . Poincare'sattackwas based in parton his rejectionof the actualinfiniteand of impredicativedefinition(see Goldfarb 1988). 15HaroldHodes suggestedthat,c onceivably,o ne couldattempta bootstrapconstruction. Firstformulate a rudimentary s ettheorywithoutseparationanddefine "thenatural numbers" to be, say, thefinitevon Neumannordinals.T hen, withthisas meta-theory, one might employ thenewly-definednotionof natural number to formulatea set theorycontaininga principleof separation,perhapsalongthelinesof Weyl 1910. However,it is not clearthatthearithmeticformulated in therudimentary s et theoryis sufficient,s ince the usualway toderivea generalinductionscheme withinset theoryuses separation . Moreover, theelementary s et theorydoes notappearto be capableof codingsome collections necessarytodo meta-theory ( e.g., thecollection o f well-formedformulas) . If an induction principlewereexplicitlystatedand addedas an axiom scheme, it wouldseem thatone is takingthenatural n umbers as fundamental afterall. Inany case, thissortof thingwas notattemptedby anyoneduringtheperiod understudyhere. I wouldsuggestthatone reasonthatsuch a set theorywas not conceivedis thattheprincipleof separationwas (and is) regardedas essentialto thenotionof set.
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Dedekind's(1888» set-theoretic formulation of thenatural numbers. Fraenkel a ndSkolem, whobeganwritingon thesubjectin theearly1920's, enjoyedconsiderableinfluence on t hedevelopmentof settheory.Although thesystems theydevelopedwereremarkablysimilar,theirattitudes toward logic, set theory, and foundations of mathematicswerequitedifferent . skepticalof thewholeenterprise Fraenkel(1922, 1922a, 1925) was rather ofmathematical logic. Heattacked Zermelo'snotion ofdefinit justbecauseit relied on the imprecisebasis of"general logic". Presumably,t hisis a reference to Zermelo's invoking the"universally valid laws of logic" theformulation in of definit property.Fraenkelt husseemed tosuggestthata sufficiently precise notionwas notcharacterized. To improve it, heintroduced an alternate : f(x) is definitorisch if it can be obtainedby thefiniteiteration of theoperation of powerset,union, and unorderedpair. Noticethat,like Weyl's version , this formulation c ontainsa referenceto "finitely many i terations" . With hindsight,it seems thatFraenkeltookthetheoryof natural numbersto be . more securethanset theory In 1922, Skolemformulated a revision ofdefinit propertysimilarto (but later1917, Skolem exindependently of) thatgiven by Weyl. Like Weyl's plicitlyrestricted t heexistential quantification to first-order variables , and he wasthefirst toformulate s eparationas a scheme-oneinstancefor each definit propertydefinable in the first-order language. 16 Skolem's proposal was, in short,to treatthe membership relationas non-logical , and, more radically, to formulate set theoryin a first-order language.In short,Skolem was moving awaybothfrom thealgebraicview ofquantifiers as conjunctions a ndlogicist view of classes as logical and disjunctions,andfrom thealgebraic entities. In thesame paper (1922) , Skolembegantheattackon theabsoluteness of set-theoretic notions, arguingthatthereis no singleintendedinterpretation toset-theoretic concepts.The L6wenheim-Skolemtheoremindicatest hatlike any first-order a xiomatization, set theoryhas a countablemodel-a model withinthenatural n umbers (if it has a modela t all). This, of course, is despite the fact thatwithinset theory, one can prove that,say, the real numbers are uncountable.He concludedfrom this, theso-called"Skolem paradox", thatset-theoretic notionsare "unavoidably relative".Forexample, one cannotclaimthata givendomain D is uncountable simpliciter,b utonly relative to a given model(containingD) of set theory. For any such D, one cannotruleoutthepossibilitythatD may be countable relative to a(richer) 16Skolem and Fraenk el (indep endently)also realizedtheneed for an axiom of replacem ent. Skolem's versionwas anaxiom scheme formulated in terms of his (first-order)notion ofdefin it property . This, of cour se , is contempora rypractic e. Fraenkelfirstgave a rather imprecise account : "If M is a set, and M' is obtainedby replacingeac h member of M with some object,thenM' is a set." Thus thename "replac e ment" . The principl e was soon modified in terms of Fraenk el 's notionof definitorisch function , butthen it was to o weak to serveits purpose. Von Neumann(1928) demonstrated t hatthis versionof theaxiom of replacementcouldbe proved in Zermelo'soriginalsystem.
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model, onethatc ontainsa functionfromthenatural numbersontoD. Skolem heldthatthisrelativity alsoappliesto theDedekindnotionsof "finite"and "simply infinitesystem".17 Given themodern trendof regardingvirtually allmathematical notionsas set-theoretic, Skolem'sconclusionswouldentail therelativity of justabouteverything.Skolem eventually held such a view , butatthetime he seemed contentto rejecttheset-theoretic foundation (see Benacertaf 1985). Skolem (of 1922) seemed toacceptWeyl's conclusion apparentpresupposition)t hatthereis no gainin formulating (andFraenkel's intuitivem athematical notions,likethatof natural number,in set-theoretic terms. Zermelo did notagree. Questionsconcerningcategoricity were oftenraisedin thiscontext.From contemporaryperspectives,thediscussioncan be read if it were full of confusionoverthedistinctionbetweenthesemanticsof first-order languages and the semanticsof higher-order languages , confusionover the rangeof logicand the rangeof categoricity, and/orconfusionover theapplicability theorems . Those whose minds concerningthese of the Lowenheim-Skolern mattersare settledmight understand it thatway. The historical assertions thatconformto one's views areto be praisedas clearwhilethosethatare incompatiblea redubbed "confused".On theotherhand, most oftheremarks figures have c ounterparts in thecurrent d ebateover secondby thehistorical orderlogic. Oncepresuppositionsarenoted,some of thepronouncements are atleastrelatively clearand, in some cases, remarkably clear,even bymodern lights. With hindsight,thereare severalsourcesof fundamental disagreement . betweensyntaxandsemanticsand, in particular , One concernst hedistinction thedistinctionbetweenprooftheoryand model theory . A closelyrelated matterconcernsthestatusof thesecond-ordervariables,involvingwhatI above callthe "logicalnotionof set". Recallthaton this conception , a fixed domain is presupposed, or determinedby context,a nd thevariables rangeover itssubdomains (or its propertiesand relations).On one elvel, thequestionof whetherthis notionis a legitimatep artof logic is amatter of decidingwhere todraw a border: Is the (logical)notionof set inthe jurisdictionof logic, or does it belong mathematics to proper? The deeper issue concernst heextentof our"int uit ive"u nderstanding ofthesecond-order thesyntax variablesa ndtheextentto whichthey can be used in formulating andsemanticsof formallanguages . The higher-order camp seems to holdthat theterminologyin questionis alreadyclearand does notstandin need of "foundation"(see my 1991) . Indeed, theclaim isthatthisveryterminology
as
17For Dedekind, a simply infinite system is a countablyinfiniteset with a successor operation . Dedekind defineda set S to be finite if there is no one-to-one corre sponden ce betw een S and a proper subset ofS. Notice thevariabl e rangingover "one-t o-onecorrespondences" . At leastconceptually , this differs from the usualdefin it ionsof finitudewhich n umbers. Her e we have aanoth er choice betweena refer to (orpresuppose) thenatural set-t heoret and ic a number-theoreti c foundation.
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is at theheartof the foundational enterprise . Any successfulattemptat formulating an axiomaticfoundation of second-orderlogic woulditselfhave terminologyinvolvingthe logicalnotionof set (perhaps in presentingthe semantics)and wouldthusbeg any questions.!"The first-order a dvocates , on theotherhand,holdthat(whatI call the) "logical" notionof set is not clearenoughfor use infoundational systems, at leastnotas it stands. To be used atall, therelevant n otionsmust first beformulated as partof an axiomatictheory. Inshort,t henotionof set must beregardedas non-logical. And anyassertionsa boutsetsmust follow from t heaxioms. This, of course, is themodel ofthemathematical school of logic.Withhindsight,t hebest candidatefor such an axiomaticformulation is the studyof the iterative hierarchy . Thus, the first-order c amp rejectsmy distinctionbetweenthe iterative conceptionandthelogicalconceptionof set. Thereare onlyiterative is non-logical. sets,and theterminology Skolem (1922) launched,in effect, a two-fold attacka gainstthecategoricthesisthatno theity of set theory. One aspectwas thegeneralrelativistic ory withan infinitemodel can be categorical. This, of course, follows from theL6wenheim-Skolemtheoremsand presupposes thatalltheoriesare firstorder.I? Skolem, of course,embracedthispresupposition.The othercritique was specific toset theory(and wouldapply to thesecond-orderversions). Skolempointedoutthateven whentheaxiom ofreplacement is added,Zermelo's system is consistentwith theexistenceof variousnon-well-founded sets and variouscollections of urelements.For example, given anydomain D thatsatisfiestheaxioms, if D containsurelements,thenthesubdomain consistingof thoseelementsof D whose transitive closures do not c ontain possibleto add urelements alsosatisfiestheaxioms. Conversely, it is always new urelements to D. One way toattackthemore generalrelativistic thesiswould be toreject thepresuppositionthatalllanguagesare first-order.T he latter a spectof Skolem'scritiqueseems to call for new axioms, some of which were suggested Bothapproacheswereforthcoming, by Skolemhimself, tryingto be helpful. some in responseto Skolem,o thersindependently. Fraenkel(1921) attemptedto formulate a categorical version of sett heory withan "axiom of restriction" , which wassimilar,butoppositeto Hilbert 's axiom of completenessin geometry. Fraenkel's"axiom" assertedthatthe only sets to be consideredarethosewhoseexistencefollows from the(other) axioms. In short, no model ofthetheoryis to have apropersubsetthatis itselfa model ofthe(other)axioms.P" This rulesoutmodels withnon-well180ne can give a set-theoretic definition of second-orderlogicalconsequencein contemporaryset theory , which is first-order.B utthispushes thequestionback one level, tothe issue of whetherfirst-order s et theoryis adequate . See my 1991, sections5.4 and 9.3. 19As far as Ican determine,thefirst explic it observationthatfirst-ord er theori es like arithmetica renotcategorical did notoccuruntilSkolem 1933. 1934, butit follows from theincompletenessof arithmetica nd thecompleten ess of first-order logic. 2oFraenkel 's axiom of restriction m ightbe calleda "minimal principle",s ince it rulesout
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foundedsetsandmodels withurelements (as wellas models withinaccessible cardinals) . Fraenkelseemed to believethattheintersection of alldomains satisfyingtherestoftheaxioms would be a model of theaxiom ofrestriction. By standards of modernlogic,theaxiom ofrestriction andHilbert 's axiom a ndmeta-language, model-theoretic ofcompletenessconflate objectlanguage semanticsin particular . The axioms of atheoryare to refer to therelevant subjectmatter,sets inthis case. It is only inthemeta-language thatone can refer to w hatfollows fromtheaxioms, or to its models . But perhaps it is notproperto impose suchstandardson historical figures." A deeper "problem" withFraenkel'saxiom of restriction is similarto the "problems" withsecond-orderlanguages . It presupposes familiaritywith sets or, as von Neumann(1925) putit, theaxiom relies on "naive" settheory.If thepurpose of theaxiomatization is to clarify thenotionof (iterative) set, and to do so while avoiding"naive" settheory(and theantinomies),thentheaxiom of restriction is circular.It might be noted,however, thatthosewho holdthat theconceptof set isintuitively clear-and does notstandin (further)need of "foundation" -couldarguethatthecircular ity is notvicious, no more so thanthe use of connectivesa nd quantifiers in a meta-language to explicate thecorresponding i tems in theobjectlanguage . On such a view,t hepurpose ofaxiomatization is not toprovidea "foundation" as such, butto codifythe reasoningin a theory, or to characterize its models. Von Neumannwent on to suggestthatatleast some oft heproblemswith theaxiom ofrestriction can be overcome by precisely formulating therelevant model-theoretic notionsin set theory.Afterall,settheorydealswiththebasic buildingblocks of modelt heory . Von Neumann'sown axiomatization, which , includesseparatevariablesforsets and classes, issuitedfor this. However thenotionof a "model" ofthis theoryrequiresa "higher" settheory,one withthesets and (proper) classes ofthe originaltheoryas elements , i.e., as sets. Of course,t hehighertheorywould have its own (superjclasses.P? Workingin sucha higherset theory , one can use therelations ofsubsetand subclassto formulate thenotionof a classsatisfyingvariousstatements , and, the existe nce ofproper substructur es thatare models of the other axioms . In contr a st, Hilbert'saxiom of completene ss is m aximal,since itrulesoutproper superstru c tur es that are models of the ot heraxioms. For an extensive treatmentof "ext rema l axioms" like Fracnkel'sa xiom of restrictionand Hilbert's axiom of complet en ess , see Carnap and Ba chmann 19:16. On the axiom of restrict ion, sec Praenkel, Bar Hillel , and Levy 197:1, pages 113-119. 21 If one allow s the conflation of object languag e and sem antics,then Fraenkel'ssyste m does provide a categoricalchara cterization . The onlymodels of first-order ZFC thathave no propersubmodelsarethoseisomorphicto theminimalconstructible model-acounta b le set. The onlysta nda rdmodels of second-order ZFC thathave no proper submode lsare those thatare isomorphic to thefirst inaccessiblerank. See Shapiro 1991 , chapter 5. 22 A "model" of von Neumann set theorywould b e a collect ion ofets s (i.e., a class) together withthecollect ion of its subclass es. But in theoriginaltheory, (proper) class es ectio ns. So, in theoriginaltheory , one ca nnotdiscuss (class) are notto beelements of coll models of set theor y. For this, we requir e a more encompassingtheoryand system.
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thus, the notionof a class being amodel of ordinaryset theory. In this context,t heaxiom of restriction has a precisemeaning-one can statein thehigher-theory thata given class has no non-trivial subclassesthatare models ofordinarysettheory . Von Neumannobservedthattheintersection (in thehighersystem) of allmodels of ordinaryset theorymight turnout to be itselfa model of settheory.If so, thenit also atisfiestheaxiom s of restriction, and noothermodel in thepostulated s ystem does. So far, so good. He thenargued,however, thateven if this process succeeds in fixing a model, theresulting t heory -ordinaryset theorytogether w iththeaxiom of restriction-may stillnot becategorical. In some cases, perhaps,a single model has beencharacterized, butonlyrelative to thechoice ofthehigherset with theoryandthesystem satisfyingit. If one had begun theconstruction an evenlarger(or still"higher") system, theremay be a differentcollection of models and , thus,a differentintersection and so adifferentmodel ofthe axiom of restriction . The lastsectionof von Neumann 1925 is a general discussion ofcategoricity. He notesthatit is easy toprove thecategoricity oftheorieslikeEuclidean geometry,b utadded, parenthetically, thattoacceptthearguments,one must "disregardthefactthattheaxioms of geometry. . . depend on thoseof set theory ." After all, the(second-order)axiom ofcontinuity refers tothe subsets of theplaneand Hilbert'saxiom of completenessrefers to models of the axioms, justlikeFraenkel'saxiom of restriction . The point is thatthe categoricity proofs arebased on higher-order, or set-theoretic premises and, thus, are relativeto aparticular system of set theory. Von Neumannthen reiterated theaforementioned conclusionthatset theory, as thenformulated , is not categorical, and he arguedthatconsiderations like the LowenheimSkolem theoremssuggestthedepressing conclusionthat · . . no categorical axiomatization of set theory seems to exist ; forat all probably no axiomatization will be able to avoid difficulties the connected w iththe axiom ofestriction r and the "higher systems ". And since there is no system for mathematics, geometry , and so forth, that does not presuppose set theory, there probably cannot be any - categor ically axiomat izedinfinite system at .all He went on to suggestthatthesame analysis mightapply to theaxiomatic treatment of basicnotionsof cardinality , since eventheseareformulated with variableswhich occur"withreferenceto theentire[set-theoretic] system". Like Skolem(1922), theDedekindnotionof finitudeis given as anexample. Von Neumannconcludest hatif thisSkolemiterelativity is sustained,t hen, of thenotionof finitude , · . . nothing but the shell of its formal characterization . .. remain would · . . It isdifficult to say whether this would militate more strongly .. . against its intuitive character, . . . or its formulat ion as given by set theory .
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We have seenthisdilemmabefore, intheguise ofwhether sets ornumbersare more fundamental. Von Neumannis ambivalenta boutwhether , say, finitude "needs" a (higher-order or) set-theoretic foundation.I f it does, he sees no way toregardthenotionas unequivocal. It mightbe notedin passingthatby thistime, Hilberthad repudiatedthe algebraicschool'sunderstanding of quantifiedformulas asinfinitary.He also came to acceptin parttheintuition i stic-andSkolemit e -claimthatinfinite sets cannotbe treateduncritically like finite ones ee, (s forexample, Hilbert 1923). Neverthel ess, he continuedto holdthatthe use of class -terminology is essentialto thefoundational enterprise(i.e., to theHilbertprogram). One cannotcharacterize theinter - relations betweentheoriesw ithoutit. The proposed solutionto thisdilemma, of course,was finitaryprooftheory, in which thelogicand themathematicsare to bedevelopedsimultan eously. Zermelo wasbeginningto move intheoppositedirection.In a lectureof 1929, he accepteda version of the restricted "logicist"thesisthatmathematics is "a systematization oftheprovableand,as such, anappliedlogic" ,b uthe 23 came to rejecttheview thatproofsmust be of finitelength. Mathematics theinfinite". was dubbed "t he logic of The same year, Zermelo returnedto his axiomatization of set theory (Zermelo 1929) . In discussinghis earlierwork, and its critics,he indicated thattheexplicationof thecrucialnotionof definit depended on thelogical resourcesavailable. v' At thetime theredid not exi st a generally recognized"mat hema t ical logic" to which Icouldappeal, any more thanit does today, where every... researcher has his ownsystem. (Zermelo 1929, page 340)
Zermelo rej ectedFraenkel 'sproposalconcerningthe notionof defin it since it relied on ac onstruction. Such "procedures"contradicted whathe took to be the essence of theaxiomaticmethod. Zermelo apparentlytook an axiomatization to be acharacterization of itssubject ,not a con s truction of it. In thisspirit, he proposed to axiomatizethe notionof definit property . The resultwas anessentially second-order formulation . Forpresentpurposes, thecrucialclauseis thatif P(g) is definit for each propositionalfunction 9 (withonly individualvariables),t henso are VfP(J) and 3fP(J). He thennotedthatthischaracterization does not rely on the notionof natural number. This remarkseems to beaimed atauthorslike Skolem and Weyl who assertedthatset theoryis less fundamental thanarithmetic . For Zermelo, set theoryis sufficient to serve as"foundation" a -as a theoryin which to (re)formulate othermathematical theories,includingarithmetic . 23Zer melo 's lectur e is publish ed in Moore 1980, sect ion 10.2. 24T he iro ny in Zermelo's rem arkis tha t o t day ther eis a (m ore or less ) gen er all y acc ept ed m athematical logic , classicalfirst-o rder predicate calculu s , but he wouldbe reluctantto ap peal to it , because he thoughtit too weak. An appeal tofirst-o rder logic wouldplay into Skolem's hands.
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Skolem (1930) was quick torespondin print,pointingoutthathis own characterization of definit (in 1922) is quitesimilarto Zermelo'snew one. The difference, of course , is thatSkolem did not allow the clauseconcerning second-ordervariables -variablesr angingover propositionalfunctions . He took this move to be obscureand, possibly, contradictory.He asked whetherZermelointendedto characterize "proposit ional functions " by composingeven more axioms. Of course, if Zermelo did this, andtheaxioms were first-order, theLowenheim-Skolemtheoremwould apply.2 5 Skolem thenrepeatedhis thesis ofset-theoretic relativity. In thiscontext,theclaimis that an unequivocal notionof definit property,as Zermelosurelyintendedit, cannot be characterized by axioms. This, it seems, is theclosestthatSkolem came to arguingfor hisinsistenceon first-order languages.P erhapstheunderlyingthesisis thatany notionthatis notcastin a first-order languageis too vague or obscureto be useful in foundational studies. In his rejoinder,Zermelo(1930) held hisground. A second-orderversion of ZF set theorywas forrnulated.P'' In describingthe theory, Zermelo presentedthecumulative hierarchy in virtually its presentform. Hethenproved allmodels of his system are characterized up to isomorphism by two numbers, chosenindependently of eachother.One is thenumberof urelements (a cardinal)a nd theotheris thestructure of theinaccessiblecardinals(an ordinal).T his is correctforcurrent second-order ZFC , but, ofcourse,not for current first-order ZFC. The dialogueended with this standoff/regress.In an expositoryarticle logic. duringthisperiod (1928) , Skolemmade a fewremarkson second-order After developing a nd explaininga languagewith first-ordervariablesand quantifiers , he showed howvariablesrangingover "propositional functions" could beintroduced ; and he raisedthepossibilityof quantifiers over these second-order variables.I repeatthepassageatthetop ofthispaper. If "all " and "t hereexists" areapplied to variablepropositionalfunctions,thequestionarises: whatis thetotality of allpropositional functions?
The latter question,of course, isthesortof thingZermelo didnot think had to be raised. Once a d omain is fixed, the range ofthe locution"all subsets" (or "allpropositionalfunctions")is determined . Afterall, similar questionsare not raised a boutthefirst-order variablesa nd theotherlogical 25In fact, aroundthis time, Hilbertand Ackermann(1928) did publish an axiomatic treatment of propositionalfunctions, an w-orderlogic. 26This versioncontainedthe axiom of replacementand the axiom of foundation . The axiom of choice wasnotincluded , however, butit seems to have been assumed as a general logicalprinciple. He did not developthe meta-theory , butit mightbe noted thatin curre nt second-orderZF , a (global ) principl e of choice can be proved if it isassumed in themetatheor y. In discussingthe models of the theory , Zermeloalsoseemed to accept a thesis to theeffectthatthecollecti on of inaccessiblecardinalsis notbounded.
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terminology.Moreover, from this perspective, theonly way toanswersuch questionsis to use thecorrespondingterminology in themeta-language . But for Skolemthequestionis legitimatea nd pressing. He asserted(in variables 1928) thatonly twoconceptionsfortherangeof thehigher-order are "scientifically tenable" . The first is, in effect , ramified type theory, in whichpropositional functionsare associatedwithformulasof anexpanding language.The secondformulation is to introduce t henotionof propositional functionaxiomatically: The axioms willthenbecome first-orderp ropositions , since the . .. "propositionalfunctions " .. . willassume therole ofindividuals . The relation betweenargument s and functionswillthenappear. . . as primitive.
This is a foreshadowing of non-standard semantics. Of course, these"firstorderpropositions " would besubjectto the Lowenhelm-Skolemtheorems andwouldthushave avarietyof "interpretat ions" Second-order . logic would itself suffer from the"relativity" ofset-theoretic notions. In sum, theclashbetweenZermeloand Skolem wasratherfundamental. to beobscureand in need of fur Skolemtookthesecond-orderterminology theraxiomatization , which would have to first-order be . Thatwould allow -theresultbeing aconfirmation various"interpretations" oftheterminology of relativism.Zermelotookthehigher-order terminology to be clear and not to standin need offurther"foundat ion" . Such terminologyis an essential partof theveryframeworkfor doingfoundations. Of course,neitherpropositional functionsnorsetsare thefirstcontrover sialentitiesto bestudiedby mathematicians . Negative , irrational, andcomplexnumberscome readilyto mind. In such cases, thereare, tospeak (very) roughly,t hreedifferentstancesthathave beentakenby proponents.Here, however,t herearetroubling circularities at everyturn.The firstorientation , is simply to postulate theexistenceof theentities.If any axioms are given theyaretakento describe thepostulated entities.One insistson some autonomybetweentheaxioms andtheentitiesthemselves , and, thus,alternate interpretations of theaxioms are irrelevant.It needhardlybe mentioned thatpostulation, by itself , is not going toconvincethewary. It begs the questionif anythingdoes. Often,postulation is accompaniedby arguments concerningtheusefulness of theentities,or thefruitfulness of theresulting theory. Much of Zermelo 's andCantor'swritingsfitthismold. However , the fruitfulness wouldapplyto any structure thatsatisfiestheaxioms. Thus, the secondpossibilityis implicit definition. One first gives axioms and thenholds thatthedefinedentities"are" or "can be " anythingthatsatisfiesthem. In arithmetic,forexample,Dedekind(1888) defines a "simply infinite system" to be anycollection of objectsthathas anoperationwithcertainproperties. He thengave acategoricity proofand defined"t henatural numbers" to be toward one suchsystem. Skolem's second "scientifically tenable"attitude
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, is an implicit definition. propositionalfunctions,t heaxiomatictreatment The problemhere isthatthevery issue athandconcernsj ustwhichsystems satisfywhich axioms. Sets, properties,propositionalfunctions, or thelike, are central items in thatveryenterprise . We can't do modeltheorywithout somethinglike settheory . If the axioms used in thedefinitionsare taken to befirst-order, as Skolemexplicitlyintended , thenmany (non-isomorphic) systems are so defined , and thereis no way todistinguisha preferredinterpretation . This is relativism.If theaxioms are(standard)second-order, as Zermelointended,t hereis no relativism,but the implicit definitionsare circular . One "defines" theentitiesin questionby usingthosesame entities . The thirdoutlooktowardproblematicentitiesis construction. One shows how theobjectsin questioncan be takenas combinationsof lessproblematicentities.ExamplesincludetheDedekindconstruction of realnumbersas setsof rationals, and the"definition"of complexnumbersas pairs of reals . Skolem'sfirstconceptionofpropositional functionsalongthelines of ramified andpropositypetheoryis of thisform. A series oflanguages is constructed tionalfunctionsare associatedwithformulasthereof . From theperspective of theadvocatesof higher-order logic,thisconstruction is inadequate.O nly countably many propositional functionsare constructed thisway. More importantly, virtually everyconstruction appealsto theintuitivenotionof set and, as Skolemacknowledged , natural number. Again, theseare the"problematic"entitiesu nderstudy.27 As we have seen, Zermelo explicitlyrejected "construction" as incompatiblewithaxiomaticmethod. 2.3.
ces«
Influenced by themethodologyof theHilbertschool,buthimselfno finitist, Godelestablished t hecompletenessoffirst-order logic in his 1929d issertation (Godel 1930) and theincompleteness ofarithmetic a yearlater(Godel 1931) . The former, of course, only appliesto thelogic offirst-order languages while thelatter concernsvirtually everyrecursivea xiomatization ofarithmetic ( and implies theincompletenessof second-order logic). Godel soon j oinedSkolem as a proponentof first-order axiomatizations . In a lectureof 1931-afterhe became aware oftheincompletenesst hearem-Zermelodevelopeda radically new perspectiveon therelationship betweenmathematicsa nd logic(Zermelo 1931, see Moore 1980). He deplored whathe called"Skolemism, the doctrinethatevery mathematicalt heory ism of . .. is satisfiablein a countablemodel", and herejectedthe relativ 27Skolem (1928) st ates thatboth of his "scient ificallyenab t le" characte rizat ions (of p ropositionalfunct ions) make essent ia l use of ethnotion of natural n umber. He then this notionwithoutusing prop osition alfuncti on s(in conced es tha t w e cannotcha rac ter ize the induction axiom ). He conclud es, "T he at te m pt to bas e the notion s of logic up on those of arithmet ic, or vice versa, seem s to me to be mi staken. The founda t ion sof both must be laid simulta neo us ly and in an interrel ated way."
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set theory.This is certainly not a case of satubbornrefusal to a cceptthe Lowenheim-Skolern t heorems . In effect, Zermelo rejectedthethesisthatall mathematical theoriess houldbe formulated in first-order languages . He derided theview as the"finitisticprejudice": . . . the true subject matterof mathematics is not , as many would have eleit, 'combinations of signs' , but conceptually-ideal relations between ments of a conceptually determined infinite manifold. Thus our system incomplete device. . . of signs is always an Zermelo went on to propose a massive infinitarylogic. In accordance withthealgebraicschool,quantifiedformulaswere to bereplacedby infinite conjunctions and disjunctions,b ut he went on to allow for each ordinal0 , formulas ofl engtho. In this context,Zermelo defined"proof" in terms of logicalconsequence , andwas carefulto formulate t herelation of implication ", as well -founded. He took it ascrucialthateverytruesentencebe "provable and proposed thatOodel'sincompletenesst heoremrelied on an overly restrictiveconceptionof quantification and proof. In fact, Zermelo arguedthat Godel's reasoningshows thatany finitisticnotionof "proof' is inadequate. In short,c ompactnessis an unacceptable propertyof logic. Over aperiodof six weeks, Zermelowroteto Godel , Codelresponded,and Zermelowroteonce more.28 After aninitialconfusionon Zermelo 's partwas clearedup, thetwo seemed to be at cross purposes. G6delinsistedon a finitisticandfirst-ordermeta-language, from whichtheessential incompletability of arithmetic could beestablished . Zermelosuggested"new methodsofproof", alongthelines of hi s infinitarysystem. He was clearon thesourceof the problem: does not sufficeto'decide' . . , a 'finitistically restricted' proof-schema the propositions of an uncountable mathematical. .system . For what presupposed or assumed in 'proof' is cannot be'proved', but must be some form... What does one understand by 'proof'? a In complete generality one understands thereby a system of propositions such that
from the assumption of the premises the validity of the conclusion can
reasonably be asserted . And still the question remains, what is 'reasonable'? In any case, not merely (and this you [Codel] have shown) the propositions of a finitistic schema . " from thebeginning,I have more been working on the basis of a general schema, .. . (translated in Moore 1980, page 128)
Oodel's proofis Zermelo's reductio ad absurdum againstfinitary,first-order languages . It is ironic that,accordingto Wang(1987), G6del acceptedZermelo's conclusions,or atleastsomethingclose tothem (see also Codel 's corresponthatthis partof logic, dencereportedin Wang 1974) . However, Godel held 28The lasttwolettersar e publishedin Grottan-Guinness 1979. Thefirstis reprinted(in slated in Dawson 1985. Zermelo'shand)and tran
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thestudyofdeductivesystems,shouldfocus on theformal . Thus, for example, proofs withinobject-language systemsshouldbe mechanically checkable . This rulesagainstthesecond-order consequencerelation andtheconsequence relation e numerated by Zermelo.ButGodel also held thatthehumanability to understand and work withmathematical conceptsgoes beyond the me chanical. As far as I know, Godel never jectedto ob higher-order Euclidean geometry , or evenhigher-order set theory, as a viable branchofmathematics . We use thetheories,work inthem, and understand w hatwe do. It is only thatthesetheoriesare notformal , andso not in the purview of logic. Zermelopublisheda fewarticlesdeveloping hisinfinitarylogic,but did not carryit very far . In 1935 he was fired for failingsalute to Hitlerand his proposalwas notpursuedby anyone else until, perhaps, the revival of infinitary languages in the 1950's. In a conference in 1938, Skolem returned to his discussion of the Lowenheim-Skolem theoremsand extendedhis relativismto mathematicsas a whole(Skolem 1941) . At thesame conference , Bernaysstill resisted b oth therelativismand therestriction to first-order languages. He suggestedthat the limitationsof the Lowenheim-Skolem theoremsare due to the "rest rictiveness" of the formalism and proposed thatthis reveals "a certaininadequacy of the methodunderdiscussion.. . formakingaxiomatizations precise" (Gonseth 1941) . Godel, of course , did notacceptthe relativism,b ut he did insist on a first-order language . This combinationprevailed. From this point, the explicitcontroversy overhigher-order languagessubsided, and most logicians began toacceptthe Skolern-Godel proposalthatonlyfirst-order languages are appropriatefortheirwork.29 I would suggestt hatthis consensus was not based on a philosophy foundational of studies. It was more of aresearch model theoryis thebest place to focus program, suggestingthatfirst-order intellectual attention . Perhapsone can saythatfirst-order logicbecame a Kuhnianparadigm. Given theapparentlack of explicit argumentsfrom the majorfigures in the first-order camp, one can only speculate on the reasons .i''' 29For example, thevastmajorityof textbook s in logicwrittenafter1940 hardlymention higher-order t erminology . Two (very) notableexceptionsare Church 1956 and Boolos and Jeffrey 1980. The debate over relativism continuedthroughout Skolem's career . See, for example, the exchangebetweenTarski and Skolem in Skolem 1958 (reprintedin partin Moore 1988) . 30Surely,GOdel's completenesstheoremmakes first-order systems attr activefor many purposes. The theoremshows, in effect ,thattheprooftheoryand thesemanticsof firstordersystems area perfe ctmatch. This entail s thatther e is butone consequencerelation , one sortof "independence"statement , etc. , and it allowsone to(automatically) transf er resultsa boutthesemanticsof a system to itsprooftheory,a nd vice versa. Thus, one can s tudysemanticsin orderto shed lighton provability . The emphasis on prooftheory reliably (andtheformulation of the completeness problem) is tracedto theHilbertprogram, which thesecond-order c onsequencerelation cannot ofcoursepredatesGOdel's work. By contrast, be axiomatized. So it may bethatfirst-orderlanguag es were adoptedin part because the resulting p rooftheoryis more fruitful, more conceptually clear , or more elegant .
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In any case,t herangeof thelocution"allsubclasses"was (andis) widely regardedas sufficiently p roblematic(or insecure)to suggestthatthecorresponding terminologybe non-logical.This amountsto aninsistencethat onlytheresourcesof first-order languages andsemanticsare reliable enough to serve logic. don'tknow I theextentto whichthis ambivalencetoward classterminologyis thecauseor theeffect oft he "triumph" of first-order languages. logic neverquite To closethissection,it is to benotedthathigher-order died out. For example, interestin type theoryremained alive, perhaps promptingSkolem (1961) toreiterate his supportfor itsinterpretation as a many-sortedfirst-order system (referringto Gilmore 1957, see alsoHazen 1983). Church 1951 is a straightforward higher-order system, and,by way of transition to thenextsection,Church 1956 containsa discussionof secondorderlogic. 3. THE PRESENT STATE
This finalsectionconsistsof a fewremarkson thecontemporary controversy concerningsecond-orderlanguages. From one perspective , the discussion continuest heborderdisputeoverwhatI callthe"logical"conceptionof set. In thoseterms, however,thequestiondoes not seem toamountto much. The best resolution, perhaps, is simply to holdthatthereis no sharpborder betweenlogicand mathematics.T he analogy, of course, is withpolitical boundariesb ut, unlike those, our lines candrawnand be moved at will (unless emotionsof sovereigntyrun high) . As we have seen,t hedeeperissues concerntheepistemic and semanticstatusof thehigher-order terminology, logic hold t hat especiallytherangesofthevariables.Advocatesof first-order second-order terminology is not sufficiently clear, or otherwise is inappropri. In some cases,theunderlying thesisis thatnotions ate toplayarole in logic used in logic s houldnot have any possible obscurityor avoidableuncertainty . Accordingly , conceptsused intheformulation and studyof correctreasoning shouldbe self-evident.A lessencompassingthesismay simply be thatterminology forsets is too obscureor uncertain to serve logic . The set-theoretic antinomiesare still used to defend thisclaim. The central o pponentof higher-order systems is W . V. O. Quine. A full elaboration of theattackon second-orderlogicoccursin his 1986, but his crit iquedatesback (atleast)to his 1941. In theearlierp aper, Quineargues thatthe intensional c haracter of the attributes in Principia Mathematica make them too obscurefor foundational work, or for any scientifically respectablestudy. He proposes thatattributes be supplantedby sets. In most contemporaryhigher-order systems, propertiesand classes areextensional (see Shapiro 1991) . In this respect, theyresemble es ts, andso thedevelopmentof higher-order logic has followed Quine'sadvice. Heargues, however , thatin invokingset-likeentities , we have crossed saignificantb order, outof
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logic" logicandintomathematics.His thesisis thatso-called"second-order is "settheoryin disguise". He pointsoutthatsome substantial set-theoretic thesescan be formulatedin second-orderlogic. Thereis, for example, a second-order s entencethatis a "logicaltruth"if and only ifthecontinuum hypothesisis true(see Shapiro 1991, chapter5). It shouldbe notedthatQuine is nota criticof set theory(unlike,say, Skolem's early1922) . He was (and remains)an advocateof thetheory,a nd has contributed to itsdevelopment . The claim here isthatset-terminology does notbelong tologic. To statetheobvious, Quine (both 1941 and 1986) follows Skolem 's laterwork intakingsetterminology and membership to be non-logical.T he idea seems to be thatlogic,properlyso-called,s houldbe topicneutral andshouldnotpresupposeanythinga boutany subjectmatter . Thereareno "logicalobjects"or, in otherwords, logicshouldnothave an ontology r'! Settheory 's staggering e xistential assumptionsare. . . hidden... in the tacitshiftfrom schematicpredicateletter to quantifiable set variable . (Quine 1986, page 68)
Noticethedivergencebetweenthisview of logic andt hatheld byboththe algebraicschoolandthelogicists,all ofw hom tooksets,or classes (orproperties, or attributes) to be underthepurviewof logic , as did many of the mathematicians . As we haveseen, historically the "shift" went inthedirectionopposite to thatsuggestedby Quine. It went fromthequantifiable variablesof higher-order logic totheschematic, non-logicalpredicateletters of first-order logic. It is curiousthatanotherprominentQuineanthemeis thelack of asharp borderbetweenrespectableacademic disciplines. He speaks oftenof our scientificenterprise as a "seamlessweb". Alongtheselines, Quineshows that mathematical concepts, involvinga "commitment"tomathematical ontology, areessentialto all of thesciences. Why shouldlogic,especiallythelogic of mathematics be different?Why is it importantto decide where logic leaves offand mathematicsp roperlybegins (see Shapiro 1991, chapters5 and 8)? The borderskirmish betweenlogicand set theorywas joined by Tharp (1975) . Noting thatfirst-orderlogic is compact and complete, whereas second-orderlogic isneither , he arguedthatotherthingsequal, first-order logicoughtto bepreferred. Argumentslikethisaresometimes bolstered by a theorem,provedin Lindstrom 1969, thatfirst-orderlogic istheonly logict hatis bothcompactand has the(downward)Lowenheim-Skolemproperty.VIt is interesting to speculat e on whytheseproperties(and completeness)a reregardedas desirable. 31A similarattackon second-order logic was recentlymade by Hartr y Fi eld (1984) , existe nce th eorems. Field embraces who suggested thatlogic shouldnothave substantial a Kantia n thesis that there should be no apriori, or purely logical, arguments for the arguments"). existenc eof anything(e.g., Critique of pure reason 8622-623 on "ontological 32Linds t r6 m'stheor em concern ssemantics. Let L be an exte ns ion offirst-ord er logic
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The completenesstheoremindicatesa perfectmatchbetweenprooftheory and semantics(see note30). It followst hatthe first-order semanticconsequence isaxiomatizable , and, thus, the calculi developedin theprooftheory t heexpressive resources of are thebestpossible. The "cost", however, lies in thelanguages(through thesemantics).First-order languages areinadequate to the taskof describing intended or standard models of varioustheories. Wang (1974, page154) agrees: When we areinterested in settheoryor classical analysis,t heLowenheim-Skolemtheoremis usuallytakenas asortof defect (often thought logic. Therefore,whatis established to beinevitable)of thefirst-order logic istheonly pos(by Lindstrom'stheorems)is notthatfirst-order sible logicbut ratherthatit is the only possible logic when we in a ... sense denyrealityto theconceptof uncountable
So does BarwiseandFeferman(1985, page 5): As logicians, we doo ursubjecta disservice byconvincingothersthat logic is first-order and thenconvincingthem thatalmostnone of the capturedin first-order conceptsof modernmathematicscan really be logic.
Somewhatconsistently, Skoleminsisted(in his laterwork)that,in effect, most mathematical theoriesdo not have unique orunequivocal"standard " interpretations (even up toisomorphism). Such is relativism . But, as far as I know,Tharp did not embrace relativism,nor do mostcontemporary advocatesof first-orderlanguages(see also Myhill 1951) . Quine, for his part,has articulated and defended qauitegeneralrelativismin ontology, but not onSkolemitegrounds. In fact, he hascriticizedtheSkolem paradoxin betweenQuinean severalwritings.An extendedtreatment oftheconnections ontological relativity, Skolemiterelativism , and Quine's views onfirst-order logic would go beyondthe scope of this paper. Meanwhile, a few voices continuedthe tradition of Hilbert,Zermelo, and languagesdo notcapturecrucialaspects Bernays, arguingthatfirst-order of pre-formalmathematical practice. A clearand forcefulexample is the lastchapterof Church 1956, devotedto second-orderlogic. Eitherexplicitly orthroughexercises, Churchshows howmany mathematical theories havenatural and straightforward formulations in higher-order systems, and thatfirst-orderc ounterparts are awkwardand inadequate . He points out thatdue toincompleteness,second-order semanticnotions,like consequence, thathas thesame classof interpretations . Then L has the downward Louienhesm-Skolem e 4> of L, if 4> has a model withan infinit e domain, property if and only if foreachsentenc then4> has amodel with(atmost) a countable domain; L is compact if andonly if for e ach set S of senten ces of L, if each finitesubset ofS has a model, then S itselfhas a model. Lindstrom's theoremis thatif L is compac t and has the downwardLowenh eim -Sko lern property , thenforeverysentenc e 4> of L ther e is a first-ord er sentenc e 4>' such that4> and 4>' have exactlythesa me models, i.e., 1= 4> == 4>' . See Shapiro 1991 , section6.5.
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and validity,must be distinguishedfrom theirproof-theoretic satisfaction, . The counterparts of deductiveconsequence,consistency,a nd theoremhood importantsemanticones arenot axiomatizabl e. He concedesin a footnote (page 326n) thatthisraisesdoubtsin some minds. Churchagreesthatthere would be cogent objectionsto a logic whose very grammaris not effective. Therewould be nomechanicalway todetermineif a givenstringis a wellconsequencerelationare formed-formula . Butobjectionsto a non-effective on a different level: Objections may indeed be made to this new point of view, on the basis of the sort of absolutism it presupposes.. . But it should be pointed outthatthis. . . is already inherent in class icalmathematicsgenerally, and it is not made more acute or more doubtful, but only more conspicuous, by its application to theoretical syntax .e., logic] [i . For our definition of the consequ encesof a system of postulates . . . can be seen to be not essentially different[those]required from for the . . . treatment of classicalmathematics . . It. is true that the noneffective notion ofconsequence , as we have introduced. .it, presupposes a certain ab. But this is solute notion ALL of propositional functions of individuals presupposed also in classical mathematics, especiallyclassical, analysis and objections against it lead tomodifications such . . . as intuitionism theaforementioned criticsofhigher-order logic favor Exceptfor Weyl, none of intuitionism . Allacceptclassicalmathematicsas itstands.P'' Anotherwidelyreaditem was Kreisel's"Informalrigourand completeness proofs" (1967).The attackis directedat first-order versionsof theories, , which have schemesw ithinlikearithmetica nd analysis(and set theory) finitely manyinstancesin place ofsecond-orderaxioms. It is arguedthat such theoriesallow only unnatural epistemologiesa ndarenottrueto theinformalmeaning-the"informalrigor"-ofactualm athematical practice(see my 1991, section5.3). One ofKreisel'sargumentsc anbe seen as anelaboration ofHilbert'sthesis thatfirst-order theoriesare inadequate to thetaskof illuminating theinteractions betweenmathematical theories . The use ofschemes depends on the language ofthetheory,a ndtheprecise set of non-logical termschosen. Thus, forexample, instancesof thefirst-orderinductionscheme onlycontainthe terminology of oneparticular languageof arithmetic,say onewiththesuccessorfunctionalone. The introduction of newterminology (such assymbols foradditionand multiplication, or variablesrangingover sets of numbers) 33C hurch uses th e word "Platonism"to describe theattit udesharedby theemployment of thesecond-orderc onsequencerelation a ndthepursuitof classicalm athematic s generally (referringto Bernays 1935) . This can be misleading. The m att e rs at handare m ethodological , concerningthingslike non-effective relations, the law ofexcludedmiddle, and impredicativedefiniti on. The issues surroundingPlatoni stic episte mologyare notraised, , are traditional m atters ofontology . nor, forthatmatter
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yield newinstancesof the axioms themselves , and, thus, constitute a "new" theory . A more natural outlookis to seetheseas theadditionof newterms analysisare to an oldt heory . Similarly , first-orderarithmetica ndfirst-order presentedas isolatedtheorieswhich areindependentof botheachotherand therest ofmathematics . This is not inaccordwiththepracticeof viewing such structures as interrelated , a practicemanifestin thecommon technique of "embedding" or "modeling"one structure in another.Kreisel(1967, page 166) wrote: . . . veryoftenthemathematical propertiesof adomain D become only graspablewhen oneembeds D in a largerdomain D' . Examples: (1) D integers,D' complex plane; use ofanalyticn umbertheory.(2) D integers , D' p-adic numbers; use of p-adic analysis.(3) D surfaceof a sphere, D' 3-dimensionalspace; use of3-dimensionalgeometry . Nonstandarda nalysis[also applies] here .. .
To follow thefirst oftheseexamples, when one realizes thatthecomplex plane (or for t hatmatter,t heset-theoretic hierarchy)c ontainsthenatural numbers- or an "isomorphic copy" of thenatural n umbers- thenone can thenatural numbers. use complex analysis(or set theory)to shed light on Thatis, since isomorphicstructures have thesame set oftruths,a theorem ofcomplexanalysisthatrefers only to the"nat ural numbers"of thecomplex plane istrue of thenatural numbers. It is well knownthattechniqueslike this yield resultst hatare notprovablein theoriginaltheories . From the first-order perspective,however,theredoes not seem to be any reasonto reor astruthsabout thenatural gardthenewtheoremsas resultsof arithmetic numbers. Why think,for example, thatthe "nat uranumbersof l complex numbers? Indeed, thecomanalysis"reallyare isomorphic to thenatural pletenesstheoremindicatesthatthereare models oft heoriginaltheoryin whichthenewtheoremsarefalse. In short, "thenatural numbers" has an in thatoutrunsw hatis capturedin first-order arithmetic , formalinterpretation and thisinformalinterpretation plays acentral role inactualm athematical practice . Alongtheselines,Montague(1965) proposedthattheinformal,b ut much used,notionof "standardmodel" ofarithmetic,analysis,a ndeven set theoryis explicatedby thenotionof "model oft he(respective)second-order theory". A secondobservationthatKreisel makes ist hateachfirst-order theoryis tied too closely theterminology to in use. Forexample,first-order arithmetic withthesuccessorsymbol alone is arather simple theory. It complete is and its models are easily characterized . First-order a rithmeticwithadditionand multiplication symbols is anothermatterentirely , as evidenced bytheincompletenesstheorems. Consequently, additionand multiplication are not definablein theformer theory. It might be added thatadditionand multiplicationhave straightforward definitionsin second-ordera rithmeticwith successoralone. Dedekind's(1888) owncharacterization, whichexplicitly in-
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, is straightforwardly secondeludesdefinitionsof additionandmultiplication order. John Corcoran(1973) and George Boolos (1975)engagedin studiesof aspectsof mathematical p racticethatresistfirst-order t reatments . Boolos went on tochallenge Quine's claim thattheontological presuppositionsof second-orderlogic are excessive . He pointedoutthatan interpretation of a second-ordertheoryis exactlythesame as an interpretation of a firstordertheory-adomain of discoursetogetherw ith interpretations of the non-logical terminology.The difference , of course, isthatonce adomain is fixed, second-ordertheorieshave variablesrangingover itssubdomains (and relations) . Thatis, in conformitywithwhatI callthe"logical conception" ofset, thevariablesrangeover thesubclassesof a fixeddomain. In two recentarticles(1984, 1985), Boolosproposed thatsecond-orderquantifiers be regardedas "pluralquantifiers"are inordinarylanguage,and has developed a semanticsalongtheselines. The claim isthat,u nderstoodt hisway, second-ordervariablesand quantifiersintroduceno ontologybeyond that of a correspondingfirst-order theory. This interpretation is employed by a numberof philosophersinterested in ontology . Even withtraditional inter pretation, however,t he"increase" inontology relative to afirst-order theory theorieshavevariablesr angingover adomain; is not severe.T he first-order the second-ordertheoriesalso havevariablesrangingover allsubdomains (and relations).Thereis no "powerset"operatorthatgets iteratedtwice, thetransfinite (unless, of course, such an operatoris partof let alone into the nonlogical terminology) . The "staggeringe xistential assumptions"that Quinementionsapplyto theiterative conceptionofset, not to logic . My own contributions (1985, 1990, 1991), on thehigher-orderside, elaborateandextendChurch'snotethatnotionsdirectlyrepresentedin secondorderlanguages areimplicitin mathematical practiceandthattheepistemic andsemanticpresuppositionsofsecond-order logic are no g reater t hanthose of classical m athematics.I proposethatcategorical characterizations are necessaryto capturetheinformalcommunicationof mathematics(see alsoCorcoran 1980) , boththemanifestlysuccessfuldescriptionof certainstructures (likethatof thenatural numbersor therealnumbers)andcertainconcepts likefinitude,well-foundedness, and minimal closure , thatresistfirst-order treatment.I also developand extendtheHilbert/Kreisel view concerning thesignificanceof higher-order languagesin capturingtherelationshipsbetweenmathematical theories . In short,higher-order logicprovidesthebest .i'" explicationof classicalm athematics This is not to saythattheconsequencerelation of second-orderlogic is completelyunderstood , or intuitively evident, nor is itas evidentas firstorderconsequence . Thereare, to be sure, deep philosophicaland mathe34 Shapiro 1991 (chapter s 3 and4) alsocontainsa detailedpresentati on of the lan gu ages, sema ntic s,anddeduct ivesystems of higher-o rder logic.
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maticalproblemsassociatedwithsecond-order logic. For example,thereare problems concerningreference and problems concerninghow thelanguage can belearned.Butthese arepresentin classicalm athematicsas well, and most oftheproponentsof first-order logic arenotwill ing to givethatup, or even to modify it or takeit atotherthanface value . To echo Church, logic,butnot theproblemsaremore perspicuousin thecase ofhigher-order .i" The thesis is thatthe languageand semantics used more troublesorne to formulate variousmathematical theoriesshouldbe in line withthe(presumed) descriptiveand communicativepower ofthe languageof informal mathematics . The contemporarya rgumentsin favor ofhigher-order logic are not successfulagainsta thoroughrelativismlikethatof Skolem (andperhapsvon t hatthesestructures and Neumann). These views, in effect , deny outright conceptsare unequivocal. Accordingly , the failure of the categoricity of first-ordertheoriesaccurately reflects the philosophicalsituation . On the languages , especiallyamong otherhand,atleastsome advocatesoffirst-order mathematicians, rejectSkolemiterelativismand, thus,presumablyagreethat we do succeed in the description(up toisomorphism) and communicationof at leastsome infinitestructures . The attitude is thatinformal mathematics (i.e., informalsemantics)is sufficient to describeand communicatethe structures and conceptsin question,but formalsemanticsmust fail where informalpracticesucceeds. Myhill (1951) develops thisthemeexplicitly . On theotherhand,Skolemiterelativismis not dead,amongphilosophers. A common responseis thatthe (semantic) "benefits"of second-orderlan. guagesaretheresultsof the(naive) settheoryhiddenin themeta-language This is an elaboration of vonNeumann's(1925) remarkthatthecategoric ity of, say,Euclideangeometry, is won bytemporarilyignoringproblems with theunderlying set theory. Some w riters , such as Weston (1976), claim thatthenon-categoricity thatpervadesfirst-order systems reappearsin the set-theoretic semanticsof higher-order languages.These objectorspropose thatwith infinite sets, the powerset operationis not unique in an absolute sense-different models of set th eory have differen t powersetoperations . If this is correct,t henthe verysemanticsfor second - orderlanguagesis itself notunique. In fact, therewould be awealthof suchsemantics- onefor each are a form of skepticism, and "powerset".Notice thattheseconsiderations it is well known t hatskepticism is difficult to refute,especially on its own terms. n atural, and more revealing , Anothersuggestionis thatit would be more 35Comp let en ess is ont acure-allfor th e sortsof philoso phicalproblems indicatedhere. First-ord er logic is not immune, nor ispropositionallogic . One groupof issues concerns the extentto which explic itor im plicit, formalor info rmal,rules for using a languag e determine whatit "mea ns". Presentattentionis focusedon theconsequen ce relation . Carnap(1943) shows ingreatdetailthateven theeminentl y completedeductive system for propositional logic si consistentwithdiffere nt "interpr etations ".
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to formulatevariousmathematical theorieswithin(first-order) set theory . This re-raisestheissue concerningtheborderbetweenmathematicsand its of set th eory. First-order set logic, and issues concerningthe understanding theory has awealthof non-standard models. No doubt, the consequencerelationof second-orderlogic is not asuitableingredientin a strictrationalist or foundationalist program(see Shapiro 1991, chapter2). First-orderlogicprobablyisn't either,becausethemost illuminating formulation of itsconsequencer elation is set-theoretic (as, for example, thecompletene ss theoremrequirestheaxiom of infinity) , butperhaps it is closer . One whotakeslogic to bethelastoutpostof foundationalism , or rationalism,might be inclinedtowardfirst-order logic. However,Quineand his followers are thoroughly anti-foundationalist, even concerninglogic (see, forexample, Quine 1960) .36 To sum up, theperiod between,say, the1930's and thepresenthas produceda detailedelaboration of thevariouspositionsand, consequently, not as much discussionis obviouslyatcross-purposes.Nevertheless, the present stand-offsare not all thatdifferent from t heirhistorical c ounterparts. The higher-order a dvocatesclaim thatwhatI callthe "logicalconception"of set -varia blesr angingover thesubclassesof a fixeddomain-is sufficiently clearfor use in logic and, moreover, theuse ofsuchterminology is essential to atleastpartof thefoundational enterprise . Advocatesof first-order logic rejecttheuse of setterminology in logic,r egarding it asproblematicor as not sufficiently topic-neutral. The alternative, onceagain,is thatmemb ershipbe treated as non-logical and thattherangesof theset-variablesbe regard ed by theunderlying semanticsas indete rminate . The taskof characterizing structuresand the "higher-order " concepts(likefinitudeand well-foundedness) is passed to first-order set theory(prettymuch as Skolemformulated it). The latter , of course, has asteep ontology(as conceivedinformally) , but it is associatedwiththemathematical t heoryof sets, not with logic . Of course, first-orderset theoryis itselfsubject totheLowenheirn-Skolemtheoremsand thushas a wealthof non-standard models. The "first-order " optionsare 36See Wagner 1987 for a lucid ela bo ra tio n of ration ali sm in logi c. The historicaland philosophical connecti on sbetw een "foundati ons of mathematics" and "found ati ona lism" are worthpursuing,butit wouldtake us toofar afieldhere (see Shapiro 1991 , chapter2) . In 1950, Skolemsketchedsevera l o p ssibl e "goals"of foundati onalresearch in mathematics. The "logicistic" desireis ". . . to obta in a way ofeasoningwhich r is logicallycorrect so that it is clearandcerta in in advan cethatcontradi ctions will n ever occur, and what we prove are truths in some sense." This , of course,is foundat ionalism . A m ore m odestdesire is ". .. to which makes it possibl e to developpresentday m athem ati cs,andwhich have a foundation is cons ist ent so far as s known i yet." This is called the "oppor t u n i stic" out look. Skol em e "unpleasa nt"eatur f e thatwe arenevercertain when the foundational states thatit has th work is complete : "We are not onlyaddingnew floor s at the top of our building, butfrom time to time it may be necessar yto make cha nges at the basi s." S kolem's third "des ire" is the "Hilbe rtprogram " , a valiantatt e mpt at "giving up the logicistic st an d po intand notbeing conte ntwiththe oppo rt unis ti co ne". Skolem 's prefer en cefor foundationalismis wareof its difficul ti es . The cont ras twithQ uin e is st riking. evide nt,buthe was well a
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eitherto embrace relativismand denythatthereis a single range to the locution"allsubclasses"(aftera domain is fixed) or to insist thatformal semanticsmust ignore the inherentsuccess ofmathematical understanding , "goals" for which practice,and communication.In short,thefoundational higher-order terminologyis invoked areregardedas unattainable in formal semantics,eitherbecause they are not attainable at all-there is nothing unequivocal to describe--orbecausecommunicationis attained"only informally". Thestand-off is particularly frustrating becausethereseems to be no common standpointfrom which toevaluate or even discuss the m atter.What one side regardsas an essential,integralp artof the whole foundational enterprise--including theenterpriseof "logic choice" - t heotherside rejects as obscureor problematic . StewartShapiro Ohio StateUniversity Departmentof Philosophy 350 UniversityHall 230 NorthOval Mall Columbus,Ohio 43210 USA Email: [email protected]
REFERENCES
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Boole, G. 1847 The mathematical analysis of logic, being an essay toward a calculus of reasoning, London. 1854 An investigation of the laws of thought on which are founded the mathematical theories of logic and probabilities, London. Boolos, G. 1975 On second-orderlogic, The Journal of Philosophy, vol. 72, pp . 509-527. a (or to be some values of some variables) , 1984 To be is to be a value ofvariable The Journal of Philosophy, vol. 81, pp . 430-449 . 1985 NominalistPlatonism , The Philosophical Review, vol. 94, pp . 327-344. Boolos, G., and R. Jeffrey 1980 Computability and logic, second edition,Cambridge UniversityPress, Cambridge,England . Carnap, R. 1943 Formalization of logic, HarvardUniversityPress, Cambridge,Massachusetts. Carnap, R., and F . Bachmann 1936 Uber Extremalaxiome , Erkenntnis, vol.6, pp . 166-188; translated by H. G. Bohnert,History and Philosophy of Logic, vol. 2, 1981, pp . 67-85. Church,A. 1951 A formulation of thelogic of senseand denotation , Structure, method and meaning (P. Henleet al., editors),LiberalArtsPress, New York, pp, 3-24. 1956 Introduction to mathematical logic, vol. 1, PrincetonUniversityPress, Princeton , New Jersey. Corcoran , J. 1973 Gaps betweenlogicaltheoryand mathematical practice , The methodological unity of science (M. Bunge, editor),Reidel, Dordrecht,pp. 23··50. , History and Philosophy of Logic, vol. 1, pp . 187-207. 1980 Categoricity Dawson, J . 1985 CompletingtheGOdel-Zermelocorrespondence,Historic Mathematica, vol. 12, pp . 66-70. Dedekind, R. 1888 The natureand meaning of numbers, Essays on the theory of numbers (W . W . Berman, editor), Dover, New York , 1963, pp . 31-115. Etchemendy,J. 1988 Tarskion truth and logicalconsequence,The Journal of Symbolic Logic, vol.53, pp . 51-79 . Field,H. 1984 Is mathematicalknowledgejust logical knowledge ?, The Philosophical Review, vol. 93, pp . 509-552. Fraenkel,A. 1921 Uber die ZermeloscheBegriindungder Mengenlehre,Jahresbericht der Deutschen Mathematiker- Vereinigung, vol.30, secondsection,p p, 97-98. der Mengenlehre , Jahresbericht der Deuischen Math1922 Zu denGrundlagen ematiker- Vereinigung, vol. 31, secondsection,pp . 101-102.
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STEWART SHAPIRO
1922a Der Begriff'definit'und die Unabhangigkeit des Auswahlaxioms , Sitzungsberichte der Preussichen Akademie der Wissenschaften, Physikalischin van Heijenoort 1967, mathematische Klasse, pp . 253-257; translated pp . 284-289. 1925 Untersuchungen Uber die Grundlagen der Mengenlehre , Mathematische Zeitschrift, vol. 22, pp. 250-273 . Fraenkel , A., Y. Bar-Hillel,and A. Levy 1973 Foundations of set theory, second revisededition,NorthHolland , Amsterdam. Frege, G. 1879 Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des in van Heijenoort 1967, reinen Denkens , Louis Nebert, Halle;translated pp . 1-82. 1883 Uberden ZweckderBegriffsschrift, Sitzungsberichte der Jenaischen Gesellschaft fur Medicin und Natunuissenschaft, vol. 16, pp. 1-10. ; translated by J . Aus1884 Die Grundlagen der Arithmetik, Koebner,Breslau tin,second edition, Harper,New York, 1960. 1893 Grundgesetze der Arithmetik 1, Olms, Hildescheim. 1895 Review ofSchroder 's Vorlesungen Uber die Algebra der Logik, Archiv fur in Translations systematische Philosophic, vol. 1, pp . 433-456; translated from the philosophical writings of Gottlob Prege (P. GeachandM. Black, editors), Blackwell , Oxford, pp . 86-106. Gandy, R. 1988 The confluence of ideas in 1936 , The universal Thring machine (R. Herken, editor),OxfordUniversityPress, New York, pp. 55-HI. Gilmore, P. -predicatecalculus,Summaries 1957 The monadic theoryof typesin thelower of talks presented at the Summer Institute of Symbolic Logic at Cornell, Institute for DefenseAnalysis. Godel, K. 1930 Die Vollstiindigkeit der Axiome des logischenFunktionenkalkuls, Montatshefte fur Mathematik und Physik, vol. 37, pp. 349-360; translated in van Heijenoort 1967, pp. 582-591. und ver1931 Uber formalunentscheidbare Siitzeder PrincipiaMathematica wandterS ystemeI, Montatshefte fur Mathematik und Physik, vol. 38, pp . 173-198 ; translated in van Heijenoort 1967, pp . 596--616. 1944 Russell's mathematical logic, Philosophy of mathematics (P. Benacerraf and H. Putnam,editors),second edition ,Cambridge UniversityPress, Cambridge,England , 1983, pp . 447-469. Goldfarb,W . : The natureof thequantifier,The Journal of Sym1979 Logic inthetwenties bolic Logic, vol. 44, pp . 351-368. 1988 Poincareagainstthelogicists, History and philosophy of modern mathematics (W . Aspray and P. Kitcher,editors),MinnesotaStudiesin the Philosophyof Science, vol. 11, Universityof MinnesotaPress, pp. 61-81.
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Gonseth,F. 1941 Les entretiens de Zurich, 6-9 decembre 1938, Leeman, Zurich. Grattan-Guinness, 1. 1977 Dear Russell - Dear Jourdain, ColumbiaUniversityPress, New York. withZermelo on his 1979 In memoriam KurtGodel: His 1931 correspondence incompletability theorem,Historia Mathematica, vol. 6, pp. 294-304 . Hazen,A. 1983 Predicativelogics, Handbook of philosophical logic 1 (D. Gabbayand F. Guenthner, editors),Reidel, Dordrecht, pp. 331-407. Henkin, L. 1950 Completenessin thetheoryof types, The Journal of Symbolic Logic, vol. 15, pp . 81-91. Hilbert,D. 1899 Grundlagen der Geometrie, Teubner,Leipzig; Foundations of geometry, , Open Court,La Salle , Illinois, 1959 . (E. Townsend,translator) 1900 Uber den Zahlbegriff , Jahresbericht der Deutschen Mathematiker- Vereinigung, vol. 8, pp. 180--194. 1900a Mathematische Problem, Bulletin of the American Mathematical Society, vol. 8 (1902), pp. 437-479. 1902 Les principes fondamentaux de la geometrie, Gauthier-Villars, Paris; Frenchtranslation of Hilbert 1899. 1905 Uber derGrundlagen d erLogik und derArithmetik,Verhandlungen des dritten internationalen Mathematiker-Kongresses in Heidelberg vom 8. in van Heijenoort 1967, bis 13 August 1904, Teubner, Leipzig; translated pp . 129-138 . 1923 Die logischenGrundlagen der Mathematik,Mathematische Annalen, vol. 88, pp . 151-165. der Mathematik,Mathematische Annalen, 1929 Problemeder Grundlegung vol. 102, pp. 1-9. . Hilbert , D, and W. Ackermann 1928 Grundziige der theoretischen Logik, Springer,Berlin. Huntington , E. 1902 A completesetof postulates forthetheoryof absolutec ontinuousmagnitude, Transactions of the American Mathematical Society, vol. 3, pp . 264-279. 1905 A completeset ofpostulates forordinarycomplexalgebra,Transactions of the American Mathematical Society , vol. 6, pp. 209-229. Kreisel, G. 1967 Informalrigourand completenessproofs, Problems in the philosophy of Amsterdam,pp. 138mathematics (1. Lakatos,editor), North-Holland, 186. Lewis, C. 1. 1918 A survey of symbolic logic, Universityof California P ress, Berkeley . Lindstrom, P. 1969 On extensionsof elementary logic, Theoria, vol. 35, pp . 1-11 .
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Lowenheim, L. 1915 Uber Moglichkeiten im Relativkalkiil, Mathematische Annalen, vol. 76, pp. 447-479; translated in van Heijenoort 1967, pp . 228-251. Montague,R . 1965 Set theoryandhigher-order logic,Formal systems and recursive functions (J. Crossleyand M. Dummett,editors),N orth-Holland, Amsterdam, pp . 131-148. Moore, G. 1978 The originsof Zermelo 's axiomatization of settheory,Journal of Philosophical Logic, vol. 7, pp . 307-329. 1980 Beyond first-orderlogic,the historicalinterplaybetweenlogicand set theory,History and Philosophy of Logic, vol. 1, pp. 95-137. 1982 Zennelo's axiom of choice: Its origins , development, and influence, Springer-Verlag, New York. 1988 The emergenceof first-orderlogic, History and philosophy of modern mathematics (W . Asprayand P. Kitcher,e ditors), MinnesotaStudiesin thePhilosophyof Science, vol. 11, Universityof MinnesotaPress, pp. 95-135. Myhill,J . 1951 On theontological significance oft heLowenheim-Skolemtheorem,Academic freedom, logic and religion (M. Whiteeditor),American PhilosophicalSociety,Philadelphia , pp. 57-70; also inContemporary readings in logical theory (1. Copi and J. Gould,editors), Macmillan,New York, 1967, pp . 40-54 . Peano,G. 1889 Arithmetices principia, Nova methodo exposita, Turin; translated in van Heijenoort 1967, pp . 85-97. Poincare,H. 1909 La Logique de l'infini,Revue de metaphysique et morale, vol. 17, pp. 461-482. Quine, W. V. O. 1941 Whiteheadand therise ofmodernlogic, The philosophy of Alfred North Whitehead (P. A. Schilpp, editor),Tudor PublishingCompany, New York, pp. 127-163. 1960 Word and object, The MIT Press. 1986 Philosophy of logic, secondedition,Prentice-Hall, EnglewoodCliffs. Ramsey, F. 1925 The foundations ofmathematics , Proceedings of the London mathematical society, series 2, 25, pp . 338-384. Rang, B., and W. Thomas 1981 Zermelo's discovery ofthe'Russellparadox',Historia Mathematica, vol. 8, pp . 15-22. Resnik, M. 1980 Frege and the philosophy of mathematics, CornellUniversityPress, Ithaca .
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257
Schoenflies, A . 1911 Uber die SteHungder Definitionin der Axiomatik, Jahresbericht der Deutsche Mathematiker- Vereinigung, vol. 20, pp . 222-255. Schonfinkel, M . 1924 Uberdie Bausteineder mathematischen Logik, Mathematische Annalen, vol. 92, pp. 305-316; translated in van Heijenoort 1967, pp . 355-366. Schroder , E. 1890 Vorlesungen iiber die Algebm der Loqik, vol. 1, Teubner,Leipzig. Shapiro,S. 1983 Remarks on thedevelopmentof computability,History and Philosophy of Logic, vol. 4, pp . 203-220. 1985 Second-orderlanguagesand mathematicalpractice , The Journal of Symbolic Logic, vol. 50, pp. 714-742. 1990 Second-orderlogic, foundations , and rules,The Journal of Philosophy, vol. 87, pp. 234-261. 1991 Foundations without foundationalism: A case for second-order logic, OxfordUniversityPress, Oxford. Skolem, T . 1920 Logisch-kombinatorische Untersuchungen iiberdie Erfiillbarkeit oderBeweisbarkeitmathematischerSiitze nebst einem Theoreme iiber dichte Mengen, Videnskapsselskapets skrifter I. Matematisk-naturvidenskabelig in van Heijenoort 1967, pp . 252-263. klasse, no. 4; section1 translated 1922 Einige Bemerkungenzur axiornatischen B egriindungder Mengenlehre, Matematikerkongressen i Helsingfors den 4-7 Juli 1922, AkademiskaBokin van Heijenoort 1967, pp . handeln,Helsinki, pp . 217-232; translated 291-301. 1928 Uber die mathematischeLogik, Norsk matematisk tidsskrift, vol. 10, pp . 125-142 ; translated in van Heijenoort 1967, pp. 508-524. 1930 Einige Bemerkungenzu derAbhandlung von E. Zermelo: 'Uberdie Definitheitin der Axiomatik', Fundamenta Mathematicae, vol. 15, pp . 337341. 1933 Uber die Unmoglichke it einer vollstiindigen Charakterisuerung der Zahlenreihem ittelseines endlichenAxiomsystems, Norsk matematisk [orenings skrifter, series 2, no. 10, pp . 73-82. 1934 Uber die Nicht-charakterisierbarkeit der Zahlenreihemittelsendlich oderabzahlbar unendlich vielerAussagenmit ausschliesslich ZahlenvariabIen, Fundamenta Mathematicae, vol. 23, pp. 150-161. 1941 Sur la portedu theormede Lowenheim-Skolem,in Gonseth1941, pp . 25-52 . 1950 Some remarkson thefoundation of settheory,Proceedings of the international congress of mathematicians (Cambridge, Massachusetts), American Mathematical Society, Providence,1952, pp . 695-704. 1958 Une relativisation des notionsmathematiquesf ondamentales, Colloques internationaux du Centre National de la Recherche Scientifique, Paris, pp . 13-18.
258 1961
STEWART SHAPIRO Interpretation of mathematical theoriesin thefirst-order p redicatecalculus, Essays on the foundations of mathematics, Dedicated to A . A. Fraenkel (Y. Bar-Hillelet al., editors),Magnes Press, Jerusalem,pp . 218-225 .
Tharp,L. 1975 Which logic istherightlogic?,Synthese, vol. 31, pp. 1-3l. Thiel, C. 1977 Leopold Lowenheirn:Life, work, andearlyinfluence,Logic colloquium 76 (R. Gandyand M. Hyland,e ditors),North-Holland, Amsterdam, pp . 235-252. Van Heijenoort , J. 1967 From Freqe to Godel, HarvardUniversityPress, Cambridge, Massachusetts. and logic aslanguage,Synthese, vol. 17, pp. 324-330 . 1967a Logic ascalculus Veblen, O. 1904 A system of axioms forgeometry , 1hmsactions of the American Mathematical Society, vol. 5, pp . 343-384 . Von Neumann, J . Journal fur die reine und ange1925 Eine Axiomatisierungder Mengenlehre, wandte Mathematik, vol. 154, pp. 219-244; translated in van Heijenoort 1967, pp . 393-413. 1927 Zur Hilbertschen Beweistheorie,M athematische Zeitschrift, vol. 26, pp . 1-46. 1928 Uber die Definitiondurchtransfinite Induktionund verwandteFragen der allgemeinen Mengenlehre, Mathematische Annalen, vol. 99, pp. 37339l. Wagner, S. 1987 The rationalist conceptionof logic,Notre Dame Journal of Formal Logic, vol. 28, pp , 3-35. Wang, H. 1974 From mathematics to philosophy, Routledgea nd Kegan Paul,London. 1987 Reflections on Kurt Godel, MIT Press. Weston,T . 1976 Kreisel, thecontinuumhypothesisand second-orderset theory,Journal of Philosophical Logic, vol. 5, pp . 281-298. Weyl, H. Mathematisch1910 Uberdie Definitionen der mathematischen Grundbegriffe, naturwissenschaftliche Blatter, vol. 7, pp . 93-95, 109-113. 1917 Das Kontinuum, Veit, Leipzig. Whitehead,A . N., and B. Russell 1910 Principia Mathematica, vol. 1,CambridgeUniversityPress, Cambridge, England . Zermelo,E. 1904 Beweis, dass jede Mengewohlgeordnet werden kann,Mathematische Annalen, vol. 59, pp. 514-516; translated in van Heijenoort 1967, pp . 139141.
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NeuerBeweis fiir dieMoglichkeit e inerWohlordnung , Mathematische Annalen, vol. 65, pp. 107-128 ; translated in van Heij enoort 1967, pp . 183198. 1908a Untersuchungen tiberdie Grundlagen derMengenlehre . I, Mathematische ted in van Heijenoort 1967, pp . Annalen, vol. 65, pp . 261-281 ; transla 199-215. 1929 Uber den Begriffder Defintheitin der Axiomatik, Pundamenta Mathematicae , vol. 14, pp. 339-344 . 1930 Uber Grenzzahlen und Mengenbereiche : Neue Untersuchungen iiber die Grundlagen der Mengenlehre , Fundamenta Mathematicae, vol. 16, pp. 29-47. 1931 Uber stufender Quantifikationund die Logik des Unendlichen,Jahresbericht der Deutsche Mathematische Verein, vol. 31, pp. 85-88. 1908
RAYMOND SMULLYAN
EQUIVALENCE RELATIONS AND GROUPS
Abstract.Ourpurposeis to show howthelogic ofrelations can be uitilized . There are some strikingsimilaritiesin certain in thestudyof group theory theoremsin grouptheoryandcertainresultsa boutequivalence r elations, and we show howtheformercan bederivedas consequencesof thelatter . This transition is accomplishedby m eansof certainismorphism theorems,p rovedin considerable generality in section2, and applied to groups in section3. In section1 we give severalmiscellaneoust heoremson equivalencer elations , which laterturnoutto have theiranaloguesin thetheoryof groups.
O. PRELIMINARIES ON RELATIONS
We letR be a binaryrelation, Dom(R) its domain, Ran(R) its range, and S its field(Dom(R) U Ran(R)), or as we say,R is on the setS. As usual,for any R 1 and R 2 on S, by theirrelative product R 1R2 (sometimes in tworelations orderedpairs (x, y) of the literature writtenR 1 I R 2 ) is meantthe set of all elementsof S such thatxRz and zRy bothhold for at least one elementz of S. Thus x(R 1R2)y iffl (3z)(xRz 1\ zRy). By R 2 we mean RR. For anyXES, by xR we mean the set of all y such thatxRy. Thus y E xR iffxRy. For any subsetA of S, by AR we shall meanthesetof ally such thataRy holds for for any at leastone a EA . Thus yEAR iff(3a)(a E Al\aRy). In particular, singleton{x}, whereXES, {x}R is xR. By the inverse R - 1 of R is meant the set of all orderedpairs (x,y) such thatyRx. Thus xR-1y iff yRx. The following propertiesare all trivial(R, R 1, R 2 arerelations on S, A and B aresubsetsof S, and x is any elementof S) . (A ~ B) :) (AR ~ BR). Thus also, (x E B) :) (xR ~ BR) .
PI
P 2 A(R 1R2 ) = (ARt}R 2 • Thus also,x(R 1R2 ) = (xRt}R 2 •
= AR U BR.
P3
(A U B)R
P4
A(R 1 U R 2 ) = AR 1 U AR2 •
P 5 (Rl
~ R 2 ) :) (RR 1 ~ RR2 ) .
P 6 R(R 1 U R 2 )
= RR 1 U RR2 •
P 7 R 1(R2R3) = (R 1R2)R3. (Multiplication of relations is associative.) P s (R 1R2)-1 = R?:l R 1 1 • 1
We use "iff" toabbreviate"if and onlyif'. 261
C. Anthony Anderson and M. Zeleny (eds.), Logic, Meaning and Computation , 261-27\. © 2001 Kluwer Academic Publishers . Printed in the Netherlands .
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RAYMOND SMULLYAN
P g (R-I)-I=R. P lO (R I U R 2)-1 = R 1 I U R"2 I . P u (R I n R2)-1 = Ri l n R"2 I . R is calledsymmetric if R ~ R-I -whichmeans thatxRy :::> yRx, for all zand y. P l2 R is symmetriciff R = R - I . 1. EQUIVALENCE RELATIONS
R is calledtransitive if R 2 ~ R-whichmeans (V'x)(V'y)(V'z) ((xRy 1\ yRz) :::> xRz) . R will be called triangular if R 2 ~ R-I-which means (V'x)(V'y)(V'z) ((xRy 1\ yRz) :::> zRx) . By Is we shall mean t herestriction oftheidentityrelation I to S x S-i.e., Isis theset of all orderedpairs (x, x) such thatxES. We areassuming thateveryelementof S is eitherin thedomain of R or therangeof R. We callR reflexive if Is ~ R-whichmeans thatxRx holds for everyxES. The following is well known . PROPOSITION
1.1. If R is transitive and symmetric, then R is reflexive.
Proof. Suppose R is transitive a nd symmetric. If x is in thedomain of R, thenxRy holds for someYES, hence yRx (since R is symmetric), hence xRx (since xRy 1\ yRx and R is transitive) . Now suppose x is in therangeof R . Then yRx for some y in S , hence xRy (by symmetry), henceagainxRx (by transitivity) . Thus xRx holds for every xES. -1 R is called anequivalence relation if R is bothsymmetricand transitive. Thus Proposition1.1 is thatevery equivalencerelationis reflexive . (Very often, reflexivitytaken is as partof the definition of equivalence r elation .) In theelegantn otationof relativeproducts,an equivalence r elation is a relation R such thatR 2 ~ R ~ R- I . (This shouldbe wellremembered,as it willfrequently crop up.) PROPOSITION
1.2. R is an equivalence relation if and only if R 2
= R = R- I .
Proof. Suppose R 2 ~ R ~ R- I. Then R = R- I (by P I2), so it onlyremains to showthatR ~ R 2 . Well,suppose xRy. Also yRy (by Proposition1.1), hence xR2y. Thus xRy:::> xR 2y for allx ,y in S, hence R ~ R 2 • Thus if R is an equivalence relation , thenR 2 = R = R- I . (The converse istrivial.) -1
1.3. R is an equivalence relation if and only if I s ~ Rand R 2 ~ R- I - in other words, an equivalence relation is a relation that is both reflexive and triangular.
PROPOSITION
EQUIVALENCE RELATIONS AND GROUPS
263
Proof If R is an equival ence relation , thenR is reflexive(Proposition1.1) andthetransitivity andsymmetryobviouslyimply triangularity. Conversely, suppose thatR is reflexivea ndtriangular. To showsymmet ry,suppose xRy. Also yRy (since R is reflexive) . Hence yRx (by triangularity) , and so R is and symmet ric. As fortransitivity , thisis obviouslyimplied by triangularity symmetry. --j PROPOSITION 1.4. A necessary and sufficient condition that R be an equivalance relation is that for every x and y in the field of R, xRy iff xR = yR
Proof (a) Necessity : Suppose R is an equivalencerelation . Let x and y be any elementof thefield ofR. We will show : (1) xRy ~ (xR = yR) j (2) (xR = yR) ~ xRy. To show (1), suppose xRy. Then y E xR. Hence yR ~ (xR)R (by Pd. Hence yR ~ x(RR) (since (xR)R = x(RR) by P 2 ) , but RR = R (by Proposition1.3), henceyR ~ xR. Also, sincexRy, thenyRx (by symmetry),so by similarreasoning,y R ~ xR. Thus xR = yR. To show (2), suppose xR = yR. Now by Proposition1.1, R is reflexive, hence y E yR (i.e., yRy) , hence y E xR, andso xRy. (b) Sufficiency: Suppose thatfor allx and y in thefield ofR, xRy iff (xR = yR). To showsymmetry,suppose xRy. Then xR = yR (by hypothesis). Hence yR = xR, and so yRx (by hypothesis).Thus R is symmetric. As fortransitivity , suppose xRy and yRz . Then xR = yR and yR = z R (by hypothesis),hence xR = z R, and so again by hypothesis, xRz. Thus R is transitive . --j DEFINITION 1.5 Partitions.A collection ~ ofsetsis called a partition if for any twoelementsA and B of E, eitherA = B or An B is empty. For anyrelation R we let~R be thecollection of all sets xR, wherex is in the domain of R. It is well knownthatif R is an equivalence relation, then~R is a partition,andthisfact is asimple corollary of Proposition1.4 as follows. Suppose R is an equivalencer elation a nd thatx and yareelementsof its domainsuchthatxR and yR have acommon elementz . Then xRz and yRz , hence alsozRy (since R is symmetric), hence xRy (since R is transitive), hence xR = yR by Proposition1.4. truethatif ~R is a partition ,then R is necessarilyan It is not conversely butwe havethefollowing . equivalance relation, PROPOSITION 1.6. If E R is a partition and R is reflexive, then R is an equivalence relation.
264
RAYMOND SMULLYAN
thefield of Proof Assume thehypothesis. Let x and y by any elements of R. We will showthatxRy iffxR = yR, and the conclusion will thenfollow by Proposition1.4. (a) Suppose xRy. Theny E xR. Alsoy E yR (since R is reflexive). Hence xR = yR (since I::R is a partition). (b) Conversely, suppose xR = yR. Sincey E yR theny E xR, hencexRy.-J We thus have the following . PROPOSITION
1.7.
(1) If R is an equivalence relation, then I:: R is a partition.
(2) If R is reflexive, then R is an equivalence relation if and only if I:: R is a partition. Here are somefurther factsaboutequivalence relations . PROPOSITION
1.8. R is an equivalence relation iff RR- 1 = R .
r thenR = R- 1 (Proposition1.2), hence Proof If R is an equivalenceelation, RR- 1 = RR, butRR = R (Proposition1.2), so RR- 1 = R . Conversely , suppose RR- 1 = R . To showthatR is symmetric, suppose xRy. Thenx(RR- 1)y, hencexRz /\ zR-1y for some z. Hencex Rz and yRz , so yRz and zR-1x, hencey(RR- 1)x, henceyRx. Thus R is symmetric. Since R is symmetric, R = R- 1 (by P 12 ) , and since RR- 1 = R, then RR = R, henceR is transitive . -J 1.9. For any two equivalance relations E 1 and E 2 on S, a necessary and sufficient condition that E 1E 2 be an equivalence relation is that
PROPOSITION
E IE2 = E 2E 1 •
Proof Suppose E I and E2 arebothequivalence relations on S. relation , thenE IE2 = (E IE2)- 1 (a) Necessity : If E IE2 is an equivalence (by Proposition1.2) = E:;l Ell(P g ) = E 2EI (by P 12 , since E I and E 2 are symmetric). (b) Sufficiency: Suppose E IE2 = E 2EI. To showthatE 1E2 is transitive, (E 1E2)(EIE2) = E 1(E2(EIE2 (by P 7 ) = E 1((E2EdE2 ) (since E 2(EIE2) = (E2Et}E2 by P 7 ) = E I((E 1E2)E2) (since E 2EI = E IE2) = (E IEd(E2E2) (by twoapplicationsof P 7 ) = E IE2 (since E? = E I and Ei = E 2). Thus
»
(E IE2f = E IE2.
As forsymmetry, (E IE2)- 1 = E:;l Ell(P g ) and Ell= E I , by Proposition1.2).
=
EIE I (since = E:;!
=
E2 -J
2. OPERATIONS AND THEIR INNER RELATIONS
In whatfollows,S will be a set fixed for the discussion and 0 will be a functionthatassigns to each orderedpair (x, y) ofelementsof S, an element
265
EQUIVALENCE RELATIONS AND GROUPS
denotedxy (not necessarily inS) . A subset H of S will be called closed (under0) if for everyx and y in H , xy is in H . We call 0 closed if S is closedunderO. For anysubsetsHand K of S, by H K is meanttheset of all elements hk, whereh E Handk E K. Also for any element XES, by xH we mean {x} H , which istheset of all elements xH, whereh E H . The sets xH (wherex rangesoverS) are called the(left)cosets of H. By H 2 we mean 2 H H. We notethatH is closed iffH ~ H . The following propertiesare trivial(H and K aresubsetsof S and x is any element of S) . P1 3
(H ~ K) ::) (xH
< xK) .
P 14 x(HuK)=xHUxK. DEFINITION 2.1 Associativity . We shall define this in a way thatdoes not presupposeclosure . We shall call associative 0 if for all x , y and z in S, if xy and yz arebothin S, then x(yz) = (xy)z . The following againtrivial. is P 15 If 0 is bothclosed andassociative , thenfor all s ubsetsH , K and L of S, H(KL) = (HK)L . Also, for any X ES, x(KL) = (xK)L . DEFINITION 2.2 Cancellation Laws. We shall call left 0 concellative if it x, y , z in S) . We obeys the left cancellation law:(xy = xz) :::> (y = z) (for all call 0right concellative if for all x, y, z in S, (xz = yz) ::) (x = y). PROPOSITION 2.3. The following three conditions are equivalent . (1) For all subsets H , K of S and every XES, H
~
K iff xH
~
xK.
(2) For all subsets H , K of S and every XES, H = K iff xH = xK .
(3) 0 is left cancellative.
Proof. Obviously (1) implies (2) . Also (3) isthatspecial case of (2) in which Hand K aresingletons . And so itremainsto showthat(3) implies (1). Suppose 0 is leftcancellative . Of course(H ~ K) :::> (xH ~ xK) (P 13 ) , and so it remains to show the converse . So suppose xH ~ xK. We must show H ~ K . Let h be any element of H . Then xh E xH, and soxh E xK, hence xh = xk for some k E K , hence h = k (by leftcancellation) , hence h E K. Thus for all h E H, h is also inK, and soH ~ K . -l DEFINITION 2.4 Right Soluble.We next define 0 to be right soluble if for all a andb in S, thereis some XES, such thatax = lr-equivalently , if for every a andb in S, b E as-or again equivalently thatfor every element a E S, S ~ as.
266
RAYMOND SMULLYAN
(The definitionthat0 is left soluble-for every a and b in 5, theequation xa = b has asolutionin 5-willplay less of a role .) We come now tothecentral notion. DEFINITION 2.5 Inner Relations.For anysubsetH of 5, we define[H] to be the set of all orderedpairs (x , y) suchthaty E xH. Thus x[H]y iffy E xH. Alsox[H] is xH. For a single element h of 5, by [hI we mean [{h}], and thus x[h]y iff y E x{ h} iffY = xh . Thus alsox[h] = xh. The relat ions [H], where H ranges over all subsetsof 5, will be called the inner relations of O. Innerrelations willplayakey role injustaboutallthatfollows . The following is an easy consequenceP14. of
P 16 For anysubsetsH, K of 5, [H U K]
= [H] U [K] .
DEFINITION 2.6 Isomorphisms. We now let7r be themapping thatassigns to eachsubsetH of 5 theinnerrelation [H] . Thus 7r(H) = [H]. PROPOSITION 2.7. If 0 is left concellative, then
7r
is a 1-1 correspondence.
Proof. Suppose 0 is leftcancellative . Suppose [H] = [K] . We are to show thatH = K. Well, letx by any elementof 5. Then x[H] = x[K] , which -l means xH = xK, and henceH = K by Proposition2.3. PROPOSITION 2.8. If 0 is closed, then the following two conditions are equioalent:
(1) For any subsets Hand K of 5, [H][K] = [HK]. (2) 0 is associative.
Proof. Suppose 0 is closed(5 is closedunder0). (a) Suppose 0 is associative . Then for anysubsets H, K of 5 and any elementx E 5 , x([H][K]) = (x[H])[K] (by P2) = (xH)K = x(HK) (by P 15 ) = x[HK] . Since for all x E 5 , x([H][K]) = x[HK] , then[H][K] = [HK]. (b) Conversely , suppose (1) holds.Thenin particular, for any elem ents y and z of 5, [y][z] = [yz], and so for any x in 5, x(yz) = x[yz] = x([y][z]) = (x[y])[z] (by P 2 ) = (xy)z, and so 0 isassociative. -l We shall sayt hat7r is an ismorphism with respect to multiplication if 7r is 1-1 and if for all subsetsHand K of 5, [H][K] = [HK] . From Proposition 2.7 and 2.8 we have the following . THEOREM 2.9. A sufficient condition that tt be an isomorphism with respect to multiplication is that 0 is closed, associative and left cancellative. Thus if o has these three properties, then for any subsets Hand K of 5 :
EQUIVALENCE RELATIONS AND GROUPS
267
(1) [H] = [K] iff H = K,
(2) [H][K] = [HK] . Proof For eachxES, the innerrelation[x] is single valued (since for any YES, y[x] consistsof the single elementyx) . Thus if H consistsof only one element , then[H] is single valued.The converse also holds if 0 is left and that[H] is single cancellative, for suppose that0 is leftcancellative valued.Thenfor anyXES, xH containsonly oneelement , so if handk are bothelementsof H, thenxh and xk arebothin xH, andso xh = xk, hence h = k by leftconcellation. -1 Let us call an i nnerrelationsingulary if it is [x], for some elementx . We now seethatevery singularyrelationis single valued, and if 0 is left thenthe only single valued innerrelationsare thesingulary cancellative, ones. We shallsay thatS is isomorphic (under7r) to theset ofsingularyinner relations if for eachx and yin S, (1) [x] = [yl iffx = Yj
(2) [xy] = [x][y] . By Theorem2.9 we have the following . COROLLARY 2.10. If 0 is closed, associative and left cancellative, then S is isomorphic to the set of singulary inner relations of O. DEFINITION 2.11 Permutations . If 0 is rightconcellative , then each singularyinnerrelation [x] is not only single valued, butis 1-1, because ifa[x]b and c[x]b bothhold,thenb = ax and b = ex, henceax = bx, hencea = b by S is finite,then[x] is a permutation. rightcancellation . If furthermore Thus by Corollary 2.10 we have the following. THEOREM 2.12. If 0 is closed, associative and both left and right cancellative, then S is isomorphic to a set of 1-1 relations.2 DEFINITION 2.13 BooleanIsomorphism. For anysubset H of S, by H' we shallmean thecomplementof H relative to S (theset of all elementsof S not inH). For anyrelation R on S, by R' we shallmean thecomplement to S x S-Le., the set of all orderedpairs (x, y) of elementsof of R relative S such thatxRy doesn't hold.Thus for anyx and y in S , xR'y iff-'(xRy) . 2From this easilyfollow s Cayley's Theorem thatevery groupis isomorphic to a tr ansformationgroup, and if the groupis finite, to a permutat ion group-d.application A 3 in section3.
268
RAYMOND SMULLYAN
We knowthatfor anysubsetsHandK of S, [HUK] = [H]U[K] (P16)' If also7r is 1-1, thenwe mightcall7r an isomorphism withrespectto union.If also [H'] = [H]' holds for every subsetH of S, thenwe will call 7r a Boolean isomorphism. And so if7r is 1-1 and [H'] = [Hl' holds for all subsetsH of S, then7r is a Booleanisomorphism (by P 16). (If 7r is a Booleanisomorphism, thenof coursealso [H n K] = [H] n [K].) We now wish to show thata necessaryandsufficientconditionfor7r to be a Booleanisomorphism is that0 is bothleftcancellative and right soluble. This will follow from thenexttwopropositions. PROPOSITION
2.14. 0 is left cancellative iff for all subsets H of S, [H'] ~
PROPOSITION
2.15. 0 is right soluble iff for all subsets H of S, [Hl' ~ [H'].
[H]/.
Proof of Proposition 2.14. Suppose 0 is leftcancellative.We show that x[H']y :J X[H]/ y for allx and y in S, and hencethat[H'] ~ [H]/ , so suppose x[H']y. Then y E xH', which meansthaty = xk for some k outsideH. Hence by left cancellation, y =1= xh for anyh E H (for otherwisewe would have xh = xk and hence h = k, hence k would be inH) . Thus y (j. xH, so x[H]y doesn'thold, henceX[H]/ y. This proves thatif 0 is leftcancellative , thenx[H']y :J x[Hl'y for all x and y in S, and hencethat[H'] ~ [Hl'. Conversely,s uppose that[H'] ~ [H]' for allsubsetsof H of S. Then for any elementa E S, [{a}'] ~ [{a}]' , hence [{a}] ~ [{a}']', hence for any X ES, x[{a}] ~ x[{a}']/, and so for anyelementy, y E x[{a}] :J y (j. x[{a}']. Now, xa E x[{a}], andso xa (j. x[{a}']. This means thatxa =1= xb for anyb E {a}', and thusxa =1= xb for anyb =1= a, or equivalently, xa = xb implies a = b, and so 0 is leftcancellative.
-j
BeforeprovingProposition2.15, let us notet hat0 is rightsolvable iff
[S] = S x S, because0 is rightsoluble iff for all x and y in S, y E xS, or equivalently, for everyx and y in S, x[S]y, whichmeans S x S ~ [S] (andof course[S] ~ S x S) .
thatfor anysubsetH of Proof of Proposition 2.15. We shall show, in fact, S, [Hl' ~ [H'] iff 0 isrightsoluble. Well, [H]' ~ [H'] iff [H] U [H'] = S x S. But [H] U [H'] = [H U H'] (by (P 16 ) = [S], and so [H]' ~ [H'] iff [S] = S x S,
which inturnis thecase iff 0 is right soluble(as alreadynoted).
-j
REMARK 2.16. We see from theproofabove thatif so much as onesubsetH of S has thepropertythat[Hl' ~ [H'], theneverysubset H of S has that property.
t helasttwopropositionsthatif 0 is bothleftcancellative It follows from and rightsolublethenfor everysubset H of S, [H'] = [H]/, and alsothat
EQUIVALENCE RELATIONS AND GROUPS
269
1r is 1-1 (by Proposition2.7), and hencethat1r is a Booleanisomorphism. Converselyif 1r is a Booleanisomorphism, thenIH'j = IH]' for everysubset H of S, hence 0 must be bothleftcancellative andrightsoluble . And so we havethefollowing: 2.17. 1r is a Boolean isomorphism iff 0 is both left cancellative and right soluble.
THEOREM
3. EQUIVALENCE RELATIONS AND GROUPS
An elementi of S is called anidentity (relative to 0) if ix = x i = x forevery x in S. As is well known,S can have atmost oneidentity(since ifi and j are bothidentities , thenij = j andalsoij = i, hence i = j). An elementb of S is called aninverse of anelementa of S if ab and ba arebothidentities (andhence alsobothequal). As is well known, 0if is closedandassociative, thenan elementa of S cannothave morethanone inverse(becauseif band c arebothinverses ofa, thenb = b(ac) = (bc)c = c). If an elementx has an inverse,thisinverse (which isunique)is denotedX-I , andfor anysubsetH of S, by H-I is meantthesetof inverses of all theelementsof H . We shall callH symmetric if H <;;;; H - I . Then H is symmetriciff everyelementof H has an inverse in H . We shall saythat1ris an isomorphism with respect to inversion if1ris 1-1 andfor everysubsetH of S, IHj-1 = [H-Ij. Now, a non-empty subset G of S is called agroup (withrespectto 0) if 0 is associativeand if G2 <;;;; G <;;;; G- I (i.e., G is closedand symmetric). These conditionsa utomatically imply thatG has anidentity,since G, being non-empty,has anelementx, and x thenhas an inverseX -I in G, and then by closure , xx-I, which istheidentity,is in G. A subset H of G is called a subgroup of G if H is also agroup(withrespectto 0) . In whatfollows,G will beassumed to be agroupwith respectto 0 and thatG x G is thedomain of O. As in section2, 1r will bethemapping that assigns to each s ubsetH of G the inner relation [Hj. Since G is a group,then, as is well known, 0 is leftand rightcancellative (since (xa = xb) :J (x-l(xa) = x-l(xb)) :J ((x-Ix)a = (x-Ix)b) :J (a = b), and similarlyfor rightcancellation) and both leftand rightsoluble(since ax = b has thesolutionx = a-Ib, and xa = b has thesolutionx = ba- l) , and so byearlierresults1r is a Booleanisomorphism and an isomorphism withrespectto multiplication. We now showthefollowing. THEOREM
3.1. 1r is an isom orphism with respect to inversion.
Proof We are to show thatfor anysubset H of G, IH-Ij = [Ht l . We first show that[Hj-l <;;;; IH-Ij. Well,s uppose X[Hj-Iy . Then y[Hjx, so x = yh for some h E H . Then xh- l = (yh)h- l = y(hh- l) = y , so y = xh :", Therefore y E xH- I (since h- l E H- I) , hence xIH-Ijy. Thus for allx and y in G, x[Htly :J x[H-Ijy, andso IHj-1 C IH-Ij.
270
RAYMOND SMULLYAN
To show that[H-Ij ~ [H] -l, suppose X[H-I]y . Then y E xH- 1 , so Y = xh- I for some h E H, hence yh = (xh-I)h = x(h-Ih) = x, hence yh = x, so x E yH, hence y[H]x, andso X[H]-I y. Thus x[H-I]y J X[H]-I y for allx and y in G, so [H- 1 ] ~ [Hl-I. --j By Theorem2.9, 2.17 and3.1 we now have the following.
3.2 Main Isomorphism Theorem. Suppose that G is a group with respect to 0 and 7f is the mapping that assigns to each subset H of G the inner relation [Hl. Then 7f is a Boolean isomorphism and an isomorphism with respect to multiplication and inversion. Thus for any subsets Hand K ofG:
THEOREM
(1) [Hl
= [K] iff H
=
K,
(2) [H][K] = [HK],
(3) [H']
[HI', (4) [H U Kl = [H] U [K] and [H n K] (5) H ~ K iff [Hl ~ [K], (6) [H- 1 ] = [Hl- I . =
=
[Hl n [K],
4. APPLICATIONS
Many results in grouptheory -standard a ndotherwise -- ·arecorollaries ofour isomorphism theorems . Here are someapplications.
Al For anysubset H of G, H is a groupiff H 2 ~ H ~ H- I, but by the main isomorphism theorem,H 2 ~ H ~ H- 1 iff [HF ~ [H] ~ [Ht l , which t hushave aneat is the conditionthat[H] is an equivalencerelation! We proofof theknownresultthata subsetH of G is a subgroupof G iff [H] is an equivalence relation.Morespecificallywe can say for any subsetH of G: (1) H is closed(H 2 ~ H) iff [H] is transitive ( [Hf ~ [H]), (2) H is symmetric (H -I
~
H) iff [H] is symmetric ([Hl - 1
~
[H]);
(3) We alsonotethatfor anynon-emptysubset H of G, i E H iff H is reflexive(i being theidentityof G). To see that(3) holds,suppose i E H. Then for anyx in G, xi E xH, hence x E xH, hence x[H]x, andso [H] is reflexive. Conversely,[H] if is reflexive, theni[H]i, hencei = ih for some h E H, butih = h, so i E H. A 2 For anysubset H of G, we letL.H be theset of all left cosets xH of H. Actually L .H is L.[H j, as defined insection1. From the fact t hatL.R is a partitionif R is an equivalancer elation(Proposition1.7, (1» it follows by Al thatif H is a subgroupof G, thenL.H is a partition(a well known
EQUIVALENCE RELATIONS AND GROUPS
271
's Theoremthatif G is finite, then fact). From thisin turnfollowsLagrange thenumberof elementsof anysubgroupH of G must divide thenumberof elementsof G (since each coset x H of H has thesame numberof elements as H, by leftconcellation). A 3 Since in agroupwe haveclosure,associativity, andbothcancellation laws, then,as remarkedearlier, Cayley 's Theoremthateverygroupis isomorphic to atransformation groupis a consequerice of Theorem2.12. Actually , Theorem 2.9 is a substantial generalization of Cayley's Theorem, which is totheeffectthattheset ofsingletons{ x} is multiplica tively isomorphicto theset ofcorresponding singularyinnerrelations [x], whereas isomorTheorem2.9 tell s us thatthesetof all subsetsH is multiplicatively phic totheset ofall innerrelations [HI.
For thenext fewapplicationswe will simplystatetheresulta ndalongside it, inparentheses,thetheoremon thelogic ofrelations from which it follows by virtueof ourisomorphism theorems. A 4 If EH is a partitionand i E H , thenH is a subgroupof G. (If En is a partition and R is reflexive , thenR is an equivalence r elation , by Proposition
1.6.) A s H is a subgroupof G iffi E A and H 2 ~ H- l . (R is an equivalence relationiff R is reflexive and R 2 ~ s:', by Proposition 1.3.) A 6 H is a subgroupof G iff for all x and y in G, (y E xH iff xH = yH) . (R is an equivalence r elation iff for all x and y in G, (xRy iff xR = yR, by Proposition1.4.)
A 7 H is a subgroupof G iffHH RR- l = R , by Proposition1.8.)
l
= H . (R is an equivalence r elationiff
As For any two s ubgroupsHand K of G, H K is a groupiffH K = K H . (For any two equivalence relations E 1 and E2 , a necessaryandsufficientcondition thatE 1 E 2 be an equivalencerelation is thatE 1 E 2 = E 2E1 , by Proposition 1.9.) We have now seen someparallels betweenthetheoryof equivalencerelationsand thetheoryofgroups. This approachmight well be p ushedfurther e.g., in thestudyof normalsubgroups,or perhapsmore complexstructures such as ringsa ndfields.
PART II COMPUTATION
HENK BARENDREGT
DISCRIMINATING CODED LAMBDA TERMS
Abstract. A coding for a (type-free)lambda term M is a lambdaterm rM' in normalform such thatM (and its p arts)can be reconstructed from rM' in a lambdadefinableway. Kleene(1936) defineda coding rM,K and a self-interpreter EK E AO such that : VM E AO EKrM,K = M .
(1)
(Kleenedid it fortheAI-calculu s, buttheresultis validalsofortheAK-calculus .) a discriminator II K E AO such that : In thisstyleone can construct
VM N E A llKrM,KrN,K = ,
{true
(== >.xy.x) false (== AXY.y)
if M == N , else.
(2)
The terms EK and llK arecomplicated.EK appearsalsoin Church 1941 under the name form. . They depend on the lambdadefinabilityof functionson the ic properties. Inspired by a construction of int egersdealingwithcoded syntact P. de Bruin(see Barendregt 1991) a differe ntcoding rM' and an efficient selfinterpreter E E AO was constructed by Mogensen(1992) such thateven
(3) This construction d oes not representsyntaxvia an encoding as numbers, but directlyas lambdaterms. This resultsin a much lesscomplex E. Mogensen's construction was simplified even furth er in Bbhm et al, 1994. In thispaper we construct a simple discriminatorII E AO such that :
VM N E AO llrM~N' = ,
{true
if M false else.
=0 N,
(4)
Notethatin (1) and (4) thestatement is onlyaboutclosedlambdaterms, while thatin (2) and (3) is aboutalllambdaterms. Moreoverin (2) syntacti c equality on termsis consideredliterary , whilein (4) we ca n dealin an easy way withthe betternotionof equalitymoduloa changeof names for bound variables.It will become clearwhy thisis so. This paper firstappea red in From Universal Morph isms to Megabytes -a Baayen Space Odyssey, printedatCWI, Amsterdam.
1. INTRODUCTION
The most importantnotations for thetype-freelambdacalculus will be given here. Backgroundcan be found inBarendregt 1984. DEFINITION 1.1. Variables and t ermsof thelambdacalculus are defined by thefollowing a bstract syntax.
var
=
term =
a I var'
var I term termI A var term 275
C. Anthony Anderson and M. Zeleny {eds.), Logic, Meaning and Computat ion , 275-285 . © 200I Kluwer Academic Publishers. Printed in the Netherlands .
276
HENK BARENDREGT
NOTATION .
(i) M, N, . . . ,P, Q, .. . rangeover A-terms. The lettersx, y, Z , .. . range overvariables.Note thatthevariablesa re {a, a', a", ... , a(n) , ...}. (ii) A is thesetof lambdaterms. FV(M) is thesetof freevariablesof M. The setof closedtermsis AO = {M E A I FV(M) = 0}. (iii) The relation denotessyntacticequality;t he relation denotes syntacticequalityup to a changeof names of the bound variables . For example
=
=0
AX.X
=0 Ay·y =i- AX.X.
(iv) The relation = denotes/3-convertibility , axiomatizedby
(Ax.M)N = M[x := N]. Here [x := n] denotessubstitution of N forthefreeoccurrences of x. E.g.,
(X(AX.X))[X
a]
:=
=
a(Ax.x).
(v) IN is thesetof natural numbers. For n E IN thetermsC n = Afx.fnx, wherefOx x and fn+l x f(fn x), denotetheso calledChurchnumerals. Notethatthec., aredistinctnormalforms; hence
=
=
C n =C m
=> n=m
by theChurch-Rosser t heorem. : the redexes wantto be A lambdaterm can be seen as anexecutable evaluated.In this sense a normalform is notexecutable a nymore. For a lambdatermM its code r M' is a normalform suchthatM is reconstructible r M,K essentially as follows. from M . Kleene (1936) defined a code 1.2. (i) By inductionon thestructure of M we define#M .
DEFINITION
#(a(n») #(PQ) #(AX.P)
, < 1, < #(P), #(Q) » , < 2, < #(x), #(P) » .
Here < -, - > denotesa recursivepairingfunctionon INwiththerecursive projections( - )0, (- h:
« no,nl »i = ni · (ii) The map r - ,K : A~A is defined by rM,K =
C#M .
Notethatfor allM EA theterm r M,K is in normalform. Moreover,
rM,K = rN,K => M
=
N.
DISCRIMINATING CODED LAMBDA TERMS
277
PROPOSITION 1.3. There is no lambda term Q such that for all MEA (0) one has Proof. Suppose Q exists. Thenfor 1== AX.X one has
Q(II) Butalso
Q(II)
= rll,K = c#(II ) = c'
= QI =
Hence < 1, < #(1), #(1) »=<
K
rl,
= c#1
= C<2, <#(x),#(x» > '
2, < #(x), #(x) », a contradiction .
--I
In spite of thisfactthatthe'quote'Q does notexists, theinverse 'evaluation'E can beconstructed. THEOREM 1.4 (Kleene,1936). There exists an EK E AO such that for all ME AO one has Proof. See Kleene 1936 or Barendregt 1984, Theorem8.1.16 .
--I
The self-interpreter E canwork only for closed terms M (or termshavingat most a fixed finitesetof freevariables).T he reasonis thatif
then
FV(M) ~ FV(EKrM,K)
= FV(E K) .
Thereforeif EK is closed, thentheM have to be closed as well.This causes one difficulty in theconstruction of EK . The closedterms do not form a E first for context-free language.Kleenesolved thisp roblemby constructing theset ofcombinatorytermsCo builtfrom thebasis {K, S} using application only;thentherealself-interpreter can beobtainedby translations between AO and Co. A different c onstruction of aself-interpreter was given by aformerstudent of mine, using ideas from denotational semantics.The equationsfortheselfinterpreter arethoseof thesemanticinterpretation and theF plays therole of a valuation ( environment). THEOREM 1.5 (P. de Bruin). There exists an Eo E AO such that for all M E A and all F E A one has
EorM'F = M[Xl ,"" Xn : = FrX 1 I, ... , Fr xn IJ (simultaneous substitution) , where {Xl , " "X n} = FV(M) .
(5)
278
HENK BARENDREGT
Proof. By therepresentabili ty of computablefunctionsand thefixedpoint theoremthereis a term Eo E AO such that Frx.,K,
Eorx.,KF
EorpQ.,KF
(Eorp.,KF)(EorQ.,K F),
EorAX.p.,K F
Ax.(Eorp.,K Frrx' ......xj),
X,
Note thatF~ can be writtenas GFx, withG closed. By inductionon the structure of MEA one canshow thatthestatement holds. -l COROLLARY 1.6. There exists an Ed B E N such that for all MEN one has
Proof (P. de Bruin). We can take dB
E
== Am.Eom'.
Indeed,for closedtermsM it follows from (5) that
2. REPRESENTING DATA TYPES Afterseeing themethodof P. de Bruin, an improved versionof it was given by Mogensen(1992) by representing d atatypes directly(i.e., not usingthe natural numbers)in thelambdacalculus as done in, e.g . , Bohm and Berarducci 1985. This approachwas improved laterby Bohm et at. (1994) by constructing a new representation of datatypes intothetype-freelambda will betreated in a slightly modified form calculus.This new representation in thissection. DEFINITION 2.1. Write
(M}, .. . , M n )
Ur
Xz.z M; . . . M n , AX} . • , Xn ,Xi ,
true
ui,
false =
U~.
DISCRIMINATING CODED LAMBDA TERMS
279
Notethat (M 1 , • • . ,Mn)U~
u;
=
truePQ
P,
falsePQ =
Q.
In particular we have(M) = >.z.zM and ( ) = >.x.x = I. Now we define the notion of lists inspiredby the language LISP (McCarthy et al. 1961). 2.2. (i) Write
DEFINITION
nil cons
=
(), >.xy.(x, y),
car
(Ui) ,
cdr
(U~),
null ? =
(U33 ,U 21 , false,true).
(ii) Define
(), cons MdM2"" ,Mn+I J.
So forexample
(In Barendregt 1984 this termis writtenas [M1 , M 2 , M 3 , IJ . At the time of writingthatbook we did not yet see the usefulness terminating of a list with a specialconstructor .) Notethat car( consPQ)
=
P,
cdr( consPQ)
=
Q,
null ?nil = null ?( consPQ) = PROPOSITION
true, false.
2.3. There exist lambda definable functions ( )i such that for
1 ~ i ~ n one has
Proof. Take
(lh (i)i+l
carI, (cdri)i.
280
HENK BARENDREGT
DEFINITION 2.4. An (algebraic) signature s consistsof a number-n E IN (thoughtof as thelist ofsymbols [ft, . .., fnD together with a list of numbers [sl, , snl (thoughtof as the arity of therespectiveNs) . We write S=[Sl, , Sn]' For examplea field hassignature s = [2,2,1,1,0,0] (thought of asthearities of thefunctionsymbols [+, x , -, -1,0, l]; so ft = +, [z = x, et cetera) . DEFINITION 2.5. If s is a signature t henterms, theset ofterms of signature s , is defined as follows.
x E var => tl, . .. , t-; E terms
=>
x E terms, h(tl , .'" t sJ E terms'
For example,in thesignatureof fieldstheterm ft(Ii (x , !J(h(y, !4(z))), f6) is usuallyw rittenas x - yz-l + 1. DEFINITION 2.6. Let s = [s., ... ,sn] be a signature. (i) A lambda interpretation of s is a list of 'constructors' C l , . . . , Cn E A. (ii) Let C l , •••, Cn be a lambdainterpretation of s. Thenwe define amap
T: terms---+A as follows.
Tx T j i (ti , .. . ,t s ,)
x, C;(Tt 1 , · . . , T t si ],
where [Ttt , . .. , T t s ; 1 is thelistoperationon lambdatermsdefined in .22. of binary trees is [0,2] . The termt = h(h(li, Ii), Ii) Example. Thesignature denotesa simple treeandt' = h(fl, h(ft, Ii)) its mirorimage. Canwe find a lambdainterpretation forthissignature in sucha waythatmirroringbecomes lambdadefinable,i.e., for some F E AO one has FTt = Tt'? The following result,due toBohm et al. (1994), will affirm this. We presentthe resultin a modified formthatwill be useful for §4. THEOREM 2.7. For every algebraic signature s = [s}, . . . , Sn] there exists a lambda in terpretation C l, .. . ,Cn such that the following hold. (i) 'VAl . .. An 3F 1< i
s n.
(6)
(ii) The C l , . .. , C n only depend on n , not on the [Sl"'" sn]' In (6) we can take F == ((AI,"" An)} .
DISCRIMINATING CODED LAMBDA TERMS
281
Proof. Define Cf; == >.le.eU?l(e} . (i) Given A 1 , •• • ,An' we see whetherF == ((A 1 , ••• ,An}} works. Indeed,
((A 1 , • • • , A n })(CI; [Tt1>' • • , TtsJ) CI ; [Tt l" " ,Tts;](A1 , • • • , An) (A 1 , • •• ,An}U?lTtl "'" TtsJ((A 1 , ••• ,An)} Ai [Ttl ' ... ,Tts; ]F. (ii) By theconstruction . 2.8. Let s = [Sl,' .. , Sn] be an algebraic signature. Let C 1, .. . . C; be the lambda interpretation of s constructed in Theorem 2.7. Then for all B 1 .. . B n there exists an F such that
COROLLARY
F(CI;[Xl "" ,Xs;n = BiX1 . .. x s;F,
1::::; i : : ; n .
(7)
Proof. Let B 1, . . . , B n be given. Define Ai = >.l.Bi(lh .. . (l)s;' Then
Adx 1, "
" x s,J = BiX1, "" xs;·
ThentheF fortheAi found inTheorem2.7 is theF satisfying(7).
--j
Now we canprogramthefunctionthat'mirrors'trees. In the signature
[0,2] forbinarytreeslet
Th
leaf tree =
)..ab.Tf 2 (a ,b)
By Corollary 2.8 thereexistsan F such that Fleaf F{treea b)
leaf,
tree(fb)(fa) .
This F has themirroreffect . E.g., F{!2(!2(/t,fd,ft})= !2{ft,!2(f1,/t)). 3. A SIMPLE SELF-INTERPRETER
In Mogensen 1992 a simple codingand self-interpreter forlambdatermsis defined, using thefactthatdatatypes (termalgebrasof asignatures ) have a lambdainterpretation . The methodwas simplified by Bohm et al.(1994) by making use oftheirlambdarepresentation of algebraics ignatur es given in §2. DEFINITION
3.1. Let s be thesignature[1,2, I]. Define const app
Ch Ch
abs
C/3
_
>'le.eUrl(e}, >'le.eU~l(e}, >'le.eU~l(e} .
282
HENK BARENDREGT
DEFINITION 3.2. For MEA define rM" as follows .
r
canstIx] , app [rp." rQ"],
rpQ" AX.P"
abs [AX.r P"].
NotethatFV(M") = FV(M) . THEOREM 3.3 (Mogensen, 1992). There exists an E E AO such that VMEA F M'
= M.
Proof (Bohm et al., 1994) . By Corollary 2.8 thereexistsa termE E AO such that
p,
E( const[PJ) E( app [p, qJ) E( abs [PJ)
(Ep)(Eq) ,
Ax.E(px).
Then
x,
Erx"
e ro:
(Erp")(ErQ"), Ax.Erp."
ErAX.P"
Now theresultsfollows byinductionon thestructure of M . Using theconstructions in §2 theself-interpreter becomes
E == «AIf.(lh, Alf.f(lh (J(lh), >,I fx.f ((Ihx))). The construction in Bohm et al. 1994 is simpler. They take const app = abs
>.xe.eU~xe, >.xye.eU~xye,
= >.xe.eU3xe.
The resultingself-interpreter thenbecomes EB = «K, S, C)). Here K == >.xy.x, S == >.xyz.xz(yz) and C == >.xyz.x(zy) . For reasonsof uniformity we have giventhedefinitionof canst,app and abs as in 3.1. This will be useful in §4. 4. A SIMPLE DISCRIMINATOR
In thissection wewillconstruct a simple termdiscriminatingbetweencoded closedlambdaterms. The discriminationis evenmoduloa-conversion.For r .,K of Kleene . open termsdiscriminationis possible only for the coding
DISCRIMINATING CODED LAMBDA TERMS
283
4.1. (i) There exists a term OIN E AO such that
LEMMA
OINCnCm
t r ue ifn = m, = { false else.
(ii) There exists a term and E AO such that and true true=
true,
and truefalse
false,
and falsetrue
false,
and falsefalse
false.
Proof. (i) By therepresent abilityof the recursive functions. (ii) Take and = Xab.a trueb. PROPOSITION
4.2. There exists a term 0 E N such that (writing
On
for
one has On r X ,rx"
OINXY,
s.:x,rP'Q"
= =
On r x ,rAx'.P"
false, false,
On r PQ,rx" s.:eo»: P'Q" s.: PQ,r Ax'.P"
false,
On r Ax.p,rz'? Onr Ax.PorP'Q" On r Ax.p,r Ax'.P"
false, false,
and (On r
r»: P")(OnrQ,r Q"),
false,
=
On+l (rP'[x := cn])(rP"[x' := Cn]).
Proof. We introducet hefollowing ad hoc notation. (i) Let A l, . . . , An E A. Thenwe write
AX![A l , . .. ,AnI == ((Ax.A l, .. . , Ax.A n}}. (ii) If B i == [Ail, ' . . , A in], thenwe write
AXl![B l , . . . ,Bn ] == ((AxlB l , .. (iii) Let for 1~ i
~
n, 1 ~ j
~
[Aij] ==
. ,
AXlBn )} .
n be given A ij EA. Then
[[All, "" A ln], [Azl , . . . , A zn]' [Anl, . . . ,AnnlJ·
ocn )
284
HENK BARENDREGT
If n = 3 we may write[Ai j ] as
Now define8 == Antt'. (
~td!~t'd'n!!
I5IN(t)J(t')J false falSe]) false and(d(t)J(t')Jn)(d(th(t'hn) false tt'n, [ false false d(tn)(t'n)(S+n)
whereS+ lambdadefinesthesuccessorfunction.Then by Theorem 2.7 this 8 satisfiesthespecification. -1 PROPOSITION 4.3. For all M ,M' E A such that FV(MM') ~ {Xl,. . . , X n } and for substitutions = [Xl := Ck l ] •• • [X n := Ck n ] with k, i- k j (for 1 ::; i <
j ::; n) one has for p
8
p
r
*
> k,
(for all 1 ::; i ::; n) that
u»: M''''*
= {
true if M==a M', false else.
of M , in each casemaking distinctions Proof. By inductionon thestructure e of M'. We treatfourinstructive cases. accordingto thestructur Case M == X, M' == x'. Then
whereX == X;,X' == Xi '. This is trueor falsed ependingon whetherX == x' (so i = i') or X ;f x' (so i i- i') . Case M == x,M' == P'Q'. Then
Case M == PQ,M' == P'Q'. Then
8p r u»: M''''* =IH
and(8p r r»: p,..,*)(8p r Q"lrQ''''* ) and(true/ false)(t rue/ false) true/ false,
as it should(= trueonly ifPQ == P'Q', i.e., if bothP == P' and Q == Q'). Case M == AX.P, M' == AX'.P'. Then
8p r M'"' M ''''* =
8p + 1 (P"'[x := cp ]) ( P''''[x' := cp ]) 8p +l r P'"' P'[x' := x]"'[x := c p ] * 8p + l r P'"' P'[x' := xj"'*',
*
DISCRIMINATING CODED LAMBDA TERMS
285
with*' = * [x := cp ] being anadmissiblesubstitution . So if P = 0 P'[x' := z], else. Now M = 0 M' iff AX.P = 0 AX'.P' (=0 AX.P'[X' := xl) iffP = 0 P'[x' := z]. Hence we are done . -j COROLLARY
4.4. Write 6.
has
= 80 .
Then 6. E AO and for all M, M' E AO one if M = 0 M',
else. Proof. Immediatefrom theproposition .
cannothold for a rbitrary M, M' E A. Forexample, Notethatt hiscorollary it is impossible to discriminaterx.., and rx'..,. Indeed takex =t x' and make [x' := z]: a substitution
a contradiction . REFERENCES Barendregt , H. P. 1984 The lambda calculus, its syntax and semantics, revised edition, Studies in Logic andtheFoundat ions of Mathematics,North-Holland , Amsterdam. 1991 Theoreticalp earls: Self-interpretation in lambda calculus,Journal of Functional Programming, vol. 1(2), pp . 229-233. Bohrn, C., and A. Berarducc i 1985 Automaticsynthesis of typed A-programson term algebras,Theoretical Computer Sci ence, vol.39, pp . 135-154. Bohrn, C ., A. Piperno, and S. Guerrini 1994 A-definitions offunction(al)s by normalforms, ESOP '94 (D. Sannella, editor),LNCS 788, Springer,Berlin,pp . 135-149 Church,A. 1941 The calculi of lambda conversion, PrincetonUniversityPress, Princeton; reprintedby KrausReprintCorporation , New York, 1965. Kleene, S. C. 1936 A-definability and recursiveness , Duke Mathematical Journal , vol. 2, pp . 340-353. McCarthy , J. , P. W . Adams, D. J . Edwards,T . P. Hart, and M. I. Levin 1962 LISP 1.5 Programmer's manual, MIT Press, Cam bridge,Massachusetts . Mogensen, T. iE. 1992 Efficientself-interpr etationin lambdacalculus , Journal of Functional Programming, vol.2(3), pp . 345-364.
KLAUS GRUE·
A-CALCULUS AS A FOUNDATION FOR MATHEMATICS
Abstract.C hurchint rod ucedthe>.-calculu s in the begi nni ng of th e thirties y, from arou nd 1992 , fulfill ed that as a foundation of mathematics, and map theor primaryaim. The present paper presentsa new version of map theory whose axioms are simplerand bett er motivatedthanthose of theoriginalver sion from 1992. The paper focuses on the semant ics ofmap theoryandexplainsthissemantics on the basis of It-Scottd omains. The new version sheds some light on the differencebetweenRussell'sand Burali-Forti' s paradoxes, and also shed s some lighton why it is con sistent to allownon-well-fou n ded set s in a ZF-s tylesystem.
1. INTRODUCTION
Churchintroducedthe A-calculus in t hebeginningof thethirties(Church 1932, 1933, 1941) as a foundationof mathematics , and map theory,from around1992, fulfilled thatprimaryaim. In the meanwhile , A-terms have been shown very useful expressing for choice of a semanticsin computerscience, but therehas been nonatural theoryforreasoningabouttheseA-terms. With thelack of such a n atural choice, computerscience hasturnedto syntacticm ethodsin which one reasonsaboutthestructure and conversion t hantheirmeaning. of A-termsrather Mathematics , however, has hadset theorywhich allows reasoningabout sets ratherthansyntax . Set theoryoffers theluxuryof referential trans parency,in which every t ermhas ameaningand everytermimplicitlydenotes thatmeaning. Map theoryresemblesset theoryin thatit assignsmeaning to A-terms and treatsA-terms in a referentially transparent fashion. Map theoryalso resembles settheorywhen comparingmetamathematical power: For every consistentset theoryZ, thereis a consistentmap theoryM more powerful thanZ andvice versa.The deepestdifference etweensetand b map theories shows up inthetreatment of infinitelooping: Russell 's sentence{x I x If. x} E {x I x If. x} is justa termthattakesan infinitely long time to compute, butset theorydealswiththissentenceby forbiddingit rathert hantaking its valueseriously . Map theory , on the contrary,assigns meaning to the • My thanksare due to the refer eefrom whose reportI copied a par agr a ph to the abstrac t. 287
C. Anth ony Ande rson and M. Zeleny (eds.), Logic. Meaning and Computation, 287 -311 . © 2001 Kluwer Academ ic Publishers. Printed in the Netherlands.
288
KLAUS GRUE
corresponding t erm (AX. -'(XX»(AX . -,(xx». Map theoryis not amarriageof conveniencebetweenA-calculus and set theory(cf. Feferman 1978). Rather , map theoryis a theorybased entirely on A-calculus in which,among otherconceps, set membership, logicalconnecover all s ets are definable, and in which all axioms tives, andquantification and inference rules of ZFC are provablewithoutresortto anysyntactical considerations. Map theoryhas thepotential to serve as a foundation ofbothmathematics (due to its power t hatis equivalent to thatof settheory)and of computer science (due to itst reatment of A-terms). Set and map theoryare both inherently difficult to learndue totheirlevel ofa bstraction, butset theory of a centuryof pedagogicalengineeringthathas made it has theadvantage easiertoapproach. In contrast, theversion of maptheoryfrom 1992 appears as asomewhatrandomcollection of axioms thataccidentally havethepower of settheoryandaccidentally describeA-terms. Since 1992 it has turnedout, however,thatmap theoryis a natural choice of atheoryof A-calculus.More specifically, it has turnedoutthat every sufficiently large so-called x-Scottdomain D for whichD ~ [D --+ D] ED1.. 1 where 1 is aone-elementset, containsa model for maptheory (Berline and Grue 1997) . When DanaScottinventedwhatis now known as Scottdomains, he was fully aware thatthe notioncould begeneralised to x-Scottdomains, butdid notpublishthis finding as he saw no application ofthem at thattime. Hence, in some sense , a A-based foundationof mathematicshas beenaroundfor a longt ime withoutanybodyrecognising it. The presentpaper presentsa new version of map theorywhich will be reMTC (Map TheorywithClassicalmaps) ; the earlierversion from ferred to as 1992 will bereferredto as MTW (Map TheorywithWell -foundedmaps). The axioms ofMTC are simpler and bettermotivatedthanthoseof MTW, .and thestep from MTW to MTC si intendedas a step in thedirectionof a theorythatis easierto el arnand teach.The paper focuses onthesemantics of MTC andexplainsthissemanticson the basis ofx -Scottdomains.
1.1. Differences from Church's approach It is apitythatChurchdid not find taheorylikemap theoryrightaway since it could have saved a lot of work computerscience. in Thereare, however, threegood reasonswhy thatdid nothappen, and thesereasonsarestatedin thenextthreesections.
1.2. Inclusion of non-functions The firstreasonis thatChurch's theorymay be seen as atheoryaboutfunctions only , andas such, is a theoryaboutonly oneconcept. Classicallogic is
A-CALCULUS AS A FOUNDATION
289
builtaroundthedistinctionbetweentruthand falsehood, i.e ., thesemantic distinctionbetween two concepts; and thesemanticnatureof classical logic and settheorystems from thisdistinction . A-calculuson , theotherhand, concept,it is imposdealswithfunctionsonly,a ndin a theorywith only one sible to make asemanticdistinction.For thatreason,A-calculus can merely deal with the provabilityor non-provability of theequivalence of A-terms, which issyntacticin nature . In map theory,this problem is solved byinsistingthattheuniverse of map theorymust containat least one non-function . Justone non-function is enough to make daistinction,namely adistinctionbetweenfunctionsand minimalisticapproachhas beentakento non-functions . In map theory, the include only one non -function . Havingbothfunctionsand non-functions in map theoryallows it to repre senttruth and falsehood, and the convention has been chosen to let functions representfalsehoodand letnon-functions representtruth.T his convention has been chosen very carefully on the basis whatmakes of definitions inside map theoryeasiest to read, butthis issue will not treated be here. Since thereis only onenon-function in map theory, it is convenientto and sincethenon-function represents introduce a name forthatnon-function, truth,thename T has been chosen . In additionto functionsa nd non-functions, theuniverse ofm ap theory containsan element which neither is a function , nor a non -function.That element isdenoted1. and representsinfinite looping . This element violates the axiom ofT ertiumNon Datur,which in thisc ontext says thateveryobject is eithera functionor anon-function. Nevertheless map theoryis still classical in naturebecause it hasa nother,similaraxiom calledQuartumNon Datur which saysthatany map is eithera function or T or 1., with nofourth possibility. The inclusion of thenon-function T in map theoryis a trivialstep, butit . is a step thatis veryimportantfor thesemanticnatureof thetheory
1.3. Set abstmction versus A-abstmction Set theoryhas setabstraction {x I p(x)}, and A-calculus has A-abstraction AX.p(X). It is temptingto identifythetwo kinds ofa bstraction a nd tryto representtheclass {x I p(x)} by AX.p(X), i.e., to representclasses bytheir characteristic functions.T his approach, however , has not succeeded , and set abstraction a nd A-abstraction seem to be twocompletelydifferent kinds of abstraction. In map theory, a function 9 does notrepresentthe class{ x I g(x) = T}. Rather,9 representsthe class{g(x) I XES} where S is a fixed class of maps . In MTW, S is theclassW of well-founded maps, and in MTC, S is {g(x) I XES} containsa tleast one element , the class C of classical maps. which allows representation of theempty set; thenon-function T is takento
290
KLAUS GRUE
representt hatset. With thisencoding, allsets of ZFC can berepresentedby well-founded maps in MTW and by classicalmaps in MTC. And, conversely , all wellfoundedmaps in MTW andall classical maps in MTC represen t sets. Classes may also berepresented, butthey arerepresented by maps thatare not wellfounded c/ lassical. In conclusion , the non-identification of set abstraction a nd 'x-abstraction has been animportantpointin turning'x-calculus intoa foundation .
1.4. Selection of well-behaved maps The thirdproblem in turning'x-calculus intoa foundationof mathematics was to find a class S of maps thatwas sufficiently well-behaved to represent thesetsof ZFC. In MTW theclassW of well-founded maps was chosenand in MTC, theclassC of classical maps was chosen. 9 S putsmany Insistingthat{9(X) I XES} shouldbe a setofZFC for all E restrictions on S, andinsistingthatallsetsof ZFC shouldbe representable by an elementof S puts strongrequirementson the size and richness of S . Nevertheless, these restrictions by no means determineS uniquely , and finding anatural S is no trivialtask.
1.5. Relation between MTW and MTC One advantageof MTC overMTW is thatelevencomplicatedaxioms and inference rules thatdescribewell-foundedness in MTW have been replaced by a singledefinitionof classicality in MTC. es thatwere Anotheradvantag e is thatMTC containssome inference rul missing in MTW. These are rules ,YM , andE in Appendix 2.5. In particular , rule Y says thatthefixedpointoperatorg eneratesa minimal fixedpoint. It is interesting t hatthetheoremof transfinite inductionis provablejn MTC from theminimalityof fixedpointscombined withthe recursivedefinitionof classicality used inMTC. The detailsare workedoutin Grue 1996. A thirdadvantage is thedistinction betweendiscontinuous andcontinuous occurrences of variablesintroducedin section3.1 which allows tareatment of equationsas terms. The syntaxof MTW specifies thesyntaxof termsand well-formed formulas . MTC is simplerin thatit does notdistinguishbetween terms and well-form ed formulas. This may turnout to be convenientin computerassistedproofsystems forMTC becauseit allows raepresentation of boththeoremsand inference rules terms. as The notionof discontinuous occurrences permits theinference rule =a T I- a, which is also valid in the systems of Feferman(1984) and Flaggand Myhill (1989). The notionof discontinuouso ccurr ences also permits the opposite rulea I- a = T . The distinctionbetweendiscontinuousa nd continuousoccurrencesof variables seems to be a new way to deal with anomalies the of equality .
A-CALCULUS AS A FOUNDATION
291
MTW and MTC may be compared bothon a syntacticand asemantic result: basis. A syntactic comparisonis made in Grue 1996 withthefollowing If theclassicality predicateof MTC is used tosimulatethewell-foundedness predicatein MTW, thenall axiomsand inferencerules ofMTW, except Axiom Well-2 inGrue 1992, are provablein MTC. Axiom Well-2 inMTW is disprovablein MTC, which isjusta consequenceof theslightdifference betweenwell-foundedness and classicality. In Grue 1992, Axiom Well-2 is only used to prove Lemma C-K and Lemma C-P, bothof which areprovable in MTC. Hence, alltheoremsofMTW proved inGrue 1992 are alsoprovable in MTC. Among otherresults , Grue (1992) proves allaxioms and inference rules of ZFC in MTW. Combining theseobservationswe have thatallaxioms and inference rules of ZFC are provablein MTC. This shows thatMTC is adequateas a foundationof mathematics(provided ZFC is considered adequate). MTW and MTC may also becomparedsemantically . This paper introduces thesemanticsof MTC by means of ax-Scottdomain V = (D , ~) where D is theuniverseof all maps and~ is a partialorderon all maps . The feScottdomain used hasthepropertythatit modelsbothMTW and MTC, so it makes sense tocompare MTW and MTC in this particular model. A class A of maps will be said to be "coherent " if any twoelementsof A have an upper bound in V. The correspondencebetweentheclassesWand C of well-founded and classical maps,respectively , may now be formulated as follows: • Any well-founded map is classical. • Any non-empty, coherentclass ofwell-founded maps has a greatest lowerboundis classical. lowerbound,and thatgreatest m ap is thelowerboundof acoherent class of well-founded • Any classical maps . as ofW under Hence, theclass C of classical maps can be seen theclosure greatestlowerboundsof non-empty,coherentsets. Animportantdifference betweenWandCis: • For allwell-founded maps g exceptT thereexistsa well-founded map h such thath < g. • For allclassicalm aps g thereexistsa minimal classicalmap h suchthat h ~ 9 (h is a minimal classicalmap if h' ~ h => h' = h for all classical maps h').
292
KLAUS GRUE 1.6. The structure and contents of the paper
Section2 describesthesemanticsofMTC basedon thex-denotational frameworkdevelopedin Berline and Grue 1997. Section3 gives arather quicktour throughthesyntax, axioms andinference rules MTC. of Section4 concludes by remarkson Russell'sand Burali-Forti's paradoxesand non-well-founded sets. Appendix 1 outlinesa model ofMTC based on themodel ofMTW in Berline and Grue 1997. Onlythedefinitionof themodel isstated . The satisfactionof theaxioms and inference rules remainsto be proved . Appendix 2 summarisesMTC. A more detaileddescriptionof theindividualaxioms and inference rules and explanations of howtheyare used may be found in Grue 1996. Note thatthesystem in Grue 1996 containsan inconsistentaxiom as pointed out by ChantalBerline. Grue 1996 plus erratamay be obtainedfrom www.diku.dkj'"grue. 2. THE SEMANTICS OF MTC
2.1. Maps over finite sets
In thefollowing , words initalicsand mathematical conceptsin boxesoccur in theindex. Figure1 shows a map overtheset I = {I, 2, 3}.
Figure1: A map over{1,2,3} In general,a map overI is a tree where each node is labelled by IT], []], or each edge islabelled by an elementof I, each nodelabelled f has one t or b has no downwardedge for eachelementof I, and each node labelled downwardedges. Maps may beinfinitelydeep. As an example, Figure2 shows amap over {I , 2}. If I is a set, ifx E I, and if 9 is a map over I,thenwe define9 applied to x, denoted g'x to bethesubtreeof 9 attached to theedge labelled x that extendsdownwardsfrom therootof g. As an example, if 9 is the map in Figure1 thenFigures3, 4, and5 show g'l, g'2 , and g'3 , respectively . Applicationis leftassociativeso that,e.g., 9'3'1 means (g'3)'1. If 9 is the map in Figure1 theng'3'1 is themap in Figure3. If h is themap in Figure
W,
I I,
A-CALCULUS AS A FOUNDATION
293
Figure2: A map over {I, 2}
Figure3: The map T 2 thenh'2'1 = h. The trees inFigure3 and4 will bedenotedIT] and Qj, respectively. No ofx edges extenddownward from the rootsof T and 1.., so the definition g' does not make sense for 9 = T and 9 = 1... To make s'x defined for all x EI and all maps9 over I, we more or less arbitrarily define:
T,
T'x .L'z
=
1...
2.2. Modelling of maps
I I
Let t, f, and b bethreedistinctobjects. For all maps9 overI we define r(g) to be the label of the root g. Hence of , reT) = t, r(l..) = b, and reg) = f for allmaps 9 overI except T and1... Let I <w denotethe set of finite lists (Xl,"" Xn) of elements of I. For all maps9 overI and all x = (Xl, . . . ,Xn) E J<w, letg[x] denoteg'x~ .. . x~ . As an example,h[(2, 1,2,1,2,1)] = h whereh is themap in Figure 2. If x is theempty tuple(), theng[x] denotes9 itself. If 9 is a map over I and ifX E I <w, thenr(g[x]) will be referred to as the label indexed by X. As an example, if 9 is the map inFigure1, thenthe labels indexed by (1), (3), and (3,3) are t, f, and b, respectively . The label
I I
294
KLAUS GRUE
Figure4: The map ..1
Figure5: A map over{1,2,3} indexedby (3,3,2) is also b as shown by t hefollowing : g[(3, 3, 2)1
g'3'3'2, (g'3'3)'2 ,
..1'2, ..l.
Hence,computation ofr(g[ (3,3,2)]) dependson theconvention t hat..1'x = ..1. In general,if r(g[ (XI, ' .. , x m)]) = b thenr(g[ (XI, ... ,X m, YI, .. . ,Yn)]) = b. Now for all X, y E I <w let x . y denotetheconcatenation of thetuplesX and y. We have
I I
r(g[x]) = b => r(g[x]) = t =>
r(g[x . y]) = b, r(g[x· y]) = t.
These twostatements may be combined into one:
r(g[x])
=1=
f => r(g[x . y]) = r(g[x]).
m= {t,
Now let
I
I
I
I
f, b}. For allfunctionsu, let dom u and rugu denote
I
I
thedomain and rangeof u, respectively. For all setsG and H, let G ~ H denotethe set of functionsu for which domu = G and rugu ~ H. If 9 is a map overI and ifx E I <w, thenr(g[x]) E L. Two maps u and v overI areconsideredequal ifr(g[x]) = r(h[x]) for all x E I": " , Hence, amap 9 may be modelled bythefunctiong E I <w ~ L for whichg(x) = r(g[x]) for all X E I <w. From now on , maps are modelledthisway whichmotivates a definitionof the setM] of maps over I by
IM] 1=
{g E I <w ~ L I VX, Y E I <w : (g(x)
=1=
f => g(x . y)
= g(x».
A-CALCULUS AS A FOUNDATION
295
2.3. Partially ordered sets A p.o V is a partially ordered set (D , ~) . As an example, [f]= (L, ~d is a p.o where
I 1=
For allnon-emptysets I , MI
(MI,~) is a p.o where
I9 ~ hi {::} 'Ix E t -- : g(x) ~L h(x). For allp.o's V = (D, <), definethep.o IV <w Iby tr> = (D <w,1 ~* ~ where For all .o's p V = (D,~) and £ = (E,~), let x E V, A ~ V and 9 E '0--+£ be shorthand for xED , A ~ D, and 9 E D --+ E, respectively. ~ E M I by For allnon-emptysets I, define~,
[!.]
..i/( (XI, , x n ) ) = b, Tr({XI, , Xn ) ) = t, F/({)) = f, Fr({U,XI,. . . , x n } ) = t for alln ~ 0 and all u , X l , . . . , X n E I . ..ir is theuniquebottomelementof MI. T randFI aretwo among many maximalelements .
2.4. K-cont inuity For allsets A and K , A is said to be «-small if A has cardinality strictly less thanK. From now on et l K be an infin i te set. An elementX of a p.o '0 = (D ,~) is said to be anupper bound of A ~ V if 'v'yEA : y ~ x. A subsetH ~ V is said to be aK-chain if allx-smallsubsets of H have anupper boundin H . In particular, theempty setmust havean upper boundin H, so anyx-chainis non-empty. A p.o '0 = (D ,~) is a K-CpO if every x-chain H ~ V has a supremum sup H in V . As an example, for allnon-emptysets I, Mr and (MI )<w areK-CpO 'S. MI is an exampleof a K-CpO witha bottomelementand (MI )<W is an example of onewithout. For allK-CPO'S V and E, a function9 E V --+ £ is said to beK-continuous if
I
g(supH) = sup {g(x) I x E H}
I
296
KLAUS GRUE
I
I
for allx-chainsH ~ V . Let [V -+ £IK denotethe I\:-CPO of x-continucus 9 E V -+ £, orderedby pointwiseordering.For allV = (D, :S;), define
We shall refer elements to of M!p as n-continuous maps over V.
2.5. Maps over R, and maps over maps over R At this point a few examplesmay be illustrative . Let R be the set of real 9 = (G,:S;) = MR be numbers, letZ ~ R be theset of integers, and let thep.o of maps overR. We havethatZ is R-smallsince Z hascardinality strictly lessthanR. For allA ~ R definethecharacteristic map XA E 9 by
XA(O)
f,
XA«(U,XI,""X n )
{
t
b
if u E A, otherwise.
As an example of anon-trivial R-chainin g<w, we have
H = {(XA) I A
~
R 1\ A is R-small}.
The supremum supH of this chain is (XR)' s et MG of maps overG = MR, i.e., ofmaps over maps Now consider the over R. For allB ~ R, define't/B E M G by
f,
{ Lx".,xn)
if 3vEB : u«(v)) = b, if't/vEB: u«(v)) = t,
otherwise .
't/B satisfies
VB'U
~{
..lG if 3v E B : u'v = ..lR, TG if't/v E B : u'v = TR, . FG otherwise
If we take T,F, and ..l torepresenttruth , falsehood , and undefined,respect hatquantifiesover B. The tively,t henVB representsa universalquantifier quantifier is strictin thesense that't/B 9 is undefined ifgx is undefinedfor some x E B. We have't/R(supH) = t and sup {'t/R(X) I x E H} = b which showsthat 't/Ris R-discontinuous . Nevertheless , it isstraightforward to provethat't/z
A-CALCULUS AS A FOUNDATION
297
is R-continuous( theproofis a somewhatlengthyproofby cases, but the pointof theproofis thatfor all R -chainsH' thereexistsa functionh" E H' such thatVnEZ:h"(n) = h'(n) where h' = supH'). In general , VB is Rcontinuousif andonly ifB is R-small. Even moregenerally, quantification over a set B is x-continuous if andonly ifB is x-small, 2.6. «-premodels
We have now seenmaps over {I, 2, 3}, maps over R, andR-continuous maps over maps overR. The maps of MTC are x-continuousm aps over maps of MTC. In otherwords,thedomain '0 of all maps of M TC satisfies'0 ~ MD' If '0 is a «-Scott domain (Berline and Grue 1997), if (J' is a strongly inaccessible ordinal (Chang and K eisler 1973), if K is a regular cardinal (Chang and Keisler 1973) greaterthan(J', and if'0 ~ M thenit follows from Berlineand Grue(1997) that'0 canbe expandedinto a model of MTW (to eV ~ M '0 ~ ['0 -+ 'OJ", EBJ.. {T D} , thenlet see that,one needs to prov A E '0 -+ ['0 -+ '01", EB-L {T D} be an orderisomorphism, letA be theinverse of A andapplyTheorem7.2.1 of Berline and Grue 1997 to ('0 , A , A)). A model ofMTC needs tosatisfymore thanthis, partlybecauseof the classical maps , partl y becauseMTC containsinferencerulesthatwere missing in MTW. From now on,assume thatther e exist stronglyinaccessibleordinals , assume that is theleaststrongly inaccessibleordinal , andassume that is a regularc ardinalg reater t han(J'. A n-premodel P of MTC is a structure ('0 , a, C, q) thatsatisfiesthefive propertiesbelow plus on e more propertywhich isstatedin section2.7.
v::}
v
0
0
• '0 = (D ,::;) is a x-Scottdomain (and, in particular , a K-CpO).
• a E ['0
-+
MD1",
is an isomorphism.
• C ~ '0 is a x-smallset of so-called x-compactelements(cr. B erline and Grue 1997 and Appendix 1.1). • q is a choicefunctionover'0, i.e., q(A) E A for A ~ '0, A =I- 0.
• q(0) E C. x-premodelsareintroduced here topresenttheintuitionsb ehindMTC. The construction of ax-premodelis outlined later.T he detaileddevelopmenta nd theverification of axioms and inference rules of MTC remains to be done. Now define a is theinverse of ,a T = aCTD), .1 = a(.lD)' ; F =a(FD).
298
KLAUS aRUE
I I
For alls ,x E V let g'x be theuniqueelementof V for which
a(g'X)«YI , .. . , Yn))
= a(g)«x, YI , · ·· , Yn))'
This definesthenotionof applyinga map 9 to anargument x , which was firstmentionedin section2.1. This concludesa circle:informalconsiderations aboutapplicationof maps led to arepresentation of maps which was refined into a model of maps which allows definition a of application . 2. 7. Classical maps
I
For allS ~ D and x, Y E D , define x =s Y
I
tr-, define x =s Y
For allx, Y E (Xl,.. .
I
,X m
)
Iby
=sY(YI, . .. ,Yn)
¢:}
m
I
¢:}
\lz E S <w: a(x)( z) = a(y)( z).
= n 1\ Xl = YI
1\ . . . 1\
Xm
= Yrn '
I
Let Pu(A) denotetheset ofo-smallsubsetsof A . Let C' be theleastsubset of D whichsatisfies
9 E C'
¢:}
\Ix E C : g'x E C'I\ 3V E P u(C')\lx ,y E C':" : (x
=v Y
=?
a(g)(x) = a(g)(y)) .
The existenceof such aC' is easy to verify. For all x E V and A ~ V , let {y E V I x y} and TA = U{Tx I x E A} . A x-premodel P of MTC (V, a, C, q) thatsatisfies the fivepropertiesin section2.6 plus is a structure theone below :
[EJ =
s
• C'
= TC. 3. PRESENTATION OF MT C
3.1. Syntax The syntaxV of variablesand T of termsof MTC reads:
V T
::= ::=
I X2 I . .. I T I TT I AV. TIP Ie 131
Xl
V
I
€(T)
IT = T .
Theconstruct TT has higherprioritythanAV. T , which inturnhas higher prioritythanT = T, so that , e.g., AXI.XIXI = X2 means (AXI. (XIXI)) = X 2 . An occurrence of a var iable v in a term t is said to be discontinuous if v occursfree in asubtermt' of t which has one of the forms E(t") or til = t'" , Occurrencest hatare not discontinuousare said to becontinuous. As an of Xl in PXIX2(XI = AXI . Xl) is discontinuous , example,thesecondoccurrence of variablesarecontinuous . whereastheotheroccurrences
A-CALCULUS AS A FOUNDATION
299
The following p urelysyntactical restriction is puton termsof MTC: Discontinuousoccurrencesof variablesare not allowed to bound. be As an example, AXl' (Xl = T) is not a well-formed term. 3.2. Proofs
A proofin MTC is a sequenceof terms in which eachterm is eitheran instanceof anaxiom scheme or follows from previoustermsin thesequence by an inferencerule. All axioms and inference rules MTC of are listedin Appendix 2.5. The interpretation of a proofis thatit provesthelasttermin theproof, i.e., it provesthatthelasttermin theproofequalsT for all values ofefre variables . 3.3 . Truth, equality and application
The termT in section3.1 denotesthevalue T defined in s ection2.6. The term a = b equalsT when a equalsb, and a = b equalsF otherwise.The value ofa = b may depend on thevalues of free variablesin a and b. As an example, Xl = T equalsT when Xl equalsT and equalsF otherwise.The inference rules truth of andequalityread:
T SA SA Sf
=rr-=
Transitivity a = b, a = c r- b = c, Substitutivitya = c, b = d r- ab = cd, Substitutiv ity a = b r- Ax .a = Ax.b, Substitutivitya = b r- €(a) = db), Equality a = T r- a, Equality a r- a = T.
a, b, c, d, g, and h denotearbitrary t erms; whereasX , In theinference rules, y, and z denotearbitrary, distinctvariables. The inferencerule a = T r- a is inspiredby Feferman(1984), Flaggand Myhill (1989).The opposite rulea r- a = T is made possible by thenotion of discontinuousoccurrencesof variables , which seems to be a new way of dealingwiththediscontinuity of equality. For allt ermsa and b, ~ denotesa appliedto b, i.e., ab denoteso'b. Since T appliedto anythingequalsT by convention,we havetheaxiom AT
Application
Ta
= T.
As an exampleof a proof, we have 1
2 3 4
AT AT 1,2,T 3, =r-
Ta=T , Ta=T ,
T=T, T.
300
KLAUS GRUE 3.4. Abstraction
For allvariablesx and termsA of ZFC, letx
1-+
A be shorthand for{(x,A)
I
x E V}, i.e., letx 1-+ A be theuniquefunction9 for whichdomg = V and g(x) = A. For all9 E [V ----+ VI/I:, let>..(g) be the uniqueelementof V for
which >..(g)( (}) >..(g)( (u, X l,.. . , x n })
=
f, g(U)({XI, . . . , Xn }) .
Finally, define >..x. A = >..(x
1-+
A)
whenever(x 1-+ A) E [V ----+ VI/I:, i.e., wheneverx 1-+ A is a x-continuous function from m aps to maps . If x is a variableand A is a term of MTC, then(x 1-+ A) E V ----+ V. If furthermore, thereareno discontinuousfreeoccurrences of x in A, then x 1-+ A is x-continuous,so that(x 1-+ A) E [V ----+ V]/I: and (>..x.A) E V . This t erm. We gives aninterpretation of >..x.A whenever>"x.A is a well-formed may nowformulate two more axioms, namely: A>.. R
Application (>..x .a)b = (a I x:=b) if b is free forx in a, >..x. (a I y:=x) = >..y. (a I x :=y) Renaming . if x is free fory in a and vice versa
I
I
Here (a I x:=b) is thetermthatarises whenreplacingall free x in a byb. See Mendelson 1987 for a definition of free for . Axiom R allowsrenaming of boundvariablesa ndAxiom A,\ expresses thattwotermsare equal if they eory are ,a-equivalent . (Note, however,thatterms may be equal in map th withoutbeing ,a-equivalent . This holds even for terms thatcontainonly in map theory variables,abstraction , and application.T he notionofequality is thesemanticnotionintroduc ed in section2.2, and this sem anticnotionis not fully capturedby ,a-equival ence.) Having abstraction and applicat ion allows adefinitionof thefixed point operatorY and thebottomelement1..:
[2]
QJ
= =
,\f. ('\x . f(xx) )'\x. f(xx), Y'\x .x.
Having thebottomelement , we may stateone moreaxiom and one more inference rule : AJ... QND
Application 1.. a = 1.., QuartumNon Datur aT = bT,aJ... = bJ... ,a>..y.xy = b'\y.xy I- ax = bx.
A-CALCULUS AS A FOUNDATION
301
Axiom A-1 expresses that-1 applied to anythingyields -1. Inferencerule QND expresses thattherootof any map is t, f, or b; thereis no fourth possibility . QND together w iththemap P describedin section3.5 allows a developmentof classicalpropositionalc alculus(cf. Grue 1992, Berline and Grue 1997, and Grue 1996). In thepresentpaper,onlytheintendedmeaning of axioms andinferencerules will be stated . The detailsof whytheyexpress theintendedmeaningandhow theyareused is statedin Grue 1996.
3.5. Selection The map P is bestdescribedby thefollowingt hreeaxioms: PT PA P-1
Selection PabT = a, Selection Pab>.x. c = b, Selection Pab.l=-1.
The map P allows one to define many auxiliaryconceptssuch as logical connectives(cf. Appendix 2.4) and a developmentof classicalpropositional calculus on thebasis of QND. For a ,b E V = (D, ~), theconstructs introduced so far allow definition a a ::; b suchthata ::; b equalsT if a ~ b andequalsF otherwise . The definition of a :::s b is statedin Appendix 2.4 and reads a::;b ~ (a = alb)
wherealb is definedon basis of P inAppendix 2.4. The definitionof ::; uses = which is adiscontinuousc onstruct in thesense thatfreeoccurrencesof variablesin a = b arediscontinuous occurrences . ::; inheritst hediscontinuity from = so thatfreeoccurrences of variablesin a ::; b arcdiscontinuous , and no >. is allowedto bind a variabl e occurrence t hatis free in asubtermof the form a::; b. We may now statethreefurther inferencerulesandone moreaxiom: Y Minimality ga ::; a I-- Y9 ::; a , M Monotonicity b ::; c I-- ab ::; ac, E ExtensionalityIgxy = Ihxy ,gxyz = gab, hxyz = hab I-- gxy if x , y, and z arenotfree in9 and h, != Equality !(a = b).
= hxy
Rule Y statesthatY9 is minimal among all fixedpoints of 9 (this does not holdin allx-premodels, but it holdsin theparticular one outlinedin Appendix 1). Rule M statesthatallmaps a are monotonicin thesense thatb ~ c implies ab ~ ac. Rule E statesthattwo maps 9 and hare equalif a(g)(x) = a(h)(x) for all x E V <w (see Grue 1996 fordetailson the interpretation and applicationsof ruleE) . Axiom!= statesthata = b is eithertrueor false .
302
KLAUS GRUE 3.6. Simple existential quantification
The quantifier 3 is a particularly primitive quantifier.3g = T if gx = T for some x E V and 3g = 1.. otherwise.C ontrary to 3, which is defined in s ection 2.4, 3g cannotbe false.€(g) is an x E V such thatgx = T if such anx exists. More formally , defineE E V --+ V by E(g) = {T1..
if 3x E V:gx = T, otherwise .
Thendefine
3
)"(E),
€(g)
q({x E V Igx = T}).
Here we usethechoicefunctionq from thepremo del (V, a, C, q). The requirementq(0) E C ~ V ensuresthat€(g) is a map even whengx #- T for all describe3: maps x. The following axioms Existence ab --+ 3a, Existence 3a --+ a€(a), Existence 3a = ?3a. Axiom --+3 expressesthatif ab is truefor someparticular b, then3a is true . Axiom 3--+ expressesthatif ax is truefor some x then, in particular, it is truefor x = €(a). Axiom?3 expresses that3a equalseit herT or 1... See Appendix 2.4 fordefinitionsof a --+ band ?a. 3.7. Hilbert 's e -operator
Define
Q(g) =
1.. if 3x E C : gx { q({x E C I gx = T} ) otherwise, )..(Q).
= 1..,
The map e is Hilbert'se -operator(Hilbert 1939) . The map e allows definitionsof many auxiliaryconceptssuch astheuniversala ndexistential quantifiers (cf.Appendix 2.4) and developmentof firstorderpredicatecalculus. Furthermore , C is rich enoughto allow all sets of ZFC tomodelledby be classicalmaps which allows definitionof set membership and proofs of all axioms andinference rules of ZFC inside MTC. The detailsmay be found in Grue 1992 and 1996. The constructs introduced so far allows thedefinitionof themap f (this definitionis statedin Appendix 2.4) . Now let C'={xEVlfx=T}
A-CALCULUS AS A FOUNDATION
303
(thisG' is actually the sameG' as theone defined in section.7). 2 Theaxioms thatdescribee read:
Q1
Quantificationfa/\ Vb -+ ba, Quantificationex : a = ex : fx /\ a, Quantificationf(ex: a) = "Ix : !a, Quantification!Vx: a = "Ix : !a.
Q2 Q3 Q4
Axiom QlstatesG ' ~ G, andAxiom Q2 statesG ~ G', so thatG = G' as in section2.7. Axiom Q2, furthermore , expressesAckermann'saxiom (Feigner 1976, page 244). The definition of e entailst hateg E G ifVXEG:gx =1= 1- and eg = 1- if 3xEG: gx = 1-. The former isexpresseddirectly by Axiom Q3, and thelatter is expressedindirectly by Axiom Q4. 4. CONCLUSION
The most immediate translation of Russell 's paradoxical sentence{x I x f/. x} E {x I x f/. x} into MTC istheterm (Ax.-,(xx))(Ax.-,(xx)) whose value is L Hence,Russell'sparadoxis essentially avoided by having tahirdtruth value1- which is the value termsthat of make a computerloopindefinitely . An importantpointin MTC is thatthisthirdtruth value can be i ntroduced withoutloosing the classical, semanticflavor of the theoryandwithoutresort to intuitionistic logic. The above translation of Russell 's paradoxto MTC translates set abstractionto A-abstraction, which givesinsightinto how Russell's paradox is avoided. As noted insection1.3, set abstraction is not the same asAabstraction, so one may also look athow Russell'sparadoxis avoided when modelling set a bstraction as in Grue 1992. This turnsout to betrivial,however, sincethatmodellingensuresthatall set s are well-founded that so all thatclassesthat sets x satisfyx f/. x . FUrthermore,thatmodelling ensures containall sets are not sets themselv es. A map x in MTC is classical fx if = T where f"':'"
-
V
.
f{ T,
-
(Vx :f(fx)) /\ 3S :fS /\ VxVy:x "'~ y
=> fx'" iv-
The precise structure and meaningof this definition is not importanthere. The importantobservationis thatthedefinition of f is recursive inthatf occurson the right h andside of ,,;, (recursive definitions are shorthand for definitionst hatuse the fixed pointoperatorY explicitly) . The twooccurrences off in thedefinitioncombined withtheminimalityof two ways. fixedpointsimplies thatclassical maps are well-founded indistinct The firstoccurrence of f makes classical m aps well-founded thesense in that for all classical maps g, Xl> X2 , . . . thereexists ann such thatgXl ... Xn = T. This kind of well-foundedness correspondsto the well-found edness of
304
KLAUS GRUE
sets expressedby theaxiom of foundation in ZFC (no infinitelyd escending E-chains). The secondoccurrence of f makes classicalm aps well-founded in a much more subtlesense. The closestanaloguein ZFC tothiswell-foundedness is thelimitation of sizepresentin ZFC. However,maps containmore structure thansets, and thesecond kind of well-foundedness is more a limitationof complexitythanjusta simple limitation of size. In any case, it is thissecond kind ofwell-foundedness thatavoids theBurali-Forti's paradox. The firstkind of well-foundedness does not avoidparadoxes. If thefirst kind of well -foundednessis abandoned,if thesecond kind iskept, and if the representation of setsused in Grue 1992 is used, thentherepresentable sets become thoseof Aczel's(1980) AFA settheory(theone in whichtheequation X = {X} has exactlyone solution) . Hence, it isconsistentto allow infinite descendingE-chainsbecausetheparadoxesare avoidedby thesecondkind of well-foundedness . share As notedby Aczel, all knowntheoriesaboutnon-well-foundedness thepeculiarity t hattheystartby constructing a well-founded universeand thenproceedto thenon-well-founded. It is an interesting t opic for further work totryto formulate a versionof map theorywhich doesnotsharethis peculiarity.Such atheorycould be atheoryaboutnon-well-founded classical maps, i.e., maps thatsatisfythesecond kind ofwell-foundedness without satisfyingthefirst.
APPENDIX A. OUTLINE OF A MODEL OF MTC
A .1. The «-denotational semantics The followingexpositionfollowsBerline and Grue 1997 except thatthe empty setis x-smalland K,-CpOS need nothave abottomelement. A p.o V is a partially orderedset, (D, :::;). We use x E V and x ~ V as shorthand forxED and x ~ D, respectively . means {v I v :::; u} . A ~ V is «-chain. if everyx-smallB ~ A is boundedby an elementof A. V is a K,-CpO if everyx-chainA has asup [i.e., leastupper bound). V is a K,-CCpO if, moreover, everyboundedA has a sup (hence, as in Berline and Grue 1997, everyK,-CCpO has abottom). A n -compact elementof a K,-CpO V is anelement U E V such that,for all x-chainsA, U :::; sup (A) implies U :::; v for some v E A. V e is thesetof x-compactelementsandIl eu = lu n »; A K,-CCpO V is a «-Scott domain iff for all u E V, U = sup l eU. A prime elementof a K,-CCpO V is an elementu such that,for allb oundedA, u :::; sup A implies u :::; v for some v E V p is thesetof prime elementsofV, and Ipu = lu n ti; A u E V, u = sup lpu. x-Scottdomain is n-prime algebraic if for all
[EJ
I I
I
A·I I
I I
305
A-CALCULUS AS A FOUNDATION
A .2. Outline of a model A modelofMTC is outlined in thefollowing. Only thedefinitionofthemodel will bestated.The satisfaction of theaxioms and inferencerulesremainsto be proved. The model construction has many similaritieswith theconstruction in section8 of Berline and Grue 1997. The construction below differs from thatin Berline and Grue 1997 in thefollowing ways: (1) The domain of the model is constructed as coherent,c omplete,initialsegmentsof x-compact elementsrather t hancoherent,initialsegmentsof x-prime elements , andthe PCS of x-prime elementsis not constructed a t all below . (2) The model construction is basedon two fixedpointconstructions rather t hanone. The first fixedpointgenerates theo-compactmaps of MTW which is aset large enoughto containthewell-founded m aps of MTW. Then thewell-founded maps areturnedinto classicalmaps by discardinginformation,and thena second fixed pointconstruction is used togeneratet hex-compactmaps of MTC. Finally , thex-premodelof MTC is constructed from coherent,initial segmentsof x-compactelements . , i.e, To simplify the exposition, assume thatK is stronglyinaccessible assume CT is theleaststrongly inaccessibleordinaland assume K is a strongly inaccessibleordinalgreatert hanCT. The assumptionthatK is inaccessible rather t hanjustregular is merelya luxury . :::5, define :::5* I by For allrelations
I
An applicative structure V is a pair (D, a) whereD is a setanda E D For applicatives tructures V = (D, a), define :Sv and M v by
I I I I
x :Sv y ¢:} a(x) :s a(y), Mv = {gEMD I Vx,yED <w : (x:s; y
~
-->
MD·
a(g)(x) :SL a(g)(y»} .
An applicatives tructure V = (D, a) is said to bemonotonic if .1 D E rnga Mt» For allm onotonicapplicativestructures V = (D, a), define
a1; E Dv
-->
Mv
av E D v
-->
M Dv
~
Dv=DUMv a(x) if xED a+(x) - { v x otherwise 9 :::5v h ¢:} a1;(g) :S a1;(h) av(g)(x) = sup {a1;(g)(y) lyE D>" /\ y:::5; x} V+ = (Dv ,av).
To ensurethesoundnessof thedefinition , we make thearbitrary convention thatsup A = b when A ~ L has nosupremum.
306
KLAUS GRUE
I
I{::}
Forapplicative s tructures V = (D, a) andV' = (D', a'), define V ~ V' D ~ D' /\ VgEDVXED<w: a(g)(x) = a'(g)(x). We haveV ~ V+ . Now define VOt for all ordinalsQ by transfinite inductionas follows :
I I
vo
VOl+! Vlj
= = =
V,
(V Ot )+ , UOtEljVOt
for limitordinalso.
Above, UOtElj has the obvious interpretaion , i.e., if VOt = (DOt, aOt) thenDs = UOtEljD Ot and alj(g)(x) = aOt(g)(x) whenevergED Ot and XED';;w . Let be an arbitrary element of the universe of ZFC which is not a ~ = (D I, ad whereD I = {l..} and wherea l E D I -+ (D~W -+ function; let
QJ
L) is defined byal(g)(x) = b. Furthermore, le~~21 = (D 2, a2) = D'[ , D2 is essentially theset ofa-compactelementsof M . of S. For allx = Let soo denote theset of infinite lists elementsof
I I
I
I
(Xl",,) and nEw, let (xln) denotethetuple(Xl,"',x n ) .
I I
Define GO = {gEV 2 I VXEDf3nEw: a2(g)(xln) = t}, and let[ !] be the leastsubsetof V 2 for whichGOo ~ clJ wheneverG ~ clJ is a-small. This clJ is essentially the set of well-founded maps introducedin Berline and Grue 1997 and Grue 1992. Now let 3 1 = (D 3,a3) where D 3 = clJ U {l..} and wherea3 E D 3 -+ (D:jw -+ L) is defined bya3(g)(x) = a2(g)(x) . V 3 is clJ from which some informationis discarded,namely, theinformationabouttheapplicativebehavior of well-founded maps when applied to non-well-founded maps . This makes V3 useful as a model of the classical .maps Now let ~~ = (D 4, a4) = V!3. D4 is essentially t heset ofx-compact elements of .CTransfiniteinductionup to II: gives thex-compactelements becauseII: is assumed inaccessible. More care would be needed the in definition of V+ and VOt if II: was not assumed inaccessible . A ~ D4 is an initialsegmentif A is non-empty and X '5:D 4 Y /\ yEA :::} X E A. A is coherentif any two elements A of have anupper bound in D 4 • A is completeif any subsetof A thathas anupper bound in V also has anupper boundin A. Let D s be theset ofcoherent,complete,initial segments,and definetherelation::;s on D s by X ::;s y {::} X ~ y . Define as E D s -+ (D~W -+ L) by as(g)((xI ""'X n}) = sup{a4(h)((YI, · .. ,Yn)) I h E 9 /\ YI E Xl /\ ... /\ Yn E z.,}. Define Cs = Ux I X E D 3 \ {l..}} and letq E P(D s ) -+ D s satisfyq(A) E A when A is non-emptyand q(A) E Cs when A is empty. P = ((D s , '5:s) , as, Cs , qs) is claimed to be ax-premodelthatsatisfies all axioms and inference rules of MTC.
IV
I
.A-CALCULUS AS A FOUNDATION
307
B. SUMMARY OF MTC
8.1. Syntax
v .. -
7
I x21··· V I T I 77 I .AV.7 I if(7, 7, 7) 1 e 13 1 /:(7) 17= 7
Xl
B.2. Priority
The priorityis as follows . Functional applicationf X has highestpriorityand appearsatthe top of thetable.Operatorson thesame line have the same priority.
fx x,y z ] y Lx x rv y XEy xEEy x rv~ y --,x Ix !x ?x x/\y xAy xVy xVy x=*y x~y x{:::}y
x{ ~ .Ax.y x=y
x~y
x~y
B.3. Associativity fx , x /\ y, x A y, x Vy and x V yareleftassociativeso that,e.g., fxy means (fx)y. X rv y, xEy, xEEy, x rv~ y, X =* y, x ~ y, x {:::} y, x = y, x ~ y, and x ~ yare"and"-associative so that,e.g., x rv y rv Z means (x rv y) /\ (y rv z) and x => y => z means (x =* y) /\ (y => z) . 8.4 . Definitions used in axioms ";,, .Ax. T
F
x y ..L
{~
";,, Pabx ";,, X] , (.Ax . f(xx))(.Ax. f(xx)) ";,, Y.Ax.x
308
KLAUS GRUE
F
-,x
~x {
lx
~x {
T
!x
~x{
T T
?x
~x{
T .1
xl..y
~x{
y F
x';;;"y
~x{
Y
T
F
T
~x{ yP
x/\y
y{ :
~x{ Y{ ~
x~y
y{ x{::}y x-+y 3 3x:a Vx:a
~
V
~
~
(x
~
~
y) /\ (y
~ ~
x)
>.j.lf(c:J) 3,xx.a -,Vx:-,a >.j.Vx :fx
~f { gP
frvg
g{
~,xX.fx
xEy
. {F = y 3z:xz
X EE Y
~
VxEEa:
~
~xl..y=xl..T
b
~
rv
rv
gx
y
3z :x E z /\ z E y (,xy.Vx:x EE y';;;" b)a
A-CALCULUS AS A FOUNDATION
309
. {y{: =x
y{ ~Z.XZ1YZ
~x=x!y
B.5. Axioms and inference rules Below,a, b, c, d, f, and9 denotearbitrary termswhereasx, y, and z denote arbitrary, distinctvariables.
T SA
AA
Transitivity Substitutivity Substitutivity Substitutivity Selection Selection Selection Application Application
A..l R
Application Renaming
SA Sf.
PT PA P..l AT
a = b, a = c I- b = c a = c, b = d I- ab = cd a = bl- Ax.a = Ax .b a = b I- f.(a) = f.(b) PabT = a PabAx.c = b Pab..l = 1 Ta=T (Ax.a)b = (a I x:=b)
if b is free forx in a. ..la=..l
AX. (a I y:=x) = Ay. (a I x :=y)
if x is free fory in a and vice versa.
= b..l,aAy.xy = bAy.Xy I-ax=bx fa /\ Vb -+ ba ex : a = ex : fx /\ a f(ex : a) = Vx: !a !Vx : a = Vx : !a ga =:; a I- Yg =:; a b=:;cl-ab=:;ac 19xy = /hxy , gxyz = gab, hxyz = hab I- gxy = hxy
QND
QuartumNon Datur aT = bT,a..l
Q1 Q2 Q3 Q4 l' M E
Quantification Quantification Quantification Quantification ~inimality
Monotonicity Extensionality
if x , y, and z are not free in 9 andh.
310
KLAUS GRUE
-+3 3-+
?3
=11-= !=
Existence Existence Existence Equality Equality Equality
ab -+ 3a 3a -+ af(a) 3a = ?3a a=Tl-a al-a=T !(a = b) REFERENCES
Aczel, P. andthenotionsofproposition,t ruth andset, The Kleene 1980 F'regestructures symposium, (J . Barwise, H. J. Keisler, andK. Kunen,editors),NorthHolland,A msterdam, pp . 31-59. Berline , C., and K. Grue 1997 A x-denotational semantics for Map Theory in ZFC+SI, Theoretical Computer Scienc, vol. 179, no. 1-2 (June), pp . 137-202. Chang,C. C., and K. J. Keisler 1973 Model theory, Studiesin Logic andtheFoundations of Mathematics , vol. 73, North-Holland, Amsterdam. Church,A . of logic I,Annals of Mathematics, 1933 A set ofpostulates forthefoundations vol. 33, pp. 346-366. of logic II,A nnals of Mathematics, 1934 A set ofpostulates forthefoundations vol. 34, pp. 839-864. 1941 The calculi of lambda conversion, PrincetonUniversityPress, Princeton, New Jersey. Feferman,S. 1978 Recursiontheoryand set theory, marriage a of convenience,Generalized recursion theory II, Proceedingsof the1977 OsloSymposium (J . E. Fenstad,R. O. Gandy, and G . E. Sacks, editors), North-Holland, Amsterdam, pp .55-98. 1984 Toward useful type-freetheoriesI, The Journal of Symbolic Logic, vol. 49, pp . 75-111. Feigner, U. 1976 Choice functionson sets and classes , Sets and classes: On the works by , Amsterdam, pp . 217-255. Paul Bernays, North-Holland Flagg, R. C., and J. Myhill 1989 A type-freesystem extendingZFC, Annals of Pure and Applied Logic, vol. 43, pp. 79-97 . Grue, K. 1992 Map theory , Theoretical Computer Science, vol. 102, no. 1 (July), pp . 1-133 . 1996 Stable map theory, Departmentof Computer Science, Universityof Copenhagen , DIKU , Universitetsparken 1, DK-2100 Copenhagen,Denmark", DIKU Report, no. 96/10 (April); available from /www.diku.dk / ;-grue.
A-CALCULUS AS A FOUNDATION
311
Hilbert,D ., and P. Bernays 1939 Grundlagen der Mathematic, vol. 2, Springer-Verlag, Berlin. Mendelson, E . 1987 Introduction to mathematical logic, thirdedition, Wadsworth andBrooks, Monterey, California. Scott, D. 1973 Models for various type-freeA-calculi , Proceedings of the IVth international congress for logic (Suppes et al., editors), Studiesin Logic and The Foundationof Mathematics , Methodologyand Philosophyof Science, North-Holland, Amsterdam,vol, 74, pp . 157-187.
DANIEL LEIVANT*
PEANO'S LAMBDA CALCULUS: THE FUNCTIONAL ABSTRACTION IMPLICIT IN ARITHMETIC Dedicated to the memory of Alonzo Church.
Abstract . We define a fragment 2>.(Peano) of Girard's second orde r Acalculus2>', in which arg ume ntsof typ e application are required to be "Peano types" , namelytypes generate d from type variables and types of the form VR.T (T qua nt ifier free of rank:::; 2) using -+ and subst it u tio. nWe show thatthe provabl y te numeric functions recursive funct ion s of firstorder arit hmet ic are precisely h thatare A-definable in 2>.(Pean o).
One ofChurch's greatachievements was th e definition of theA-calculus and his proof, withKleene,thatallcomput able fun ctionsare A-definabl e.This suggest s hat t isgnificantclasses of comput able fun ctions m ight arise from imposing on the A-calculustyping disciplines, therebyiden t ifyingthe functionalabst raction inh erentin such classes . However ,the connections between natural type disciplines and nat ural classes of comp utable functio ns ave h turned ou t to be somewha ttricky. The mos t rudimentary type discipline, the simply typed lambda calculus1>', yields onl y the extended polynomials er odd as definablefunctions(Schwichtenberg 1976, Sta tman 1979), a rath class that does not includeeven the predecessor function . If differentArepresenta tionsare allowed for i nput and output, then the exponentiation andpredecessorfunctionsbecome definable , but subtra ctionremains outs ide the clas s (Statm an 1979).1 A powerfulextension of 1>' is Girard's second order typ e syst em, 2>' (Gimrd 1971, 1972) . Withinthatcalculus Girardconsidereda polymorphic variantof Church's numerals:fi =d f At . Ast -+ t , z': s[nJ (z ), which was laterrediscoveredindependentlyby Reynolds(1974) , Fortun e (1979), and O'Donnell (1979). From Girard(1971, 1972) it follows thatthe functionsdefinable in
* Resear chp artiallysupported by NSF grant CCR-9309824. T he author is since rely gra tefu lto Thie rryCo quand,J ean-Y ves Marion, and NormanDanner for usefulcom me nts aboutea rlier versions of this paper. lThese anomaliesare repaired when the A-ca lcul usis defined on top of a free algeb ra with consta nts for zero, s uccessor , predecessor, and discr im ina t or . One obtains then the numeric funct ionscomp utable in linear space, by m od ifying thecha rac te rizat ion ofypol time in Leivant 1993 to an a lgebra witha singles uccessor. (This duali tylinear space / po lynomial time is the same as in Handl ey 1993 and Leivan t 199..0 313 C. Anth ony Anderson and M. Zeleny (eds.), Logic, Meaning and Computation, 3 13-329. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.
314
DANIEL LEIVANT
2A modulothesenumeralsare precisely the provably recursivefunctionsof secondorderarithmetic.However,thiselegantcorrespondence characterizes a rathervast class of computablefunctions. 2A so as tocharacterize tamerclasses One natural approachto restricting of functions is typestratification . We showed inLeivant 1991 thatthe funcs tratified form tions definable in the variantof 2A wheretypes are finitely recurrence preciselyGrzegorczyk's class£4 (i.e. thefunctions defined by one from elementary functions) . Type stratification can beextendedintotransfinitelevels, yielding characterizations of theprimitiverecursiveand theprovable recursive functionsof arithmetic.While thoseresults(which we may prove elsewhere) are useful corroborating in well-known correspondences between function classes and ordinals,they do notprovide insightinto the functional natureof thesefunctionclasses.? The system 2A can also be confined by syntacticr estrictions on permisbecauseit corresponds(via a sible type arguments.This is most natural, Schonfinkel-Curry-Howard style"formula-as-type" homomorphism, see Leivant 1990) to restrictions on thecomprehensionprinciple,which have been fundamental in prooftheoretic studies. Some preliminaryresultsin this directionwerenotedalreadyin Fortune 1979, O'Donnell 1979, and Leivant 1981, butthese fell shortof characterizing classes offunctions . Here we exhibitsuch acharacterization for the class of provably recursivefunctionsof Peano'sArithmetic,PA. This class of functions has also natural a computationaldefinition , as theclass offunctionsg eneratedvia primitiverecursion in all simpletypes (Gadel 1958), which is why wedenoteit byRec'". Our main resultprovides two characterizations of Rec'" by definability to in subsystemsof 2A. One is minimalist: type argumentsa rerestricted thetypes needed to p ermit..\-definability forprimitiverecursionin all simple allows abroaderclass oftype arguments, types. The othercharacterization which we dub"Peanotypes". These may containtype parametersin the under(type-correct) substitution of Peano scope ofV, but are still closed types for typevariables.This formalism is thestrongestone thatwe can handlewhen showingthatall..\-definablenumericfunctionsare provablyrecursive functions PA. of The proofof thelatter resultuses aformalization of Girard'sstrongnormalization prooffor2A: when all t ypespresentarePeano types, the formalization can becarriedout in secondorderarithmeticwith comprehensionand inductionrestricted to arithmetical formulas, atheory uses the well-known observation conservative over PA. Our formalization thatformulas of t heform Vii.3x. tp (cp primitiverecursive)areequivalent to formulas." A completestatementof our mainresultis given as existential the MainTheoremat theend of §2 below. 2 A related approach,using infinitetypeconjunctions , is explored inLeivant 199Ga. 3Such formulasare known as strict (Barwise 1969, 1975), or computational
(Leivant 1989) .
nl
PEANO'S LAMBDA CALCULUS 1. THE SECOND ORDER LAMBDA CALCULUS
315 2~
1.1. Peano Types
We refer toGirard'ssecondorderA-calculus , 2A, in its original, ontological (i.e. Church -style),formulation.tThe types of 2A (2-types forshort)are generatedinductivelyfrom a countablecollection of type variables bythe closureconditions : if 7 and a are types,thenso is 7 -+ aj and if7 is a type and R a typevariable,thenVR. 7 is a type. Parentheses are used inconcrete syntax,and -+ is assumed to associateto the right , e.g. 7 -+ a -+ p stands for 7 -+ (a -+ p). We write71 , " . , 7 r -+ a for 71 -+ 72 -+ . . . -+ 7 r -+ a, and r" -+ a if alltypes7 i arethesame typeT . We denoteby {a/ R} thesyntactic a for every free occurrence of the operationof simultaneously substituting variableR in theargument , modulostandardvariablerenamingto protect intendedscoping. A type is functional if its mainconstruct is -+ , and universal if it isV. A type is first order if it has no occurrence of V. We use Q,Qi, R , Ri, S . . . as syntacticvariables for formal type variables . A type VR 1 . ,. VRr.7 will be abbreviatedby VR 1 •• • Rr.7 (or evenVRs, if R1 . • . R; are evidentor irrelevant). If 7 = VR 1 ••• Rc,«, wherek > 0 and a is not universal , thena is thematrix of 7. Given a set oft ypes T we write*T and VT forthetypes generated from T using only-+ and onlyV, respectively. Thus a E VT iffa is of the form VRs with7 E T . Morerestrictively , VIT willdenotetheset oftypesof the form VR.7 , with 7 E T. If 7 is a firstordertype, thenits rank, rnk (7), is the count of negative nestingof -+ in 7 j thatis: rnk(R) = 0 fortype variablesR, and rnk(7 -+ a) = max(l+rnk(7), rnk(a)). For eachk > 0 we letIs. denotetheset of first ordertypesof rank:::; k. The Peeno types aregenerated inductively by thefollowing clauses . (1) Everytype variable is aPeanotype. (2) If 7 and a are Peanotypes, thenso is 7-+a. (3) If 7 is a firstordertypeof rankj; 2, thenVR.7 is a Peano type. (4) If 7 and a are Peanotypes, thenso is {a/ R}7 (correctsubstitution) . For example, all types in *V'J. are Peano, as are ((R -+ R) -+ R) -+ Rand VR.((7-+R), (R-+a) -+ R) , for any7,a E *'v''J. in whichR is not free. On theotherhand, 'v'R.(((R -+ R) -+ R) -+ R) and 'v'R.((VQ.(R -+ Q)) -+ R) are notPeano(forthelatter this follows from Lemma 4 below.) 4For furth er detailsabout2~ see e.g., Girard 1971 , Reynolds 1974, MacQueen 1982, and Fortune, Leivant, and O 'Donnell 1983.
316
DANIEL LEIVANT 1.2. A Closed Definition of Peano Types
We give an alternative, closed,definitionof Peanotypes. This definitionis an aside, andwill not be used elsewherein thepaper, exceptfortrivialuses of Theorem 1. We referinitially to types satisfyingthecloseddefinitionas cPeano,andproceedto provethatcf'eanoandPeanotypesarethesame. We say thata 2-type r is separated if it has nonestedV blocks witha variableb oundby theouterone occurringfree intheinnerone. For example, VR . (VS.(S -+ R) -+ R) is notseparated,b utVR , S. (S -+ R) -+ R is. For a type r E *T define itsrank over T, rnkT(r), to be thecountof negative nestingof -+ in generating r from T; thatis: rnkT(r) = if rET, and rnkT(r -+ o ) = max(l+rnk T(r),rnk T(u)) when r -+ o is notin T. For k = 0,1 . . . we writek(T) fortheset of typesin *T of rank~ k over T . For a tupleii = (R I . • • Rk) of type variableslet a[ii] be theset of types with no R; free, and let a* [ii] =df *( {R I . •. Rk} U a[ii]). In particular, if r is separated,t henforeverysubtypeVii.u of r we must have a E o" [ii]. Call a type r cPeano if it isseparatedandforeveryuniversal s ubtypevii. o of r thematrixa is of rank~ 2 over a* [ii] .
°
LEMMA
1. A subtype of a cPeano type is cPeano.
Proof. We need onlyprovethattheimmediatesubtypesof a cPeanotype r arecPeano. This is trivialif r is a functional type,or ifr is VR:a witha the matrix. If r is VR. VS I . . . Sk. a (k > 0), it suffices too bservethattherank of o over a*[SI ... Sk] is at most therankof o over a*[R, SI . .. Sk], since a[B, R] ~ a[B] . Since r is cPeano, thelatter is ~ 2, andso theformeris also 2. --I
s
LEMMA 2 . If no variable among ii is free in a , then the rank of {u j S}r over a*[ii] is the same as the rank ofr overa*[ii] .
Proof. Straightforward structural inductionon r. LEMMA
3. If rand a are cPeano types, then
so
--I
is {u j S} r.
Proof. By structural inductionon r. The cases wherer is S , or S is not free inr, are trivial.If r is p -+ ~, then[zrj S} p and {u j S}~ arecPeano, by inductionassumption;from thedefinitionof cPeanotypes it follows then that{u j S}r = {u j S} p -+ {u j S}~ is cPeano.Finally,suppose thatr is vii. P with p thematrix. Then {u j S}r is vii. {u j S} p. By inductionassumption {u j S} P is a cPeanotype. Since r is cPeano,therankof p over a* [iiI is ~ 2, whence,by Lemma 2, therankof {ujS}p over a*[ii] is also~ 2. Therefore {ujS}r = vii. {ujS}p is cPeano. --I
PEANO'S LAMBDA CALCULUS LEMMA
317
4. A 2-type is cPeano iff it is Peano .
Proof. The forwardimplicationis proved bystructural inductionon types. Type variables are all Peanoby definition.If T == U -+ P is cPeano,thenso are a and p, by Lemma 1. By inductionassumptionboth types are then Peano, which impliesthatT is Peano. Finally , suppose thatT == VR.u is cPeanowherea is thematrix. Since therank ofo over a*IRI is ::; 2, we have o == {(jQ}uo for some types ( E a*IRI and freshtype variablesQ, and whereUo is a firstordertype of rank::; 2. Since ( are all s ubtypesof T, theyare all cPeano,by Lemma 1, and b~ inductionassunwtiontheyare therefore Peano. Since novariableamong R appearsfree in~, we also have T == {(j R} (VR.uo), which isPeanoby definition . The backwardimplicationof the lemma is proved by inductionon the derivationof a type as a Peano type. All type variables are cPeano by definition . If T == U -+ P is Peano, whereo and parePeano,thena and p are cPeano,by inductionassumption. From the definition cPeanotypes of it immediatelyfollowst hatT is cPeanoas well . If T == VR.u, wherea is a firstordertypeof rank::;2, thenT is cPeanoby definition . Finally , suppose T == {pi R}u, wherep and a are Peano; by inductionassumptionp and o are cPeano,so T is cPeanoby Lemma 3. --j 1.3. Terms
The terms of 2A are generated from typed variables using o bjectand type abstraction andapplication,as follows.We writeE : T, or ET , for "E is of type T". (1) For each2-type T thereis a countable collection of variables of type T, which are defined to be terms of type T . We use xT ,yT ,xI . . . as syntacticvariablesrangingover formal variables ; (2) If E : a, then>'x T • E : T-+Ui (3) If E : T-+U and F : T, thenEF : a , (4) If E : T, thenAR.E : VR.r, providedR is not free in the typeof free objectvariablesin E . (5) If E : VR.T anda is a 2-type,thenEo : {ul R}r (modulo usual variable renaming). As usual, we useparenthesesin concretesyntax,and associateapplication to theleft (i.e.E FG standsfor (EF)G) . If E and x are ofthesame type, we write{E I x} forthesyntactico perationof substituting E for every free occurrence of x in theargument,modulo variable renamingin theargument
318
DANIEL LEIVANT
to preserveintendedscoping. The notionof subterm is defined asusual , wherewe takeeachtermto be asubtermof itself. A term(>.xT.E)F {3-contractsto {Fjx}E, and (AR.E)r type-contracts to {r/R}E. For eachcontraction thesourceis calledtheredex and thetarget thereduct. We say thatE {3-reduces (respectively , t-reduces) to F , andwrite E => 1 F (respectively,E =>~ F) if F arisesfrom E by replacinga {3-redex (type-redex)subtermby thecorrespondingr educt.The relation = {3t is the symmetric,reflexiveand transitive closureof (=> 1) U (=>f). E is normal if none of itssubtermsis a redex, and strongly normalizable if everysequence of reductions starting with E terminates . Peano's >'-Ca.lculus, 2-\(Peano),is thesub-calculus of 2-\ whereeverytype argumentis Peano.P More generally, Tifis a set of types, we write2-\(T) forthesetof termswhere everytype argumentis in T. In addition , we say thata term E is confined to T if thetypeof everysubtermof E is in T .6 LEMMA 5. Suppose that E is a normal term of 2-\(Peano), where the types of E and of every free (object) variable are all Peano. Then E is confined to Peano types. Proof. By structural inductionon E. If E is a variable,thenits type is Peano by assumption. Suppose thatE == >'x T.F<7 : r-+(J. By assumptionr-+(J is Peano ,andso bothrand(J arePeano. Thus F satisfiestheLemma 's conditions, and by inductionassumptionthetype of everysubtermof F is Peano ,and so the same is trueof E. Suppose E == ASI • • . ST.FT : VB.r . By assumptionVS.r is Peano, andso r is Peano, by Lemma 1. Thus the presentlemma's condition s hold forF, and thetypeof everysubtermof F is Peano. The remainingsubtermsof E are oftheform ASi ... ST.F, whose type, VSi ... Si:» , is a subtypeof VS.r, andis therefore Peano, by Lemma 1. Finally,s uppose thatE is an application.Since E is normal,it must be of the formz" Al . . . A r , where x is a variablea ndAI, ... , A r consistof types and terms. By theLemma's condition(J is Peano. It is easy to prove , by inductionon i, thatthetypeof everytermE, =df xA I •.• Ai (i ~ r) is Peano, andso thelemma appliesto each suchE i • In particular , E == E r is confined to Peano. --I 5T he phrase "calculu s" is justifi ed by the fact that2>..(Peano) is closed under {3-fr reductions . This is trivial for ,I3-reductions, because these do not gen erate new type arguments. A type-reducti on m apping a term (AR.E )T to {T/R}E might cha nge atype argumentCT withinE into {r/R}CT. However , since the inputterm is in 2>..( Peano), both r and CT arePeano, and ther efor e he t new typeargument{r/ R}CT is also Pean o. 61f E is confinedto Peano , then E is in 2>..(Peano), except poss iblyif the re si a vacuou s typ e applicationin E ; for instance, xVR, S - Sr is confined ot Peanofor any typ e r. Such ty, by requiring thatvacu ou stype anomalies may be elim inated, withoutloss of generali applica t ions have aypte varia bleas arg ume nt .
319
PEANO'S LAMBDA CALCULUS 2. RECURRENCE SCHEMAS
Recallthattheprimitive recursivefunctionsare generatedfrom thezeroconstant , successor, andproject ion functionsby compositionandtheschema of primitiverecursion,
1(0, x) = g(X)
(1)
I(sn,x) = h(n,x,/(n,x))
G6del (1958)considereda generalization of primitiverecursionto all sim inductively from a base ple types." As usual,thesimple types aregenerated type0 by thebinarytype constructor - t, i.e. if T and a aresimple typesthen so is T - ta. We adoptthe same notational conventionforsimple types as forquantifier freetypes of 2..\. Note thatevery simpletype T is of theform Tl,· . . , T r --+0 for somevectorof types 1I"(T) =df (Tl . . . T r ) . Primitiverecursionin simple typesis oftenstatedas a readingof schema (1) above withtheoutputof 1 and theparametersx permittedto have any simple types. However , the use of highertype permits a streamliningof thisdefinition . Say thata function1 : 0 - tT is defined by closed primitive recursion from functions9 : T and h : 0 , T - tT if
1(0)
=9
(2)
I(sn) = h(n)(f(n))
Schema (1) is easilyreducibleto (2): If I(sn,x) = h(n,x,/(n,x)) then I(sn) = h'(n)(f(n)), whereh'(a)(b)(x) = df h(a, x, b(x)). Our proofs below will be simplified by theuse of analternat ive definitionof primitiverecursionin simple types. We say thatfunctionsh : 0--+ Tl , .. . , I» : 0 - tTk are defined by simultaneous recurrence from functions gl: Tl , · ·· , gk: T k and h i : (TI . . . Tk-tTI) , ... , h k: (TI . .. Tk--+Tk), if
1i(0) Ii(sn)
= gi = hi(ft(n)) · · · (!k(n))
(3)
(i = 1. .. k) LEMMA
6. Primitive recursion is reducible to simultaneous recurrence.
Proof Suppose that1 : r --+ 0 is defined by (2). Define h by thefollowing s imultaneous recurrenc e.
12 : 0 - tT
h(O) h(sn)
o sh(n)
Then, by inductionon n, h (n)
12(0)
12 (sn)
=
:
0 --+ 0
and
9
h(h(n))(h(n))
= nandh(n) = I(n).
-J
7The generalizati on of primitive recursion to second orde r fun cti onal s see ms to have been not iced first by Hilb er t (19 25 ) , where Ackermann's functionis defined by primitive recursion over type 0 -+ o.
320
DANIEL LEIVANT
In fact, an even simplerdefinitionofsimultaneous recurrence suffices. Say thatan instanceoftheschema(3) ofsimultaneous recurrence is type-uniform if thetypes Tl .. . Tk are oneandthesame type T . LEMMA 7. The schema 01 simultaneous recurrence is reducible to type-uni[orm simultaneous recurrence.
Proof. Withoutloss ofgenerality, considerthe case of two simultaneously defined unctions. f Suppose thatIi : O->Ti (i = 1,2) are defined by
h(O) h(sn)
= =
12(0)
gl hl(h(n)Hh(n))
=
12 (sn) =
g2 h 2(h(n)Hh(n))
Let T =df 1l"(Tt},1l"(T2) ->O. DefinethefunctionsUi : Ti->T and Ds : T->Ti (i = 1,2) by
u, D1 D2
=
A T. A_7r(T,)_7r(T2) (_7r(T;)) . a Xi a Xl X2 T AZ Ax7(T,).Z(Xl)(e""(T2))
=
AZT AX;(T2). z (e""(Td)(X2)
=
where, for at ype <7, e" is some canonicalfunctionof type <7, e.g. AX""(al. o. Then I i = Di(fI) , whereI~ , 12 aredefined by thet ype-unif orm simultaneous recurrence
I[{O) = Ui(gi) I[{sn)
with
h~
= h~(f;(n))(f2(n)) =d f
Ayry; · Ui( hi(D I (yt}HD 2(Y2)))
Let Rec" be thesystem of functionsin all simpletypes generatedfrom theconstant-zero and successorfunctionsusing typedA-abstract ion andapplication(andhenceprojectionsa ndcomposition),and theschemaof simultaneousrecurrence in allsimple types. We are nowready tostatepreciselyourmain result . THEOREM 8 M a in T heorem . Let conditions are equivalent .
I be a numeric function. The following
(1)
1 is a provably recursive function
(2)
1 is definable
of PA.
by (closed) primitive recursion in simple types.
(3) I E R ecw • (4)
I is A-definable in 2"X(*Vl £).
(5)
1 is A-definable in Peano 's A-Calculus, 2"x(Peano).
PEANO'S LAMBDA CALCULUS
321
The implication(1) => (2) is Codel's(1958) "Dialectica"T heorem. (2) => (3) is Lemma 6 above. We shallprove (3) => (4) as Lemma 9 below. (4) implies (5) trivially. Finally, we provetheimplication(5) => (1) as Theorem 19 below. 3. LAMBDA DEFINABILITY OF SIMULTANEOUS RECURRENCE IN SIMPLE TYPE
The canonical r epresentation of natural numbersin 2,). is by type-abstracted Churchnumerals:fi =df AR.>.sR-+R , zR. slnJ(z), wherethe superscript[n] standsfor n-folditeration . We denoteby v thetypeVR.((R->R),R -> R) of thesenumerals . For asimple type 7 we write7[V] forthe2-type {VIO}7 thatresultsfrom replacingthebase type 0 in 7 by u, Note thatfor every simple type 7 thetype 7[V] is in *V1~. A functionf : ~r -> N is defined in 2,). by a term F, if F : t/" -> v, and Ffi1 " . fir =(3 ,t f(n) for alli i = (n1 . . . n r) E Nr . More generally, thenotion f ofsimple type7 is given bystructural ofdefinability fornumericfunctionals recurrence on 7: If 7 == 71 , . . . ,7 r -> 0, thenf is defined by a term F : 7[V] if for all t ermsG 1 .. . GT! defining overN functionals g1 .. . gr of types 71 .. . 7 r, respectively, we have FG l . · · G; = {3t fgl ... gr' Given termsE l , ... , E k , all oft ype 7, let
(E 1 • • • E k )
=df
>.xr k -+ r • xE 1 • • • E k ,
and let(k7) standforthatterm'stype, namely(7 k->7)->7. For i = l .. . k let i kr =df >.q(kr). q(>.ur . . . Uk ' Ui) ' Thus i kr (E 1
•••
Ek}
= (3
Ei .
LEMMA 9. Every function f E Ree'" is definable in 2,),(*V1~).
Proof. By Lemma 7 a functionf E Rec'" is definable from zero and successor usingabstraction, application,a nd type-uniformsimultaneousrecurrence. We proceed by inductionon the lengthof such definitions. The constantzero functionis defined by>.av .O, and thesuccessorfunctionby >.av.AR.>.sR-RzR .s(aRsz). Closureof thedefinablefunctionsu nder>.-abstraction a ndapplicationis trivial. Suppose thath .. .!k : 0 - t 7 are defined bytype-uniformsimultaneous recurrence over 7, as in (3) above. Byinductionassumptionthefunctions gl·· ·gk, hI ... h k are defined by someterms Gl .. .Gv, HI . . . H k, respectively . Then Ii is defined by where and
Fi
=df
H
=df
>.a V • i kr (a (kT[v]) n g), >.q(kr [v]) .(H l (hrq) ... (lfkrq), . . . , Hk(lkrq) . . . (kkrq)} ,
=df
(G l ,
g
...
,Gk)'
322
DANIEL LEIVANT
Notethatthetypeargument(kr[vJ) is in *'11£, so eachF; is in 2~(*Vl£). -l 4. COMPUTABILITY OF DEFINABLE FUNCTIONS
4.1. Normalization of Terms in
2~(Peano)
We shownextthateveryfunctiondefinablein 2~(Peano) is a provablyrecursive functionofPA. Thesalientobservation is thattheTait-Girard's proofof strongnormalization for2~ (Girard 1972) , whenrestricted totermsconfined to Peanotypes, can be formalizedin secondorderarithmeticw ithcomprehensionand inductionrestricted to arithmetical formulas. Sincet helatter theoryis conservative overPA, thiswill implythatallfunctionsdefinablein 2~(Peano) are provablyrecursivein PA. While manyexpositionsof Girard'smethodhave been given since its inception,we outlinehere anexpositiontailoredto theapplicationin hand. Considera languageL AMBDA withobjectvariablesM , .. . intendedto range overuntypedA-terms,andset (i.e.unaryrelation) variablesintendedtorange over sets of such t erms. For setvariableswe re-usethetype variablesof 2~. The vocabulary of LAMBDA has identifiersfor the basicsyntactico perations over A -terms, as follows.T he constantidentifiersof LAMBDA are the(unare abinary* (used in infix) , and typed)A-variables.T he functionidentifiers for each A-variable x a unary1mb x and a binarysbtx ' These are intended to denoteapplication,A-abstraction, and(correct)substitution, respectively; thatis, if M and N have thevalueE and F, thenthevalues ofM * N, Imbx(M) and sbtx(N,M) are EF , Ax.E and {F/x}E, respectively,where E arise fromE by canonically renamingvariablest hatare free inF . Also, LAMBDA has aunaryrelation identifierV ar, intendedto holdexactlyof the A-variables. We construe2-typesas conveyingsemanticpropertiesof untypedA-terms (as in Leivant 1986) . To start,letFrame[B, SI be theconjunction of the following p ropertiesof setsBand S. We shalleventually set S to betheset S of stronglynormalizable terms, and B theset VS* oftermsof theform zEl ..• E k (k ;::: 0), with z a variableand E, E S. However, we needBand S to beparameterized for now. (1) {A-variables}~ B ~ S :
'1M. (Var(M)
->
B(M))
1\
(B(M)
->
S(M)).
(2) If Ez E S (z a variable)t henE E S : '1M, N . S(M * N) 1\ Var(N)
(3) BS
~
->
S(M) .
B: '1M, N. B(M) 1\ S(N)
->
B(M * N).
323
PEANO'S LAMBDA CALCULUS of thefollowing two conditions. Let Adeq[R, B, S] be theconjunction
(1) B<;R<;S. (2) If {F/x}E E R withFE S, then(>.x.E)F E R:
VM,N. R(sbtx(M,N»
1\
S(N)
->
R(lmbx(M)
* N).
(It suffices totaketheformulaabove forjustone >'-variablex; other cases follow moduloo-conversion .) Adeq [R, B, S] is intendedto statethatR is "adequateas a type" withinthe frame B,S. For each2-type T we define a formula sp" of LAMBDA, witha single free variable.
cpR[M] cp"-
=
R(M) W(cp"[Y]-> cpP[M * Y]) VR.Adeq[R,B,S] -> cp"[M]
Note thattheboundset variablesin ip" arethetype variablesb oundin T, and thefree setvariablesarethefreetypevariablesin T , plus perhapsBand
S. LEMMA 10. The formula cp{pJ R}" is syntactically identical to {cpP / R}cp".8 structural inductionon T . Proof. Straightforward LEMMA ll. Assume Frame [B, S], and let T be a 2-type. Assuming Adeq[R,
B,S] for every variable R free in
T,
we have Adeq[cpT,B ,S] .
--
Proof Structural inductionon T . For atermE of 2A letE be theuntypedform ofE. 9 LEMMA 12. Assume Frame[B, S]. Suppose E : T is a term of 2A, with free variables Xfl... X~k. Assuming Adeq[R, B, S] for every variable R free in T, we have -(4)
SRecallthatthesecond ordervariablesR are unarypredicates,and cpP is a formula with a singlefree variabl e. {cpP / R} denotesthe syntacticoperationof replacingeach formula-occurrence of theform R(t) in theargumentby cpP[tj. 9The precise definitionis by recurrence on E . Use Xi -r (i = 1,2 .. ., T a 2-type) as thecollection of untypedA-variables . Let xi =df Xi r , Axj.E =dfAXiT 'fl., EF =df E.E., AR.E =df E., and ET =df E.. -lOStrictly speaking, fl. shouldbe replacedby thetermof LAMBDA thatdenotesfl., if (4) is to be aformulaof LAMBDA.
324
DANIEL LEIVANT
Proof By structural inductionon E . Forthecase ofapplicationuse Lemmas -I 11 and10. (See e.g., Lemma 2.3 in Leivant 1986 fordetails.) COROLLARY
13. For every term E of 2A, E is strongly normalizable.
Proof Let ET be a termof 2A. Withoutloss ofgenerality, E has no free objector type variables . It is easy to seethatFrame[VS*,Sj holds, where S is theset ofstrongly-normalizable untyped-X-terms,and VS· is theset of expressionsof theform zEl •• . Ek where z is a variableand E; E S. So, by cpT ~ S. By Lemma 12 we Lemma 11, Adeq [cpT, VS*, s], and inparticular -I havecpT(E), andso E E S.
14. Every term of2A is strongly normalizable, i.e . every reduction sequence terminates.
COROLLARY
Proof Note thatif E reduces toF by a type-contraction, thenF is identicalto E. Thus, if E = Eo , E 1 . •• is a reductionsequencein 2A, then ~ =df ~,El . .. is a sequence of ,a-reductionsand identities . Also, type reductionsdo notgeneratenew redexes , and therefore everystretchof consecutivetypereductions in E must be finite . By Corollary 13 thesubsequence is finite. So~ is finite, andtherefore E is finite. of~ consistingof,a-reductions -I
4.2. Formalization of the Normalization Proof
Let usoutlinea formalization of theproofabove withinsecondorderarith, metic, SOA.l l SOA is a theorybased on secondorderlogicwithequality variablesfornumbersand setsand quantifiers overthem,and comprehension of SOA to consist of theconstant for all formulas. We takethevocabulary o andan identifier for each primitiverecursive function. The non-logical axioms are Peano'sthirdandfourthaxioms for the successor function,defining equationsfor all primitiverecursivefunctions,and theinductionaxiom 'v'X.(X(O) 1\ 'v'z.x(z) -->X(sz))
-->
'v'z .X(z).
Sincecomprehensionis postulated for all formulas , it follows thatallinstances thelanguage are derivable. of inductionfor formulas in Fix some canonical a rithmetization ofthesyntaxof theuntyped-X-calculus, withprimitiverecursivenumericcounterparts of theLAMBDA identifiers . It follows thatthepredicatesFrame andAdeq areexpressedby firstorderuniversal formulas (with parameters).This set is in contrast to thepredicates ip"; whose complexityin theanalytical hierarchygrows withthesyntactic 11
NB: This formalization is partial , in a sense to beexplainedmom entarily .
325
PEANO'S LAMBDA CALCULUS
complexityof T. 12 However, if only a finite collection types of is of interest, thena formalization withinSOA becomes possible, along t hefollowing lines.13 Let T be a finite set of types closedundertakingsyntacticsubtypes. thatthetype variables are renamedso Assume, withoutloss ofgenerality, thatno type variable boundin T is free inT or bound byanother occurrence of 'If. Whenrestricted to typesin T, Lemma 11 amountsto verifying a finite number of cases, usingmetamathematical inductionon the complexityof typesin T, and this can be proved SOA. in SinceT is finite, thepropertystatedin Lemma 12 can be defined explicitly fortermsconfined toT, as follows . Set
cpT[M]
=df
'ljIT[M]
=df
V
cpT[M] TET ('If sequenceN 1 ... .Ni; of A-terms) ('If A-variablesx) (
Adeq[T,B,S]
=df
1\ cpT (Nil
-t
cpT({N/ x} M )
1\ Adeq [cpT , B,S].
TET
ThenLemma 12, fortermsconfined toT, states Adeq [T,B, S] -t'ljlT[M], which is provedwithinSOA by induction.The formalization in SOA ofthe remainingof the proof leading Corollary to 14 is routine. 4.3. Peano A-Definability Implies Provability
The formalization above oftheproofof strongnormalization uses thefull prooftheoretic power ofSOA in essentialways. However, for termsconfined to Peano types theproofis formalizable in the subtheorySOAo of SOA, obtainedby restricting comprehensionto arithmetical formulas. (Note t hat this impliesthattheinductionaxiom can beinstantiated to such formulas only.) As usual, a formulaSOA of is arithmetical if it has no set quantifiers. Recallthatan arithmetical formula is~~ (II~) if it is prenexwithn alternating blocks ofquantifiers, starting w ithan existential (respectively,universal) block. 12This is an essentialfeatureof the proof: iftherewere a formularp[t, m] such that rp[#r, #M] expresses rpT[M] (where#r and #M arecanonicalnumericcodes for rand M) , thenthenormalization p roofabove couldbe formalizedwithinSOA, from which it instanc e by a formula-as-type homomorphism, as in Leivant 1990) that wouldfollow (for SOA provesits ownconsistency . 130ne may takeinsteadthecollection of alltypeswhose syntactic c omplexityis bounded by some fixed upper limit.
326
DANIEL LEIVANT
The following o bservation seems to be due to Georg Kreisel: LEMMA 15. Each SOA formula tp of the form 'VR3x. tpo , with CPo quantifier free, is equivalent (in the standard model) to a ~? formula. Moreover, this fact is provable in SOAo.
Proof. To avoidcluttered notation, assume thatCPo == tpo[x, y, R, QJ, i.e. with singletonR and X, and where cp has one freenumber variabley and one free set variableQ. Fixing valuesP, 5 E P(N) for Rand Q, the truth value ofcpo[m, n , P, 51 depends only on a finite number of numericvalues r emains unchangedwhen membership in the d1 < .. . < de, and therefore sets P,5 is modified at values>de. Thus, for each choice of P thereis a valued(P) such that3x .cpo[x, n , P, 51 is determinedby thevalues ofthe characteristic functionXP only fora rguments< d(P) . By Konig's Lemma, thereis an upper bound d to all values d(P), and therefore a bound b = 2d on binarycodes oftheinitialsegmentsof XP for all P 's neededto verifythe truth of cp[n, 51. Thus, cp[n, 51
...
3b 3x 'Vr < b. cp~
wherecp~ arises fromCPo by replacingeach subformulaof theforma R( t) by r@t= 1, with r@t denotingthet + l-stbitof thebinarynotation of r , Since tp~ is primitive recursive,so is 'Vr < b. tpo, and cp is therefore equivalent in SOAo to a ~? formula . The formalization oftheproofabove inSOAo is straightforward, including (see e.g., Konig'sLemma, whose informalproofis easily formalized SOAo in Friedman 1969, Theorem3). -\ LEMMA 16. If T is a first order 2-type of mnk ::; k, then SOA by a formula.
rrZ
ip"
is rendered in
inductionon k . Proof. Straightforward LEMMA 17. If T is a Peano type, then ip" is rendered in SOAo by an arithmetical formula T. That is, the formulas T satisfy, modulo numeric encoding and provably in SOAo , the equivalences that define the formulas ip" ,
Proof. By inductionon thedefinitionof Peanotypes. 1. For atypevariableR we havecpR
==
R, which istrivially arithmetical.
2. If T = a --> p, wherea and parePeano,thencpu and tpP are rendered by arithmetical formulasu and P , by inductionassumption,andso tpu-+p is renderedby
u-+P[m]
==df
'Vy. "y codes aA-term"
1\
U(y)
-->
P(app(m,y»
where app is the primitive recursivefunctionrenderingthe LAMBDA identifier*.
327
PEANO'S LAMBDA CALCULUS 3. Suppose thatr = VR I 2. By definition,
s
cpT[M] ==
. . .
Rc.o , where (J is a firstordertype of rank
vii. AAdeq [~, B , 8]- cpU [M].
ny
The formulaAdeq, by its definition,is renderedby a formula,a nd by Lemma 16
ng. ng
ng
4. If r = {pi R}a (correctsubstitution), where(J and parePeanotypes, thencpT is syntactically identicalto {cpP I R}cpu, by Lemma 10. Since bothcpu and cpP are renderedby arithmetical formulas
328
DANIEL LEIVANT REFERENCES
Barwise,J . 1969 Applicationsof strict-IIIpredicatesto infinitarylogic, The Journal of Symbolic Logic, vol. 34, pp . 409--423. 1975 Admissible sets and structures, Springer -Verlag B , erlin . Feferman, S. 1977 Theoriesof finitetype relatedto mathematicalpractice,Handbook of mathematical logic (J . Barwise, editor),N orth-Holland , Amsterdam, pp , 913-971. Fortune,S. 1979 Topics in computational complexity , Ph.D . Dissertation, Cornell University. Fortune,S., D. Leivant,a nd M. O'Donnell 1983 The expressivenessof simple andsecond-ordert ypestructures , Journal , vol. 30, no. 1, pp . of the ACM (AssociationforComputingMachinery) 151-185. Friedman,H. 1969 Konig'sLemma is weak,unpublishedmimeographednotes,Stanford University . Girard, J .-Y. 1971 Une extensionde l'mterpretation de GOdelit l'analyse,et sonapplication a 'elimination l des coupuresdans l'analyseet latheoriedes types, Proceedings of the Second Scandinavian Logic Symposium (J . E. Fenstad, editor),North -Holland , Amsterdam,pp . 63-92. 1972 Interpretation Fonctionnelle et Eliminationdes Coupuresde I'Arith, These de doctoratd 'etat,UniversiteParis metiqued'OrdreSuperieur VII (June). GOdel,K. 1958 tibereine bishernoch nichtb enutzte Erweiterung des finitenStandpunktes, Dialectica, vol. 12, pp. 280-287. Handley, W . G. 1993 Bellantoni and Cook's characterization of polynomialtime functions, , Universityof Leeds. manuscript,D epartmentof PureMathematics Hilbert,D . 1925 tiberdas Unendliche,Mathematische Annalen, vol. 95, pp. 161-190; translated in van Heijenoort 1967, pp. 367-392. Leivant, D. 1981 The complexityofargumentpassingin polymorphicprocedures , Proceeding of the thirteenth symposium on theory of computing, Associationfor ComputingMachinery,New York, pp . 38-45. 1986 Typing and computational propertiesof lambdaexpressions, Theoretical Computer Science, vol. 44, pp. 51-68. 1989 Descriptivecharacterizations of computational c omplexity , Journal of Computer and System Sciences, vol. 39, pp. 51-83.
PEANO'S LAMBDA CALCULUS
329
Contracting proofstoprograms, Logic and computer science (P. Odifreddi, editor), Academic Press, New York, pp. 17!f-327. 1990a Discretepolymorphism, Proceedings of the sixth ACM conference on LISP and functional programming, pp . 288-297. 1991 Finitelystratified p olymorphism, Information and Computation, vol. 93, pp.93 -113. 1994 Ramified recurrence and computational complexityI: Word recurrence andpoly-time,Feasible math ematics II, Perspective in computer science, (P. Cloteand J . Remmel, editors), Birkhauser-Boston, New York, pp. 320-343. Leivant,D., and J.- Y. Marion 1993 Lambda calculus characterizations of poly-time,Fundamenta Informatiandtypetheory cae, vol. 19, pp. 167-184; Specialissue: Lambda calculus (J . Tiuryn,editor). Macqueen,D. B., and R. Sethi 1982 A semanticmodel of types forapplicativelanguages,A CM Symposium on LISP and functional programming, pp . 243-252. O 'Donnell , M. 1979 A programminglanguaget heoremwhich isindependentof PeanoArithmetic, Eleventh annual A CM symposium on theory of computing, AssociationforComputingMachinery . Reynolds,J . C. 1974 Towardsa theoryof typestructures, Programming symposium (colloque , Berlin, pp. sur la programmation, Paris), LNCS #19, Springer-Verlag 408-425. Schwichtenberg, H. 1976 DefinierbareFunktionenim Larnbda-Kalkul mit Typen, Archiv mathematische Logik u. Crundlagenforschung, vol. 17, pp. 113-114 . Statman,R . recursive , Theoretical Computer 1979 The typedA-calculusis not elementary Science, vol. 9, pp. 73-81. Troelstra, A. S. 1973 Metamathematical investigations of intuitionistic arithmetica nd analysis, Lecture Notes in Mathematics, vol. 344,Springer-Verlag , Berlin. Van Heijenoort,J . 1967 (editor),From Freqe to Godel, A source book in mathematical logic, 18791931, HarvardUniversityPress, Cambridge, Massachusetts . 1990
RALPH LOADER·
THE UNDECIDABILITY OF A-DEFINABILITY
Abstract.In thisarticle , we shallshow thatthePlotkin-Statman conjecture (Plotkin 1973, Statman 1982) is false.The conjecture was that , in a modelof the simply typedA-calculus withonly finitelymany elementsa teachtype,definability . This conjecture h ad been shown (by a closedtermof the ca lculus) isd ecidable to imply many things, for example, Statman(1982, see also Wolfram 1993) has shown it implies thedecidability of pure higherorderpatternmatching(a problemthatremainsop en atthetimeof writing)a ndis equival e nttohigherorder patternmatchingwitha-functions.T he proofof undecidability given hereuses encodingsof semi-Thue systems as definabilityproblems. It had been thought thatA-definabilitym ight be characterized by invarianceunderlogical relations, which would imply the Plotkin - Statmanconjectur e. We give a re latively s imple counterexample to this, using ourencodingof wordproblems.
1. PRELIMINARIES
We shallconsiderthesimply typed "\-calculuswith a single g roundtype o. All models Mconsideredhere will be finite (in thesense of finitely m any clementsateachtype) and full (for any types T and U, MT~ U is theset of shallbe allfunctionsfrom MT to Mu) . Terms of thesimply typed.A-calculus assumed to be intheChurchform, i. e. allvariablesare typed, butwe shall usuallyo mit thetypes fortypographical convenience . We shall onsidertermsin c .a-normal,1]-longform. .a-conversion isgeneratedby thefollowing rules:
(.Ax.s[x])(t) s --+fJ S' t --+fJ t' S
--+fJ
S
,
s[t], s(t) --+fJ s'(r), s(t) --+ fJ s(t'), .Ax . s --+fJ .Ax . S',
--+fJ =} =} =}
wherebound variablesmay have to berenamedappropriately.1]-expansion is theoppositeof theusual7]-cont r action : tA ~B
--+'1
S --+1'/ S'
=}
t'
=}
t
--+1'/
.AX A . t(x) , s(t) --+1'/ s'(t), s(t) --+'1 s(t'),
(1) (2)
• This researchwas supportedby the Com monwea'l t h S c holar ship Commission in the UnitedKingdom. I would like to t hankAllenStoughton,Lincoln Wall en (my supervisor), and thereferee , for usefulc onversations and advice. 331 C. Anthony Anderson and M. Zeleny (eds.}, Logic, Meaning and Computation, 331-342 . © 2001 Kluwer Academic Publishers. Printed in the Netherlands.
332
RALPH LOADER S - ' I S'
=>
AX. S
-'I
AX. S',
where, in (1),X is not free int. To retainnormalizat ion, we make the restriction that'Tl-expansioncannotcreatenew,B-redexes; in otherwords, (1) AX. t', and (2) may not may not beappliedif theterm t is a A-abstraction be appliedif s - ' I S' is obtainedas an instanceof (1). Given a simplytypedA-termt and avaluation v defined onthefree vari of t with respectto v will bedenotedby [tJv. The ables oft, thevaluation subscriptv may be omittedif theintendedvaluation is obvious fromthe contextor ift is closed. If t[x] is a term, and a E M have theappropriate of t w.r.t. types, thenwe shallwrite[t[a]] orjustt[aJ todenotethevaluation thevaluation sendingx to a. The valuation of atermis invariant u nderboth ,B-conversionand rrexpansion. A closedterm t is said to definea E M if [tJ= a. A term t[x] is said to defineb w.r.t . a if [t [aJ] = b. The (absolute)A-deflnability problemis: Given a full andfinite model M anda member a E M T for sometypeT, is therea closedtermthatdefines a?
The relativeA -deflnability problemis: Given a modelM, againfull andfinite, a, b E M, is therea term t[x] thatdefinesb w.r.t. a? These twoproblemsarerelated as follows : LEMMA 1. The A-definability problem is decidable if and only if the relative A-definability problem is decidable. 1
Proof. Suppose thattheabsoluteproblemis decidable . We reducetherelativeproblemto this as follows : Given ai E M T i , i = 1 · . . nandb E M T , look forf E MT, - ...-Tn -T such that: 1.
f is A-definable by a closed term ¢,
If such af is found, thentheterm¢(Xl) .. . (x n ) definesb relative to theai. If a term t[XI .. . x nJ defines b relativeto al ... an, then1 and 2 above are satisfiedwith ¢ = AXI ... Xn . t, and f = [¢]. Further,thesearchforf satisfying1 and 2 is effective (by ourassumption thattheabsoluteproblemis decidable), and terminating as we can list the finitely manyobjectsof a giventype in M . Thus therelativeproblem is decidableif theabsoluteproblemis. The converseimplicationis trivialas theabsoluteproblemis a specialcase of therelative problem. -I 1 I assume a sensiblecodingof finit e A-mod els,encodingfunctionsas Godel numbers of their graphs. Notethat thislemma applies bothto the problems in a fixed model,or the problems rangingover all (full and finit e) mod els.
THE UNDECIDABILITY OF >'-DEFINABILITY
333
The statement t hatthe>.-definability p roblemis decidableis known asthe Plotkin-Statman conjecture.We will showthat,in fact, theseproblemsare undecidable , by encodingwordproblemsas relative >.-definability problems. 2. ENCODING WORD PROBLEMS
We willencode word problems (specifically,s emi-Thue systems) over the alphabet{A, B} as members of themodel M with(atleast)seven elements {A, B, L, R, Y, N, *} atgroundtype. Given a wordW o and rulesC, --> Di, i = 1· .. n, wheretheC, and the D, are words, aderivationis a sequenceof wordsW o, . . . , W n such thateach Wj+l is obtainedfrom Wj by replacinga sub-wordC, by Di. The word W is derivableif it isthelastword of somederivation. The wordproblemswe considerare as follows: Given initial an word W o, andsome rulesC, --> D i , i = 1· . . n, can we derive a word W?
C, --> D, such that the problem of determining if a word W can be derived is undecidable.
PROPOSITION 2. There is a word W o and rules
Proof. Davis (1977) encodes auniversal T uringmachineas a semi-ThuesysPi and a word Wo such thatthe problem tem so as to give a set of rules "Givena wordW, can W o be derivedfrom W?" is undecidable.By reversing alltherules ofthis problem, we get theproposition,exceptwith alarger -j alphabet.It is easy toencodesuch aproblemin thealphabet{A, B}.
We nowencodewordsandrules aso bjectsin M. A word W of lengthn will be codedby an objectrW 1 of type 0 -> ... -> 0 -> o. If words C and D have ~
n
lengthsm and n respectively , thentheruleC --> D will beencodedby an object'c --> D 1 of type (0 -> .. . -> 0 -> 0) -> (0 -> ... -> 0 -> 0). ~
~
m
n
The encodingof a wordW is defined bythefollowing clauses: WI. If W has c E {A, B} in theithposition(i = 1 ··· n), then
rW1 (*) . .. (*)(c) (*) . .. (*) = Y,
-------- -------i- I
W2. For i = 1 · . . n - 1,
n-i
' W 1 (*) ... (*)(L)(R) (*) . . . (*) = Y,
-------i- I
-------- n- i - l
W3 . rW 1(x I) · · · (x n ) = N unlessstatedotherwisein WI or W2.
334
RALPH LOADER
It may help ther eader'sunderstanding to considerrW 1 as answeringquestionsaboutthe wordW, with the ithparametercorrespondingto theith letter of W. W2 may thenbe thought of asencodingtheorderingof thelettersin W j withoutthis axiom, anypermutation of W would have itsencoding for ourpurposes. A-definablefrom rW 1 , which isundesirable The characteristic xS E M o -. ... _ o_o of a setS C M o x . . . x Mo is defined by
Thus Wl-3 above may bestatedas: The encodingof a wordCI characteristic of the union of the two sets
••• Cn
is the
{(~,C,~ li=l···n} n-i
i- I
and
{(~,L,R,~ I i= l·.. n-l}. i -I
n-i- l
A ruleF = C - - D, whereC and D havelengthm and n, is encodedas follows :
n
m
R2. rF 1(X{( R ,*, . .. , *)}) = X{(R,* , ... ,*)},
---...-m- l
---...--
---...-n-l
R3. rF1(X{(*, .. · ,*,L)}) = X{(*, ... ,* ,L)} , m-l
---...-n -I
R4. rF 1( rC 1) = 't» , R5. rF1(g) = x(0) in all cases not covered above . The encodingof rules is designed so that :
3. If WI = gl'" gk ChI' .. hI and W 2 = gl'" words, and F is the rule C __ D, then rW 2 1 is given by
LEMMA
gk
D hi .. . hi are
where X, ii, z and ii. arevectors of k, n, I and m distinct variables respectively .
THE UNDECIDABILITY OF A-DEFINABILITY
335
Proof. An easybuttedious verification. For example,WI by and W3, AU. ' WI1(gd (*) .. . (*)(UI) '" (Urn) (*) ... (*)
-------k-l
-------I
is thecharacteristic of the set {( *, . .. , *)} and thusby RI, f(gd (*) . . . (*) = Y.
-------k+n+I-I
4. If a word W is derivable from Wo by rules F';, then ' W 1 is A-definable from rW o1 and the ' Fi 1 .
COROLLARY
Proof. By an inductionoverthederivationof W, using the lemma. 3. FAITHFULNESS OF THE ENCODING
In thissectionwe shallfixa wordWo andrulesFi = C; --+ D i • We showthat if rW 1 is A-definable r elat ive to ' Wo1 andthe' Fi 1 , thenW is derivablefrom W o by therulesFi, and thusconcludethatthePlotkin - Statmanconjecture fails. Our proofs will be by inductionovertermsin the following form : DEFINITION 5. A preword-term is a termt of type0, containing c onstants among ' W0 1 and [-F I 1 . .. rFn 1 such that:
• t is in .a-normal , 1J-Iong form, • t is not a variable, are oftype o. • All free variablest in If a preword-terms has a closureA XI . •• X n . s thatencodes a word,t hen Xl'.. X n must all occur frees in (as by WI and W3 a function encodinga n must dependon all of its n arguments)and theorderingof word of length Xl ... X n (and thusthewordencoded)is unique (by W2 and W3). lemmas about In preparation forthemain proof, we prove twotechnical preword-terms.Althoughlemmas 6 and 7, and proposition9, are stated aboutpreword-terms,theyare really used to give informationabouttheir closures, as we wish to identifywhich closures of preword-terms encode words . It issmootherformally, however , to provethingsby inductionover prewordterms. LEMMA
orN.
6. If s is a preword-term and v any valuation, then [s]v is either Y
RALPH LOADER
336
Proof As s is in normalform and not avariable , s must be intheform K(td'" (t i ) , whereK is rW o1 or one oftherl'i 1 . That[s] E {Y, N} is now obvious fromWI -3 and RI-5. --I LEMMA
7. Let s be a preword-term:
1. If x is free in s, and v is a valuation with vex) E {A , B} such that [s)u = Y, then v (y ) = * for all yother than x free in s.
2. If [AUI .. . 'Un . s)u encodes a word, then v(x ) = free in AU. s,
* for
every variable x
Proof We prove the lemma by inductionover the preword-terms. We t heinductionhypothesis . Considerthe case whens is first derive I from
rW ol (tl ) " ·(t n ) . • If one ofthe ti is not avariable,t hen[tilu E {Y, N} by theprevious lemma, so that[s]u = N by W3, and hencethereis nothingto prove. • If all oft het; are variabl es, thenI is immediateby WI-3.
Suppose s is r Pil (AU. t)(t 1 ) 1.
., • (tn)'
For anyvaluation v such that[s)u = Y:
Eachof theti is a variable,as else[ti)u E {Y, N} and [s]u = N.
ii. By R5, t cannotbe a variable , and thust is a preword-t erm. iii. By R5, (AU. t)u ::J x(0) . We can now verify I for this s: • If x is one ofthevariabl es tj , thenexaminingRI-5, theonly way we can havev(x) E {A, B} , [s)u = Y, is if [AU . tl u = re i 1 (so thatby of theword Di, and R4, rPil ([AU. t]u) = rDil), vex) is thejthletter V(tk) = * for k ::J j. Now by theinductionhypothesis2 fort , v(z) = * for allvariablesz free inAU . t also.
• Suppose thevariablex is free inAU .t, and vex) E {A,B}, [s)u = Y. By iii thereis a valuation v' with v'(y) = v(y) for ally free inAU.t and [tlu' = Y. Applyingtheinductionhypothesisto t and v', we see thatv(y) = v'(y) = * for eachy ::J x free inAU.t , and that[AU. t]v = X{(* · · · *)}, so by RI, we must haveV(tk) = * for eachtk also. We now derive 2 from1. If [AU. s)v encodesa word, theneitherfor c= A or for c= B we have v' by Define avaluation
• v'(y) = v(y) for y free in AU . s,
THE UNDECIDABILITY OF A-DEFINABILITY
• v'(ur) = c and V'(Ui) =
* fori =
337
2 ·· -n ,
so that[s]v- = [Au.s)v(c)(*)· ··(*) = Y, and thusby 1 we havev(x) -1 v'(x) = * for anyvariablex free inAU . s. COROLLARY
8. If a preuiord-ierm s has a closure AU.s that satisfies
whenever rW1(XI ) ' " (x n ) = Y
for some word W, then each of the variables U occurs free at most once in s. Proof. Suppose thata variableUi occursmore thanonce in s, so thatwe can find apreword-termtry, z] such thatboth y and z occurfree intry, z], and s is the term t[Ui' ud. Now by part1 of thelemma above, [t)v = N wheneverv(y) = v(z) E {A , B}, so that[s]v = N wheneverV(Ui) E {A, B} . Butby thehypothesisof thelemma, and WI, we must have avaluation v withV(Ui) E {A, B} and v(s) = Y, a contradiction. -1 PROPOSITION
9. If a closure AU.s of a preword-term s satisfies
whenever rW-1 (XI )'" (x n ) = Y for some word W, then the word W is derivable, and [Au. s] = ' W -'.2 Proof. The proofis by inductionon s. If s is rWol(t r). . . (tn), thenthe t, must be distinctvariablesby WI-3, and AU . s must encodetheword Woo If sis rpi l(AXI" " , Xm . t )(t r) · · · (tn), thenby R4, R5 and WI, allthe t, must be distinctvariables.Then by W2 and RI-5, theclosureof s that satisfiestheconditionof thepropositionmust be
for some sequencesU and ii of variables. Let thelengthsof U and ii be k and I respectively.By corollary 8, none oftheti occursfree inAX . t , so that t' = AU, X, ii . t is a closedterm. The inductionpredicatefors will follow from theinductionhypothesisand lemma 3 if we showthat
[t'](Xr)· ·· (x n)
=Y
wheneverrW 'l (x r) · · · (x n)
=Y
(3)
2The readerwill nodoubtnoticethatthispropositionis stronger t hanneeded. This is a - the author's good exampleof savingwork bychoosingan inductionpredicatecarefully originalproofof whatis actually needed( "if [.xu. s] = rW' , then W is derivable ") required severalpages of tedioustechnicallemmas similar to lemma 7 in orderto make a proof virtually identicalt o theone herework.
RALPH LOADER
338
for some wordW' whosesub-wordatpositionsk+l ·· · k+m is Gi . For each j= 1 ··· k , thereis a j E {A, B} such that s' (*) . . . (* )(aj) (*) . . . (*) = Y ,
----....-.-- ----....-.-k-j+n+l j- I
andso by Rl and R5 we must have
(4)
t' (*) . . . (*)(aj) (*) . . . (*) = Y
----....-.-- ----....-.-k-j+m+l j - I
also. Similarly, for= j 1 ... l, thereis bj E {A, B} such that
-------
s' (*) . . . (*)(b j ) (*) . .. (*) = Y,
----....-.-k+n+j-I
l-j
so by Rl andR5 we have
-------
(5)
t' (*) . .. (*)(bj ) (*) . . . (*) = Y. k+m+ j - I
----....-.-l-j
We will showthat(3) is satisfied whenW' is the wordal. .. ak C, b, ... bi. For one of c= A or c= B , we must have
= Y. ----....-.-----....-.-k n-I+l
s' (*) . . . (* )(c)( *) . . . (*)
InspectingRI-5, this can onlyoccurif AX. t[u,v := *] encodes the word Gi , and c isthefirstletter of Ds, so thatR4 applies. Thus for j = 1 · · · m, we have (6) t' (*) . . . (*)(Cj) (*) . . . (*) = Y
----....-.-----....-.-k+j -l m-j+l
where Cj is thejthletter of the wordGi • Similarlyto the above, for j= 1·· · l+m +k - l , t' (*) . . . (*)(L)(R) (*) ... (*) = Y.
----....-.-j- I
----....-.--
l+m+k-j-l
Togetherwith (4), (5)and (6), this gives (3) withW' as required. THEOREM
=
al · · · a k Gi bl· · · bl
--I
10. The A-definability problem is undecidable.
Proof. By lemma 1, it suffices to show t hattherelative problemis undecidable. LetW o and F, be a word and set of rules such thatthe problemof
THE UNDECIDABILITY OF A-DEFINABILITY
339
whetheror not a given word is derivable undecidable is , as given by proposition2. As ourencodingis effective, it now suffices to show thatrW 1 is 1 1 W is derivable from Wo definablerelative to rW o and therF; if and only if by therulesF;. One directionis providedby corollary 4. Fortheotherdirection, suppose thatrW 1 = WW0 1, rF l 1 , .. . , rF n lj. By takingthe /1-normal, rr-long form we may assumethatt is theclosure of pareword-term and hence apply thepreviousproposition. -l The proofgiven is notoptimalin the sensethatthe size oft hemodel used t hatdefinability ofobjectsofranktwo may be reduced . It is however known or less is decidable, a decision procedurebeing given by logical relations , as is definability in themodel withjustone elementateachtype (this latter case isjustthedecidabilityof intuitionistic logic). Inparticular, theproofs given are opt imalwithrespectto therankof theobjectsconcerned . Recently JungandTiuryn(1993) proved apartial decidability resultby restricting the numberof boundvariables in ,ax-term,by usingmodificationsof thenotion of logical relation . 4. THE FAILURE OF LOGICAL RELATIONS
One ofthemain reasons for formulating t henotionof logical relations was to usethemto characterize ,x-definablility. For example, Plotkin(1980) uses a Kripke logical relation to characterize definability in infinite models . However, as will be shown in this section,logicalrelations must fail to give such acharacterization in finite models, as otherwisethis would yield a deci sion procedurefordefinability . As ourproofofundecidability is constructive , we could in principle extractan exampleof this failure from the proof , by diagonalizing the appropriatealgorithm . Insteadwe shallconstruct a reasonablysimple exampleof how logical relations fail tocharacterize relative definability . DEFINITION 11. Given a modelM of theA-calculus , an n-arylogicalrelationis a collection of relationsRA C M,4 for eachtype A, such that (ft, .. ·, In) E RA ..... B if and only if
Given a Kripke frameK (i.e. a partialorder (K, s» , an n-ary Kripke logicalrelation over K is a collection of relations RA,i C M,4 fori E K and A a type, suchthat
• RA,; C RA,j wheneveri ::; j, • (ft ,· .. , In) E RA.....B,i if andonly if(ft (Xl),' . . ,In (X n )) E RB ,j wheneveri ::; j and (Xl,... , X n ) E HA,j.
340
RALPH LOADER
Notethata (Kripke) logicalrelation is completely d eterminedby itsbehavior atgroundtype. We say thata E M A is invariant in a (Kripke) logical i E K). relation R if (a, . . . ,a) E RA «a, .. . , a) E RA ,i for all The followinglemma is originally from Plotkin(1973), and is easilyproved by inductionoverterms. LEMMA 12. Let R be a logical relation, t be a term and Vi (i = 1··· n) be valuations such that (VI (x), ... , vn(x)) E R for each variable x free in t. Then
([f]u» ... , [t]Vn)
R also.
E
13. If a is A-definable (relative to bl . . . bm ), then a is invariant in every logical relation R (such that the b, are invariant) .
COROLLARY
These also holdusing Kripke logicalrelations . For classicalratherthan Kripke logicalrelations, thefollowing is straightforward. 14. As a predicate of a, A, M and n , "a E M A is invariant in every n-ary Kripke logical relation" is decidable. There is a member of our seven-element model which is invariant under every Kripke logical relation, but not A-definable.
PROPOSITION
Proof. Given M andn , define to to be
{
IK
R t
is a Kripkeframe, i E K and R is an } n-aryKripke logicalrelation over K
{ RB ,d B :Type, andset R ~ R' iffRB c R'a for whereR, means thecollection alltypes B. Now we can define a logical relation R i over by x E Rk,R i iff x E RB. Obviously,a E MA is invariant in R if andonlyif a is invariant in everyn-aryKripke logicalrelation. Fixing a E M A , we nowreduceKi to a finiteKripke frame by amethod similarto thatof filtrations in modal logic (d. Chellas 1980). Define an equivalencerelation= on K i .by R = R' iff RB = R'a for allsub-types B of A. Let K A be thequotientK i I =, and for i, i' E KAput i ~ i' iff thereare REi , R' E i' such thatR ~ R' . Now letR A be theuniquenaryKripke logicalrelation s uchthatR:'R /= = R o for eachequivalence class (RI=) E KA. It is now easy to verify byinductionthatfor anysub-typeB of A , and (RI=) E K A , we have R~ ,R/= = R B so thata E MA is invariant in R A iff a is invariantin R i iff a isinvariantin everyn-aryKripke logical relation. Further,K A is finite, asthe equivalence class ofR E K i is determinedby thecollection (R B : B is a sub-typeof A) , andthusthesize IKAI is bounded by N = TIB IMBln whereB rangesoverthesub-typesof A , which isrecursive in everyn-aryKripke logicalrelation if in A, M and n. Thus a is invariant
«:
THE UNDECIDABILITY OF -X-DEFINABILITY
341
and only if it invariant is in everyn-aryKripke logical relation over a Kripke . frame of sizeatmost N. This showsthatthe givenpredicateis recursive ion" is It followsthat"a E MA is invariantin everyKripke logicalrelat co-r.e. in a, A andM . -X-definability is obviously .e., r andso cannotbe co-r.e. implies invariance, as this would c ontradict theorem10. Since A-definability the converse implicationmust fail, sothatthereis a member of the seven element model t hatis invariant b utnot definable. --j We finish byconstructing an example where relativeinvarianceholdsbut relative-X-definability fails.The example is given bytheencodingof a semiThue system in ourseven element model. For n 2: 1, letW n be the word consisting ofn letterA 's. Let F I be theruleA ---> AAA and F2 be the WI if and only ifn is odd. ruleAAA ---> A , so thatW n is derivable from However , we will seet hat(theencoding of)W 2 (in fact anyW k) is invariant in any logical relation such thatF I , F 2 and WI are. PROPOSITION
1. If k
15. Let R be an n-ary (Kripke) logical relation.
> 3n and
rW k_ I 1 is invariant in R , then so is rWk1 .
2. If rF I 1 , rF2 1 and f WI 1 are invariant in R , then so is rW 2 1 • But rW2 1 is not -X-definable from rF I 1, r F2 1 and rW I 1 . Proof. We shallassume R is a classical logical relation, although the proof works for Kripkerelations also. For the first part,suppose thatk > 3n, and rWk_I 1 is invariant in R . We showthatrWk 1 is invariant in R also. Consider suchthat(xl , .· . , xi) E R o fori = 1 . .. k, We must show thatalso
xi
(7)
For eachj = 1 . . . n , we shall choose a set COJ{I, . . . , k} withatmost three members as follows :
xi
• If thereis io E {I, ... ,n-2} such that o = L, X io+ 2 = Rand xt = for all o theri = 1 ·· · n, then let OJ = {io, io+1 , i o+2} .
*
xi =I- *, thenlet
• If thereare morethanthreei E {I, . . . , n} such that OJ be theset containingt hefirstthreesuch i ,
• Otherwiselet OJ= {i
I xi
=I-
* }.
The OJ have been chosen so thatif i E {I , . .. , k} - OJ, then
342
RALPH LOADER
Since k > 3n, and theCj haveatmost threemembers, thereis i E {I, . . . , k} thatis not in any of theCj • Now
rW k_ 1 1 is invariantin R, and so by (8) we seethat(7) holds also . For thesecondpartof theproposition,suppose thatrF 1 I , rF2 1 and rW 1 1 areinvariant in R . Take an evennumberk > 3n. Now rW k _ 1 -1 is derivable as k-1 is odd, sothatby corollary 4, rW k _ 1 1 is definable,andthusinvariant by corollary .13Now by thefirstpartrWk1 is invariant also. rW 2 1 is derivable from rWk1 by therulerF2 1 , so thatsimilarlyrW2 1 is invariant.T hatrW 2 1 is not definable is obvious from proposition9. -j
as
REFERENCES
Chellas, .BF. 1980 Modal logic: An introduction, Cambridge UniversityPress, Cambridge, England . Davis, M. 1977 Unsolvableproblems, The handbook of mathematical logic (J . Barwise, editor),North-Holland, Amsterdam. Jung, A., andJ. Tiuryn 1993 A new characterizat ion of lambdadefinability , Typed lambda calculi and applications, LectureNotes in Computer Science, vol. 664, SpringerVerlag,Berlin,pp. 245-257. Plotkin , G. D. 1973 A-definabilityand logicalrelations , Universityof Edinburgh,School of ArtificialIntelligence , MemorandumSAI-RM-4 (October). 1980 A-definability in thefulltype hierarchy,C ombinatory logic, lambda calculus and formalism (J . P. Seldin and J.R. Hindley,editors),Academic Press, New York. Statman , R. 1982 Completeness,invariance,a nd A-definability , The Journal of Symbolic Logic, vol. 47, no. 1, pp. 17-26. Wolfram, D. A. 1993 The clausal theory of types, CambridgeTractsin Theoretical Computer Science, vol. 21,C ambridgeUniversityPress, Cambridge, England .
PER MARTIN-LOF
A CONSTRUCTION OF THE PROVABLE WELLORDERINGS OF THE THEORY OF SPECIES* Dedicated to the memory of Alonzo Church.
1
Introduction .
1.1 It seems to be very difficultconstruct to a "natural" systemofordinal notationswhich would play therole fort hetheoryof species thateo plays for number theoryand Takeuti'sordinaldiagramsof finiteorderplay for the theoryof iteratedinductivedefinitions. However if one has the more modestaim of: (1) generating certainsyntactical objectsto be called ord inal notations , (2) defining a recursively enumerablepredecessorrelation between theordinalnotations,(3) proving in thetheoryof species thateach ordinal , and (4) showing notation is accessible with respecttothepredecessorrelation theunprovability in thetheoryof species oftheuniversal s tatement t hatall ordinaln otations are accessible , thenquitea simplesolutioncan beobtained as follows. 1.2 Considerthe system of terms introducedby Howard (1963) which containsamongotherthingsa typesymbol 2 for the t ypeofBrouwerordinals. Extendthis system a. la Girard (1970) by addingtype variablesand the term formation.Let the closed terms of type associatedrules of type and 2 in theextendedsystem serve asordinalnotations . Of two closedterms of type 2 say thatone precedestheotherif theBrouwerordinaldenoted by the one isintensionally a predecessorof the Brouwer ordinaldenotedby theother . This predecessorrelationis recursivelyenumerable . Provethe t erm of type 2 with respectto thepredecessor accessibilityof each closed relationin thetheoryof species by themethodof computability . Finally , wellordering of thetheoryof species, a closed show that,given a provable term of type 2 can be found whose associatedwellordering dominatesthe of given one.The lastgoal can be achieved by means of a kindrealizability interpretation which may be of someindependentinterest . At least,it can , as establishedby Girard be used to give an a lternative proofof thefactthat
* Postscript, Nov. 1997. Accordingto my memory, which agrees with thatof JeanYves Girard, this paper was writtenin September 1970, just beforeI began workingon type theory.It had some circulation a tthetime, but was neverpublis hed. int uitionist ic I have correctedthespellingof "int ent iona l(ly)" "intensional(ly)" to . Also, the styleof therefer ences has been changedby theeditorsso as to a gree withtherest ofthevolume. Otherwise,this printedversionagreeswiththeoriginaltypescript. 343 C. Anthony Anderson and M. Zeleny (eds.), Logic, Meaning and Computation, 343-351. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.
PER MARTIN-LOF
344
(1970), everyprovablyrecursive function thetheory of of species is definable in his system. 2
A simplified formulation of Girard's systemof terms.
2.1
Type variables : These will bedenotedby €, TI, . . . .
2.2
Types.
2.2.1
A typevariableis a type.
2.2.2
If a and r are types, thena
2.2.3
If a(€) is a type, thenso is TI€a(€) .
---+
r is a type.
2.3 For eachtype we allow ourselves introduce to as many variables x, y, . .. of thattype as we please. If it is absolutely necessary , we shall . indicatethetypeof avariableby a superscript 2.4
Terms.
2.4.1
A variableof type a is a termof type a.
2.4.2 If x is a variableof type a and a(x) a termof type r, thenAxa(x) is a termof type a ---+ r. 2.4.3 If a(€) is a termof type a(~) and € does notoccurfree inthetypeof a variablein a(€), thenA~a(€) is a termof type TI~a(~).
If a and b are termsof type a termof type r.
2.4.4
---+
r and a, respectively,thenab is a
If a is a termof type TI€a(€) and r is a type, thenar is a termof type a(T) .
2.4.5 2.5
Rules ofcontraction:
ha(x)b contra(b), A~a(~)T contre fr). A terma reduces to a t ermb (abbreviated a redb) if b can beobtainedfrom a by repeatedcontractions of subterms.
2.6
Pairing: For any twot ypes a and T put
a x
T
= TI~((a ---+ r ---+
0
---+
~),
and definethecorresponding pairingand projectionfunctionsby
(a,b) = A~AXIT-+T-+{xab, Pc = ca(Ax IT AyTX), Qc = cr(AXIT AyT y).
THE THEORY OF SPECIES
345
Then
P(a, b) reda,
Q(a, b) redb
as desired. 2.7 Natural numbers: The type ofthenatural numbers,which we shall denoteby 1, can be defined by p utting 1 = n~(~ If
01 =
81 =
-+ (~ -+ ~) -+ ~) .
)..~)..x{)..y{-+{x,
)..ZI)..~)..X{)..Y{-+{Y(z~xy)
are used torepresentthenatural numberzero andthesuccessor function, respectively,thenwe have access to o rdinaryrecursion, because 01rab reda,
8 1 crab redb(crab) .
In thepresence of apairingoperationit is, of course, sufficient to have recursionin this simple formwithouta side argument . Brouwerordinals : In analogy with thewaythenatural numbers were represented, thetype2 of Brouwerordinals,the zero element (Urelement) of type 2 and thegeneralized successoroperationof type (1 -+ 2) -+ 2 can be defined byputting : 2.8
2 = n~(~ -+ ((1 -+~) -+~) -+~), O2 = )..~)..X{Ay(I-+0-+{X,
82 =
)..zl-+2AOX~)..y(I-+{)-+{y(AWZW~XY) ·
The schemaof transfinite recursion in the form 82crab redb(Awcwrab)
thenbecomes satisfied. 2.9 Although, as hasjustbeen shown,thetypes1 and2 can be defined, we shall prefer to treatt hemas primitivein the sequel. This makes it necessary to addthesix constants : 01 of type 1, 8 1 of type 1 -+ 1, R 1 of typen~(~ -+ (~-+ 0 -+ 1 -+ ~), 02 of type2, 82 of type (1 -+ 2) -+ 2, R 2 of type n~(~ -+ ((1-+~) -+~) -+ 2 -+ ~) ;
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PER MARTIN-LOF
andthe fourcontraction rules: RlrabO l contra,
R 1 rabi S, e) contrb(R l rabe), RzrabOz contra, Rzrab(Sze) contrb(>.wRzrab(cw)) to thebasic system. 2.9.1
a A termof the formSl( ' " Sl(SlOd. . . ) will be callednumeral.
3
Definition of the equalityand predecessorrelations .
3.1 The terms a and b are said to beintensionally equal and we write a = b if thereis a term e such thata rede and bred e. The transitivity of theequalityrelation is guaranteed by thewellknown propertyof Church and Rosser,thatis, thatif a redb and a rede thenthereis a term d such thatb redd and e redd. The proof of thisp ropertyfor thesystem we are consideringfollows the usual pattern . 3.2 Let a and b be two closed t ermsof type2 and n a numeral. We shall say thata is thenthpredecessor of b if thereis a closedterme oftype1 ...... 2 and anumeraln such thata = en and b = Szc- If a is thenthpredecessor of b for somen, thena is a predecessor of b. Clearly,thepredecessorrelation so defined is recursively enumerable. 4 The accessibility of a closed termoftype2 withrespecttothepredecessorrelation is defined by the following generalized inductivedefinition . 4.1
If a = Oz, thena is accessible .
4.2
If a has predecessors which are all accessible , thena is accessible .
5 Definition of computability : For the sake of simplicity, we shall define whatit means for atermto becomputableonly if it is clo sed, and this we do by inductionon its type. 5.1 Type 1: As in Tait 1967, thatis, a closedtermoftype1 is computable if and only if it reduces to a numer al. 5.2
Type 2.
5.2.1
Oz is computable .
5.2.2 If ab is computablefor all c omputableterms b of type 1, thenSza is computable . 5.2.3
If a reduces toband b is computable,t hena is computable .
5.3
Type u""" r: As in Tait 1967.
THE THEORY OF SPECIES 5.4
Type
n~(7(~):
347
As in Girard 1970.
6 For every closed termit can be proved in the theoryof species that it is computable.Almost all of theproofcan be takenfrom Tait 1967 and Girard 1970. The only novelty that is we have to prove the computability of 02, 8 2 and R 2 • 6.1
O2 is computableby 5.2.1.
6.2 Suppose thata is a computableterm of type 1 ---+ 2. By 5.3 this means thatab is computablefor allc omputableterms b of type 1. Using 5.2.2 we can conclude t hat8 2 a is computable . But a was arbitrary a nd, . consequently , 8 2 is computable 6.3 In orderto showthatR 2 is computableit suffices to show t hatif r is a type, aT a computabilitypredicateof type r , a a computabletermof type r , b a computabletermof type (1 ---+ r) ---+ randc a computableterm of type 2,thenR2rabc is computable(with respectto aT) ' We do this by . inductionon theproofthatc is computable 6.3.1
R 2rab0 2 is computablebecauseit reduces to a which iscomputable.
6.3.2 R 2rab(82c) reduces to b(AwR2rab(cw)) and, since b was assumed thatAwR2rab(cw) is computable,t hat computable,it thussuffices to show is, that(AwR 2rab( cw)) d is computablefor all c omputabletermsd oftype 1. Now, (AwR 2rab(cw))d is indeedcomputablebecauseit reduces to R 2rab(cd) which is computabl e by inductionhypothesis.
6.3.3 If c reduces tod, thenR 2rabc reduces toR 2rabd. By inductionhytermis computableand, consequently, so is the former . pothesisthelatter 7
Consequencesofcomputability.
7.1 Comparisonof thedefinitionof accessibilityandthedefinition of comThus, putabilityfor closedtermsof type 2 shows thattheyare equivalent. we have proved the accessibilityof every closed t ermof type 2 in thetheory of species. If thedefinitionof computabilityis extendedso as to coverterms 7.2 which are not necessarily closed(thevariablesfunctioning essentially as constants),t henit can be provedthatallcomputabletermsare normalizable. Moreover, it does not requiremuch extralaborto showthateach term is normalizable.T his has computablein theextendedsense and hence also importantconsequences.
7.2.1 For anypair of terms a and b it can bedecidedwhetheror nota = b by checkingwhetheror nottheirnormalforms areidentical(neglecting differences in the namingof theirboundvariables) .
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PER MARTIN-LOF
7.2.2 If a is a closedtermof type 2, theneithera = 02 or a = S2C forsome closed term c of type 1 -+ 2. This is seen by reducinga to normalform, becausea closednormaltermof type 2 must be eitherO2 or oftheform S2C. 7.2.3 If a and b areclosedtermsof type 2 and n is a numeral,t henit can be decidedwhetheror nota is thenthpredecessorof b. Indeed, a is thenth predecessorof b if and onlyif thenormalform ofb is of theform S2C and a=cn. 8 Ourfinal goal is to show thattheuniversal s tatement "all closed t erms of type 2 areaccessible"is unprovablein thetheoryof species. This will be of thetheoryofspecies can achievedby showingthata provablewellordering alwaysbe dominatedby thewellordering associatedwitha closedterm of type 2.
8.1 The language ofthetheoryofspecies which weshallconsidercontains variablesforindividualsa ndspecies andthefollowing c onstants. 8.1.1 Functionconstants:Eachfunctionconstant denotesa primitiverecursive function . Thereshouldatleastbe a constant 0 forthenatural number zero, aconstant s forthesuccessorfunctionand a binaryfunctionconstant f . Thinkof fxy as theythpredecessorof x . 8.1.2 Predicatec onstants:= forequality a nda unarypredicateconstant A foraccessibility . 8.1.3 Logicalconstants:- + for implicationand 1\ for universalq uantificationof firstandsecondorder. 8.2
Axioms and rules ofinference.
8.2.1
The definingequationsof theprimitiverecursivefunctions .
8.2.2
Peano'sthirdand fourthaxioms:
I\xl\y(sx
8.2.3
= sy -+
1\ x 1\ y 1\ X(x =
y
--+
Xx
--+
Xy).
Induction :
I\X(XO
8.2.5
1\ x(O = sx --+ 0 = sO).
= y),
Equalityaxioms:
I\x(x = x) ,
8.2.4
x
-+
I\x(Xx
-+
Xsx)
-+
1\ xXx) .
Axioms of accessibility :
I\X(XO
--+
AO, I\x(l\yAfxy -+ Ax), I\x(l\yXfxy --+ Xx) -+ I\x(Ax
--+
Xx)).
THE THEORY OF SPECIES 8.2.6
349
Rules of inference for second orderminimal logic :
G
F-+G G
F(x) I\xF(x)
I\xF(x) F(t)
F(X) I\XF(X)
I\XF(X)
F
F(T)
8.3 Definition oft hetypeassociatedwitha formula or species term: By inductionon its logical complexity, 8.3.1
The type associatedwitht = u is l.
8.3.2
The typeassociatedwithAt is 2.
8.3.3 The type associatedwith Xt is { where{ is a type variableuniquely associatedwith X . 8.3.4 CT -+ T
If CT and T are thetypesassociatedwith F and G, respectively, then is thetypeassociatedwithF -+ G.
8.3.5 If CT is thetypeassociatedwithF(x) , then1 -+ CT is thetypeassociated with1\ xF(x). 8.3.6 If CT({) is thetype associatedwith F(X), thenII{CT({) is thetype associatedwith1\ X F(X). 8.3.7
The type associatedwith F(x) is also the type associatedwith
AxF(x). 9 Realizability predicates : Let T be a closednaryspecies termand T its associatedtype. A relation Q:T between a sequence n ofclosedindividual termsand a closedtermof type T will be called ealizability a r predicateof conditions . type T provided itsatisfiesthefollowing two 9.1
Ift= u and Q:T(t,a), thenQ:T(u,a).
9.2
If a redbandQ:T( t, b), thenQ:T( t, a) .
10 Definition ofrealizability: Let T(X) be a species term which containsno freeindividualvariablesand all of whose free species variablesare conta ined in thesequenceX = Xl, . .. , X n and letaT = Q:T" . . . , Q:T" be a
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PER MARTIN-LOF
corresponding sequenceof realizability predicates.By inductionon thenumber of logical signs T(X) in we shall definecertain a realizability predicateof d enotedPT(X)(or). If F is a closedformula , then typeT(T) which will be PF(a) may be read"a realizesF " or "a is a proofconstant for F" .
=u
10.1
If t
10.2
The definitionof PAt (0 r) is by generalized i nduction.
and a red 01 , thenPt=u(OT )(a).
10.2.1 If t = 0, thenPAt(or)(02)' 10.2.2 If t f. 0 and PA/tu(oT)(au) for all closed individualterms u, then PAt(or)(S2 a). 10.2.3 If a reducesto band PAt(or)(b), thenPAt(or)(a). 10.3
If PTi(t,a), thenpXit(or)(a).
10.4 If PF(x)(or)(b) implies pc(x)(or)(ab) for allb, then PF(X)_C(X )(OT )(a). 10.5 If PF(t,X) (or)(at) for all closed individualterms t, then PI\xF(x,X)(OT )(a). 10.6 If PF(X,X)(aT , 0T )(aT) for all T andaT whereT is thetypeassociated withT , thenPl\xF(X,X)(oT)(a) . 10.7
If PF(t ,X)(oT)(a) , thenP>.xF(x,X)(or)(t,a).
11
Propertiesof PT(X)(or).
11.1 PT(x)(or) is a realizability predicateof typeT(T) . To establishthis thefollowing two properties . we have to verify 11.1.1 If t = u and PF(t,X)(oT)(a) , thenPF(u,X)(or)(a) . Proofby induction on then umber of logical signs intheformulaF(x, X) . 11.1.2 If a redband PF(X)(OT )(b), thenPF(X)(OT )(a). Proofby induction F(X). on thenumberof logical signs in the formula 11.2
Substitution p roperty:
The verification of thispropertyis immediateby inductionon thenumberof logical signs in t heformulaF(X , X). 12 Let F be a closed formula and (J its associatedtype. Then, from a proof ofF we canfind a closedterma of type (J such thatPF(a) is provable in thetheoryof species.
THE THEORY OF SPECIES
351
12.1 The proofof this proceedsby inductionon thelengthof thegiven once a strongerinduction derivationof F and is entirelystraightforward hypothesishas been made. For example, theaxiom ofinductionis handled by means of R 1 and the axioms of accessibilityby means of O2 , 82 and R 2 , respectively.Also, thesubstitut ion propertywas establishedbecause it is preciselywhatone needs inorderto handlethesecond orderrule of /\-elimination, recursively 13 We can now piecethingstogether.Let < be an arbitrary enumerableb inaryrelation.Find a primitiverecursivepredicateP suchthat y
< x ......
VzPxyz.
Let (y, z) be a primitiverecursivepairingfunctionlike(y, z) = 2Y(2z + 1)-1 anddefine aprimitiverecursivebinaryfunction/ by putting sy if Pxyz, /sx(y, z) = { 0 otherwise,
and /O(y, z) = 0, say. Then it is easily proved in t hetheoryof species that, for allx , x is accessiblewithrespectto < if andonly ifsx is accessiblefrom o withrespectto /. Moreover, thewellordering determinedby an accessible closedindividualterm t withrespectto < is dominatedby thewellordering determinedby st withrespectto[ , Suppose nowthatit can beprovedin the theoryof species thattheclosedindividualterm t is accessiblewithrespect to / fromtheinitialelementO. We can thenfind a closedterm a of type 2 such thatPAt(a). But PAt(a) means precisely thatthewellorder ing deter determin ed mined by t with respect to / is isomorphic to thewellordering by a. Therefore , everyprovablewellordering of thetheoryof species can be dominated by thewellordering determinedby a closedtermof type 2, and thisis allthatthereremainedfor us to prove. REFERENCES
Girard, J.-Y. 1970
Une extensiondu systeme de fonctionnelles recursivesde GOdelet son applicationaux fondementsde l'analyse,to appear. Howard,W . A. 1963 Transfiniteinductionand transfinite recursion , Reports of the sem inar on the foundations of analysis, sec. VI, vol. 2,Stanford . Tait, W. W . 1967 Intensional interpretations of funct ionalsof finite ype, t Th e Journal of Symbolic Logic, vol. 32, pp. 198-212.
COLIN MCLARTY
SEMANTICS FOR FIRST AND HIGHER ORDER REALIZABILITY
Abstract.F irstorder Kleenerealizability is given a semantic interpretation, includingarithmeticand other types. These types extendat a stroketo full higherorderintuitionistic logic. They arealso usefult hemselv es, e.g., as models , for which see Asperti and Longo 1991 and papers on PERs for lambdacalculi and polymorphism (IEEE 1990) . This semanticsis simpler and more explicitthanin Hyland 1982, giving the logicalcontentof Freyd, Carboni, and Scedrov's (1988) assemblies. Category theoryhere isonlyan organizingdevice and can be skipped, except in Lemmas 11-12 verifyingthehigherorderlogic. We verify some higherorderconstructive recursiveanalysis , and prove two metatheoremsby generalizing the construction to othertoposes.
1. ASSEMBLIES
We enumeratet hepartialrecursivefunctionsand writen. m for thevalue of then-thfunctionappliedto m. Of coursen. m may be undefined.We use a surjectiverecursivepairingfunction( , ) with recursiveprojections,l and r, read "left"and "right". So we havelen, m) = nandr(n, m) = m, and n = (In, rn). An assembly (A) consistsof a set A calledthe carrierand an infinite A. Each An is sequenceof its subsets (AI, A 2 , ' ..) whose union is all of calledthen-thcaucus. An arrow betweenassemblies, writtenI : (A) --+ (B) , is a functionbetweenthecarriersI : A --+ B with atleastone modulusthatis a number e such thatwhenevera E An thene. n is defined and [a E Be. n- Arrowscompose, withthecompositeof modulias a modulusfor thecomposite. So we have a well defined category . Assembliesappearin Freud, Carboni, and Scedrov 1988, Preyd and Scedrov 1990, and McLarty 1992, andas w-sets in Asperti and Longo 1991. Let theassembly(1) have asingleton set 1 ascarrier,a nd thesame 1 for numbers every caucus . Thereis an assembly (N) with N theset of natural and each caucusN n thesingleton{n}. An arrowI : (N) --+ (N) must be recursive,with its own codes as moduli. Categorists note(1) is terminala nd (N) a natural numberobject. Assemblies (A) and (B) have aproduct(A x B) whose n-thcaucusis Aln X B rn. The projections(A x B) --+ (A) and (A x B) --+ (B) are well arrowsfrom (A) to (B) as defined. The exponential( B A ) has the set of carrier , and then-thcaucuscontainsthosearrowswithn as modulus. E.g., 353 C. Anthony Anderson and M. Zeleny (eds.), Logic, Meaning and Computation, 353-363.
© ZOO) Kluwer Academic Publishers. Printed in the Netherlands .
354
COLIN MCLARTY
thecarrierof (I~N) is thesetofrecursivefunctions,each one inthosecaucuses which code it.T he categoryis cartesianclosed. A subassembly of (A) is a monic arrowI : (P) >-> (A), i.e., any arrow one-to-oneon thecarriers.T his subassemblyis containedin g: (Q) >-> (A) if thereis an arrowh : (P) -+ (Q) withgo h = I. Subassembliesareequivalent of (A) to asubset8 of thecarrier if eachcontainst heother.The restriction is thesubassembly(8) >-> (A) with carrier8 and witheach caucus8 n the intersection of 8 withAn. Given arrowsI and g bothfrom (A) to (B), the restriction (E) >-> (A) of (A) to thosex such thatf x = gx is an equalizer for I and g. The categoryhas all finite limits. 2. ASSEMBLY REALIZABILITY
The firstorderlogic ofthecategoryof assembliesis multi-sorted, witha sort for eachassembly (B). Each variableof sort(B) is a termofsort(B). For eachb : (1) -+ (B) we have atermb ofsort(B), a constant . For anyarrowI : (A) -+ (B) and term t of sort(A) thereis a term It of sort(B). Since a closedtermof sort(B) is a constant or suitablysortedarrowsappliedtoconstants, it determinesan elementof thecarrierB. Ournotation confuses closed t ermswithelements whenthecontextmakes it clear. For eachsubassembly(P) >-> (A) thereis a predicatesymbolP applicable to termsofsort(A). For eachsubassemblyof aproduct,(R) >-> (A x B) , we have a two place relation symbol R, applyingto pairs of termsof sort(A) and (B). Similarlyforn-aryrelations for anyn. For any(A) thediagonal of (A x A) to pairs (A) >-> (A x A) is the equality relation,therestriction (a, a). We write= for itsrelation symbol. We definerealizability only forsentences . For any i : (R)(A x B x C) and closedterms tA, tB, tc of sorts(A), (B), and (C) respectively , n realizes theatomicsentenceRtAtBtc iff some r inR n has i(r) = (tA , tB, tC). is obvious. Generalization to n-aryrelations variableis Let 8 and S' be sentencesa nd 8y any formula whose sole free y of sort(B). Then n realizes : 8 & Sf iffIn realizes 8and rn realizesS' , 8 V 8 f iffeither:In = 0 and rn realizesS or In = 1 and rn realizesS',
S => S' iff, for any mthatrealizes8 , n. m is defined and realizesS', '" 8 iff no m realizes8 . (Vy)8y iff, for every b in B m , n. m is definedand realizesSb. (3y)8y iff for someb in Bin, rn realizesSb.
FIRST AND HIGHER ORDER REALIZABILITY
355
REMARK 2.1. Just asking whethera sentenceis realized by somenumber gives standardtruthconditions,exceptfor V. I.e., S ::::> S' is realized iff : if S is realized so isS' . (3y.(B))Sy is realized iff someSb is. Similarly for &, V, and r-«, Notice (Vy. (B)) rvrvSy is realized iff Sb is realized for allb in B . (Constructively , throughout the "if' clauses for=> and V read "not-not- realized" .)
variablesbut x, y , and z, of sorts Let Pxyz be a formula with no free (A), (B), and (C) respectively. We write[xyz I Pxyz] for theextension of P overthatlist ofvariables.The n-thcaucus[xyz I PXYZ]n containsthose triples(a, b, c) such that(a, b, c) is in (A x B X C)ln and rn realizesPabe. The inclusion makes this subassemblyof a (A x B x C) . Even if y, for example, does notoccurin Pxyz, so b does notappear in Pabe, we define[xyz I Pxyz] as a subassemblyof (A x B x C) as above. x- ofdistinctvariablesincluding We define anextension[x- I P] for any list allthosefree inP. A subassembly(P) >-+ (A) need not be[x I Px] but is equivalent to it We write"x. (A)" or "x- . (A x B x C)" to showthesortsof variables. 3. THE LOGIC OF ASSEMBLIES
These extensionsadmit an intuitionistsequentcalculus,agreeing with the firstorderpartof topos logic (seeBell 1988 or McLarty 1992). So this interpretation agrees with the categorical one. P A sequent is an expressionr : P with r a finite set of formulas and a formula . We use a new formula "fa" , read as"false" . The extensionof fa over any list of variablesis theempty subassemblyof thecorresponding productof assemblies. Let »: listthevariablesfree inr : P . Let [x- I I'] be theintersection of in or equivalently the extensionof theextensionsover x: of all formulas r, the conjunction of formulas in r. The sequentr : P is trueif [x- : I'] is containedin [x- I PI. In thatcase, we write
r I- P. So we haveI- P ifftheextensionof P is theentireproductit is defin ed over. Calculation shows thesequentsabove anydoubleline here are trueiff the one below is. Here r is any finite set of formulas and P , P', Q, and Q' any r or Q formulas , and x any variablenot free in
r: P
and r: P'
r:P&P'
r,Q :p r:Q=>p
r, Q : P
and r, Q' : P r.ov cr . P
r ,Q : fa r . rvQ
356
COLIN MCLARTY
r :p
r ,p:Q
r . (Vx)P r, (3x)P: Q We use acutrule: If r , Q : P and r : Q are trueso is r : P, unlessthere a freevariableover theempty assembly in Q but none in r : P. Every
is sequentwith a free variableovertheempty assemblyis true. This logic does not prove the lawexcludedmiddle, of IIp V '" P". Any formula"P" implies "", '" P " buttheconverseneed not hold. 4. ON NEGATION , AND EQUIVALEN CE RELATIONS
A subassembly (C) >---+ (A) is complemented, or decidable, if the formula (Vx .(A))(Cx V ",Cx) is true. The terminologyis natural here since a subassembly of (N) is complementediff it isequivalentto the restriction of (N) to some recursiveset ofnumbers. The diagonal(N) >> (N x N) is numbers. It is not for complemented-e-i.e ., equalityis decidablefornatural (NN), becauseequalityof recursivefunctionsis notrecursively decidable. A subassembly (C) >---+ (A) is double negation closed, or d.n . closed for short,iff f- '" '" Cx => Cx .
x.(A)
So it is d.n. closediff it isequivalent to arestriction of (A). Everydecidable subassemblyis d.n. closed , as is equality on any assembly. Call a formula thelanguage in ofassemblies almost n egative if all its pr edicatesand relations are d.n. closedandits only conn ect ivesare &, =>, "', and
V.
4.1. Let P be any almost negative formula with free variables (A x . . . x K) . Construe the predicates and relat ions in P as subsets and relations of the earners A· · · K. Then [x- I P] is equivalent to the restriction of (A x . . . x K) to the set theoretic extension of P over X- .
THEOREM X- .
Proof. By thelogic of assemblies,almostnegativeformulas are .n. d closed. And everyn realiz es r- ' " S if some numberrealizesS . Thenuse Remark2.1. -j
An equivalencerelationis a relation (R) >---+ (A x A) satisfyingtheusual sequentsfor reflexivity, symmetry, and transitivity.A quotientfor (R) is definedcategorically as a coequalizer forthetwoprojections(R) -> (A) with thoseprojectionsas kernel pair. In termsof logic it is an a rrowq : (A) -> (Q) such that: f- (Rxy => qx = qy) & (qx = qy => Rxy), f- (3x. (A))qx = z.
x ,y. (A)
z . (Q)
4.2. An equivalence relation (R) on (A) has a quoti ent assembly iff it is d.n . closed.
THEOREM
FIRST AND HIGHER ORDER REALIZABILITY
357
Proof. Equalityis always d.n . closed, so given the first sequentabove, (R) is also. Conversely, if (R) is d.n. closed, it gives an equivalence relation on the set A witha projectionq : A --+ Q totheset of equivalence classes. Define (Q) by making each Qn the set of equivalence classes of members An.of Then q : (A) --+ (Q) is a quotientof (R). -j REMARK 4.3. For any function f : N --+ N, define an assembly(T) where eachnatural numberm occurs only in thecaucusT(m , fm) . The inclusion defines asubassembly(T) >-+ (N) whose doublenegationis all of(N). And f is a well defined arrow f : (T) --+ (N). I.e., every function from N to N is an arrow from some double negation dense subassemblyof (N) to (N). But (T) >-+ (N) is equivalent to all of (N) ifff is recursive.
5. ARITHMETIC IN ASSEMBLIES
Theassembly(N) with successor arrow s verifies thePeanoaxioms. By trivial realizability :
x,y. (N) z. (A)
f- "-'(0 = sx), f- sx = sy => x = y,
P(z,O), (3x)(P(z,x) => P(z , sx)) f- (3x)P(z ,x) where(P) >-+ (A x N) is any relation to (N). Alltheclassicaltheoryof primitiverecursive functions applies in assemblies, since all theproofs inKleene 1952, chapterIX, work in the logic of assemblies. Furthermore, primitiverecursive definitions are almostnegative . So the extensionof anyn-aryprimitiverecursivepredicatein assemblies is the restriction of (N x . . . x N) to thatpredicate'sset theoretic extension. This includes all graphsof primitiverecursive functions . The classicalt heoryof general recursive functions Kleene in 1952, chapter XI, works inthelogic of assemblies exceptfor thediagonala rguments . Those use excluded middle, which fails for the claim "functionf is defined for t heundecidability of thehaltingproblem. argumentn" in assemblies due to The classical p roofthatthereare non-recursive functions from t henatural numbersto themselves fails here , andin fact for assemblies, all such functions are recursive. To show this, define Kleene 's T-predicateT(e , x, y) and outputfunction U (y) in the logic of assemblies. The usualtheoremsfollow inthatlogic. So justas in the classical case, we read T(e, x, y) as "e is G6delnumber of a definition(or, e is a code for partial a recursive function) which when applied to argumentx gives a calculation with G6delnumber y", and readU (y) as y" . "thefinaloutputfrom calculation THEOREM
5.1. Church 's Thesis is true for assemblies:
f- ("If. (NN))(3e)(Vx)(3y)(T(e, x , y) & U(y)
= fx) .
358
COLIN MCLARTY
Proof. Since theT-predicateand U are primitive recursivetheyagree in extensionwiththeclassical ones. Letn,c(m) be theGodelnumberfor the completedcalculation (if thereis one) obtainedfrom definitionnumber n applied to argumentm. Church'sThesis is realized by any code for the functiontakingn to (n , h(N)) whereh(N) codes the function takingany k to (c(n, k), ((n, k, c(n, k)), (n . k, n . k))) .
This is easier to check thanto read . THEOREM
5.2. First order Markov principle. For any relation (P)
-j >-+
(N x
A) :
(Vy . (N))(Pyx V
rv
x . (A)
Pyx) , rvrv(3y)Pyx I- (3y)Pyx.
Proof. It suffices to find partial a recursivey such that:For alla in A , if k realizes(Vy)(Pya V rv Pya) and some Pra is realized , then1/Jk realizes (3y)Pya. Define ¢Jk to be thesmallesti such thatl(k. i) = 0 (so r(k. i) realizesP ia) . For 'l/Jk take(¢Jk, r(k . ¢Jk)). This uses Markov'sprinciplein the metalanguage, since constructively a realizer for rv rv(3y)Pyx only implies (3y)Pyx is not-notrealized . -j
Since (N) is isomorphic to(N x N), Markov'sprincipleextendsto two or morequantifiers 3 over (N) before a decidable formula. countsuch We formulas asalmostnegative,a ndTheorem4.1 still holds . The axiom of choice is not truein assemblies, buttheaxiom of choice from (N) to any assembly is. 5.3. For any assemblies (A) and (B) and relation (P) B x A) we have:
THEOREM
(Vy)(3x. (B))Pyxz I- (3f. (BN))(Vy)P(y , fy , z) .
>-+
(N x z , (A)
Proof. Given a in A , letn realize(Vy)(3x. (B))Pyxa. I.e., for all mthere exists b in B I(n. m) suchthatr(n. m) realizesRmba. So (assumingcountable choice in ourmetalanguage) thereis an arrowf a : (N) --+ (B), with the functiontakingm to l(n. m) as modulus,such thatr(n . m) realizesRm(fam)a. This easily gives a realizer (3f for. (BN))(Vy)P(y,fy, z) recursive inn . -j
An assembly ismodest iff each caucus An hasatmost one member. These amountto the"strictly effectiveobjects"of Hyland (1982) . An arrow between them isdeterminedby any ofits moduli. They are thequotientsof sub-objectsof (N) by closed equivalence relations and appearin othernotations asPERs (see IEEE 1990, A sperti and Longo 1991). Noticethatfor modest(B), theproofof choice from(N) needs no choice principlein the metalanguage . Ordinaryarithmetized analysisdefines the integers (IT) andrationals (iQ) as quotients ofproductsof(N) by decidableequivalence relations . The assembly
FIRST AND HIGHER ORDER REALIZABILITY
359
(QN) of sequencesof rationals, allrecursiveby Church'sThesis, has asubassembly (CS) of Cauchysequences. Hyland(1982) shows thatif werequire a certainrateof convergencefor Cauchysequences, thentheirequivalence is closed. Thus, theCauchyreals formtheassemblyof recursivereals. In fact, all thesearemodest. Hyland(1982) says more on this. His"canonically separatedobjects"amountto assemblies. 6. THE EFFECTIVE TOPOS
This construction of cff was firstsketchedcategorically in Freyd, Carboni, and Scedrov 1988. Furthergeneralities are in Freyd and Scedrov 1990 and McLarty1992. The objectsof cJj are equivalence relations of assemblies. When we take (R) >-+ (A x A) as an objectin cJj we may call it( AIR) to suggestthe quotientof (A) by (R). An arrowin cJj from (AIR) to (B I R') is given by a relation ( F) >-+ (A x B) which isequivariant for (R) and (R') in thissense: Rxy,Fyz f-- Fxz , Fxz, R' ZtlJ f-- Fxw ;
x,y. (A) Z,w . (B)
andfunctional in that : Fxz, Fxw
I-
R' zw,
I-
(3z)Fxz .
Relations(F) >-+ (A x B) and (F') >-+ (A x B) give thesame arrowfrom (AI R) to (B I R') if theyareequivalent sub-objectsof (A x B) . Given such ancJjarrow(F) andanother (G) from (BI R') to some (CI R"), therelation ( 3y. (B))(Fxy&Gyz) defines anarrow(GoF) . This is associative, and (R) is theidentityfrom (AIR) to (AIR). SO cJjis a category . THEOREM
6.1. The category cJj has all finite limits:
(i) The object (11 =) is terminal. (ii) Any (AIR) and (BIR') have a product (AxBIRxR') with RxR'(x,z) (y , w) defined by (Rxy & R'zw) .
(iii) Any arrows (F) and (G) both from (AIR) to (BIR') have an equalizer,
[z, (A) I (3y)(Fxy&Gxy)] modulo the equivalence relation on it induced by R .
Proof. The natural proofs oftheseforquotient s etsaresoundfor assemblies. --I Some resultsfitassembliesintocfJ:
360
COLIN MCLARTY
LEMMA 6.2. A relation (F)
>-+
(A x B) satisfies:
Fxy& Fx z I- y = z, I- (3y)Fxy
z , (A) y, z , (B)
in the logic of assemblies iff there is a unique arrow of assemblies f : (A) (B) su ch that I- Fxy +---> fx = y .
-+
In other words, F is equivalen t to the graph of f. Proof. By realizability, if thesequentsare true , (F) is isomorphic to (A). --j Composing withtheprojectionof (F) to (B) gives f.
So thereis a fulland faithful f unctorto cff takingeach assembly (A) to (AI =) andeacharrowto itsgraph. We treatassembliesand theirarrowsas objectsandarrowsof cffviathisfunctor. 6.3. (i) Each (R) : (A) -+ (AIR) is a quotient of the two proj ections from (R) to (A). It is their coequalizer and they are its kernel pair.
LEMMA
(ii) If (F) : (AIR) assembly.
>-+
(B) is monic in cjJ, then (AIR) is is om orphic to an
assemblies. For (ii), (F) induces a relation Proof. For (i), use the logic of between(A) and [y o(B) I (3x. (A))Fxy] equivariant for(R ) andfunctional in bothdirections , thusan isomorphism in cff. --j For anyequivalencerelation(R) equivariance as defined by
>-+
(A x A) it is easy tosee that R-
Rxy, Hyz I- Fx z
x , y. (A)
z, (B)
( H) >-+ (A x B) has thesame pullbackalong is equivalent tosayinga relation thetwoprojections(R x B) -+ (A x B) .
6.4. Relations in cff from (AIR) to an assembly (B) correspond to R-equivariant relations (H) >-+ (A x B) .
LEMMA
Proof. By Lemma 6.3(i) anyrelation from (AIR) to (B) pulls back to such an (H). Conversely, let (H) be R-equivariant and (R') theequivalence relation on theassembly (H) inducedby therelation (R) on (A) . Then (HIR') is the unique(up toequivalenc e) relation from (AIR) to (B) withpullback(H). --j
Thereis a kind oftruthvalueassembly (W) with each cau cus Wn the powersetof thenatural numbers. For asubassembly(M) >-+ (W) counting in thesets of natural numbers which includen. as tru e, let eachMn conta Given any(C) >-+ (B), define f : (B) -+ (W) with eachfb the set ofrealizers thepullbackof (M) alongf , but forCb. Then (C) >-+ (B) is equivalent to not onlyf. We say f weakly classifies (C) >-+ (B) .
FIRST AND HIGHER ORDER REALIZABILITY
361
Thereis an equivalence relation E (E) >--+ (W x W) such that : Any (f, g) : (W x W) factorsthrough(E) iff f and g weakly classify thesame subassemblyof (B) . Each En containst hosepairs (B, B') ofsetsof numbers such thatfor every m inB thevalueIn. m is defined and is in B', and for every m in B' thevaluern. m is defined and in B. Intuitively, n realizes (B => B') & (B' => B). (B) -
LEMMA 6.5. Every £jJ object (BIR') has a power object. Proof. Arrows(A) _ (W B I E B ) uniquelyclassifyrelations from (A) to (B) . A relation from any(AIR) to (B) amountsto anarrow(A) _ (WBI E B ) coequalizing t heprojections(R) - (A) . This induces anarrowfrom (AIR) to (W BI EB) classifying theoriginalrelation . So (B) has powerobject (W B IEB ) . The equalizer (B) >--+ (W B) ofthetwoinducedarrowsfrom (W B) to (W R ' ) has anequivalence relation ( E') inducedby therelation ( E B ) on (W B ) . And (BI E') is a powerobjectfor(BI R'). -l
So £jJ is a toposand has its own higherorderlogic,related to thelogic of assembliesby: LEMMA 6.6. A formula with all assembly sorts is true in the logic of £jJ ijJ it is true in the logic of assemblies. An £jJ formula Qx, with x a variable over the power object of an assembly (B) , is true ijJ every Q(P) obtained by replacing x with a relation of assemblies (P) >--+ (A x B) is tru e. Proof. Bothclaims follow from L emma 6.3(ii) . For thesecond, every £jJ objectis surjectiveimage of an assembly, so we need only know Qs forevery generalized elements : (A) - P(B) defined over an assembly, thusevery -l subassembly(P) >--+ (A x B) .
REMARK 6.7. Any (AI R) is d.n. separatedin £jJiff it has .n, d closedequality, so iff(R) is d.n. closed in£jJ. By Lemma 6.6 thatis iff (R) is d.n. closed in assemblies. And so iff( AIR) is isomorphicto an assembly.T he categoryof of assembliescontaining t hose sets, Set, is isomorphicto thefullsubcategory (A) witheach An equal tothecarrierA . Call these assemblies codiscret e. They are d.n. sheaves in t hecategoryof assemblies. Each (B) has a d.n. dense monic tothe codiscretebased on the set B, and is a sheafiff this is an isomorphism. So thed.n. sheaves in£jJ are (up toisomorphism) the codiscretes,i.e., theyare (up tofunctorequivalence)Set embedded in £jJ. 7. ARITHMETIC IN EFF
The assembly (N) is a natural n umber objectin £jJ. To see this, use the Peanoaxioms wheretheinductionaxiom has P, a variableoverthepower objectof (N):
362
COLIN MCLARTY f- rv(O = sx) , f- sx
= sy
P(O) & (Vx)(P(x)
~
~
x
= y,
P(sx)) f- (Vx)P(x).
The first two deal only withan assemblyandso remaintruein £jJ. Lemma 6.6 proves theinductionaxiom from theinductionscheme for assemblies . Church'sThesis also deals only with assemblies . Lemma 6.6 gives theMarkov principle withP a freevariableoverthepowerobjectof (N): (Vn)(Pn V rvPn) ,
rvrv(3n)Pn f- (3n)Pn.
Considerchoice from(N) to any(BI R), with P a variableoverthepower objectP(N x (BI R)) : (Vy. (N))(3x. (B 1R))Pyx f- (31. ((BI R)N))(Vy)P(y,fy).
Certainly this istruefor (B 1R) if it istruefor (B) . Then use Lemma 6.6. And again, if(B) is modest, we have choice from (N) to any(BI R) without assumingchoice inthemetalanguage . In £ff we can define the Dedekind realnumbers by Dedekindcuts. As Hyland(1982) remarks,theCauchyandDedekindrealsagree(up to isomorphism) in £ffsincechoice from(N) to (Q) shows everyDedekindcutgives a Cauchysequence. We have goodconditionsfor re cursiveanalysisin full higherorderintuitionistic logic. 8 . METATHEOREMS
The above reasoningis constructive, exceptforparticulars of arithmetic , so it works in any t opos. This quickly gives two standardr esultsin ourcontext : 8.1. Realizable realizability is realizability. I.e ., realizability in £ff of sentences with all terms of sort (N) agrees with classical realizability.
THEOREM
Proof. Via G6delnumberingof thosesentenceswe regardrealizability as a relation betweennatural numbers. Its definitionsarealmostnegative , except numbers is decidable , so thatclause is for S V Sf. And equalityof natural equivalent to thealmostnegative : If In = 0 thenrn realizesS; and if not In = 0, thenIn = 1 and rn realizesSf. --I
8.2. More arithmetic is realized classically than intuitionistically; or than follows intuitionistically from the Peano axioms plus Church's Thesis.
THEOREM
Proof. The second claim follows fromthe first asthe Peano axioms and Church'sthesisare intuitionistically realized. For t hefirst we give atopos where Markov 's principlefails inordinaryarithmetic . By theproofof Theorem 5.2, Markov'sprincipleis alsonotrealizedthere.
FIRST AND HIGHER ORDER REALIZABILITY
363
Take presheaves(Kripke models) onthis poset: The set ofpairs (n,O) and (n,l) for alln atural numbers n, where each(n,O) precedes (n,l) and each (m,O) and (m, 1) with n < m , while each(n , 1) is a dead-endpreceding nothing. The natural numbers here are the p resheafassigningN at every node. Markov 's principlefails fort hesub-presheafS which isempty ateach (n,O) and at each(n, 1) is thesingleton{n}. -1 Van Oosten(1991) gives severalotherhigherorderrealizabilities , all of which can bepresentedin this way, using a first orderlogic forvariants of assemblies andthengettinga topos and higher orderlogic byadjoining partsof arithmetic,which is much of the quotients . They realize different point. And Remark6.7 fails for some. REFERENCES Asperti,A., and G . Longo 1991 Categories, types, and structures, MIT Press, Cambridge, Massachusett. Bell, J . L. 1988 Toposes and local set theories, An introduction, Oxford. Freyd,P., and A. Scedrov 1990 Categories, allegories, North-Holland , Amsterdam. Freyd, P., Carboni, and A. Scedrov 1988 A categorical approachto realizability and polymorphictypes, Proceedings of the third ACM workshop on mathematical foundations of programming language semantics (Morin et al., editors),LectureNotes in ComputerScience, no. 298, Springer-Verlag, Berlin. Hyland 1982 The effectivetopos, The L. E. J. Brouwer centenary symposium (A. S. Troelstra a nd D. van Dalen, editors),North-Holland , Amsterdam. IEEE 1990 Fifth annual IEEE symposium on logic in computer science, Computer SocietyPress. Kleene, S. C. 1952 Introduction to metamathematics, North-Holland, Amsterdam. McLarty,C. 1992 Elementary categories, elementary toposes, OxfordUniversityPress, Oxford. Van Oosten, J . 1991 Exercisesin realizability, Dissertation, Universityof Amsterdam.
JOHN C. SHEPHERDSON
LANGUAGE AND EQUALITY THEORY IN LOGIC PROGRAMMING
Abstract.The underly ing language of symbols used in logicprogrammingis usuallytakento be theone consistingof thesymbols occurringin theprogram, w iththequery,butsometimes infinitelym any function or theprogramtogether symbols are included . As far asPrologor SLDNF-resolution a reconcerned, it makes no differencesincetheseoperateentirely w ithintheformerlanguage . Butit does make a differencet otheClarkcompletionof a programandto 'const ru ct ive negation',a recentlyproposed extensionof negationas failure . The difference via theequalitytheory . We considertheeffect oft heunderlying is transmitted language on theClarkcompletion,t heequalityt heory,c onstructive negationand 3-valuedsemantics.
1. INTRODUCTION
The theoryof logicprogrammingis sometimes developed in full generality E of symbols. Butoften withrespectto some fixedbutunspecifiedlanguage is takento be £(P), consistingof allsymbols (e.g., Lloyd 1987) thelanguage occurringin theprogramP. Most people whotalkaboutHerbrandmodels assumed in databaseapplications.I f one use thislanguagea ndit is usually wishes toconsideran arbitrary queryQ, thecorresponding minimal language thoseappearingin P or Q. But in is £( P U {Q}) whose symbols are all his work in 3-valued semanticsfor logicprogramming,Kunen (1988, 1989) uses a languagew ithinfinitely many functionsymbols of allarities, and in theirwork onconstructive negationand equalitytheory,Chan (1988) and Przymusinski(1989) assume infinitely many constant or function symbols. As far asPrologand SLDNF-resolution areconcernedit makes no differencewhatlanguage E is used sincebothoftheseoperatewithin theminimal £ (P U {Q}), butit does make a difference to Clark the completion, language comp,e(P), of a programP which isthemost widelyacceptedsemanticsfor P and to theanswersgiven by Chan's (1988) constructive negation. For example,if P is P
+---+-.q(x)
q(a) +---+, thencompc(P) consistsof CETc., Clark'sequality t heoryforL, together with thecompleteddefinitionsof p and q, viz. p q(x)
+---+ +---+
3x -. q(x) x = a,
365 C. Anthony Anderson and M. Zeleny (eds.), Logic, Meaning and Computation, 365-392. © 2001 Kluwer Academic Publishers. Printed in the Netherlands .
366
JOHN C. SHEPHERDSON
which imply p
~
3x(x
i= a).
If I:- has onlythe constanta thenneitherp nor --, p is a consequenceof comp.dP), butif I:- containso therconstant or functionsymbols, thenp is a negation.On consequenceof comp.dP); alsop succeedsunderconstructive theotherhand,theHerbrandmodel ofcomp.dP) based on thelanguageI:of theprogramhas onlytheelementa, so p is false inthismodel. A technical reasonfor usinginfinitelanguagesI:- is thattheysimplify the treatment becauseCETr. is thena completetheory.Butit is alsoclaimedthat theyare closer to Prolog , wheretheuser may posequeriescontaining symbols not intheprogram. However,theuser presumablyhas in mind some model or class of models of P or ofcomp(P), and it seems absurdto suggestthat theseareequippedwithdefinitionsof infinitely m any irrelevant functionsa nd constants.And it is notsimply a questionof extendingthemodel by giving thesefunctionsa 'free' interpretation, since as the above example shows, theadditionof newterms may changethetruthvalues of old formulas. It seems more reasonable to saythatPrologoperatesw ithinthefinitelanguage I:-(U{ Q}) defined bytheprogramand thequerytogether.One peculiarity of using I:-(P) is thatit causesan 'instability'.If we add to theabove program theclauser(b) +-- , thenp becomes aconsequenceof comp.dP) withthenew I:- (andtruein allHerbrandmodels, whereasbefore it was false in Herbrand all models). As Kunensays (privatecommunication),' ... we do notexpecta truth value toswitchfromtrueto falseu pontheadditiontothedatabaseof a statement in acompletely disjointlanguage' . Using alanguage withinfinitely but commits one many functionsymbols of allaritiesavoids this difficulty to the existenceof infinitelymany irrelevant and non-existentt erms. In practice,one surelyhas in mind some particular (usuallyfinite)language 1:-, possiblyincludingsymbols notexplicitly o ccurringin theprogram,so the idealdevelopmentwould be tostateresultsfor such ageneral fixedI:- ~ I:-(P) as far as possible,notingwherefurther a ssumptionsaboutI:- are necessary. In section2 we considertheeffect oncomp.dP) of extending1:-. If I:- is infinite(strictly:has infinitelymany functionor constant symbols), it has no effect (on consequencesof comp.dP) exressiblein 1:-); if I:- is finite, it has theeffect of a ddingstatements assertingtheexistenceof finitely or infinitely many 'roots'or isolatedelements , i.e., elementsnot in the range of any 1:-function . In section3 we give(Theorem3.9) a normalform forequalityformulasin thetheoryCETr. forgeneral1:-. This generalizes thework ofChan (1988) , Maher(1988), and Przymusinski(1989) who gave suchresultsforthecase of infiniteZ and (in Maher 1988) for finiteI:- plus thedomain closureaxiom DCAr. . It canbe regardedas a refinementof anormalform given by Malcev (1971). Itshows thatChan'sconstructive negationcan beperformedwitha finiteZ, butthatthe normalform ofanswersto queriesand thealgorithm
LANGUAGE AND EQUALITY THEORY
367
forobtainingthembecome morecomplicatedthanwith infinite E: However, if thedomain closure axiom isassumed, thena simplernormalform exists. The resultson thedomain closure axiom also show thatSato andTamaki's (1988) replacement overtheHerbranduniverse based on £.(P), ofquantified inequationsby predicatesgiven by definite clause programs, extends(Theorem 3.13) toarbitrary equality formulas.Withoutthe domain closure axiom, one canobtain(Theorem3.14) a hierarchic normalprogramwith 2 or 3 levels whichrepresentsa givenequalityformula. In section4 we show howtheresults of Kunen (1988, 1989) are affected by the language . His semanticsbased on the 3-valued analogueof thefamiliar T-operator correspondstoconsideringall 3-valued consequencescompdp) of if E is infinite, and all -valuedconsequences 3 of com pdP) U {DCA.c} if E is finite(Theorem4.2). The 3-valuedsoundnessof SLDNF-resolution with completenesswithrespectto compdP) respectto com pdP) andits 3-valued for allowed programs(Kunen 1989) hold for all E. I am verygrateful to thereferees for making corrections and suggesting improvements. 2 . HOW comp,e(P) DEPENDS ON C
NOTATION: A language E is a set offunctionand predicatesymbols of assigned arities . Constantsare treatedas O-ary function symbols. Since the predicatesymbols in E play no role, we shall ignore them; thuswhen we say a languageis infinite, we mean it has infinitely many function symbols (includingthose ofarity0). Clark 's equality theory for L (CETr.) , consists of the axioms :
distinctf, g in L , (1) f(XI,"., x n ) =f g(Y I, "., Ym) for all (2) f(XI, . .. , x n) £.,
= f(YI, . . . , Ym) - - Xl = YI /\ . . . /\ Xn = Yn for allf
in
(3) X =f t(x) for all t ermst(x) distinctfrom X in whichX actually occurs.
We refer to this lastiom axas the'occur axiom' . (4)
;f. XI
denotesa tupleof variablesXI , • • . , X n , and ;f. = J!. standsfor = YI /\ • • • /\ X n = Yn'
£.-Programs are finite sets of clauses of the form A+--
whereA is an £.-atomand
368
JOHN C. SHEPHERDSON
If alltheclausesof theprogram P which havethepredicatep in their headare p(td +--
thenthe completed definition of p is r
p(;r)
+--->
V 31li (;r =
ti
1\
i= 1
where;r are new variablesand lli are thevariablesin P(ti) +--
x = Y ---> Y = x, x
= Y 1\ Y = z ---> x
;r = II ---> (p(;r) ;r = II ---> (J(;r)
and = z,
+--->
p(ll)) ,
= f(ll)),
are alsoassumed). For f in .c, we defineNf(x)-'x is not in the range of f'-by Nf(X)
=def
\;jYI, . . . , Yr (X :f:. f(YI, .. . ,Yr))
where r is thearityof f , and if .c has finitely m any functionsymbols, we define Rc(x)-'x is an .c-root'-, by Rc(x)
=d ef
/\
Nf(x) ,
fE.c Gl-'there are at least k .c-roots'-by GI=def3XI,,, .,Xk(Rc(xdl\ .. . I\Rc(Xk) 1\ /\Xi:f:.Xj), iii GZO-'there are infinitely many .c-roots'-by GZO
=d ef
{Gl, Gi,·· .},
and DCAc-the Domain Closure Axiom,-by DCA c
=def .,
3xRc(x).
This ClarkcompletioncompC
LANGUAGE AND EQUALITY THEORY
369
(1) if pu{ +-- Q} has an SLDNF-refutation with answer B, then comp.c
\I.Q.
1=
This holds whenE = £(P U {Q}), thelanguagecontaining onlythefunction and predicatesymbols actually occurringin theprogramp and queryQ. SO it obviously holds also for language any containingthis,since comp.c, (P) 1= comp.c 2(P) if £1 ~ £2. However,SLDNF-resolution is usuallyincomplete for comp.c
~
£1
~
£(P).
(i) If £1 is finite and £2 - £ 1 contains no function symbols of positive arity but a finite number k of constants, then comP.c2(P) is a conservative extension of comP.c i (P) U {GfJ. (ii) If £1 is finite and £2 - £1 contains a function symbol of positive arity,
or infinitely many constants, then comP.c2(P) is a conservative extension of comP.c, (P) U {GZ'} . (iii) If £1 is infinite, then comP.c2 (P) is a conservative extension of comp.c, (P).
Proof Throughout, cp denotesan £1-sentence.Then comp.c
Conversely, if
GET.c, U {GI,} ~ cp, thenthereis a modelM satisfying
GET.c, U {GIJ U {.cp}.
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JOHN C. SHEPHERDSON
This can be extendedto a model ofGETc2 withoutextendingits domain by takingthe k new constantsto be k of theLl-rOOtSof M and, if there e.g., alltrue. (The are any newpredicatesin L2, definingthesearbitrarily, only axiom ofGETc2 involving the predicatesis theequality axiom {f. = y ---> (p({f.) ...... p(y)), which isautomatically satisfied,since we areinterpreting = as equality in our models . So from now on we shall ignore any predicates new .) m any constants follows (ii) The case whereL2 - Ll consistsof infinitely from (i) bycompactness. If L2 - Ll containsa newfunctionsymbol j, then takingg(x) = j (x, . . . , x), GETc2 implies that,for anyelementx and any k, g(x),g2(x), ... , gk(x ) aredistinctLl-roots,hence impliesGI, . This shows thatGETc2 is an extensionof GETc, U GZ: . We must show it is aconservative extension,i.e., thatif
GETc2 F
ip ,
thenGETc, U GZ:
F cp.
Suppose thenthat
GETc2 F cp but GETc , U GZ:
~
cp.
Then
GETc, U GZ:
U {-,}
is consistentand countable , so it has acountable model M . By compactness, GETc2 F cp implies GETc2, F cp whereL2' is a finitesublanguage of L2 which (byadjoiningLl if necessary)we may suppose conta ins Ll . If we could extendM, withoute xtendingits domain D , or changingthe interpretations of the function andpredicatesymbols inLl, to a model of GETc2, by suitably -, ip true defining the new functionsof L2 over D, thenwe would have left and producedthedesiredcontradiction. This is notalwayspossiblebecause thenew functionsnot only have to be Ll-roOtS but have to the values of satisfythe'occur'axiom of GET which saysthatif t(x) is a termof non-zero depthcontaining x, thent(x) f= x. Suppose, forexample,thatLl consistsof a binaryfunctionj, and M has rootsPI, P2, ... and elements00 , 01 , 02 , . . . satisfying0 ;-1 = j(Pi,Oi) for i = 1,2 ... as in Figure 1. Then if L; has a
Figure1 a new unaryfunctionsymbol g, theng(oo) must be defined to be root,but it cannotbe PI becausethatwould givePI = 9 (J(PI , 01)) a violationof the
LANGUAGE AND. EQUALITY THEORY
371
occuraxiom. Similarly , it cannotbe any offJ2 , P3," . becausethatwould give fJ2 = g(J(PI ,f(P2' Ct2)) , . . . . The problemis thatalltherootsare ancestors of CtQ. To getaroundthisdifficulty, we use compactnessagain. If CETc 2 F ep, thenfor somed CETf,2 F ep, wherethesuperscriptd indicatesthattheoccur axiom is restricted to terms t of depth ~ d. We now showthatM can be extended,w ithoutextendingits domain, or changingthe interpretations of functionor predicatesymbols of £1 , to amodel of CETf,2 . If gl, .. . ,ga arethefunctionsin £~ - £1 , we have to define thecountably many valuesgl (Ctl'. .. , Ct r.) , i = 1, ... , a forCtl, . . . , Ct r, rangingovertheclementsof D. To satisfyCETf, thesehave to be all distinctfrom eachother , 2 all£1-rootS, and have tosatisfyt(x) =J x for allt erms t of (non-zero)depth ~ d. Let VI, V2, • . . be an enumeration of thesenew functionvalues to be defined. If Vn is g(!3I' ... ,!3r) where9 is one ofgl, ... , ga and!31 , .. . ,!3r ED, thenwe defineV n to bethefirst oft he£l-roots PI , fJ2, ... of M which has not alreadybeen used, and which isnota (d - 1 )-ancestorof any of!31 , . . . , !3r . Here ak-ancestor of !3 is an elementCtl of D such thatthereis an £~-term t(XI, . .. ,xm ) of depth~ k containingXl andelementsCt2, ... , Ct m of D such thatatthepresentstage, i.e., beforeVn is defined, t(CtI," " Ct m) is defined and equalto!3. It is easily proved by inductionon k thatthere areonly finitelymany k-ancestors of any el ement!3. We now showthatif the £~ d-occur axiom was satisfiedbefore V n was defined,thenit remainssatisfiedafterwards.By the£~ d-occur axiom , we mean theaxiom which says thatif t(XI , . . . ,xm ) is a termof non-zerodepth ~ d containing Xl, and if t(Ctl' . . . , Ct m ) is defined,thenit is notequalto Ct l. Since the£~ d-occuraxiom is satisfiedin thissense inM (since the£1 occur axiom is satisfied), and since eventually allfunctionsin areeverywhere defined, this will completetheproof. Suppose thenthatthedefinitionof V n , i.e., puttingg(!3I,'" ,!3r) = P, createsa violationof the£~ d-occuraxiom, Xlandofnon-zerodepth~ d and i.e., thereis a termt(XI ' . . . ,xm ) containing elementsCtl,. .. ,Ct m of D such thatt(Ctl , ' . . , Ct m) becomes definedandequal to Ctl.Take anypathfrom theroott to this leafand terminateeach side , branchattheend of thefirst edge,labelingthenew leafwitha newvariable as in theexampleof Figure2 (ignorethethirdcolumnforthetime being). This new termalso hasthepropertyused to definet(XI, . . . , x m ) above, so let ust akethisto be t(Xl' ... , x m ). Since the£~ d-occuraxiom was satisfied before g(!3I' . .. , !3r) was defined to beP, at leastone ofthefunctionsin t must be 9 and occurringin a context9 (tl (Xl, .. . , x m ), . . . ,tr(XI , . . . ,xm )) where (afterVn is defined) ti(Ctl, ""Ctm) = !3i, for i = l,... ,r. Take the highestsuch occurrence in thetreeof t, andreplaceit by a newvariabley as in thethirdcolumnof Figure2. This gives atermt' of depthdl < d in which y occursbut Xl does notoccur,becausefor Xl= Ctl, .. . , Xm = Ct m, Y = P thevalue oft' is thesame as thevalue oft, i.e., Ctl, and for thesevalues t' is defined before g(!31 , .. . ,!3r) is defined, so if Xloccurredin it, there would have been an existingviolationof d-occuraxiom. So we may write t' = t'(X2, . .. ,X m, y).
c;
372
JOHN C. SHEPHERDSON
til = f(X4 ' Y)
d=3
Figure2 Becauseof theway we haveprunedtheside branchesof the the t tree, d, is the depth in t of this highestoccurrenceof the new value . Now take thelowest suchoccurrenceof the new value, at d epth da , say. This takes theform g(tl (Xl,... , Xm ), . . . , tr(XI, . . . , x m )) where ti(OI ,' .. , om ) = !3i for i = 1, ... ,r, andsince it isthelowestoccurrence , thesevalues of the ti , are defined beforeg(!3l' . . . , !3r) is defined. Atleastone ofthe ti, say tl,must containXl, and it hasdepth ~ d - (d2 + 1). Now
where
t"(X2,'" , x m,Y) = tl(t'(X 2"" , xm, Y), X2 , ' " , xm) is a termof depth ~ dl + d - (d 2 + 1), which is ~ d - 1, since d, ~ da. Also til (02, . . . ,Om, p) was defined before g(!3l' ... , (3r) was defined,and Y occurs to thehypothesismade when in til, so p is a (d - I)-ancestorof !31l contrary it was chosen ast hevalue ofg((3l ' .. . ,!3r)' (iii) If GET£;2 F ip, thenby compactnessGET£;; F cp for some finitesubset L; of L2. Let L~ = L; n LI . By (i), (ii) appliedto L~ ~ L~ we have
GETc.'\ U G;?,; '- \
F sp,
By (ii) appliedto Ll ~ L~, we obtain
GET£; , The converse is obvious .
F cp.
LANGUAGE AND EQUALITY THEORY
373
3. NORMAL FORM S FOR EQUALITY FORMULA
The normalforms given here are based on those usedChan by (1988) in his constructive negationprocedure . This is a way ofextendingnegationas t henegationof answers failure to non-groundnegativeliterals by returning to queryQ as answers to-, Q. If Q has variablesX I, . • . ,Xn and succeeds with answersubstitution () for theprogram P, we can write the answer in equational form as
t heRHS otherthanXl,. • • , X n . where 3quantifiest hevariables on If thereare a finite numberof answers to the query Q and no infinite fair executions,then Q f---> Al V .. . V A k is a consequenceof compdP) ; so if compdP) is theintendedsemantics,it is legitimate to return -,(A I V . . . V An) as theanswer to-, Q. If thereare infinitely many answersQto thisprocedur e fails,butwhen it does succeed returns it as answer to a query Q an equality formulaE such that
Qf--->E
is a consequence of comp.c
374
JOHN C. SHEPHERDSON
infinite) . His Theorem8.1 statesthatthereis an algorithmforreducingany equalityformulaE to astrictly normal equality formula NE suchthat
F VeE +------+
CETe
NE).
A strictly normal equality formula is a disjunctionof strictly simple equality formulas each of which has theform :JYI," . ,Yx (Xl
= tl
/\... /\ Xm
= t m /\ VeX; =f s r) /\ ... /\ V(x'm =f sm)),
where eacht i, s, is eithera non-variable t erm or one ofthefreevariables of this formuladistinctfrom X i , x~ respectively,wheretheV in V(x~ =f s.) universally quantifies some (perhapsnone) ofthevariablesin s., theYI,·· . , Ys are distinctfrom Xl, . . . , X m , X~, . .• ,x'm andeach Yi occursin atleastone of the terms tl, "" t m . Chan does notdescribe his normalform explicitly, butfrom hisexamplesand his reductionalgorithm,it wouldappearthathe achievesthefurther r estriction corresponding to theuse ofidempotentmgu, thatthe occurrenceof X i on theLHS of X i = t i is theonlyoccurrenceof X i in theformula . Butin orderto achieve this he has toadmit quantified inequat ions of theform V(Yi =f r.) as well.T his is theform wegeneralize to arbitrary I:- in theDefinition3.7 andTheorem3.9 below. The only difference in the normalform is thatif I:- is finite, we need to add closedformulas GI., .., G{ assertingtheexistenceof atleastk, less thank', I:--roots. So if one is workingwithmodels for whichthenumberof I:--rootsis known, the same normalform isachievable,butouralgorithm is more complicated ' t han Chan's. in a finitelanguageI:- which issupposed to contain We workthroughout thelanguage 1:-( {E}) consistingof thefunctionandpredicatesymbols which actually occurin theequalityformulaE which is to bereducedto normal form. We usetheterminologyof Przymusinski(1989) but witha different meaning; we startwitha weak defin ition of normalequalityformulawhich is strengthened in Definition3.7. DEFINITION
3.1. A simple equality formula has theform S = 3(EI
/\ ... /\
En),
n~O
where3 quantifiessome (perhapsnone) ofthevariablesin E 1 , • • • , En and each E, is either : (i) an equationtl = t2 betweentermsof 1:-, or (ii) an inequationt l
=f t 2, or
(iii) Rc(v) for somevariablev , or (iv) ..,GI. for somepositiveintegerk.
375
LANGUAGE AND EQUALITY THEORY
A normal equality formula is a disjunction 8 IA .. . A8M,
m~O
ofsimple equality formulas. Anempty conjunction (disjunction)is identified withtrue(false). We first prove some basicr eduction . lemmas. From now on weassume
GET£. so thatwhen weasserta resultip (e.g., theequivalence oftwoformulas) we mean CET£. f-- cpo 3.2. There is an algorithm for reducing any conjunction G of equations to equivalent solved form 8 ':
LEMMA
where VI, ... ,Vn are distinct variables which do not occur in any of the terms t l , . . . .t.; Proof. This is justa version oft heusualunificationa lgorithm . See Lassez, -j Maher, and Marriott 1988. 3.3. If all the variables Xl,• . " Xn (and possibly other variables as well) actually occur in t(XI, , x n ), then
LEMMA
t(XI, . . . ,xn) = t(YI
,Yn) ~
Xl
= YI A . .. A x n = Yn'
Proof. By inductionon thedepthof t. If thedepthis 0, thent(XI) is simply For theinductivestep GET£. implies
Xl.
f(tl (Xl, . . . , x n), . . . ,tr(XI, ~ tl(XI, . . . ,xn) = tl(YI,
,xn» = f(tt{YI, , Yn), ' " ,tr(YI, ,Yn) A ... A tr(XI, ,xn) = tr(YI ,
,Yn» ,Yn),
so theinductionhypothesisgives theresultbecauseeach Xi must occurin sometj(xl,' '' 'x n), -j LEMMA 3.4. If z', Ji. = Xl, " . ,X m , Y = YI," . ,Yn, &: = variables, and Ji., J!.. are all the variables in t(Ji., J!..), then
.., 3~(X'
= t(Ji., J!..) A K)
~ {'v'J!..(x'
Zl,"" Zp
are distinct
i= t(Ji., J!..») V 3J!..(x' = t(Ji., J!..) A .., 3~K)} .
Proof. The LHS logically implies theRHS andthefirstdisjunctof theRHS implies theLHS. ThattheseconddisjunctoftheRHS implies theLHS follows -j from Lemma 3.3. LEMMA
'v'~(X'
3.5.
i= t(tl, .. "
t r») ~ {'v'J!..(x' i= t(yl, . .. ,Yr»)V 3J!..(x' = t(YI," ., Yr) A 'v'~(YI, i= tt A . .. A Yr i=
tr»)},
where Y = YI, • • . ,Yr are variables not in the LH8, and the variables ~ do not occur in t(YI, . . . ,Yr) '
376
JOHN C. SHEPHERDSON
Proof. WritetheLHS in theform
-dlb~(x' = t(Y1 , ' " , Yr ) /\ Y1 = t1/\"'/\ Yr = tr)
and apply Lemma 3.4. The hypothesisabout~ means thattheironlyoccurrences in t( t1 , ... , t r) arewithint1 , . . . , t r • 3.6. Th ere is an algorithm for reducing disjunctions, conjunctions, and existential quantifications of normal equality formulas to equivalent normal equality formulas.
LEMMA
Proof. For a disjunctionthereis nothingto do; for aconjunctionwe use the distributivelaw towriteit as adisjunctionof conjunctionsof simple equalityformulas,a ndconvertthelatter intosimple equalityformulas using 3yA /\ 3y' A' +----> 3y, y'(A /\ A') (provided y does notoccurin A' , and y' does notoccurin A) . For existential quantification we use 3y(A V H) +-----+ 3yA V 3yH to movethequantifi er in front of thesimple equalityformulas, which still leaves themsimple equalityformulas. By replacingA --+ H, A +----> H, VxA by .AvH, (.Av H)/\(.HV A) and • 3x • A respectiv ely,we canreplac e E by an equivalent formulacontaining to includeboth/\ andV). An onlytheconnectives/\, V," 3 (it isconvenient atomic formulat1 = t2 is a normalequalityformulaand Lemma 3.6 deals with /\, V, 3 so it remains to show how tow ritethe negationof a normal equalityformulaas a normalequalityformula . Using .(81 V . . . V 8 m ) +----> .81 /\ , • . ' 8m , it isenough to do t hisforthenegation• 8 of asimple equality formula 8.Ourfirststep is to simplifythisstillfurther.
DEFINITION 3.7. A strictly simple equality formulawithfree variables = X l, • . • , X r has theform
;!;.
n ~O
where (a) eachE, is either: (i) an equationXi = t, wheret is a non-variable termor is Xj for j =1= ij and thereis at most one of thesefor eachXi, and calling such Xi solved variables, no solvedvariableoccurselsewhere in t heformula , or (ii) an inequationXi =1= t or Yi =1= t, or (iii) Rc.(Xi ) or Rc.(Yi) , or
(iv) Gt foratmost one positiveintegerk, or (v) •
GX for atmost one positiveintegerk'; and
(b) each ofY1,' .. , Yn occursin the RHS oftheequations(i).
LANGUAGE AND EQUALITY THEORY
377
A strictly normal equality formula is adisjunctionofstrictly simple equality formulas. simple equality formula isequivalent to a simple equalNotethata strictly ity formulabecause is 3YI,. . . ,Yk (Rc(yI) /\ . . . /\ Rc(Yk) /\ 1\ Yi :f:. Yj)'
aJ.
i#j
We show later,in Lemma 3.11, how to replace a simple equalityformula by a strictlynormalequalityformula. Assuming this, we now dealwith negation . 3.8. There is an algorithm for replacing the negation of a strictly simple equality formula F by an equivalent strictly normal equality formula.
LEMMA
operatesrecursively on thenumberof equationsin F . Proof. The algorithm If this is zero, thenF is a conjunction of inequationsXi :f:. t(J<), Rc(x;), and-, F is equivalent to adisjunction ofequationsXi = t(J<), -, Rc(Xi), which are all strictlysimple equalityformulas except-, Rc(Xi), which isequivalent to thestrictly n ormalequalityformula
aJ.,
-, at; -, aJ., at
V 3Yi, ... ,Yr(Xi =
J(YI, . .. , Yr)) .
IEL
If thenumberof equationsin F is non-zero , we write F as 3lL,~(XI
= t(J<'lL) /\ K)
wherethevariablesin t are all of the Y andsome (or none) of the J< butnone of theg , Using Lemma 3.4, we replace-, F by
\7'lL(XI :f:. t(J<'lL)) v 3lL(XI = t(J<,lL) 1\ -'3~K) . By Lemma 3.10 below, the firstdisjuncthere can be reduced to strictly normalform, and, by inductionon thenumber of equations,so can the simple equality formula,and Xl does not second,becauseif 3yK' is a strictly occurin K', so is 3lL3Y(XI = t(J<,lL) 1\ K'). Assuming Lemmas 3.10 and 3.11 below, we can nowstate : 3.9 . There is an algorithm for replacing every equality formula by an equivalent strictly normal equality formula .
THEOREM
3.10. There is an algorithm which will produce an equivalent strictly normal equality formula F* for any formula F of the form
LEMMA
\7'~(x' :f:. t(J<,~))
where J< = Xl,···,Xn , ~ = Zl , . . . , Zm are the distinct variables in t(J<,~). Each of the strictly simple disjuncts of F* either has no solved variables or has z' as the only solved variable.
378
JOHN C. SHEPHERDSON
Proof. Ift is x' , thenF* is false; if t is a non-variable termcontaining x ', then F* is true. Otherwisethealgorithmwill bedefinedrecursively on thedepth of t. If t has depthzero, .e., i is a variable,t henF is eitherx' =I- Xl whichis leftaloneor 'v'ZI (x' =I- zd, which isreplacedby false . If t is f(tl , . . . , t r ) , we firstreplaceF by
(whereYI, . . . , Yr arenew variables),which isequivalent to it byLemma 3.5. Now replaceNf(x') by thestrictly n ormalequalityformula
Rc(x')V
V
3YI , . .. ,Ys(X'=g(YI , .",ys))
9EL ,g#!
to which, byCET.c, F is equivalent . Now by Lemma 3.4,
is equivalent to
where~l arethevariablesin ~ actually occurringin tland ~2 theremaining variablesof~. Since tl,.'" t r have lowerdepth thant, we can callthe presentalgorithmto put 'v'~I(YI =I- td into strictlynormalform; and by an algorithmoperatingrecursively on r, wecan put thewholeformulainto strictly n ormalform, with Yl, ... , Yr theonlypossible solvedvariables . To s imple equalityformulawith see this, we observethatif 3yK is a strictly Y2, . . . ,Yr as theonlypossiblesolvedvariables( and notcontain ing yd then
is a strictly simple equality formulawithYl, . . . , Yr as theonlypossiblesolved variables . A similarargumentshows thatreplacing'v'~(YI =I- iI v... VYr =I- t r ) by thisequivalent strictly n ormalequalityformulain
and eliminatingby substitution a ny of YI , . . . , Yr which aresolved, gives a formulaF* withthedesiredproperties.(If RC(yd occursand Yi is replaced by a non-variable term, thenRc(Yi) and thewholedisjunctcontainingit is replacedby false) . (The formulas k do notoccurin F*.)
a
3.11. There is an algorithm for replacing a simple equality formula by an equivalent strictly normal equality formula.
LEMMA
LANGUAGE AND EQUALITY THEORY
379
Proof We achievetherequirements (a) (i)-(iv) of Definition 3.7 in o rder.
(a) (i) Letthegivensimple equalityformulaS be
Firstuse Lemma 3.2 toputtheequationsin S intosolved formVI = tl/\' .. /\ Vk = tk : Replaceeach Vi by ti in therestof S . If one oftheu's is a Y, say VI is YI, thendeletetheequationVI = tl anddelete3YI from thequantified prefix. If one ofthet's is a Y, say tl = YI, thenreplaceYI by VI, deletethe equationVI = YI, anddelete3YI from theprefix. (ii) If thereis an inequation t =1= t' whereneithert nor t' arevariables,then put theequationt = t' intosolved formVI = tl /\ . .. /\ Vk = tk and replace
S by
k
V S, whereS,
i=l
is obtainedfrom S by replacingt =1= t' by Vi
=1=
t.,
(iii) If any Rr.{t) witht a non-variable is now present(as a resultof the substitutions of (i)), replaceit and hencethewhole formula by false . (iv) If thereis more thanone justtaketheone withgreatest k; if there is more thanone ..., taketheone with least k', (b) Let YI,' .. ,Yh be the irrelevant variables, i.e., those ofYI," . , Ys which YI, does not have do notoccurin any oftheequations.I f any of these, say a correspondingc onjunctRr.{yd in 5, thenwe may simply delete3YI from theprefixand deleteallinequationsc ontainingYI. This is becauseLemma 3.3 implies thatan equationt(YI' Y2, ... Ys Xl, ... Xr ) = V has atmost one solutionforYI, so theonlyconditionYI has tosatisfyis to be different from a finitenumberof elements: such YI a existsbecauseCETc implies thereare infinitelymany elementsin thedomain. At least it does providedthereis at leastone functionsymbol of non-zeroarityin E; the case where L consists only ofc onstants ai, .. . ,ad needs adifferent t reatment . In this case, we a dd thevaliddisjunctionRr.{YI) V YI = al V . . . YI = ad; move thedisjunction sign totheoutsideof theformula;and in thedisjunctcontainingYI = aj, replaceYI by aj and delete3YI from theprefix. If thisreplacement gives rise to Rr.{aj), thisis replacedby false; if it resultsin an inequationaj =1= t, this is left alone tifis a variable,replacedby false if t is aj and true if t is a non-variable termotherthanaj (i.e., some ai with i =1= j). So we mayassume S containsR r.{yd/\ . . ./\Rr.{Yh)' We cantherefore delete any inequationYi =1= t where t is not avariable,since thisis a consequence of Rr.{Yi)' So theinequationsinvolvingYI, .. ·, Yh are oftheforms Yi =1= Yi> Yi =1= v, wherei, j :s: h; or V =1= t(;f', y'), whereV is a relevant variable,i.e., one ofthevariablesYh+l Ys XI~"" Xr , where s', Y' denotesubsetsof ;f, Y and t is a non-variable term. The nextstepis to remove the last typeof inequation ; let us call them indirect inequations. We may writeone ofthese as V =1= t(YI, ... , Ya 1l.) where YI,···, Yh are renumberedso thatYI Ya are thoseactually occurringin t and 1l. is a vectorof relevant variables.By Lemma 3.5 thismay be replacedby
at,
at',
I
I
I
I
I .. . I
I
I
I ••• I
JOHN C. SHEPHERDSON
380 VY~ , ... , Y~ (v
# t(y~, . . . , y~, Q)) V 3y~ , .. . ,y~ (v = t(y~, .. . , y~, Q) 1\
a
V (yj # Yj))'
8=1
We may move thedisjunctionsigns totheoutsideof thewholeformulaso t hereplacement of v # t(Yl' ... ,Ya, Q) by we canconsiderseparately A: Vy~, . . . ,y~ (v # t(y~, . . . , y~, Q))
and by
B: 3y~, . . . , y~ (v
= t(y~, . .. , y~, Q) 1\ yj # Yj)'
In thefirst case we use Lemma 3.10 toreplaceA by a strictly n ormalequality formula S;V .. . vs~ wherethe s~ are strictly simple equalityformulas. Moving t hedisjunction signs totheoutsideagainwe canconsiderseparately t hereplacementof A by each Afterchangingto newvariablesif necessarywe can move the existential quantifier prefix3~ of tothebeginningoftheresulting formula. The only solvedvariableof S; can be v, so its solved form v = t'(Q,~) must variables).Since v occurs containallthe~ (and a subsetQ of therelevant in an inequationof S, it is not a solved variableof S and neitheraretheQ. SO v is eitherone ofthex which is not a solved variableof S, or one ofthe relevant Y, i.e., one whichoccursin at leastone oftheequationsof S . In bothcases wesubstitute t'(Q, ~), for all occurrences of v. In thefirst case we retaintheequationv = t' (Q,~), making v a new solvedvariable;in thesecond prefix. Inboth case wedeletethisequationanddelete3v from thequantifier cases thevariables~ are still relevant, since theyoccurin anequation,so we have not introducedany newirrelevant variables . The replacementof v by t' (Q, ~), if t' is not avariable,wouldconvertRdv), if it werepresent,into R.c(t'(Q,~)), which isreplacedby false, and itconvertsan inequation v # tl intot'(Q,~) # tl ' If t 1 is a relevant variable,t his canbe left alone, t if 1 is an irrelevant variableVi, it can be replacedby false, as a consequenceof R(Yi), otherwiseit isreducedas in (a)(ii) above to adisjunctionof inequations with variableson theLHS and thedisjunctionsign takento theoutsideof the whole formula.E ach of the resultingd isjunctscontainsjustone ofthese inequations,which can only involve irrelevant an variableif tl does, i.e., if v # t 1 was an indirectinequation . So thisdoes notincreasethenumberof indirectequationsa nd thenet resultis thatwe have got rid of one these of . When B insteadof A is thereplacement , theargumentis similar. So by proceedingin thisway, we can get rid of all theindirectinequations. oftheform Yi # v wherev is a relevant variWe nowturntotheinequations able andYi an irrelevant variable.Since RdYi) is already in theconjunction, we can use
Sr
RdYi)
S:
1\ Yi
#v~
(RdYi)
1\
Rdv) 1\ Yi # v) V (RdYi) 1\..., Rd v))
LANGUAGE AND EQUALITY THEORY
and --'Rc.(v)+---+
381
V 3UI, . . . ,ur(v=f(UI, . . . ,Ur)) IEL
to replace the formula by disjunction a of formulas in which Yi =f:. v is replaced by u. =f:. v/\Rc.(v) or by3UI, ... ,ur(v = f(UI" " ,Ur))' In thelatter case, we replacev by f( UI, . .. , u r ) , add v = f( UI, ... , u r ) as a new solved equ ationif v is an z, or delete this equationand theexistential quantifier 3v if v is a y. Inequations v =f:. tlwhich becomef( UI, . . . , u r ) =f:. tl aredealtwith as above (no newindirectequationscan result) . We have nowreducedthepartof theformulacontainingthe irrelevant variables to
which weabbreviateto
whereY is the conjunction of allinequationsof the formYi =f:. Yj between irrelevant variablesa ndallinequalities of the form Yi =f:. Vj betweenirrelevant and relevant variables,and V is the conjunctionof allinequationsof the form Vi =f:. Ve betweenthoserelevant variablesVI , .•• , Vk which occur in such inequationsYi =f:. Vj ' The finalstep is to banish theirrelevant variablesto contexts of the formGt (i.e., 3YI,. ", yk(Rc.(ydf\·.. . /\Rc.(Yk)/\ /\ Yi =f:. Yj))' if.)
It is convenient to express this step in termsof graph-colorings . Let Q be thegraphwhose vertices are YI , .. . , Yh , VI, .. . , Vk which has an edge between verticesU and u' when U =f:. u' is in Y /\ V, and Qv thesubgraphof Q with verticesVI , . • • , Vk. Considereach coloring C Qv of (two colorings which differ only in the names of the colors being considered identical) . Let Vc be the conjunction which for each i, j ~ k containsVi = Vj if Vi, Vj have the sam e color in C and containsVi =f:. Vj if Vi, Vj have different colors . Let n( C) be the
leastnumberof colors needed for a coloring C' Q extending of thecoloring C of Qv. Then
This is truebecause a model for theLHS gives a coloring of Q which is an extensionof C by giving two vertices the same color theyare iff equal in the model, so it needs at least n(C) colors, .e., i has atleastn(C) different elements, all of them roots. Conversely, if the RHStruein is a model of CET.c, thenwe get a model of Rc.(y) /\ Rc.(JlJ /\ Y /\ V /\ Vc if we choose YI, ... ,Yk so thattheyarerootsandso thatYi and Yj or Yi and Vj are equal iff they have the same color C'.inThis is possiblebecauseG2(C) says there
382
JOHN C. SHEPHERDSON
areenoughrootsto dothis. Now
v
<--+
VVc c
wherethedisjunctionis takenover all colorings C of 9v. This is so because if C is a coloringof 9v, thenVc implies V, andif V is truein a model, then Vc is trueforthecoloringC which givesVi , Vj different colors Viiffi= Vj is truein themodel. So weobtain
3!L( R.c.(y)
1\
Rc(y.) 1\ Y
1\
V)
<--+
V(G;(C) c
1\
Vc
1\
R.c.(1l)),
and replacingtheLHS by the RHS andtakingthedisjunctionoutsidegives the requireddisjunctionof strictly simple equalityformulas, since we have -j noweliminatedirrelevant variables . This completestheproofof Lemma 3.11 and Theorem3.9. We now relateournormalform of Definitions.1,3 3.7 to thoseof Chan,Przymusinski, Maher, and Malcev . The formulaRc(x) used above is aconjunction of quantifiedinequations of thesimplestpossible form
so thetype (iii) conjunctin our simpleequalitya ndstrictly simple equality formula could be replacedby 'a quantifiedinequation V (v
i= t)
for somevariablev'
or 'a quantifiedinequation V(Xi
i= t)
or V(Yi
i= t)',
respectively, as in C han's normalform. Przymusinskidoes not allow the inequations Yi i= t and quantifiedinequations V(Yi i= t) in his strictly simple Yi must occurin equalityformula. We can also get ridthesebecause of some equationx = t'(Yi,{f,li), so thatwhen conjunctedwith this, V~(Yi i= t({f'!L'~)) is equivalent to V~(x i= t'(t({f'!L,~),{f'!L)) by Lemma 3.3. After doing this, ourstrictlynormalform givesthatof Chan and Przymusinski describedabove, when thereductionof E is made on thehypothesisCETc' for some infinitelanguage£ ' containing£ ( {E}). This is becauseif wecarry out ourreductionusing E = £( {E}), the formulas Gi aretruefor all k , since CET£,' implies theexistenceof infinitely many £-roots. Indeed, allthatis necessary for thisreduction to be possible ist hat£' - £( {E} ) shouldcontain onefunctionsymbolofaritygreater t hanzero or at leastk o constants , where ko is thelargestk for whichGi or ..., Gi occursin theequivalent s trictly normalequalityformula . In thelatter case, thek o new constantsprovide
LANGUAGE AND EQUALITY THEORY
383
the ko roots; in the former case , if the new function symbol has arityr and 0 is any element,t heng(o) , g2(0), . . .. ,gko(o) provide ko roots,where g(x) = f(x, . . . ,x). If, for example,£ ' wasthelanguage determinedby a logic programand E arose fromthesolutionof a particular query usingChan's constructive negation , it would berathereasy to checka teachstagethat therewas asparefunction in£' . However iftherewas not, and one had to rely on ok spare constants,it would behardto make use of this simplified reduction procedure,unlessthereis a way ofdeterminingko without carrying outthemore complicatedreductionbased on£({E}). For thisreductionof anequalityformulaE on thebasis of GETC' for an infinitelanguage £ ', Maher (1988, Theorem6) gives anormalformconsisting of a booleancombinationof basic formulas . Theseare oftheform
whereYI,... ,Ys and Xl,... ,X m aredistinctsets of variables and Xl, . • • , X m do not occur in t l , . .. , t m ; so theyare theanswer formulas corresponding to theusual answer substitutions given bySLD-resolution usingidempotent mgu. This can be convertedto theChan/Przymusinski strictnormalform by puttingit intodisjunctivenormalform andreducingnegationsof basic formulas to s trictly normal form as in L emma 3.10 above. E, a normalform Malcev (1971,Theorem3) gives, for the case of finite differing fromthestrictly normalform of Definition.73 in allowing N f(Xi), Nf(Yi) insteadof Rdxi) , RdYi) andin notimposing condition(b), thateach of YI, . .. ,Ys shouldoccur in the RHS of an equationXi = ti, i.e., shouldbe whatwe have called relevant variables . Theelimination ofirrelevant variables makes theChan/Przymusinskistrictnormalform of an answer much more intelligible; isitthecrux(Lemma 3.11(b)) of theproof of ourTheorem3.9. , and In two cases it is easier ; whenE is infinite,t heycan be removed at once whereE is a closed formula, in which case thegraphgv is empty. This last case wasalreadydealtwith by Malcev (1971, Theorem4). THEOREM 3.12. The theory GET£, is complete iff £ contains infinitely many function symbols. If E contains only finitely many function symbols, a complete theory is obtained by adding to GET£, the axiom GE 1\..., Gr l which says there are exactly k £-roots. In particular, adding DGA£" which says there are no Ei-roots, gives a complete theory.
Proof. Let E be a closed£-equality formula and£0 = £( {E}) . The strictly normalform ofE obtainedusing£0 containsonly formulas of the form GEo ' . ., GE~. If £ contains£0 and has infinitely many function symbols then, as . If E notedabove inTheorem 2.1(ii), GET£, implies thatallGEo are true containsonly finitely many function symbols,strictly the normal form of E obtainedusing E containsonly formulas theform of GE ,.. , G{ . -l
384
JOHN C. SHEPHERDSON
These Note thatwe arecountingconstantsas O-ary function symbols. completenessresultsare alsoimmediate consequences of Malcev'sTheorem 4. Otherproofs of the completenessof CETc for infinite Z have been given by Kunen(1988) and Maher(1988) . The latter also givesanotherproofof the completenessof CETc plus theDomain Closure Axiom for finite I:- (except I:- is singular , i.e., containsonly one function symbol , which in the case where .was left as an open problem,butis covered bytheargumentsabove). The case whereI:- is takento be I:-(P) and theDomain ClosureAxiom is assumed is of interestbecauseit includes the usual Herbrandmodels whose domain is the set of groundtermsbuilt up from the function symbols P .in (This is studiedin some detailby Malcev, 1971.) In this case,t hestrictly normalform and the a lgorithm forobtainingit can begreatly simplified. The Domain Closure Axiom impliest hatRc(p)(x) is false for all x and all Ct,(P) are false . So thestrictly simple equalityformulas take the form 3YI , . . . ,Ys(/\ i EA
Xi =
ti / \ Xj
jEB
# Ti
/\
Yk # Sk)
kEG
whereXI, ••• , X n arethevariables,A ,B ~ {l, ... ,n}, C ~ {l, . .. , n}, t, is eithera non-variable term(in Xl,... , X n , YI , . . . , Ys) or is Xj forj # i, theXi fori E A do not occur anywherein theformula except on theLHS of X i = t i , theTj, Sk aretermsin Xl, . . . , X n , YI, .. . , Ys and each ofYI , .. . , Ys occurs in one oftheequationsX i = t.. (This could also be derived describedabove as from thereductionto booleancombinationsof basic formulas , given (for 1:-) by Maher(1988, Theorem11) for the case when the Domain non-singular Closure Axiom is assumed .) This reduction can be used toextendtheresult used by SatoandTamaki (1988) . They give amethod for transforming a logicprogramwith first orderformulas inthe bodies of clauses to a definite programwhich is in some sense equivalent to it Herbrand for models. In doingthis,theyneed to give definiteprogramsforrepresenting quantifiedinequationsVJ!. (s(;f., J!.) # t(;f., J!.)) . They do this by usingthefactthatthis is a recursive relation in ;f. over theHerbranduniverse,so thereexists a definite programwitha predicate p(;f.) such thatforgroundterms to, p(t o) succeeds (fails) iff V'J!. (s(to , J!.) # t(to, J!.)) is true(false) over the Herbranduniverse. Buttheyseem unwilling to make anyassumptionsaboutwhathappens fornon-groundtermssince theform Vy(x # t) theyinserta proviso'it is assumed thatevery goal of has no free variables at thetime of failure' . However, we can showthatfor any relation defined by anequalityformulaE(x) thereis a definiteprogram whichrepresentsit completelyforgroundtermsand anyresultsit gives for non-ground t ermsare alsocorrect.More precisely: 3.13. If I:- is a finite language and E(;f.) is any I:--equality formula, there exists a definite clause progmm PE with predicates PE, equal,not-equal such that
THEOREM
385
LANGUAGE AND EQUALITY THEORY
(i) for ground to, PE(to) succeeds in PE if GET.c U {DGAd (ii) for ground to, PE(to) fails in PE if GET.c U {DGAd
r- E(to) ,
r- • E(to),
(iii) the sentences obtained by replacing PE, equal, not.equal by E, =, i- in the completed definitions of PE, equal, not.equal in compdPE) are consequences of GET.c U {DGAd, (iv) if PEW succeeds in PE with answer (), then GET.c U {DGAd
(v) if PEW fails in PE, then GET.c U {DGAd
r- VE(tO),
r- V.EW.
Proof Firstnotethat(iv), (v) followfrom (iii), thesoundnessof SLD-resolutionwith respectto P and thefactthatcomp(P) r- P . If I:- has function symbols Ii of arityai fori = 1, . . . , l, a suitableprogramPE may be defined as follows : equal(x, x) f - noi.equoltx, x')
f--
equal(x, fi(Yl , . . . , Ya ;)), equal (x' , h (y~, . . . 'Y~l/))' for all i,i' = 1, .. . ,l,i i- i'.
noLequal(x,x') +- . equal(x, fi(Yl, .. . ,Ya,)), equal (x', ft(y~,· ··, Y~J), noLequal(Yj, yj) , for all i = 1, ... , 1 with ai ~ 1, all j = 1, . . . ,ai. If E is thestrictly simple equalityformula
::lYl"" ,ys(A Xi = ti
iEA
A Xj i- ri AYk i- Sk) kEG
JEB
thenthe clause defining PE is
PE(Xl,' . . ,xn)O
f--
A not.equal (Xj, rj) , Anot.equal (Yk, Sk), kEG
JEB
where0 = {X;/ti : i E A}. If E is a disjunction
E1 V E2 V
.. . E
m
of simple equalityformulas,t hereare m clauses for PE obtainedin this way that(i), (ii), (iii), (iv) aresatisfied from E 1 V E 2 V . .. Em . It is easily verified for =, i-, and finally for E. -j It is not possible to get completenessfor non-groundqueries, i.e., to get (i) fornon-groundto, e.g., X i- f(x) is a consequence ofG ET but not.equal (x,j(x)) succeeds with answer () only for 0 whichgroundx.
Withoutthe Domain ClosureAxiom, it is not possible to deal with iin this way andobtaina definite clause program, but by treatingx i- Y as .(x = y) one canobtaina normalprogramwhichrepresentsE (x) in a stronger sense.
386
JOHN C. SHEPHERDSON
3.14. If E(x) is an .c-equality formula , there exists a normal progmm PE with predicate PE such that
THEOREM
PE may be taken to be hierarchic with 2-levels if .c is infin ite, with 3-levels if
.c is finite.
Proof The existenceof such a PE follows bythe usualtechnique(Lloyd 1987, chapter4), fortranslating firstorderlogicprogramsintonormalones, starting from thefactthatthe programequal(x, x) <-- represents= in this sense. The reductionto a low level hierarchicp rogramis obtainedby using the normalforms above. If .c is infinite , thesimplestone to use isMaher's, a booleancombinationof basic formulas . WriteE(;f) in disjunctivenormal form as adisjunctionof basic formulasa ndnegationsof basic formulas.The disjunctionis dealtwithas above bytakingclausescorrespondingto each disjunct.If E(x) +-----+ b1 /\ . .. /\ br /\ -, b~ /\ ... /\ -, b~" whereb1 , ••• , b. , b~, ... , b~, arebasic formulas,t henPE has a clause
andone clausefor each of Pb\ , .. . , Pb r , Pb'I' , . . . ,Pb'r' i if
bi(;f)
+-----+
3U(Xl
= t)
/\ . . . /\
= tm ) ,
Xm
then theclauseforPb. is
Pb.(tl , . .. ,tm,X m+l, . .. ,X n)
<--.
If we use insteadtheChannormalform
E(;f)
+-----+
3Yl,'''' Ys(
1\ Xi = ti 1\ 'iY j(Xj '1: Tj) 1\ 'i:!!.k(Yk '1: iEA
jEB
Thenletting () = {Xdtl : i E A} ,cpj = {Xj/Tj},,pk PE is
PE(Xl, . . ., x n)() <--
/\ -,
jEB
kEG
= {Yk/sd,
Sk»,
theprogram
qj (XI, . . . , Xn,Yl, · .. , Ys) /\ -, Tk(Xl, . . ., Xn, b1 , • . •, bs) kEG qj(Xl"" ,Xn ,Yl" ",Ys)CPj <-Tk(Xl, ... ,xn,b1 , . .. ,bs),pk <-- .
If .c is finite,our normalform above needs athirdlevel to deal withthe closedformulas-, -1
Gt.
This shows thatwhenChan's constructive negationgives ananswer for all queriesfor aprogramP [i.e., producesan equivalent equality f ormula ), there
LANGUAGE AND EQUALITY THEORY
387
is a 2-levelhierarchic p rogramwhich isequivalent to P in the sensethatits completionis a conservative extensionof comp(P). Since it is easy to show thatfor ahierarchicp rogramthecompleteddefinitionof eachpredicateis equivalent to anequality formula, it follows thateveryhierarchicprogramis in thissense equivalent to a 2 or 3-level one. If E containsno function symbolsofaritygreater t hanzero, i.e.,consistsof al,. .. , ad, thentheconjunctsin a strictly simple equalityformula constants can bereducedto Xi = Xj, Xi = aj, Xi I- Xj, Xi I- aj, c-, --. c-, whereG k says thereareatleastk elementso therthanal,. . . , ad. 4. 3-VALUED SEMANTICS
Kunen(1988) uses a 3-valued logic introducedby Fitting(1985) based on theKleenetruthtables . Here abooleancombinationof thetruthvaluest, J, u is t (resp. f) iff all possible ways replacing of the u by t or f gives similarlyfor thequantifiers, the valuet (resp. f) in ordinary2-valued logic; treating these as infinite conjunctions or disjunctions;t hus 3cp(x) gets the f if valuetif thereis at leaston elementfor whichcp is t, it gets value ip is f for all elements,and the value u otherwise . Thereis one exception +------+ used in forming the completeddefinitionsof to this, the equivalence predicatesin comp(P) gives cp +------+ 'ljJ the valuet iff sp, 'ljJ have thesame truthvalue andf otherwise . Equalityis takento be the2-valuedrelation many functionsymbols of identity . He uses a language£00 with infinitely of allarities . A fundamental resultwhichKunengives withoutp roofis that SLDNF-resolution is soundforthecompletionof aprogramin 3-valued logic, i.e., that if P u {+-- Q} has anSLDNF-refutation withanswer0, then compC
388
JOHN C. SHEPHERDSON
The main resultof Kunen 1988 is that3-valuedsemantics based on compc(P) is equivalent to one defined by iterating a 3-valuedanalogue ofthe familiarT p operator of vanEmden and Kowalski. At stagen , forn = 1,2, ... , some sentencesbecome true(are given the value t) and othersbecoming false f), therest havet hevalueu. At eachstage, truthand (are given the value falsity are assigned first groundatoms for and extendedto othersentences usingtheKleenetruth tablesa ndinterpreting quantifiers as ranging over the Herbrand universe, i.e., the universe of £'-terms. This makes thenotionof truthand falsity at s tagen dependenton E; when necessary, we show this dependenceby thesubscript£'-'true.catstagen '. At stage0, allground atoms have valueu and allequationss = t betweengroundtermsare true if s, t areidentical,falseotherwise.At stagen+1 , a groundatom a becomes trueiff for some clause {3 <--
(2) sp becomes true.c at some stage. He provesthat(1) implies (2) by constructing one 3-valuedmodel N for compc(P) , such thatfor everysentence
(a) for ground terms tl,' " , t r ,
, r)
r,o(tl ,
, tr) is false at stage n iff Fn( r,o)(tl,. . . , t r ) is true;
t
is true at stage n iffTn(
and (b) for elements aI,
, ar of any 3-valued model M of comp( P),
if Tn(r,o)( al ,
,ar ) is true, then
, a r ) is true,
ifFn(
,ar ) is true, thenr,o(al ,
,ar ) is false.
LANGUAGE AND EQUALITY THEORY
389
Proof Tn (cp), Fn(cp) aredefinedby inductionon n, and, for each n, by inductionon thestructure of cpo If cP is an equations = t, thenTn(cp) is s = t and Fn(cp) is s =f. t. If sp is an atomp(~) andtheclauseswithp in theirheads are p(h) -- CP1,'" ,p(h) -- CPk and -y1 . arethefreevariablesin theithclause,thenTo(cp),Fo(cp) arebothf and k
V 31L;(~ =
Tn+l(cp) is
t ; A Tn(cp;»)
;=1
k
Fn+l(cp) is
1\ 't1LJ~ =f. tvFn(CPi») '
;=1
The booleanconnectivesa ndquantifiers aredealtwithin theobviousway:
Tn(--'cp) is Fn(cp), Tn(cp A 'l/J) is Tn(cp) A Tn('l/J) , Tn(cp V 'l/J) is Tn(cp) V 'I'; ('l/J) , Tn(cp - > 'l/J) is Tn(cp) V Fn(cp), Tn('txcp) is'txTn(cp) , T n(3xcp) is 3xTn(cp) ,
Fn(--'cp) is Tn(cp), Fn(cp A 'l/J) is Fn(cp) V Fn('l/J), Fn(cp V 'l/J) is Fn(cp) A Fn ('l/J) , Fn(cp - > 'l/J) is Tn(cp) A E; ('l/J) , F':('txcp) is 3xFn(cp) , Fn(3xcp) is'txFn(cp).
It is easilyproved,by inductionon n and on thestructure of sp thatTn(cp) , --j Fn (cp) definedin thisway satisfytheconditionsof thelemma. THEOREM
4.2.
(a) If sp is an Ei-sentence then, if.c is infinite, cp is true in every 3-valued model of compc(P) iff sp becomes true.c. at some stage, (b) If E: is finite ip is true in every 3-valued model of comp.c. (P) U {DCA.c.} iff sp becomes true.c. at some stage.
Proof (a) The 'onlyif' halffollowsb ecause,if cp is truein every3-valuedmodel of comp.c.(P), it is truein Kunen's model N, hencetrueatsome stage. If cp becomes trueatstagen, thenby Lemma 4.1(a), Tn(cp) is trueatstage n. Now Tn(cp) is an equality s entence,so it iseithertrueor falsealreadyat stage0, andit is trueatstage iff it istruein theHerbrandmodelof CET.c. whosedomainconsistsof all£'-terms. Now 3-valuedmodelsof CET.c. arealso = as 2-valued,so by thecompletenessof 2-valuedmodelssince we interpret CET.c. , Tn(cp) must be truein all3-valuedmodels of CET.c. , hencein all 3valuedmodels of compdP). So, by Lemma 4.1(b) cp is truein all3-valued models of comp(P). (b) The 'onlyif' halffollowsb ecauseDCA.c. is trueatstage0, hencetruein Kunen'smodel N. If cp becomes trueatstagen, thenas in (a), Tn(cp) is true
°
390
JOHN C. SHEPHERDSON
in theHerbrandmodel ofCET£:. whosedomain consistsof all -£terms. This modelsatisfiesDCA c, and compdP)U{DCAd is complete,so theargument -1 is completedas in (a). COROLLARY
4.3. If sp is an £(P) sentence, then
(a) if £-£(P) contains a function of positive arity, or infinitely many constants, then cp is true in every 3-valued model of compC(p )(P) U Gl!(P) iff ip becomes truer. at some stage, (b) if £-£(P) contains only k constants, then cp is true in every 3-valued model of com PC( P) (P) U {Gt(p) , --, G~(F!,} iff ip becomes true£:. at some stage.
Proof Since theequalityr elation in our3-valued models is 2-valued,Theorem 2.1 applies to3-valuedmodels. So if£ is infinite, (a) follows from (a) of Theorem4.2 and (ii) of Theorem2.1. If E is finitebut £-£(P) contains a functionof positive arity , then(a) follows from (b) of Theorem 4.2 and (ii) of Theorem 2.1 together w iththefactthattheextensionof M defined in the proofof (ii) does, inthecase where£2' is takento be £2, satisfy DCA c2 since all£ l-rootSareused up intheconstruction. (To see this, take g(x) = f(x, .. . , x) where f is one ofthenew functions. By i nductionon d, one showsthatatno stagecan Pi be a d - 1 ancestorof gd(Pi) ' so unlessPi has alreadybeen used as a value of a new functionit will betakenas the value ofgd+I(Pi) whenthatis defined.) Part(b) follows from (b) of Theorem4.2 and thefactthatin thiscase CETc U {DCAd is, by an argumentlikethat that of Theorem2.1 part(i), a conservative extensionof comp£:.(p)(P) U {Gt(P)' --'G~tA}. -1
It shouldbe notedthattheseresultsonly hold for ep which are builtup using theKleenetruthtables,which havethepropertythatif thevalue of a truth functionis t or /, it does notchangeif one of itsa rgumentsis changed from u to t orf This is nottrueoftheconnective+----> used incomp£:. (P) and indeed thesentencesof comp£:. (P), which are obviously truein all3-valued models ofcompdP), may notbe true£:.at anystage. For example, if P is p(O),p(s(x)) t - - p(x), thecompleteddefinitionofpis V'y(p(y))
+---->
y = OV 3x(y = sex) Ap(x)),
which is false at each stagen becausefor y = sn(o) theLHS is u and the RHS is t. In fact, the n otionof truth atsome stagedoes not make sense for sentencesusingnon-Kleeneconnectivesbecausethey canswitchfrom trueat some stageto falseata laterstageand vice-versa. Take, for example, the instance p(s(O)) +----> s(O) = 0 V 3x(s(O) = s(x) A p(x))
LANGUAGE AND EQUALITY THEORY
391
of thecompleteddefinition of p. This is trueatstage0, falseatstage1, and trueatalllaterstages. Thereis alsosomethingstrangeabouttheclauses of P itself.I f these arewrittenwiththe Kleene - > thentheymay not betrue in all models of comp(P) , e.g., in this examplethereare modelswithelements a for whichp(a) and p(s(a)) are u so thatthe clause Vx(p(x)) - > p(s(x)) is alsou. If you wantP to be a 3-valued consequencecomp(P), of you have to write the clausesPofwith 2-valuedimplicationp :J q, 'if p is t, then q is t' (you can add'and if q is f, thenp is J' if you like).But then,like the sentences of comp(P) theyare not covered by Theorem4.2, indeed, the notionof trueatsome stageis not sensible. Following F itting,K unenextendsthe notion of truthatstagen into the transfinite where iteventually stabilizesa tsome closureordinal;if we call truthat thisstage'truthat 00 ' thenKunen'sTheorem6.6 statesthatsentence
f=3
REFERENCES Chan,D. 1988 Constructive n egationbased on the completeddatabase , 5th International conference and symposium on logic programming, Seattle, MIT Press, Cambridge, Massachusetts, vol. 1, pp. 111-125. Clark,K. L. 1978 Negationas failure , Logic and data base (M. Gallaireand J . Minker, editors),Plenum,New York, pp. 293-322 . Fitting , M. R. 1985 A Kripke-Kleenesemanticsfor logicprograms, Journal of Logic Programming, vol. 2, pp. 295-312. Kunen,K. 1988 Negationin logicprogramming, Journal of Logic Programming, vol. 4, pp. 289-308 . 1989 Signed datadependenciesin logic programs, Journal of Logic Programming, vol. 7, pp . 231-245. Lassez, J.-L., M. J . Maher,and K. Marriott 1988 Unificationrevisited , Introduction to foundations of deductive databases and logic programming (J . Minker,editor), Kaufmann,Los Altos, California, pp. 587-625.
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JOHN C. SHEPHERDSON
Lloyd, J . W . 1987 Foundations of logic programming, second edition, Springer-Verlag , Berlin. Maher,M. J. 1988 Completeaxiomatization of thealgebrasof finite,infiniteand rational trees, Srd Annual symposium on logic in computer science, Edinburgh, pp . 348-357. Malcev, A. 1971 Axiomatizable classes of locally free algebrasof varioustypes, Metamathematics of algebmic systems: Collected papers, ch. 23, North-Holland, Amsterdam, pp. 262-281. Mancarella, P., S. Martini,a nd D. Pedreschi 1988 Completelogicprogramswithdomain closure axiom,Journal of Logic Programming, vol. 5, pp. 263-276. Przymusinski,T . C. 1989 On constructive negationin logicprogramming,Proceedings of the North American logic programming conference, Cleveland, Ohio, Addendum, MIT Press, Cambridge,Massachusetts. Sato,T ., and H. Tamaki anddeterministic synthesis,Proceedings of 1988 Deterministict ransformation the 2nd Franco-Japanese symposium on programming of future generation computers II (K. Fuchi and M. Nivat, editors),North-Holland, Amsterdam,pp . 307-327. Shepherdson,J . C. 1989 A sound and completesemanticsfor a version of negationas failure, Theoretical Computer Science, vol. 65, pp. 343-371.
PART III PHILOSOPHY, MEANING, AND INTENSIONAL LOGIC
C. ANTHONY ANDERSON
ALTERNATIVE (1*): A CRITERION OF IDENTITY FOR INTENSIONAL ENTITIES
Abstract.The problem of formulating an adequatecrite rion of entit id y s and otherintensionalia was atone time considered the princifor proposition of an acceptabl egeneralintensionallogic. The ple obstacleto the construction objectionwas urged time and again, principallyby W. V. Quine, and st illhas its influence . I AlonzoChurchresponded directlyto thechallenge a nd attempted to incorporate v ariouscriteria ofidentityintofullyformalizedinten sionallogics, differentimplementation s of his logic ofsense and denotation . The logisticsystem whichwas to bebased on the m ostimportantof Church'sideas, synonymous isomorphism , remains unfinished . In thispaper I att e mpt to clarifythe pointandnature of demands for crite ri a of identityand to eva luate var ious objectionsagainst Church' scri te rio n. M y conclusion is that a modificationsugges ted yb certainallegedcount erexamples appearsto fit th e data bett er andso m er itsfurth er st udy.Importan tp art s of the e the developmentof a detailed logistic tre at me nt m e eting overallprojectrequir curre ntstandards of rigor. This will n o t be att e mpted her e, butis reserved for a technicalsequel.
1. CRITERIA OF IDENTITY
To fix ideas,z define a(potential) one-level criterion of identity for being an p I to be arelation R o satisfying:
(CI I ) pi (zd & pi (zz) :::>.
Zl
= Zz == Ro(ZI ' zz ).
As an exampleof this, we may takethefamiliarone, the Axiom of Extensionality of set theory. Here pi is being a set and'Ro(ZI' zz )' means that all th e members of Zl are members of Zz, and vice-versa . In thiscase and analogous ones to bediscussedbelow weshallspeak of criteriaof identityof thissortas ontological criteriaof identity. Now suppose thatG I , pZ, and R" satisfy:
(CI z) G 1 (x) & Gl(y) o . pZ(x, zt}& FZ(y, zz) o ,
Zl
=
Zz == R,,(x, y),
I
where pz is a function(on G 's) . Thatis, pz obeys thetwoconditions: (E)
GI(x):::> 3ypZ(x,y),
(U) G I (x) :::>. p Z(x, u) :::>. F Z(x, v) :::> u
= v.
ISec, for exa m ple, Grim 1951, page 20. 2 And
followingWilliamson 1990, pages 144- I 48. 395
C. Anthony Anderson and M. Zeleny (eds.}; Logic, Meaning and Computation, 395-427. © 200I Kluwer Academic Publishers. Printed in the Neth erlands.
396
C. ANTHONY ANDERSON
Then therelationR u will besaid to be a (potential) two-level criterion of identity. If certainotherconditionsare satisfied,we might be willing to say thatR u is a (two-level) criterion of identityforthe-range-of-F'<with-itsdomain-confined-to-C" . Many of the examples in the literature, proposed as illustrative of the generalidea of a criterionof identitycan be made to fit one or theotherof thesepatterns.O ne much discussedtwo-level c riterionis thecase wherethe GI'S arepredicatesor open sentences(ofsome given language)and F2 is the relation b etweentheseand theirextensions. Here R u is therelation which holdsbetweentwoopen sentenceswhen theyare true-of, or satisfied by, the same objects.Since we will be c oncernedwiththisexampleandits analogues forintensions,we callsuchcases of two-level criteriaof identityby themore suggestivename, "semanticalcriteriaof identity'l .i' The most pleasantcase occurswhenwe havecriteriaof bothsortsand if F I is related to F 2 and G I in thisway: ("Totality C ondition"). Therearevariousinteresting logicalrelationships betweenthesethings." If R; is wellunderstood,we may even be ableto usefullydefine anappropriateR o . In some cases, if otherthingsarefavorable,we can "construct" the FI'S as equivalence classesof G I 's o Here weareinterested in obtainingfurtherconditionswhich canguide us in oursearchfor criteriaof identityfor intensions. Considerfirstourparadigmof an ontological criterion,theAxiom of Extensionality justmentioned. Therearevariousways ofchoosingan appropriatecompaniontwo-level c riterion,butwe takeourcue fromQuine: Observe, in contrast,how wellthecorrespondingrequirementis met in the individuationof classes. Ibegan by saying thatclasses are identical when theirmembers areidentical;butwhatwe nowwantis a satisfactory formulation of a relation betweentwoopen sentences' Fx' and 'Gx' which holds if and only if'Fx' and 'Gx' determinethesame class. The desired formulationis of courseimmediate: it is simply '(x)(Fx == Gx)'. It does not talkof classes; it does not use class abstraction or epsilon,and it doesnotpresupposeclassesas values of 3The terminologyof semantical/ontological criteriaof identityis notappropriatefor many of theexamples discussed in the literature.J ohn Myhillalmostrecognizesthe distinctionbetweensemanticaland ontological c riteriaof identityfor intensionsin his discussionof Church's Alternative (0) : This [Church'scriterion]d oes notdeterminetheidentityor diversityof those senses which are not senses of expressions, but it does suggesta series of axioms concerningthe identityand diversityof senses from which various interesting c onsequencesc an be deduced. (Myhill 1958, page 82) 4Timothy Williamson(1986) has begun a studyof logicalquestionsaboutcriteriaof identitywhichshouldbe emulated .
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variables. It is as pure as thedrivensnow. Classes, whatevertheir on thisapproach. foibles, arethevery model ofindividuation (Quine 1975, page 7)
Unfortunately, thereis a use-mentionerrorhere whichobscuresan importantpoint. Quinemerely cites a formula of -order first logic as specifying the relation betweentheopen sentences'Fx' and'Gx', buttheformulaitselfdoes notspeak of thesesyntactical entities.Ratherthe(semantical)criterionin which holds between theseopen sentenceswhen questionmust be therelation '(x)(Fx == Gx)' is true . Equivalently, we may speak of therelation which holdsbetweentheseopen sentenceswhentheyare true-of or satisfied by the same objects. It is evidently required thattruth-of (or satisfaction) either be takenas primitivein themetalanguage, or be defined inset-theoretical termsthereinforthegiven language .P And thereis a hiddenparameterin thecriterionas statedby Quine, namelythelanguage whose opensentences are inquestion . 1 is being an open So let ust akeas ourinstanceof (CI2), thecase where G formula (of such-and-such a language L), F2 is has as extension (in L) (or determines- or, perhaps, denotes-in-L), and He, is therelation which holds true-of-in-L (or satisfied-bybetweenx and y when they are open formulas in-L) forexactlythesame objects. Thus, in semi-English: (CI2)S If x and yareopen sentencesof languageL, thenif Zl is the extensionof x (in L) and Z2 is theextensionof y (in L), then Zl = Z2 if andonly ifx and yaresatisfied(in L) by the same objects. The superscript'S'is supposed to suggestthatwe areultimately interested in a theoryof sets. Here, alas , if wetakeF 1 as being a set in (CId, thenthe TotalityC onditionwith respectto G1 and F 2 does not hold. For Lif is a , therewill besetswhich are not theextensions language ofthefamiliar kind of any open formulas L. in Still, this seems to be the semanticalcriterionof interest ." The semanticalprincipleis supposed to have a kind of epistemic priority over thecorrespondingontological principle." We imagine thatopen senPerhapsin thegeneralcase we tencesandtheirpropertiesare familiar to .us 5This observationmight lead one to questionthe "purity" of(CI2)S below. I am awarethatone might try to fin ess e the semanticalconcepts by m eans of some sortof "d eflat iona ry" tactic . Buteven apartfrom doubtsaboutthefeasibilityof such projects, we areseekinga geneml criter ion ofidentityfor se tsas extension s of predicates, and this seems torequiretheuse of sem anticalconcepts. 6Thereareinteresting "int ermed iate"p rinciplesthatmightserveas criteriaof identity forsets. For exa m ple, wemight see somethinglike Frege's Basic Law V as tellingus that conceptshave thesame sets as ext ensions iftheyare "materially equiva lent" . This does ogicalin oursense. It does rightly e mp has ize the notquitequalifyas semanticalor as ontol epis temologica lpriorityof attr ibutes or concepts ov er set s. It s uffers fromthefactthat w e curr e ntlyhave nodetailedtheoryof concepts. 7Compare Anderson 1980, page 220.
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C. ANTHONY ANDERSON
might justrequirethatthe entitiesin questionbe treatedby a systematic and well-understood theory. And it supposed is thatwe eitherunderstand directly, or by way of a theory,relation the ofsatisfaction or being-true-of for a givenlanguage . Extension-of-in-L is justa restriction of theidea of a set tothosesupposed tocorrespondto opensentencesof the given language and criterionitself, is to beantecedently understood,or atleastclarified by the perhapswiththeaid of anaccompanyingtheory. In the presentcase, the epistemicprioritycomes out clearly when we considerthepossibilityof proving thattwo sets are identical. Typicallythis is done by showing, in aparticular language,a certainbiconditional-the constituent formulas of which are takenas determiningas extensionst hesets in question . between (CI2)S and (CIt)s? Well, it is Nowwhatis thedesiredrelationship evidently t hatwe are to find R o , formulated in settheoretical terms,using E if desired, sothat(CII)S has thesemantical criterion ( Ch)S as "observational consequence".Of course this will be little of assistanceif we do not have a well-developed theoryaboutsets-butwe in fact do, namely, set theorywith the Axiom ofExtensionality functioning as ontological criterion . Here theremay arise thethreat of a regress. Quine (1975) observes that theidentityof sets isrelative to theidentityof theirmembers. Butin a set theoryallowing individuals,or non-setsof some kind,thequestionmay arise as to theontological criterionof identityfor these things. Somewhere the regressmust haltand explanations must come to an end . To illustrate thepoint, let us consider thesimplercase oftheextensional entitiescorrespondingto sentences . The desiredcriteriaof identityin that case would seem to be : (CI 2)T If x andyaresentences of language L andif Zl and Z2 are the truth-values determined(ordenoted)by x and y, respectively, thenZl = Z2 if andonly ifx and yarematerially equivalent in is, x andyarebothtruein L or bothfalse inL. L-that and (CII)T If Zl and Z2 aretruth -values,t henZl = and Z2 = t, or Zl = f and Z2 = f.
Z2
if andonly ifZl = t
betweenthecriteria -we could (and Here we havethedesiredrelationship often do) even define the relation of materialequivalence by reference to the truth -values. Suppose we continueto press ourSocraticquestion : "But whatis the criterion ofidentityfor beingeithert or f?" Heretheonly possible answer is thatthesetwothingsaredistinctand we can tell this in some cases. If we take t and f as being the values somethingknown of to betrueand something known to befalse-sayas of thepropositions'v'x(x = x) and 3x(x :f. x), respectively-then in some sense we knowwhatthe truth -valuesare and thattheyaredistinct.
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INTENSIONAL ENTITIES 2. CRITERIA OF IDENTITY FOR INTENSIONAL ENTITIES
Presumably,t hedesiredanaloguesforattributes are oftheseforms:
(Ch)A If x and yareopen formulas(of a languageL), thenif Zl is theattribute expressedby x and Z2 is theattribute expressed by y, thenZl = Z2 if andonly ifR~(x, y), and (CIdA If Zl and Z2 are attributes,then Zl R~(ZI' Z2),
=
Z2
if and only if
whereTotality may not beassumed and therelation R~ is well-understood . Using ouranalogy, it is apparently permissiblethatourtheoryof intensional entitiesemploytherelation R~ which is notnecessarily definableby appealto R~. Butthetheoryofintensionsusing R~, likethetheoryofextensionsusing E, shouldmeet variousformaland informalcriteriaof adequacy. (CI 2 )A is u nfamiliar to be "pureas thedrivensnow"- it may notemploy theinitially theoretical primitivesinvolved inR: and thequantification involved inR~, if any,shallnotbe explicitly''overintensions. The analogywith the extensional case of sets is notquiteexact. The ontological criterionfor setsappealsto the identityof theirelementsand theseelementsmay all be(but need not be) of a new kind. The expected relation betweencomplex intensionsmay wellappealto acomparisonof "constituents" , some of which arethemselvesnecessarily intensional entities. We shouldcertainly d emand thatthetheorygive a detailedaccountingof theallegedconstituent relation, buttheontological criterionis notdefeated simply by thefactthatit involvesreferenceto intensions. No regress isforthcomingif we suppose thatin some fundamental cases we can justtellwhethertwo simple expressionsexpressingintensionsare synonymousor, whatis thesame, thatthemeaningstheyexpressareidentical or distinct. Butof course we can,and when we do, we may suppose that thesynonymyis statedby, or is aconsequenceof, thesemanticalrules oft he language-the rules which codify theknowledgeattributable to acompetent, perhapsideallycompetent , user ofthelanguage. To my knowledge no one has offered compelling, a or evenplausible,a rgument thatif .aputativekind is notaccompaniedby criteriaof identity,t hen thealleged kind unacceptable is on groundsof obscurityor otherdefect. Until andunless a clear, careful, completesetof and conditionsforwhatmakes somethingan acceptable(actual,insteadof potential)criterionof identity is given, thereis scanthope for such anargument . But thedebatewillat leastbe advancedif we cansupplywhatwe taketo becriteria of identityfor intensions , closelyparallel to thetwo principlesaboutsets and extensions,
R:
BIn thepromised logis tic treatment, we propose to quantifyover everyt hingwhatev er. However, theconditionon quantification is designedto avoid circularity and it does not appearthatanyconceptual or epis te mologicaclircularity is reintroduced by ourprocedure .
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andsubjectto the(vaguebutsuggestive)epistemological constraints. Then it shouldbe possible foropponentsof intensionsto explain the respectsin whichtheproffered principles are inferiortheirset-theoretical to analogues andtherebyclarify the questionof theneed forandvalue of such principles. Forsimplicityin the discussion to follow, we confine attention tothequestionof criteriaof identityforpropositions, buttheunderstanding is thatthe more general case of attributes is also inquestion . Thus we seek a clear relation among sentences and, ultimately, a theoryof propositionsembodyinga relation R~, such thatthefollowing holds for any given language L:
R:
(CI 2 t If x and yaresentences (of languageL) and ifZl and Z2 are then thepropositionsexpressed (inL) by x and y, respectively, Zl = Z2 if and only if R~(x, y), And (ChtIf Zl and Z2 are propositions,then Zl = Z2 if and only if R~(ZI,Z2)' is to governpropositions.The latter is to yieldtheformer as aconsequence whenappliedto theparticular propositionsexpressed bysentencesin L. In thepresentpaper, we devoteconsiderableeffort to the formulation of the firstsortof criterion,and only briefly indicatetheformaltheorywhich will accompanythelatter. 3. SYNONYMOUS ISOMORPHISM
RudolfCarnap(1947) proposed thattwo expressions, say sentences,are synintensionally isomorphic. Considertwo exonymous ifand only if they are pressionsin a singlelanguage . Assume, for simplicity,thatbothare written in primitivenotation."I f thesemantically relevant p artscan bematchedup in such a waythatthepartsareL-equivalent, thatis, necessarily equivalent, isomorphic-otherwise not.!" thentheexpressionsareintensionally So, consider ahypothetical arithmetical languageutilizingboth Arabic and Roman numerals . In such alanguage'V > I' might be intensionally 9For most formalizedlanguagest hathave actually b een used, this willalmosteliminatethedistinctexpressionswhich are intensionally isomorphic. Really, ihorderto have anythingsignificant,we wouldconsider,followingChurch(1954), especia llytreatments thatallow"defined" notationin the objectlanguage . Thatis, theformalizedlanguage is supposed to have rules of definitionwhich allowaugmentation of theobjectlanguage by addingnew definednotation . This is in contrast to themethod, now common among logicians , of regardingalldefinitionsas being metalinquistic abbreviations . Cf. further t he discussionin Church 1956, note168. Natural l anguagesynonymies, forexample between 'procrastinates' and 'putsthingsoff', areprobablymost usefully seen asinvolvingonlythe objectlanguage . 10 Actually , if boundvariablesa represent , thematteris a little more complicated.Carnap in factallowst hatalphabetic c hangeof boundvariablesp reservesintensional structure . We shallhave occasionto discuss thetreatment of variablesbriefly, innote20 below.
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isomorphicto '5 > 1', butnot intensionally isomorphic to theL-equivalent '2 + 3> 1'. Strangely,Carnapdoes not go on to c onstructa theoryof intensional entitiea'!correspondingto intensional i somorphism-as his workand that of his successors in possible worlds semanticshas producedin thecase of necessaryequivalence as criterion of identity.B utthe idea isn earathandand in his critiqueChurchevidentlyhas such a goal in mind . So let usconsider intensional isomorphism as a criterionof identityforthepropositions(of a certainsort)expressed bysentencesin a given language . Notice thatthis is not anontological criterionwhich is directlyabout intensionsor meanings; it's aboutexpressions in aparticular languag e-whatwe have calledsemantical a criterion . But,as we have urged, one might striveto construct a theoryaboutintensionsdirectlyand use as one of the conditionsofadequacyfor thetheorythatthe ontological principleentail,in applicationto anyparticular case, thecorresponding semanticalprinciple. Church(1954)12 provided acounterexample toCarnap's criterion , atleast if it isintendedto be used inconnectionwith a logic of belief . Church's exampleis a language in which'Qn', with'Q' a primitivepredicate,expresses thatn is a positive integer greater t han1 for whichtherearepositiveintegers x, y, and z, withz" + yn = z", In thesame language,"Pn' expresses that n is less than3. Again'P ' is a primitive predicateof the language. If Fermat'sLastTheoremis true , andwe now knowthatit is, '3n ("" Pn & Qn) ' is necessarilyequivalent to, indeed,intensionally isomorphic to'3n("" Pn & Pn)'. But,Churchsays, surely one could believeproposition the expressedby thefirstwithoutbelievingtheexplicitcontradiction expressed bythesecond. In thelight of recent developmentsin numbertheory, one might change the argumenta bit to avoidunnecessaryside issues: Andrew Wilesrefutedthe propositionexpressed by the first of these sentences. Why not say abouthis -"Well, big deal, we've knownthatatleast since Ari stotle, alleged discovery andprobablybefore. The propositionthat3n( '" Pn & Pn) is obviously false. So why all the fuss aboutWiles?" Surely this would nottaken be seriously . It is relevant to observe in response thatwe now have aproof-andcertainly we owe this to Wiles . But it is still correctto saythatwe now know the 11 Rather , he employs the notionof inte nsiona lsomorphism i in the m etalangu a ge and attemptsto analyz e beliefin thoseterms without everconsideringthe p ostulation of propositionscorrespond ing to t hisidea. Kinichi Fukui made theoddityof this , given C arnap's other work, clearto me. Carnap's (1946) use of the "Looking-aroundMethod", so called by Ign a cio Angelelli (1984), is reallyjust an ap plica t ionof the generaluse of crite ria of identitywe have described. Certainly A ngelelliis rightto criticiz e them ethodif it si considered a echniqu t e " , butit need notbe thuschara cter ized. Matt e rs here are more for "definingby abstraction m pts in the natur al sciences to find theori eswhich willexp la in or le ss analogousto the atte thedata . 120ther s have re -discover edor re-utiliz ed Carnap 's crite rion,butthey do notrespond to Church 's criticism . See, forexa mp le, Cresswell 1985 and its review by Anderson (1991) .
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C. ANTHONY ANDERSON
propositioncommonlycalled"Fermat'sLastTheorem", whereasbefore we did not. Various moves arestillpossible for the Carnapian. The most immediatesuggestionis thatwe ascendintothemetalanguage and requirethatthe expressionsinvolved inthesemanticalrules for t hepredicatesof theobject language also beintensionally isomorphic. Of course if we formalize themetalanguage,it toomight pass the test,even whenapplied to the example before us-ifit usesprimitive predicatessynonymouswiththeobjectlanguagepredicates"Pn' and 'Qn'. Then the questionmust be asked again aboutthemeta-metalanguage, andso on, up thehierarchy. It reasonable is to insistthatthethreatened infiniteregressof metalanguages must come to explicit,one willp robablyrequiretheideathat a halt . To make thecriterion some meaningsareintrinsically simple-" and,ifthecorresponding s emantical criterionis to serve itspurpose,simplicitymust be somehow clearly p resent or absentin particular cases. The more seriousdifficulty, it seems, that is thecriterion,when applied to twosimple expressionsor, really, to two expressionswithsimple meanings, seems utterly arbitrary .l" Thattwosynonymouscomplexexpressions shouldhave acorrespondence betweentheirmeaningfulpartshas acertain plausibility-every meaningexpressedby theone has to beexpressedby the other(although it may do so by way of definitions) . Butwhy thinkthattwo simple meanings,if they(or theirexpressions)are logically equivalent, must 13Fregemight haveobjected.In his discussionof negation,he pointsout(Frege 1919, in Geach and Black 1966, page 125) thatit isoftenquitedifficult to saywhichof twosentences expresses a negativeand (presumably)more complex propositionand which expresses merelysome pure affirmative. Church (1954, note8) also expresses some reservations aboutthenotionof simplicity. 141 have urgedthiscriticismbrieflyelsewhere : in Anderson 1987, page 160, note14, and in Anderson 1991. DonaldDavidson (1963) alsocomplainsaboutthe ad hoc character of intensional isomorphism, buthe is consideringmainlyits merits as a solutionto theParadoxof Analysis. Given thesemi-Fregeannatureof Carnap's solutionto Frege'sPuzzle , "How can a = b, if true,differ inmeaningfrom a = a?", and its applicationto thefailureof thesubstitutivity of identityin modal contexts,D avidsonargues(followingChurch,1946b) thata parallel treatment is calledfor inthiscase and in applicationto substitution in beliefsentences . c riticism: We quiteagree,butwouldatleastqualifyhis further Equivalenceis firmly based on sameness of extensionalreference; Lequivalence is providedwitha similarlyimpressivesemanticground,s ameness of intensional reference.B utintensional i somorphism has no such factual or (Davidson 1963, page 342) theoretical justification. "Intensional r eference " is justa name for apostulated b inaryrelation involvedin Carnap's two-level c riterion of identity,a nd thetheoretical basis involves aone-levelp rinciple for Carnapianintensionsand a correspondingequivalencerelationbetweenthem. What is reallyd oing thework is(i) thefactthatL-equivalence is a relation b etweenexpressions epistemic access, and (ii) Carnapprovidesa mathematically to which we have certain a workabletheoryof intensionsin termsof possible worlds(for him, state-descriptions) and functionsinvolvingthem. Of course,similarthingsare requiredfor intensionali somorphism and its descendants .
INTENSIONAL ENTITIES
403
be identical?T rue, clearcount e rexamplesare hardto come by, buttheidea itselfseems to have noreasonable basis. 15 Church' s emended criterion,synonymous isomorphism, requiresthatin orderfor twoexpressionsto besynonymous,thereplacement ofpartsis to inthesynonymyandnon-synonymy volve only synonymousexpressions-where ofsimple expressions,andthesynonymyofsimple expressionswithcomplex ones, viadefinition,is supposed to builtintothelanguage.Such thingsare to bejustgiven bythesemanticalrules oft helanguage. Now some will find t hislast ideapuzzlingor, even, acheat. We arelaboring mightilyto give aclearcharacterization of synonymy (suitable , remember, for use in asemanticalc riterionof identity) . Churchseems to beproposing thatwe cuttheGordianknotby justsupposing thatthesemanticalrules alreadytell usw hatis synonymouswithwhat! Well,thesituationis not reallyas justdescribed.!" The thoughtis, I believe,thattwothingsareclearaboutsynonymy,pre-analytically conceived. languagesor technicalextensions One is that,in the case ofconstructed languages,one may just stipulate thata simple expressionshall of natural henceforth be synonymouswitha complex(or simple, iftherewere anyreason for it)expressionalreadypresentandunderstood in thebase language . Even Quinefindsthiscase unproblematic (Quine 1953a). Thesecondthingis that synonymsmay be substituted forsynonymswithpreservation of meaning. I literally truefor some ofthenatural hastento addthatthislast isnotreally languagest hatwe knowand love. Substitution fails topreservemeaning, or evendenotation a nd truth , in quotation c ontextsa nd others . Churchis supposing, I conjecture, thatthefailure of substitution is suchcases, likethe failure of thesubstitutivity of identityin modaland beliefcontexts,is to be t alkingaboutthesame things explainedin theFregeanway-oneisn'treally in thosecontexts . Such anomaliesare to beeliminatedin themore precisely formulated languagesto whichthesemantictheoryis to bedirectlyapplied, andtheapplicationto othercases will be am atterof approximation . 17 Notice alsothatthereis a pretensethata languageis "given" by giving its semanticalrules,includingperhaps extensional rulesaboutdenotation. We hope thattheseidealizations will provetheirworthin a finishedtheory, butthiscertainly cannotbe firmlypredicted.Here we arejustemulatingt he natural sciences and, especially,t heiruse ofmathematicsas appliedto the physicalworld. Church'soriginalstatementof the motivatingidea behind his criterion 15It is worthnoticingthatourgraspof meaningsand of synonymyis in factepi stemologically p rior to ourknowledg eof logicalnecessityor equivalence--and notthe otherway around . 16Recallalso thedicsu ssion at theend of section 1 aboutthe possibilityof a vicious regress. 17The generalassumptionthatcertainirregularities must be removed is reallyalread y presentin the pretensethatextensionalsemanticalrulesand principle s, e.g., Tarski's Schema T, fitactualn atural languages,as applied to, say,a mbiguou sexpressions.
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C. ANTHONY ANDERSON
appearsin an abstractdescribinghis initialwork onthelogic of senseand denotation (Church 1946a) and was apparently formulated i ndependently of Carnap. He says therethat(further) axioms aboutthenotionof sense (in one sense) are to beo btainedby using as aheuristicprinciplethat"two names are assumed to havedifferentsenses in all cases where it is not alreadya consequencet hatthesenses arethesame." Laterclarifications led him to see this as based on theintuitionsb ehind synonymousisomorphism-andthe heuristicprincipleinvolved came to be called "Alternative (0)" .18 A similar thought is expressedby JohnMyhill, It appears howeverthatin view ofthephilosophicalapplicationwe
intendto make ofmodal [better:intensional]logics, we should never identify intensions unless we are forced to; forthefeweridentifications we make themore flexible will ouranalysis be of belief -sentences." (Myhill 1963, page 306. My insertion .)
This is provocative,butwhatdoes it really mean? Churchdoes not here tell us whatwe are allowed assume to in orderto seewhatconsequencest here are aboutidentityof sense,and Myhillsimilarlygives us noguidanceas to whatcould force us to identifyintensionsor senses.l? The answerto this questionwhich is presupposed by Church'ssynonymous isomorphism is this: we assume onlythatthe stipulationsinvolved in definitionsproducesynonymy, orsamenessof sense, andthatsynonymsmay be interchanged withpreservation of sense.20 Thus explained,Church's suggestionhas an initialintuitiveappealand seems to depend only on thepreservation of theessentialand cl e arfeatures of synonymy(slightly ealized) id . l8T he bett er terminology is to ca llsy no ny mou s somorphism i the sem a nt ica lcr ite rio n a tive (0)" for the strategywhich involvesformulating an and to erserve the name "Alte rn ontological rite c rionand accom pany ing theo ry insuch a way as to entail, when applied to particularlanguag es , the sema ntical crite rion for thatlanguag e. 19In fact,Chur ch o d es not reallyadher e to theprincipl e asstated in his furtherwork. See the discussionin Anderson 1998. 20We have been ignoringthe factthatbothCarnapand C hurch allowtha talphabe tic cha nge ofboundvariabl es, in the usualsorts offormalized langu a ges , preserves sense. One languag e ana logues ofsuch things , supp oses thatthey wouldsay the sa me aboutnatural e.g., "he" and "thatman (or m al e) ". This is an interestingexcept ion tothe intuitiv e motivationand illustra tes thatwe may be "forced" to identifysenses in orde r topreserve the appearances . It appearsthatthe m eaningof a bound variabl e is not to be expla ined by , say, simply referringto thesemantical rule which speci fies (thesense which gives) its range. Rather , the m eaningof such varia blesalso involv es the relations (of their occurrences)to one ano t he r incert a incontexts. Those relati on smust be preserved when subs t it u ti ng anoth er (bound) var iab le. Inthis connection , see also Church 1956, page 10, note 24, Quine, page 70, andthe discussion of "wiring diagram s" n i Kaplan 1986, page 244. (T he Quine-K aplan idea need s to be supplem ented to take acco unt ofthe meaning contributed by the kind of the vari abl .)e T he re is neverth eless an analogy with the inte rc h ange of sy nony mo usconst ants which gives some suppor t ot the except ion.
INTENSIONAL ENTITIES
405
4. ALLEGED COUNTEREXAMPLES TO SYNONYMOUS ISOMORPHISM
A numberof different sortsof counterexamples have been urged againstsynonymous isomorphism as criterion of synonymy. The most telling cases'" fall into two main classes which 1 shall call commutation (1) and conversion of non-symmetricals, and (2) commutation of symmetricals. Undertype(1) are includedsuch cases involving verbs (or relations) as:
'a 'a 'a 'a
lovesb' 0 'b is loved bya' , is taller thanb' 0 'b is shortert hana', is greater t hanb' 0 'b is lessthana', precedesb' 0 'b succeedsa' (or 'b follows a').
(I use single quotes insteadof cornerquotes, and '0' to signify alleged synonymy.) A plausible case involvingnon-symmetrical a connective,r ather thana relation, is: 'Q if p' 0 'P only ifQ'. It seems clearthatno explicitdefinitions,even pretendedto be involved in thesemanticalbasis of English, will have theconsequencet hatsuchthings are synonymies. It is possible to argue , with some show of plausibility, thatnone ofthese are really exactsynonymies, but ratherthatthey arejustobvious logical equivalents which nevertheless have some (slight) differenceinformation in content.P To considerthe first and easiestexample,represent'a lovesb' as '(a L b)'. 23 If weintroduce' AX(YL x )Ay' as alambda-expression of a special kind which allowslambda-conversion with'AX' operatingto its left and 'Ay' operatingto its right, the form of'b is loved bya' might betakento be'b(AX(yLx)Ay)a'. Perhapsthefirstlambdaoperator correspondsto 'is__-ed' and the second to 'by'. So we have toperformperhapstwosteps of lambda-conversion to get fromtheactive to the passive and these logical steps, thoughelementary, 21I have dealtelsewhere(Anderson 1987) withGeorgeBealer's proposed count e rexample, 'v outweighs z' 0 ' t heweightof y is greater thantheweightof z' . Diana(now Felicia) Ackermannproposed in conversation the example (or a close elat r ive thereof) 'J ones is believed by Jones to be bald' 0 'He him selfis believedby Jones to be bald'. This is of a rather differentsortand prettyclearlydep ends on meaningswhich arerelativeto conte xt. Thatis, exceptforthefactthat'He himself 'conveys masculinity while 'Jones' does not(a defectof theexamplewhich is easilyremedied), in theconte xt these two expressions are synonymousand we do notherehave a realcount e rexample . Initially, and forsimplicity, age we aresupposing thatgenerallysuch conte xtsensit ivity iselimi na ted fromthelangu beforethetestis applied. Butsee thediscussion of variablesin note20. 22See, for example, King 1996, pages 503-504. 1 do notendorse the principleP which he discusses andrejects, butnevertheles s continueto be troubledby cert a in oft he alleged erexamples to synony mousisomorp hism. count 23David Kaplaninventedthis, or a simila r,operatorand used it inlecturesa t UCLA long ago .
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C. ANTHONY ANDERSON
are sufficient to changethemeaning.P" In thesecond example it seems natural to see 'taller'and 'shorter'as c omplex-themodifier 'er' even has akind of logic.This being so, logically thestepfrom 'a is taller thanb' to 'b is shorter t hana' is veryplausiblytaken step.25 Or so one mightargue. to be a real logical More difficult to resist examplesof are commutedsymmetricals:
'a equalsb' 'P and Q'
0 0
'b equalsa',26
'Q and p'.27
Here we have arelation or connectivewhich isnecessarily s ymmetric. Other examplesof thissortof thingwillreadilycome to mind.28 Again, the determinedSynonymy Isomorphistcan offer somereasoned thesethingsseems to resistance.PAnd maybe he is in theright. None of 240f coursesomeone mightmaintainthatlambda-conversion p reservesmeaning. See the discussionof Alternative (1) below. 25BertrandR ussell(1903) arguesthequestionand rejectstheidentity,p artlybecause: [Ilf weare to holdthat"a is greaterthanb" and "b is less thana" arethe same proposition,we shallhave tomaintainthatbothgreater and less enter intoeach of thesepropositions,whichseems obviouslyfalse... (page 228) This might be seen as a "distinct-denotation" argument,of thesortI offer belowagainst associativeandreflexivem eanings. Such argumentsa renotconclusive,since it may always be claimed,withmore or lessplausibility,thattheconceptis conveyedimplicitlyand the relevant d enotation is determinedthereby . PeterGeach (1957) might be seen assupportingtheopposite view: For notonlyis theconceptof a pair of converserelationsa singleand indivisiblementalcapacity(eadem est scientaia oppositonJm); theexerciseof theconceptin judgmentalsobringsin thetwo relations equallyand simulto b is theverysame act taneously, fortojudgethata bearstheone relation as judgingthatb bearstheconverserelation to a. (page 52) Buthe does notofferany independentsupportfor thisclaim. . 26JohnWallaceproposed thisexamplein conversation 27This example is due to Carnap (1963, pages 899-900), wherehe observesthatthe commutativity of conjunction is commonlytakento preservemeaningin ordinarylanguage "even in itsmost carefuluse ofindirectdiscourse,forexample,in thestatement of awitness underoath."A replyalongthelinesof King 1996 (op . cit .), thattheequivalence d epends on specialpropertiesof "sa id that",t akessome of thestingoutof theseexamples. 28In thesame generalc lassareexampleswherethecommutationinvolvedis notconfined to sentencesor singulart erms,e.g., 'The yellowcarrandown thestreet'0 'T he car,which . (Ri chmond Thomason proposed thisexampleto me in was yellow,r andown thestreet' conversation .) Butherethereis alt e rationin therestof thesentenceandhencesome room to arguenon-synonymy . Thereare howeversimilarexamples which do notinvolvesuch additions : 'AlmightyGod createdtheuniverse' 0 'God Almightycreatedtheuniverse'( d. also Richard 1990, page 31) . We do nottreatthesethingsexplicitlybelow, but it will be airlyclearhow f wewould dealwith them, once questionsof logicalform have been settled.Some such examples, anyway,might be takenas havingtheform: ' . . . (£x)(<jJx & 'l/Jx) .. .' 0 ' . . . (£x)('l/Jx& <jJx) . . .' , so thatthetreatment shouldfollowthatof conjunction . 29See Richard 1990 and King 1996 for some considerationssupportingthis view. I remainsuspicious of argumentsthatrely onthedifferencesb etweensuch thingsas 'John
INTENSIONAL ENTITIES
407
havetheforce of asynonymybased on astipulation-our paradigmcase. The plotthickens a bit when we cons ider alsotechnical extensionsof Enformalized glish, mathematical Englishsay, and examples of (interpreted) languages . Thereone finds such possible counterexamples as these: 'a < b' 'P::> Q' and
'a=b' 'P&Q'
0 0
0 0
'b> a', 'Q C p'30 (commutationandconversionof non-symmetricals) , 'b= a', (commutationofsymmetricals). 'Q & P '
The technical extensionsa ndformalizedlanguages areactually more diffit hanexamplesfrom theinformalnatural languagebecause cultto deal with stipulation is entirelypermissible in thosecontexts . Even if forEnglishwe can somehowestablishthat'P and Q' and 'Q and P ', say, arenotreally , that'P & Q' is nonsynonymous,does it have to be so, in everylanguage synonymouswith 'Q & P', no matter what concept of conjunction we assign t hatin ourtechnical extensionof Englishor to '&'? Can'twe juststipulate 31 in my constructed language,order does not count in this case? And if the principle shouldjusthappento hold for English,thatfact may us be bestseen as psychological or even accidental . Whatdoes this tell about propositionsand othercomplex concepts? uses ofnotation. We arethuswell advisedthentoconsidersome technical Some may plausiblybe seen as involving the synonymy of expressionsrelatedby commutationwith conversion. One commonly occurringc andidate , therelations of less and greater amongnumbers: mentionedalready, involves (la) But when we wish toindicatethata precedes andb follows inthe natural scale, we employ one of theinequalities: a < b, read"a is lessthanb"; b » a, read"b is greater t hana." (Fine 1906, page 8) (Ib) Definition : In an ordered domain, the two equivalent statements a < b (read "a is less than b") , b > a ("b is greater than an) both mean that b - a is positive. (Birkhoff and MacLane 1944, page 8)32 deduced thatP and Q' and 'John deduced thatQ and P' , and the like, to conclude non-synonymy . These argumentscome quite close to begging the questionand dep end on an assumptionaboutthe intensional op eratorin questionwhich does not holdin all cases. Namely, thatthe proposit ion expressed does not depend on synt act ica leatur f es of theoperand-andthis assumption fails incases not nec e ssarilyamountingto hidden quotation . An argumentin Linsky 1967, page 34, similarlydepends on such anassumptionabout thesyntactical insensitivityof 'Jones deniedexplicitlyt hat ' . Church's"Translation A rgum ent" does not seem to m e to settlet hem att e r (d. Linsky op. cit., note 1, page 34) . 30This lastis Church' s notationforconver se implication . See Church 1956, page 37. 31No, we cannot , as willeme rge fromourdiscussion below. 32Essent ia lly s imilarto this lastis:
408
C. ANTHONY ANDERSON
Of course, we supposethatthedefinitionsare notintendedforjustthecases of the particular constantsor variablesexhibited. And grantedthatthe examples aresomewhatdifferent, we may ask, whattheoretical accountare question , we will be we to give of these procedures?If we can answer this in a positionto betterdecide theirforce as proposedcounterexamples to synonymousisomorphism. A fairly clear candidateforcommutationofsymmetricalsinvolves anotationcommon in settheory: (2a) A set havingjusta few elements is usually denotedby puttingbraces arounda list of symbolsdenotingits elements . No significance is attached to the order in which the symbols of the list are written down, andseveral symbols in the list may denotethe sameobject. (Warner 1965, page 5, my emphasis) Again, butless clearly referringthemeaning to ofnotation: (2b) For a set, th e orderof succession of its elements shallmatter, not pro. (Kamke 1950, page 1) vided thatnothingis said tothecontrary If similarthingsare not actually often explicitly said connection in with identityor conjunction , it is easy to imagine t hatthey might have been. 5 . CONTEXTUAL AND EXPLIC IT DEFINITIONS ; CONNECTIVE S AND PROPER SYMBOLS
To see our way clearly throughthesematters,we must distinguishbetween contextual definitions and themore usualsort,let us call the latter, explicit definitions. For clarityon such topics , one naturally turnsto thewritingsof Alonzo Church: An expressionA introducedby contextual definition -i.e., by a definikinds of expressionscontainingA, as tionwhich construesp articular abbreviationsor substitutes forcertainexpressionsnot containingA itself,butprovidesno suchconstruction for A itself -is an incomplet e symbol in thissense. (Church 1942)
In Church 1956 he adds: In such a case, where complexnotation a i ntroducedby definitioncarries thefalseappearanceor suggestionthatsome partof thenotation Now we can define the symbols <, >, ~ , ~ , called, espectively, r less than, greater than , less than or equal to, and greater than or equal to, as follows :
x
< y > x
m eansthaty - x is positive; m eans thatx < y ; x ~ y me ans thateither x < y or x = y ; y ~ x m eansthatx ~ y. (Apostel 1967, page 20)
y
INTENSIONAL ENTITIES
409
is to betakenas denoting(orotherwiseas havingmeaningin isolation), definition. . . . it is usual tospeak of contextual On theotherhand, D6 [[A == B] -+ [A ::) B][B ::) All (for example) would not o rdinarily be called caontextual definition ofthesign == . , .. The difference isthat,in thenotation introduced by D6 .. . thereis nothingto suggestthat.. . the sign == standingalone is significant in any way (e.g., as an a bbreviation of a formula of a logistic system). . The arrow'-+ ' is Church'ssign (Church 1956, page 323, my insertion forequalityby definition.) For lackof a betterterm, I shallincludesuch thingsas D6 as contextual definitions.S uchdefinitionsa reentirely p ermissiblebecausetheyarein fact, if notin motivation,s tipulations . And this is so even if we regardthedefinitionas extendingtheobjectlanguageso as to includethenew notation . In Church'sterminology(Church 1956, page 12, note30) we may say that a contextual definitionis a stipulated sense-concurrence. The authorgiving thedefinitionhas stipulated t hatthis expression(and its alphabeticvariantsand instantiations by terms) is in everycase to mean thesame as the corresponding e xpressiongivenas thedefiniens. to therelateddistinctionbetween We shallalsohaveto attendcarefully proper symbols and connectives, again as explainedby Church. A proper symbol is a symbol . . . having meaningin isolation, the primitivenames as denoting(or at leastp urportingto denote)something, the variableshaving (or at leastpurportingto have) anon-emptyrange. (Church 1946, page 32) Improper symbols do not have meaning in isolation;t heyare syncategorematic, buttheycanbe combined withpropersymbols to form longerexpressions whichdo havemeaning. Connectives arecombinationsof impropersymbols which may be used together with one or more c onstantsto form orproducea new constant . ... [I]f we replace one or more theconstants of each by a form of thatconstantamong its values , the rewhich hasthedenotation sultingexpressionbecomes a form(insteadof aconstant);and the free variablesof this resulting form arethefreevariablesof all the forms (one or more) which were unitedby means oftheconnective(with each otherand possibly also with some constants)to producethe resulting form. In orderto givecompletelythemeaning-producing c haracter of a particular connectivein a particular language, not only is it necessary of thenew constantin everypermissible case to givethedenotation thattheconnective is used together with one or more constants to form such a newconstant , butalso, for every cas e thatthe connectivemay be used with forms or forms and constants to producea resultingform, it is necessary to give the completescheme of values of thisresulting form for values of its free variables . (Church 1956, pages 32-33)
410
C. ANTHONY ANDERSON
Observethat,on thisaccount,contextual definitionsmay in all cases be seen thisterminologyis not as introducing (one or more)connectives-although . actually used intheliterature and emphasize these distinctions,let usconsidera simple To illustrate form of a possibledefinition : (CD) F(x)
=df G(x)
& H(x) .
This is a more or lessstandardform, among philosophersa tany rate,of presentingdefinitionsof predicates. But thedifferencebetweenthis, seen as a contextual definition,and as anexplicitdefinitionof a predicateis not consequence.P usuallynoted-indeed,it is often of no real Taken as acontextual definition,this does not i ntroduce a meaning,strictt hatif juxtaposition of a predicate ly so-called, forpredicate a . Note well with avariable(or otherterm) enclosed inparentheseshas alreadybeen assigned ameaning (say applicationof functionto argumentor converse thisas astipulative definition elementhood), thenwe arenot atlibertyto offer that'F' has a of a predicate'F'. We cannot,by fiat, make it the case meaningwhich, whencombinedwithournotation shallpreservethemeaning of juxtaposition in 'F(a)', say, and also makethatmeaning identicalwith thatof 'G(a) & H(a)'.34 Butwe cancertainly introducea newpredicateby a purelystipulative, explicitdefinition in some such way as: (D) F =dfAX(G(X) & H(x)) . In (CD) theexpression'F' is best seen as aconnective (admittingthe c ase-apredicate oddityof theterminology).In (D) we have aquitedifferent is explicitlydefined as apropersymbol and has been assigned meaning a in isolation . Both of theseare of courseentirelypermissible methods of conflated. definition,b utin thepresentconnection,they must not be Synonymousisomorphism, accordingto itsintuitivemotivation,should be bothexplicitandcontextual understood to allow s ubstitutions on the basis of definitions . 33Thedistinction b ecomes importantwhenintensional meaningsareunderconsideration . GeorgeBealer,forexample, does notseem torecognizethedistinctionin Bealer 1982 and thusthereseems to arisefor histreatment a problemdue toF . P. Ramsey (1925, in Mellor 1990, pages 14-15) . (Cf. Anderson 1987, page 120, for details.) Once thedifference betweenthetwotypesof definitionis clearly s een, thepuzzledissolves. Bealerpartlysets mattersr ightin Bealer 1994, butglidesoverthefactthathis previousfailuret odistinguish thetwo types of definitionactually leadsto problemsfor theapplicationof his suggested his note3, page 165. analysesof variousnotions-seefurther For therecord,I do not,anddid not(in Anderson 1987), accepttheargumentwhich he uncharitably attributes to me (Bealer 1994, page 142). I was using somethinganalogous to theschema he calls,in thelaterpiece, "highlyintuitive " (Bealer 1994, page 154), viz. "If Fx iff df .. . x . . . , thenthepropositionthatFa is thepropositionthat.. . a . . ." . 34AlonzoChurchbroughtme to see theproblem aboutthepossible clashof meanings in suchcases.
INTENSIONAL ENTITIES
411
6. COMMUTATIVE MEANINGS
Armed withthesedistinctions,let usreturnto theexamples. Considerfirst thecommutationwithconversion cases , (la)and (lb). Thereare twoimmediatelyplausiblereconstructions for(la),it being indifferent to the case whetherwe seethemas explicitorcontextual definitions . On theone, 'a < b' is being setsynonymouswith'a precedesand b follows' and 'b > a' is being setsynonymouswith 'b followsa nd a precedes'-the not-implausible -assumptionbeing thattheselast two Englishexpressionsare synonymous. On thisshowing,the"readings" arejustthat,suggestivebut not necessarily intendedas themselvessynonymies. In this case, thematter devolves to the natural language,thestipulation havingsucceedingonly in thequestionof identityof meaning. Similarlyif wetakethe distransferring playedreadingsin Englishtoliterally give meaningtotheformalexpressions, thequestionis thenwhethertheexpression'a is lessthatb' is synonymous t hana' , as itarguablyis not. In eithercase the in English with ' b is greater stipulation has onlysucceededin makingthesynonymyof formalexpressions dependon thatof natural language c ounterparts. The example (Ib) is most naturally seen as involving c ontextual definitions. Both 'a < b' and 'b > a' are being treatedas connectivesa nd are stipulated to besense-concurrent with'b - a is positive'. We might regard'a > b' as asserting thata is greaterthanb and see 'b < a' as assertingthesame, exceptthattheconvention is thatthelatter is to be read from rightto left . (Of course, we might also see bothof themas from assertingthatb is less thana, with the first formula being read right to left .) Thereis here really no difference fromcorresponding the c ontextual definitions . legitimate,butobservethatthecorresponding Bothof theseareperfectly explicit definitions wouldequirean r additional assumption if we are to con cludethesynonymyof 'a < b' and'b > a'. In effect,t heassumptionis that thereare relational attributes or meaningswhich can erveas s simultaneous "solutions " for'< ' and "> ' in thetwoequations'a < b =df b - a is positive' and 'b > a =df b - a is positive'.35 And this does not go withoutsaying. Withinthecontextof mathematics,the difference between obvious logical equivalence and synonymyis oftenlittle significance.36 So wemightview, in all these cases, thedefinitionsas onlyrequiringt hatrelation between definiens and definiendumand thusas not really bearingdirectlyon thequestionof synonymy. Given these observationsand consideringthe intuitivecharacter of the 35Such a solutionis po ssible if we regardlambda-conversion as preserv ing sense and l c itdefinition s to amountto esse nt ia lly t he sa me thingin (hence) takecontext uaandexpli these cases. See the discussion of Alternative(1) below. e and obvious logicalequiva lence is 36It cannotbe thesame since synonymy istransitiv not.
412
C. ANTHONY ANDERSON
allegedcounterexamples , it does not seem to be requiredthatthetheoryof meaningmake specialprovisionsfor such cases . The allegedcounterexamples in cases ofcommutativityof relations and connectiveswhich are(necessarily)s ymmetricalare lessdearlydefeasible . 'a equalsb' and This is partlybecause oftheintuitiveforce of the examples: 'b equalsa' justdo not seem to differ meaningatall in - even recognizing thefactthatnot all relationsare symmetrical.And thematteris equally troubling in thecase ofconnectives,forexample'P and Q'. Let usconsiderfirst the more difficult case of a formalized language with a connectiveforconjunction.One ofourquestionsis whetherone canproduce by a definitiontogether with a stipulation that order is not significant, say, a "commutative"conceptof conjunction.One cannot. Once we have assigned meaning a to, forexample, "&" , or tocontexts in which it occurs, it is not clearly withinour power to make t hatmeaning is defined (orexplained commutative.Suppose, forexample,thatconjunction using implication(ordisjunction)and negation -along in themetalanguage) thelinesindicatedby thecontextual definitionr'"
[A & B]
=df
,,-,[B :J "-'[B :J All.
Surelythereare ways of giving a ppropriatemeaningsto ',,-,' and ':J' so as to make thecorrespondinglaw ofthecommutativityof conjunct ion quite significant. So we must askinsteadwhetheror notthere is a permissibleconnective (or conceptof conjunction)which we couldintroducewith (orexpress by way of) a symbol , say, '&', and such that'[A & B] == [B & AI' is notonly a truth,b ut is a "meaningidentity "-thetwo sides oftheequivalence are synonymous. thereare entirelypermissible "symmetUpon reflection, appearsthat it rical ways" of introducing such aconnective.Togetherwith anappropriate explanation forcomputingvalues, we cantakeit thatthetruth-table specification of aconnectivegives its "meaning-producingcharacter ", to which Churchalludes in the quotation above-ofcourse itdependson exactlyhow thetruth-table is given. To introduce a meaning-symmetrical connective , we might give twotruth-tables : -
&
t
f
trtfl
f~ 37
1 am being a bit lax here about thediffer en cebetween conte xtual definition s of cons of correspond ingtruth-fun ctions. If such a commutative nectivesand explicitdefinition conceptof the truth -function is possible, thenboth kinds of definitioncan be used in connectionwithit.
INTENSIONAL ENTITIES &
413
f t
frrrl t~
The explanation is to beaddedthattheseareentirely on a par, eventhough theother.Betterstill, we could givesymmetrical a one is here given before truth-table r'" t f & tftf'lt -
f~f t
f
'&
Or we might moredirectly j ustconstruesuch atableas defininga truth . This may not be function , the "associatedfunction"of the connective entirelypersuaslve.i'? But considerthefollowing . Firstobservethatit is possible to introducea commutativeconnective. Considera system of the propositionalcalculuswhich hasdisjunctionas a connective,introducedin any wayatall-not necessarilyas havinga commutativemeaning. For simplicitysuppose thatall wffs are eithersentential constants,PI , P2, . . . , or (¢ V 1/J), where ¢ and 1/J are wffs . Let therebe given aninterpretation, so thateach sentential constantexpressesa propositionand thecomplex wffs express propositionsinvolvingwhateverconceptsof disjunctionarechosen forthepurpose. Choosea particular effectivee numeration of thewffs ofS. Now consideran extensionof S, S*, whose wffs ared eterminedby therules of S together w iththerule: If ¢ and 1/J are wffs,t henA¢1/J is a wff, where 'A ' is a new binaryconnective.A ssign to each wff ¢ of S* , a wff¢* of S determinedby thefollowing recursiverule: If ¢ is a wff ofS, then¢* is just ¢ . (A¢1/J)* is thewff(¢* V 1/J*) or thewff(1/J* V ¢*) , whichevercomes first in theeffectivee numeration . The meaningsof thewffs ofS* aredeterminedby therulethat¢ is to besynonymouswith¢* . By any plausiblecriterion,'A ' is a connectiverepresenting d isjunctionand A# is fullysynonymouswith A1/J¢. It might be maintainedthata genuinecounterexample to Alternative (0) must involve apropersymbol actually denotingthetruth-function Disjunction,and not merely caonnectivewhich has such as its associatedfunction. 38Even the repetition of theconnective couldbe eliminatedif we coulduse thr ee dimensions. 39(Ad ded in proof) Indeed, it doesn't p ersuade me since I presented and discussed this materialat a Philosophyof Languag e workshopatUCLA. The counte r exa m ple to follow of that workshop,expec ia llyDavid was addedin proofand owes much to the participants Kaplan, Terence Parsons,Louis deRosset, and CalvinNormore.
414
C. ANTHONY ANDERSON
butinstead Well,consideragain a simplesystemof propositional calculus, Disjunction of a connectivefordisjunction,let'8' denotethetruth-function (undersome concept).Then 8(¢,1/J) is a wff if¢ and 1/J are, and thismeans theapplicationof thetruth -functionDisjunctionto thetruth-values of the wffs enclosed in parentheses . We canproceedas before by effectively enumeratingthewffsandextendingto asystemcontaining wffs oftheform8[¢, 1/11explainedas beingsynonymouswiththatone of8( ¢* ,1/1*) and8(1/1* , ¢*) which comes first intheenumeration.In thiscase we mayregard'8' in theextended system as denotingDisjunction,a nd underthe conceptexpressedby '8 ' in thesystemextended.Whatwe have done here is to define connective a representedby squarebracketsin termsof Disjunctionand ordinaryapplication of functionto argument,hererepresented by parentheses.And we may, if we choose,regardthewffs oftheextendedsystemas actually containing wffs of theform 8[¢,1/J], rather t han,as is more usual, construing t hedefinitionas a metalinguistic abbreviation. If it isobjectedthatthesquarebracketsstillrepresenta connective,then thereply isthatthisis correctb utthatit is inprincipleimpossible to eliminateallconnectives.t"And in any casethe observationdoes nothingto If somethingmore is rediminish theforce oftheproposedcounterexample. quiredfor acounterexample to synonymousisomorphism, thenit is difficult to seewhatit mightbe. Similarly, we could introduceor defineothermeaning-symmetrical con, § , nectivesand functions,e.g. meaning-symmetricmaterialequivalence andaccommodatetheothercounterexamples (takingthemas genuine). and
a~ b
=df
(F)[F(a)
{a-;b}
=df
{x : x
=
§ F(b)], a V x = b}
are definitions(one contextual, one explicit)of meaning-symmetricalnotationsforidentityand pair-sets. This lastyields aplausibleand satisfyingaccountof the "definit ions" introduced in theinformalexamples(2a) and (2b). Of coursethereare otherways ofreconstructing thoseexamples. As alreadynoticed,we might seethemas employinga notionof definitionwhich does notrequirestrictmeaning identity,obvious logical equivalencem ight be enoughfor manymathematical p urposes-ifcertainotherconditionsare met. The matteris different if we tryto embed thenotationinto a logic of belief, say. some generala rgumentwill be Given theidea ofsymmetricaltruth-tables, neededto showthattheremust be a difference in meaningconveyed byorder andwhich is somehowlurkingin thetruth-table methodwe haveused.v' 40For a compellingargumentfor this , see Church 1956, page 35, footnote 87. 's preferencewouldpresumablybe to have thevalues of the "t ru t h-ta bles" for 41Church theconnectives given by way ofsema nt ica l rul es (see Church 1956). On some ways of
INTENSIONAL ENTITIES
415
Once weadmitsymmetricalconjunction (or disjunction),it becomes possible to provet hattherearesymmetricalconcepts of otherdefinablebinary functions-not confined totruth-values . To illustrate, suppose thata necessarilysymmetricalbinaryfunctionf is set-theoretically defined, say by:
f
=df
{(x, y, z) : A[x,y, z])},
where A[x, y, z] is a well -formed formulawiththeindicatedfreevariables. Then a meaning-symmetrical conceptof thesame functionis expressedby the definition:
7 = df {(x, y, z) : A[x, y, z] V A[y, x, z]}.
The idea obviouslyextendsto functions of otheraritiesand withotherpermutations.The binarycase, and possibly eventheternary,seems to have language.I t is very unlikely thatsuch thingsappear applicationto natural explicitly in anynatural languages even forquaternary functions, for obvious reasonsof computational complexity. Still , the generaltheorymight have some intrinsicmathematical interest. considerations , we propose to formulatea On thebasis of the foregoing semanticalc riterion of identitywhich allows for symmetricalor commutative meanings. 7. QUINE'S NOTATION
Therearesome interesting observations by Quineaboutthepossibilityof various kinds ofnotation which might bethought to swiftly s ettlethequestions we have beendiscussing. Let us tentatively suppose, contraryto Whiteheadas of 1898, that 'x + y = y + x ' does hold as agenuineidentity;i.e., thattheorderof summands is whollyimmaterial . A notationof additionthatis more suggestivethan'x + y', then, wouldconsistin simply superimposing 'x' and 'y ' in themannerof amonogram. This notation q uiteproperly refrainsfrom suggestingany orderof summands; and thereceases to be any analogueof 'x + y = y + x ', thenearestapproachbeing ofthe vacuoustype 'z = z'. Now theone objectionto thisprocedureis the expenseof castingmonograms; and thusit isthatwe revertto alinear notationwhich imposes anarbitrary n otational orderon summands. Theprinterfoistsupon us a redundant notation , issuingin synonyms. One andthesame sum can now beexpressedin two w ays, 'x + y ' and 'y + x'. The law 'x + y = y + x ' comes to be needed as a means of neutralizing thisexcess ofnotation oversubjectmatter . The lawthus preservesa consequential role, evenwith '=' construedin thestrict sense ofidentity . doing this, it is a bit difficultto "trivialize"c ommutativity,butit stillwill bepossible.
416
C. ANTHONY ANDERSON It is not easy to imagine a notation capable of absorbing various other laws,e.g., 'x(y+z) = xy+xz', in the manner in which the monogram 'x + y = Y + x'; nor is it easy to imagine a notation notation absorbed capable even of absorbing 'x + y = Y + x' and yet leaving 'x + x = x' unprejudi ced. For most purposes, notations teeming with synonyms are forcedupon us by circumstances yet more compelling than the cost of monograms . Hence the utility of the identity concept. (Quine 1941, pages 128-129)
If wedistinguishthemeaningof anequationfrom itstruth, we need not be so generouswithsynonymy.V And wemust objectthatfor certain concepts of addition, thenew notation will simply leave us withno way ofexpressing a significantlaw ofcommutativity . Suppose thatwe defineadditionby the recursionequations:
(al) x+O=x, (a2) x+y'=(x+y)'
(or better,by an explicitsecond-orderdefinition) . If additionis introduced in thisway,theequation'x + y = Y + x' is bothinformativea nd needing of proof. For other(symmetrical)conceptsof addition,it may be thata monogram notation is perfectly a ppropriate,considerations of thepublisher'sand typesetter's lotsbeing set aside. We couldpreventautomaticreflexivity by making themonogramof 'x' withitselfdarkerthus: x. No doubtlong sums will be ofd oubtfullegibilityand such a notationwould not work for nonassociativeoperations,b uteven thesedefectsmight be overcomewithmore notational ingenuity. thefactthatsuch notations arepossible does notsettle , or Nevertheless, eventendtosettle,ourquestions.Not everyaspectof themeaningassigned to a givennotationneed be made explicit.We might avoid theexplicitexpressionof conceptsin theobjectlanguage,even if such are clearlypresent nevertheless-as indeedmight be expressedexplicitlyin givingthemetalin guisticsemanticalrules forthe notation . So the exerciseof constructing notations whereinvariousthingscannotbe expresseddoes notseem to offer any independentevidencefortheallegedsynonymies. 8. ASSOCIATIVE MEANINGS ?
One may see aparallel in thecase ofassociative functionsa ndconnectives. Giventhata certainoperationis associative , oneoftenjustdropsparentheses or brackets . Here is atypicalexample: 42Note thattheWhiteheadvolume firstappearedin 1941-thesense-denotation distinc. tion was nota tthattime commonlyrecognized
INTENSIONAL ENTITIES
417
Heretheorderofcombiningthenumbershas noinfluence on theresult . This law of the (sequence) of combination also holds for all natural numbers.
== 12. Similarl y, additionsof more thanthreesummands may be writtenwithoutbrackets . (Gellert, Kiistner, Hellwich and Kastn er 1977, page 20)
It means thatbracketsmay be omitted: 5+3+4
One finds asimilarsortof thingin connectionwithassociativeconnectives in thepropositional calculus . For example: On accountof thecommutativeandassociativelaws, conjunctions and disjunctionsw ithseveralcomponentsmay be writtenwithoutparentheses. (Hilbert and Ack ermann 1950, page 7)
Here it is mostnatural to allow thatthenew notation,with noparentheses, reallyconveys a newconcept. In spite of thefactthatcommutativity , associativity ( andreflexivity) areoftenintroduced a nddiscussed together in elementary mathematicst exts, we have herereally a very different sortof case from commutativity . The examplesare plausiblyseen asrequiring,i nstead of a specialsortof meaning,onlytherecognitionof concepts(of functions) s, one might see which takea variablenumber of arguments.In such case 'x + y + z' and 'x + y + z + w' as applicationsof a singlesummationfunction,perhapsconstrued as applicabletosequences. Bettert hentowritesuch thingsas 'E(x ,y, z)' and ' E(x, y,z , w ),' where 'E' is "anadic" (d. Gmndy 1976) , or applicableto sequence s of argumentsof arbitrary length.Notice thatif '+ ' is assigneda binary functionconceptas itsmeaning, it will not be up to us tostipulate t hatthetrueequation : x + (y + z) == (x + y) + z
expresses a synonymy. This is slightlymore evidentif we use anotation wherebythefunctionis writtenbeforethearguments , say:
Plus(x, Plus(y, z))= Plus(Plus(x,y), z ). Indeed, it is to beobservedthatsomethingis explicitlydenotedv'on the left side of t heequationwhich need not be explicitlydenotedon theright, namely, y plus z. It could still maintained be thattheentityis "implicitly denoted"by way of being determinedby a conceptwhich must begrasped if therighthandside is understood,and hence thatwe do notreallyhave distinctnessof denotations.Buttheclaim isstrain ed , atbest.44 On theaccounthere beingurged,thetruth (andproof)of such anidentity can motivateus tointroduce a conceptof thecorresponding a nadicfunction 43Really, th e entity is the valueof an expressionon the left for an assignme nt ofvaluesof the variabl es ,butthe pointhold s fi we use constants ins te adof varia bles inour exam ple. 44The m atter is of coursedifferen tif "Plus" has been introdu ced by context u a l defini t ion and is notreall ya denoting exp ression.
C. ANTHONY ANDERSON
418
E. And of course such conceptcouldcertainly a be introduced,using set theoryor along the lines suggestedby the"recursion"equations: E(x, y) = x
+
Yj E(Xl,X2,·
..
,xn+d = (E(Xl, X2,·
.·
,xn)
+
xn+d ·
Commonly, onesuspects,thenotation 'x + y + z' isjustused ambiguously.V' Withinmathematical contextsit is anambiguitywhich makes for very little dlfference .t" But we aresupposing generally t hatequivocationsare to be beforethetheoryis applied. eliminatedin thebase language I concludet hatthephenomenonofassociativity does notrequireany modificationof ourcriterionof synonymy, butpointsinsteadto therecognition of functionsof variablenumbersof arguments,andconceptsthereof. 9. REDUNDANT MEANINGS ?
Suppose that'8' denotesa reflexiveo peration(in some domain), so that: x 8 x = x holds for all x (in thatdomain). Is it possiblethatthereis, correspondingto this, ameaning for 8' ' of such asortthatthis equation expresses a meaning identity? Even thoughone can find inthe informal mathematical discussionsof suchthingsexamples which (vaguely)suggest thattheauthortakesthisto introducea synonymy, it does not seem to be a If '8' expressesa certainfunctionconceptin thecontext 'a 8 real possibility. b' andthesame functionconceptin 'a 8 a', it isimpossible to seethislatter as synonymouswith'a' (exceptby theintroduction of some ambiguity)-one must graspthefunctionconceptto understand t heformer, not so t helatter. somethingis denotedby an expression Indeed,as in thecase ofassociativity, in theformer which is not (explicitly)mentionedusingthelatter , namelythe function8 .47 A related case arises inconnection w iththeidentityfunction,thatfunction which yields as value theentitywhich isassignedas argument.In treatments ofthelambda-calculus (such as Church 1941) , one finds suchthingsas: Ix = x , whereI is theidentityfunction,representable using thelambdaoperator as (>.xx). Let 'a' be a constant and considertheidentity(>.xx)a = a It is very difficult to see thisas expressingan identityof meaning. The conceptof a function,eventherelatively trivialidentityfunction,requiresa mathematical abstraction of some difficulty . 450ne can of courseintroducea conventionof, say, associationto the leftand drop the parentheses-but this introducesno newconcept. 46This is notso obvious in theconte xtof a proof Observe thatFrege (1884, as translated in Austin 1978, page 7 e ) faultsL eibniz' purportedproofof 2+ 2 = 4 exactlyon theground thatit implicitlyassumes theassociativity of addition . The repl yis not very plausiblethat ther e is being used her e an "associa t ivemeaning" for the functionof additionand hence thatsuch a step is notreallyan inference . 471 am ignoringhere the possibility that'8' is a non-denotingexpression, yet havinga definit e functionconcept as meaning. Here onlythe first point holds. Also it is obvious thatthe matteris quite differentif '8' has been conte xt ua lly defin ed or is otherwise syncat e gorematic .
INTENSIONAL ENTITIES
419
Anotherpossible counterexample we must consideris producedby the 48 iteration of functionsor connectives . Let ustakeas ourexampletheiteration of negation . Thereis disagreementin the li terature as to whetheror not adoublenegation , say, '"" "" P ' is to be regardedas synonymouswith ' P '. F. P. Rams ey and HerbertHochberg(Hochberg 1977)49 say thatthese are mere symbolicvariants . ArthurP rior(1963)50 and Churchregardthem as distinct . If thisformeropinion is correct , thenther e is stillanothercase which must be accommodatedin theformulation of a correctcriterionof synonymy. From thepresentpointof view,51 thedisputedquestionis not well-posed . Therecertainly are conc e pts of negationwhere itsi very difficult to see their doubleiteration as producingtheverysame proposition. For example, in a formulation of thepropositional calculus in whichtheprimitivesare ':::>' and constantfalsehood'1', one may define aconceptof (orconnectivefor) the truth-function of negationthus:
""A
=df
(A:::> f)
Thenthecorresponding law ofdoublenegationis
A == «A :::> f) :::> f), and for natural ways ofassigningmeaning to ':::>', this need notexpress a synonymy. Therealquestion is this: Isthereany concept of negation whereby thecorrespondinglaw ofdoublenegationexpressesan identit y of meaning? I know of only one argumenton thematter.F. P. Ramsey arguesthus: We might, forinst ance,express negationnotby insertinga word' not,' but by writingwhatwe negate upside down. Such a symbolism is onlyinconveni entbecause we arenottrainedto perceivecomplicated symmetryabouta horizontal axis, butif we adoptedit weshouldbe rid oftheredundant ' not-no t ', fortheresultof negatingthesentence 'p ' twice wouldsimply be thesentence'p ' itself. 52 It seems to me, therefore , that'not'cannotbe a name (for if it were, 'not-not-p'would have to be abouttheobjectnotand so differe ntin meaningfrom 'p'), butmust functionin a radically d ifferentfashion. (Ramsey 1927, in Mellor 1990, pages 42-43) 48The case of theidentit y function , alread ydiscussed, m ay be usefullyseen as a degenerate case of the presen t sortof exam ple. 49H ochber g there ant icip ates some of the propertyidentiti es we endo rse, butoffers no justific ations for his choices. 50Prior states as a g en er alprinciple that ". . . no prop osition be a logical complicat ion of itself. .." (page 192), and thus also rejects the reflexitivity of conj unct ion as meaning preserving. t which semantics and inte ns ion al logic sho uld o nt 51T hat is , the approach accor di ng o be construct ed insuch a way to give prefer en ce tone o conce ptof a function overany ot her. See furth er, Anderson 1989, section VII.
420
C. ANTHONY ANDERSON
This argumentmay show something, but it does not show thatthereis a conceptof negationsuch thatthe doublenegationof a proposition, formed thepropositionitself . One might by way ofthatconcept,is ident ical with concedethatsymbols for negation,'not'or '''.'', say, may be, and perhaps typicallyare, treated as connectives-non-denoting, buthavingonly an associatedtruth-function (see further,Church 1956, section5). The argument certainly has notendencyto showthatthereis no truth -functionwhich takes t into f, and f into t. If we acceptthe existence of suchfunction a , the possibility arises of various conceptsof it,expressedby definitedescriptionsor (perhaps) names. Indeed, formalizedlanguageswith suchinterpretations have beenactually c onstructed . If we considerthe notationproposed by Ramsey and assign to 'p' a proposition,say thatsnow iswhite,thena negation thereofwill beexpressedby 'd'. Operatingon this latter symbol, we thatsnow is white.The only get back'p'-which, we suppose, still expresses conclusion -" to be drawnseems to be thatone need not have notation a - theanalogue of thesituation with Quine 's forexpressinga doublenegation monogramnotation. I concludethatwe need not countenance a conceptof negationwhose doubleapplicationto apropositionproducesthatveryproposition.If 'Neg', say, expresses aconceptof negationin a givenlanguage,thenit would seem that'Neg-Neg-p == p' will express someinformation,p erhaps obvious, but not onthataccountentirely w ithoutcontent.So too, it would seem, for the generaliteration of function-concepts . Here noamendmentsare suggested for ourformulation of acriterionof identity. Can ithappen that'e(a , a) = e (a)', where'e' expressesa function cone is a correspondingontocept, expresses ameaning identityso thatther logicalpossibility? Given thatwe allowfunctionsw ithvariablenumbersof ruledoutas ill-formed or as imarguments,t hethingis notautomatically themost difficult of mediatelyconceptually incoherent.This case isactually the "redundant meaning" examplesto decide. Many oftheargumentsused to help decideo thercases do not apply here. On balance , I believethatwe should not grantthe real possibilityof an additionalexceptionto synonymousisomorphism. First,it is not obvious how we could specify such redundant meanings by semanticalrules, orotherways 52It is interestingto recallAugustus De Morgan's (1860) notation for producingthe "cont ra ry of aerm"-anexpress t iondenotingthe com pleme nt ofthe class denoted by the term. It con sistsofrepla cing a capi t a l etter l , ' A' for example, by its sma ll ett l er counte r pa rt 'a'. If we give no particul arpriorityto eit he r oft hese , thenwe have a notationto which a version of Ramsey's argumentcouldbe applied . 53The argumentdocs, though,highlighta relation, Neg, say, which holds only between propositions aboutwhich we would normallysay that the one is the negationof the ot he r. At a cert a in evel l of abst ract ion, on e couldeven maintain that'tp3!qNeg(p,q) and 'tp'tq(Neg(p,q) J Neg(q ,p» . We can view this as involvinga notionof proposition which abstractsa way from thecommon elem entof st ruct ureetweenwhat b is expressed by a sentenceand an explicitnegationof it.
INTENSIONAL ENTITIES
421
of specifyingvalues, so as to have something analogous to our(truth)tables forsymmetric meanings. Also, thereareso far nointuitiveexamplessupportingtheproposedmodification,as thereare inthecase of thesymmetry exception . thatthecase could have any mathMoreimportantly, it is difficult to see ematicalor philosophicalinterest . Even if this is a real possibility, nothing significantseems to be lost if we restrict our theoryto ignorethemso as to deal only w iththosekinds ofmeaningswhichmight have someindependent interest . This mightbe due to a failureimagination, of b utif some argument has been overlooked or some mathematical or philosophical i nterest develops forthiscase, thenone can see how to modify (andcomplicate)thecriterion which we shall propose. 10. OTHER CRITERIA AND ALTERNATIVE (1*)
In his work onintensional logic,Churchused threedifferent heuristicprinciof the logic of sense and denotation . As we have ples to guidet heconstruction said, Alternative (0) is theapproachwhich usessynonymousisomorphism as (semantical)criterion . Churchattemptsto incorporate a(n ontological) criterioninto a formalized language corresponding to this- in justthe sense we d enotation itselfdoes notspeak of haveexplained . The logic of sense and expressionsand semanticalnotionsP"buttreatsintensionsdirectly in such a shallbe entailedin any waythatit is intendedthatthesemanticalcriterion semanticsof particular case where the t heoryis used to givetheintensional a language. Alternative (1) is a heuristicprincipleaccordingto whichexpressions to which arelambda-convertible are to besynonymous.5 5 It is interesting notethatAlternative (1) entails,undernatural assumptions,the synonymy of non-symmetrical examples. Thus it may ourcommutationandconversion of theextentthattheexamplesare forceful. be regardedas being confirmed to thedistinctionbetween theexplicitdefinitionof a prediOn thisalternative cateandthecorresponding c ontextual definitionessentially disappears-and with it anysupposed advantageof thelatter for this case. The argument used inRamsey's Problem(cf. note 33) is a simplep roofthatthisalternative is incompatiblewiththeidea thatcomplexconceptsarebuiltup in a natural setof constituents . way from a finite Unfortunately itentailstoo much. The numeralsa ndtherules for multiplicationin arithmetic,forexample,can be codedintothelambda-calculus (see , Church 1941) . Thus equationsof the form'k = m x n' becomes trivialities numbers. (If desired,you may imagine even where mand n are fairly large 54 Cont a r ry
towhatthename suggests, as RulonWells(195!!) points out. Anderson 1998 for more det ails n o thevariouscrit eria . Church (1999) proposes a slightmodificationof Alternative(1) as being possibly worthyof investigation. 55 See
422
C. ANTHONY ANDERSON
thatwe learnournumeralsdirectly from thelambdacelculusj .P" This is not thecorrectconclusionif propositionsare to betheobjectsof belief. Alternative (2) is basedon theideathatlogically equivalent expressionsare to besynonymous. Churchclearly recognizedthatthislast,whileadequate of modallogic, would not do appliedto if suchthingsas the fortreatments logic ofbelief . t hepresentinvestigation, I proposethefollowing (as applied In thelight of to a givenlanguage): (C12 t If x is a sentenceand y is a sentenceand if x and y express Zl and Z2, respectively, thenZl = Z2 if andonly ifx and yare synonymouslyisomorphic modulo permissible permutations. . In orderforthisto begenerally applicable , we must suppose thattheintensional s emanticalrules oft helanguagemay specify rules of synonymyin sucha way as to allow permutations to in certaincases. Or, ifthesemantics is given by rules of sense, thenit shall be p ermittedthatcommutativemeanc ertainexpressionsof the ings ofthesortwe havediscussedare assigned to language . The supportingtheoryhereenvisionedwillemploy a notionof thecomposition of concepts, complex conceptsbeing capableof being constructed by way ofthisoperation .57 It will besupposed thatevery complexconcept has aprincipal function-concept, thatis, thatthereis a certainfunctionconcept involved in itscomposition, which may betakento be thatoperation (concept)applicationof whichimmediatelyyieldsthecomplexconcept. It is alsosupposed thata function-concept carrieswith it apermutation set, a set of operationswhich it ispermissibleto apply to thesequenceof argument conceptswithpreservationof identityof theresulting complex concept. In orderto accommodateour observationsa boutassociativity,it is assumed thatfunctions(and theirconcepts)are permittedwhich areapplicableto a variablenumberof arguments . The correspondingontological criterionis then: (CI1)P If Zl and Z2 are propositions,thenZl = Z2 if and only ifZl and Z2 havethesame principalfunctionconceptandthesame permissiblep ermutasequenceof argumentconcepts-modulo tions. Let us usecapitalGreekletters as variablesr angingoverfunctionconcepts and lower case Greek l etters withan arrowabove asrangingover sequences 56 Alo~zo Churchpointed outthis consequen ce of Alte rnative(1) to me (in corres pondenc e) and ther eby persuadedm e thatit won't do for belief and the like. 5 7 Compositionis the inten sionalanalogu e of the exte nsiona lelation r of set memb ership. David Kaplan(1975) has used a versionof this idea in connectionwithAltern ative (2) . e by com posit ion, anotionof On the assumptionthatallcomplex intensions areobtainabl sim plicityis easilydefined.
INTENSIONAL ENTITIES
423
of concepts.Then, with boldcurlybracketsto expresscomposition,we can stateourcriteriona bit more precisely : (CI 2 t {cp, a} = {\11 , l} if andonlyif (i) cP = \11 , and (ii) where 81 and 82 arethepermutationsetsforcP and \11 respectively, thereare permutations P E 8 1 and P' E 82 such thatPea) = p'(l). (The appropriateidentitypermutation will bepostulated to belongto every permutationset.) The adequacyof this as a generalontological criterion of identityfor propositionsdepends on the plausiblepostulatet hatevery propositionis a composition. The analogous criterion of identityforcomplexfunctionconceptscP and \11 may be suppliedby analogy . For simple functionconcepts(and, as a special case, simpleattributes), thethingsareidentical if thereis justone, otherwise not. Semantically, as appliedto aparticular language, it will be q auestionof whethertwo simpleexpressionsaresynonymous- aquestionwhich weregard to be (ideally) settledforanyonewho hasmasteredthelanguage.( Compare thediscussionof theidentityof truth -valuesat theend ofsection1). Because of itsr elationship to theothercriteriaChurchproposed, we call this "Alternative (1*)".The correspondingcriterion for Alternative (0) , which may still toe btenable,r esultsin thespecial case wherethe only permutat ion involved istheidentity.Alternative (0) has simplicity'" in its favorandgiventhealternative explanations available for itsapparentfailure , it does not seem to be definit elyrefuted. 59 The important(and difficult)r emaining taskis to supply adetailedlogistic treatment of theintensionalentitiesin questionto embody andclarify thislast,and toinvestigateits consequences. Any furt herdiscussionreally seems to call for a precise formulation of thatunderlying t heory . Technical detailsof a formalized t heoryof intensionsembodying theseideas and the correspond ing formalized ersionof v thiscriter ion of identityare plannedto be forthcomingin full. It is hoped thatthe present discussion will serve to expose the projectto anyremaininggenerala nd principledphilosophical objections.P''and to clarify the generald ebateaboutintensional logic.P! 58K ing (1996) em phasi zes this advantage. is orphis m) fail sagainstAlternative 59T he objectionof Linsky (1949) (to intensiona l om (0) once conte xt ua lefiniti d on s are cle arlyrecognized. And the object ion ofMyhill(1958, p age 82) is based on assumption s thatwe reject-briefly,thatther e is suchathing as the set of allp ropositions, so thatone can const ructt he Canto r di agonalset involved. 6°1 have been ca refu l to oid av engaging thequestionof interlinguisti c criteria,some t imes y dealwiththese, requires , I believe ,a welldevelop ed withconside ra bledifficulty.To full theory of (possible) languages as well as co n side rab lefurth er discussion. And quest ion s p erta i ningto variousforms of p hysic a lis mor behaviorism as applied to theo riesof meaning have not been here discussed. I believethatsuch objection s can be full y ans weredand hop e to address them in ano t her pl ace. 61 1 am ind ebted to Tyler Burge for an objecti on to an ar gumentof mine abo ut commutative m ean ing s. T he objection for ced m e to re -think the con ceptual foundation s of ( 1° ) . Alternative
424
c.
ANTHONY ANDERSON REFERENCES
Almog, J., J. Perry,and H. Wettstein 1989 (editors),Themes from Kaplan, OxfordUniversityPress, Oxford. Anderson,A. R., R. B. Marcus,and R. M. Martin 1975 (editors),The logical enterprise, YaleUniversityPress, New Haven. Anderson,C . A. 1980 Some new axioms forthelogic of sense and d enotation: Alternative (0), Noiis, vol. 14, pp. 217-234 . 1987 Bealer'sQualityandConcept,Journal of Philosophical Logic, vol. 16, pp. 115-164. 1989 Russellianintensionallogic, in Almog, Perry and Wettstein 1989, pp. 67-103 . 1991 Review of Cresswell 1985, Philosophical Review, vol. 100, pp. 476-479. 1998 AlonzoChurch'scontributions to philosophyand intensional logic, Bulletin of Symbolic Logic, vol. 4, no. 2. Angelelli,1. 1984 Frege andabstraction, Philosophy Natumlis, vol. 21, pp. 453-471. Apostel,T. M. 1967 Calculus, vol. 1,secondedition,Xerox College Publishing , Lexington . Austin,J. L. 1978 (translator), Gottlob Freqe, The foundations of arithmetic, Basil Blackwell,Oxford; Englishtranslation of Prege 1884. Bealer , G. 1982 Quality and concept, OxfordUniversityPress, Oxford. 1994 Propertytheory : Thetype-freea pproachv. theChurchapproach, Journal of Philosophical Logic, vol. 23, pp. 139-171 . Birkhoff, G ., and S.MacLane 1944 A suroey of modern algebm, Macmillan,New York. Black, M. 1945 The "paradoxof analysis"again: a reply,Mind, n.s. 54, pp. 272 -273. 1946 How cananalysisbe informative?,Philosophy and Phenomenological Research, vol. 6, pp. 628-631.
Carnap,R. 1947 Meaning and necessity, Universityof ChicagoPress, Chicago. 1963 Replies andsystematicexpositions,in Schilpp 1969, pp. 859-1013. Church,A. 1941 The calculi of lambda-conversion, Annalsof mathematicss tudies,vol. 6, PrincetonUniversityPress, Princeton,New Jersey. 1942 Incompletesymbol, in Runes 1942, p. 143. 1946a A formulation ofthelogic of sensea nddenotation ( abstract),The Journal of Symbolic Logic, vol. 11, p. 31. 1946b Review ofWhite1945a, Black 1945,W hite1945b, and Black 1946, The Journal of Symbolic Logic, vol. 11, pp. 132-133. 1951 A formulation of thelogic of senseanddenotation, in Henle, Kallen and Langer 1951, pp. 3-24.
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Intensional isomorphism andidentityof belief, Philosophical Studies, vol. 5, pp. 65-73. 1956 Introduction to mathematical logic, vol. I, PrincetonUniversityPress, Princeton,New Jersey. 1993 A revised formulation of thelogic of senseand denotation,Alternative (1), Now, vol. 27, pp. 141-157. Cresswell , M. J . 1985 Structured meanings: The semantics of propositional attitudes, MIT Press, Cambridge, Massachusett . Davidson, D. 1963 The methodof extensionand intension,in Schilpp 1963, pp. 311-349. De Morgan,A. 1860 Syllabusof a proposed system of logic, inHeath 1966, pp . 147-207. Frege, G. 1884 Die Grundlagen der Arithmetik, WilhelmKoebner,Breslau . 1919 Negation,Beitriige zur Philosophie des deutschen Idealistnus 1, pp. 143in Geach and Black 1966, pp . 117-135. 157; Englishtranslation Fine, H. B. 1906 College algebm, Ginnand Company; reprintedby Dover, New York, 1961. Geach, P. 1957 Mental acts , their content and their objects, Routledge& Kegan Paul, London; HumanitiesPress, New York. Geach,P., and M. Black 1966 (editors), Translations from the philosophical writings of Gottlob Freqe, Basil Blackwell, Oxford. Gellert, W. , H. Kiistner,M. Hellwich , and H. Kastner 1977 The VNR concise encyclopedia of mathematics, Van NostrandReinhold, New York. Grandy,R. 1976 Anadiclogic, Synthese, vol. 32, pp. 395-402. Grim, P. 1991 The incomplete universe. Totality, knowledge, and truth, MIT Press, Cambridge, Massachusetts . Hahn,L. E. 1986 (editor),The philosophy of W. V. Quine , Libraryof LivingPhilosophers, vol. 25,Open Court,La Salle . Heath,P. 1966 (editor),On the syllogism and other logical writings by Augustus De Morgan, YaleUniversityPress, New Haven. Henle, P., H. M. Kallen , and S. K. Langer 1951 (editors),Structure, method and meaning, Essays in honor of Henry M. Sheffer, The LiberalArtsPress, New York. Hilbert , D., and W. Ackermann 1938 Principles of mathematical logic, Englishtranslation of thesecond German edition, Chelsea,New York, 1950.
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Hochberg,H. 1977 Properties,a bstracts,and theaxiom ofinfinity,Journal of Philosophical Logic, vol. 6,pp . 193-207. Kamke, E. 1950 Theory of sets, Englishtranslation of thesecond Germaneditionby F. Bagemihl,Dover, New York. Kaplan,D. 1975 How toRussella Frege-Church,The Journal of Philosophy, vol. 72, pp. 715-729. 1986 Opacity, in Hahn 1986, pp . 229-289 . King, J . 1996 Structured propositions , Journal of Philosophical Logic, vol. 25, pp. 495521. Linsky, L. 1949 Some noteson Carnap'sconceptofintensional isomorphism andtheparadox ofanalysis,Philosophy of Science, vol. 16,pp . 343-347. 1967 Referring, HumanitiesPress, New York. Mellor, D . H. 1990 (editor), F. P. Ramsey, Philosophical papers, Cambridge University Press, Cambridge, England . Myhill,J. 1958 Problemsarisingin theformalization ofintensional logic, Logique et Analyse, vol. 1, pp. 78-83. 1963 An alternative to themethodofextensionandintension , in Schilpp 1963, pp . 299-310. Prior, A. N. 1963 Is theconceptof referential opacityreally ecessary?, n Acta Philosophicia Fennica, vol. 16, pp. 189-198 . Quine, W . V. 1941 Whiteheadand therise ofmodernlogic, inSchilpp 1941, pp . 127-163. 1953 From a logical point of view, HarvardUniversityPress, Cambridge,Massachusetts ; Harper& Row, New York. 1953a Two dogmas of empiricism, in Quine 1953, pp. 20-46. 1970 Mathematical logic, revised edition, HarvardUniversityPress, Cambridge, Massachusett s. 1975 On the individuationof attributes , in Anderson, Marcus, and Martin 1975, pp . 3-13. 1981 Theories and things, HarvardUniversityPress, Cambridge, Massachusetts. Ramsey, F. P. 1927 Universals,in Mellor 1990, pp . 8-30. 1927 Factsand propositions , in Mellor 1990, pp . 34-51. Richard,M. 1990 Propositional attitudes, An essay on thoughts and how we ascribe them , CambridgeUniversityPress, Cambridge, England .
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Runes, D. D. 1942 (editor),Dictionary of philosophy, PhilosophicalLibrary,New York. Russell, B . 1903 The principles of mathematics, Allen and Unwin, London. Schilpp, P. A. 1941 (editor), The philosophy of Alfred North Whitehead, Libraryof Living Philosophers,vol. 3,Northwestern University , Evanston:second edition, Thdor,New York. 1963 (editor), The philosophy of Rudolf Carnap, Libraryof Living Philosophers, vol. 11,Open Court, La Salle;C ambridgeUniversityPress, London. Warner,S . 1965 Modern algebra, Prentice-Hall , Englewood Cliffs; reprintedby Dover, New York, 1990. Wells, R. 1952 Review ofChurch1951, The Journal of Symbolic Logic, vol. 17, p. 133. White,M. G. 1945a A noteon the"paradoxof analysis",Mind, n.s. 54, pp . 71-72. : A rejoinder , Mind, n.s. 54, pp. 357-361. 1945b Analysisand identity Williamson,T . 1986 Criteriaof identityand theaxiom of choice, The Journal of Philosophy, vol. 83, pp . 380-394. 1990 Identity and discrimination, Basil Blackwell, Oxford& Cambridge.
JOHN P. BURGESS
NOMINALIST PARAPHRASE AND ONTOLOGICAL COMMITMENT For Alonzo Church on the occasion of his ninetieth birthday.
Abstract.Severalnominalist parnphroses, in the sense ofmethodsby which assertionsof conventional mathematicst hatgive theappearanceof implyingthat therearesuch thingsas numberscan be systematic allyreplacedby otherassertionsnotgiving thisappearance , have been proposed in theliterature . Can the success of such methodshelp showthattheoriginalassertionsare notreallyontologically committed in thesenseof notreallyb eing incompatiblewiththedenial thattherearesuch thingsas numbers? I argueagainsttheclaim thatit can.
1. COMPATIBILIST NOMINALISM
The debateoverintuitionism,which reachedits climax intheearly1930's, justas AlonzoChurchwas beginninghis career,is sometimes said to have been aboutexistencetheorems. This is notquiteaccurate,since therewas no debateover such existence theoremsas: (1) Thereare numbersthataregreater t han10
10 10
and are prime .
It would be morea ccurate to saythatthedisputewas overexistencetheor ems with nonconstructive proofs; mostaccurateto say it was over nonconstructive proofs. This debatereached aresolution of sorts: The mathematical communitydid not cease to a cceptnonconstructive proofs, asdemandedby theintuitionists, butdid learn,t hroughChurch's Thesis, how todistinguish withinclassicalmathematicsbetweenconstructive and nonconstructive, and . so to partially addresstheconcernsthatmotivatedintuitionism A debatethatcan accurately be said to have been a boutexistencetheorems, and to whichChurchalsocontributed , began in the 1940 's, namely, the debateover nominalism in its modernsense thatwas introducedin an articleby Goodmanand Quine(1947) in thejournalfounded andeditedby Church.Nominalismin this sense denies: (2) Thereare (suchthingsas) numbers. In denying(2), nominalistsare notdenyinganythingexplicitlyassertedin conventional mathematics , since, as pointedout byCarnap(1950) , conventionalmathematicians spend little time speaking(and presumablylittle time 429 C. Anthony Anderson and M. Zeleny (eds. ); Logic, Meaning and Computat ion, 429-443 . © 2001 Kluwer Academic Publishers. Printed in the Netherlands.
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thinking)a boutsuch characteristically philosophicalthesesas (2). Nonetheless,GoodmanandQuinedid notdoubtthatin denying(2) theyweredenying somethingimplicitin assertionsof conventional mathematics , such as (1). This is not tosay thattheirpositionwas straightforwardl y revisionist like thatoftheintuitionists: Theydid nottakeit to be ataskformathematicians to seek anominalisticr eformation of mathematical theory . On thecontrary , Goodman (1956) , in his positionpaper in a volume onnominalismedited by Churchandothers,devotesa section(§3) to answeringobjectionsand in answeringone ofthem (viii) explicitlyrejectsrevisionism: Objection: Nominalismwould hamper the development of math ematics and theothersciencesby depriving them of method s they have used and are using to achieve some of their important most results . Answer: Not at all. The nominalist does not presum e to restrict the scientist . The scientist may use platonistic class constructions, complex numbers, divination by inspection of entrails, or claptrappery any that he thinks may help him get the results he. wants
Indeed, insofaras Goodman and especiallyQuine themselvesworked in mathematics(mathematical logic), theyspoke like conv e ntional m athematicians whiledoing mathematics;thoughwhen doingphilosophytheyspoke, as above, of such "platonistic c laptrappery " as, e.g., complex numbers being comparableto "divinat ion byinspect ion ofe ntrails " . This is notto say theirpositionwasstraightforwardly instrumentalist likethatoftheformalists. They did nottakeit to besufficient merelyto deny whil e doing philosophy whatis assertedwhiledoing mathematics , saying, "We don't reallybelieve whatwe said," and nothingmore. Rather,t heytookit to be ataskfor philosophersto seek anominalistic reformulation of mathematical l anguage,so thattheywould beableto say while doingphilosophy , "We don't reallymean whatwe said whiledoing mathematics, " and to saysomethingmore, "Whatwe reallymean is rather thefollowing . .. ," wherethereformulation or paraphraseof whattheysaid would follow . Thus Goodmancontinues: But what he produces then becomes raw material for the philosopher , whose task is to make sense of all: this to clarify , simplify, explain, interpret in understandable .terms The pract icalscientist does the business but the philosopher keeps the books . Nominalism is a restraintthata philosopher imposes upon himself , just becausehe feels he cannot otherwise make real sense of what is eforehim. put b Goodman and Quine did not find more thana few of theparaphrases theywereseeking, and Goodman gradually became a less vocalproponent of nominalism, whileQuine soon became a quitevocalopponent. Already in thepaper justquoted,writtenwithina decadeof their firstnominalistic
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collaboration, Goodman writesof Quine's "recentas stillsomewhattentative defection from nominalism". Buttheoppositionvoiced inQuine'slater writingshas tendedto inciterathert handeterlaternominalists.Thus the Introduction to Chihara 1973 begins byquotingan anti-nominalist passage from one of Quine'S works, and continues: When I first read these words in Quine's Word and Object several years ago, I wrote in the margin : "This philosophicaldoctrine should be soundly refuted ." By thefinalchapterV and technicalAppendix of the book, astrategyof paraphraseis been suggested,which has since been f urther developedin Part I of Chihara 1990. PartII, a surveyof recentliterature, mentionsseveral competingstrategies . The competingstrategies in therecentliterature tendto sharetwo common features distinguishing thelaternominalistsfrom Goodmanand Quine. First, theytendto useapparatusthe originalnominalistswould not have allowed themselves. ThusChiharauses modality, a distinctionbetweenwhat is and whatit would be possible in principle,if notfeasible thereactually in practice,to make. While for specialists,the mostinteresting a spectof Chihara'swork is hisdevelopmentof amodalparaphrasefor mixedcontexts expressionsand forcontextsininvolvingbothphysicaland mathematical volvinganalytical as well asa rithmetical expressions. Forpresentpurposes it will beenoughto indicatehow he treatstheselast.Assertionslike (1)are systematically replacedby assertionslike: (3) Therecouldbe constructed numeralst hatweregreater t han10 wereprime.
10 10
and
Whereas(1) appearsto imply (2), whichChiharadenies, (3) merelyappears to imply: (4) Therecould beconstructed (such thingsas) numerals. whichChiharaasserts. Second, laternominalistst endnot to beexplicitlyincompatibilist. They tendnot to concede thatthethesisthattheyas philosophersdeny, viz. (2), is implicit in theoremsthatmathematiciansassert,e.g., (1). This is not to. say thatthey tendto be explicitlycompatibilist, insistingthatdespite appearances(1) does notreally imply (2).T hey tendrathersimply to be inexplicit . It is unclearto me, forinstance , whetherC hiharawishes to deny critical of thoserival (1) or to denythat(1) implies (2).1 In his survey, he is nominalistswho tendtowardsmore explicitcompatibilism,butalso of those 1 The presentnotederives(very indirectly) from (thefirst half of) theauthor'sc ontributionto asymposium on "Modalityand Existence"held atPrincetonUniversity,A pril, 1992. The authoris grateful to theothersymposiast, ProfessorCharlesChiharaof the
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who tendtowardsmore explicitincompatibilismand oftheincompatibilist argumentsof theanti-nominalist Burgess (1983). My aim in thisnotewill be tosubjectcompatibilismto explicitscrutiny. The issue is notwhethernominalistslikeGoodmanand Quine orC hiharacan dissociatethemselves from t heimplicationsofwhattheymay say while doing mathematicsby subsequentretractions while doingphilosophy(or perhaps silentreservations even while doing m athematics). No one deniesthat(in some sense) theycan. The issue is, rather,whatthoseimplicationsare. The issue iswhetherconventional mathematical theory,including(1), implies (2). 2. ONTOLOGICAL COMMITMENT
Questionsof this kind are often discussed undertherubric"ontological commitment",whichQuine introduced(Quine 1948) andexplained(Quine 1951). Chihara(in the §III.3 of his first book) has been criticalof Quine's commitment",and he is nottheonly explanations ofthejargon"ontological thenotion one to have found themobscure. No one has done more to clarify of "ontological commitment"thanChurch(1958). The notionof "ontologito theissue ofcompatibilism,and roughly calcommitment"thatis relevant the oneadoptedby Church,whose formulation not evenChiharahas been thefollowing:T is ontologically comable to criticize pointof in clarity," is by "Thereare (suchthingsas) mitted to S's if the following is implied T: S's, [or thereis (such a thingas) an S]". In theinstancerelevant to the mathissue ofcompatibilism,T is a body of scientifictheory(conventional ematics), involving adistinctivegeneralkind ofexpression, S expressions (numberexpressions),and it isdebatedamong philosopherswhetherthere arecorrespondingly thingsof adistinctivegeneralkind, S's (numbers). Now, thereis no difficulty ( apartfrom Church'sTheorem) in determining theontological commitmentsof a theory,when thetheoryis formalized. But in connectionwiththeissue ofcompatibilism,we areconcernedwith Universityof California a tBerkeley,a ndto participants (toonumerousto listindividually) in thequestionand answersession thatfollowedourtalksfor usefulc omments. Professor Chihara'scontribution willappear elsewhere . The questionwhetherhe wished to deny (1) or to denythat(1) implies (2) was put to ProfessorChiharaduringthequestionand answersession afterhis symposium talk.One schoolof thoughtd oubtswhetherit is appropriateto quotein printwhata philosophersays on such an occasion, given thatit may representno more thanan initialreactionto an unexpectedobjection , ratherthan a consideredopinion. Butsince ProfessorChihara 's practicein his criticalsurvey plainly , it may be reportedthathe said he was shows thathe is not ofthis schoolof thought undecided. 2Chiharadoes notethefollowing : Suppose some mathematical or linguistictheoryinvolvingsuchscientificterminology as "number" or"proposition", butnotthephilosophical termof art"abstracturn ", is ontologically committedto numbersor propositions . Suppose some philosophicalt heoristsdefines "abstractum" , as many do, to mean "number, proposition,or thelike". Still,a ccordingto theabovedefinition,t hemathematical or linguistic theorycannot,s trictly s peaking, be said to be "ontologically committed"to abstracta.
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an unformalized b ody of theory.The issue is one as towhetheror not a certaintheoryin unformalized languageimplies a certa in thesis also in unformalizedlanguage , with thefurther complicationthatthetheoryconsists of assertionst hatare ordinary in thesense thatthey might well be made by a philosophically unselfconscious scientist,while the thesis is an assertion thatis extraordinary in thesense thatit wouldhardlybe made exceptby a philosopher . NowChurchstateshis definit ion for formalized theories,b utalso wishes to applyto it tounformalized bodies of theory, namely , to theviews ofcertain philosophersof theOxonianordinarylanguage schoolabout"propositions". w hatmost concernedCarnap,who (The statusofpropositionshad also been fearednominalistobjectionsto suchthingswouldhamper thedevelopment of linguistics .) Churchquotesassertionsof the kind : "there is apropositionthat. . ." , from several a uthors . Do suchassertionsimply: "thereis (such athingas) as proposition", whichtheauthorswhom Churchquotes deny?Certainly the answer is yes if it isappropriateto formalize the "thereis" in bothcases as "3". For: 3x(Px 1\ . .. )
implies: 3x(Px).
Butshouldthe"thereis" here be formalized as "3"? In thisconnection , Churchenunciates(atthe end of his paper) a principlethatmight be called "Church'sThesis", were notthatlabelalreadyapplied toanotherand more famousdoctrine : [T[hose philosopherswho speak of "existence,""reality ," and thelike areto beunderstoodas meaningtheexistential quantifier, and are to be condemnedas inconsistent if on thisbasis inconsistencyappearsin theirwritings. The justification is thatno otherreasonablem eaning of "existence"has been provided (which would fit into thecontexts of thekind thatwere quoted),and the burdenof providing such a second meaningof "existence"restson thosewhose writingsor whose philosophicalviews require it.
I interpret thispassageroughly as follows:P erhapstheauthors Churchquotes don'treally mean w hatthey say when they ,say "Thereis a propositionthat .. .": In thatcase, theburdenis on them to provide a araphrase p indicating whatthey do really mean . Or perhapstheauthorsin questiononly mean to denythatthere"are" (suchthingsas) propositionsin some extraordinary
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w hatthis sense sense of "are" : In thatcase, theburdenis on themto clarify may be. The principleto be found in t heearly writings of Quine , and more clearly in thepassageabove, seems to bethatthereis a presumptionto theeffect that : (5) Thereare S's that. . . [or: thereis an S that. . .j, implies: (6) Thereare (suchthingsas) S's [or: thereis (such athingas) an Sj. and thattheburdenis on a would-becompatibilistto show how denial of (6) can beconsistentwithassertionof (5). ButQuine's critics, includingChihara,havenotedthatthis principleprovides no fully reliable criterion.Consider, forinstance: (7) Thereis a strongchancethatMoriartyis responsiblefor thisoutrage . Surely this does not imply : (8) Thereis (such athingas) a strongchance. This is surely an ceptional ex or idiomaticuse of"thereis" , in contrast to a normalor paradigmaticuse as in: (9) Thereis a sturdychairthatMoriartywas reposingin thisafternoon , which surely this does imply: (10) Thereis (such athingas) a sturdychair. Can a fully reliable criterionbe obtainedby amending theprincipleabove to exclude idiomaticexceptions?Thereis some difficulty in formulating the amendmentin such a way as to provide a widely applicableand genuinely usefulcriterion . A formulation in termsof adistinctionbetween occurrences of "thereis" thatareappropriately formalized as"3" andones thatare not would be useless, since it would leave iththequestion one w whichoccurrences of "thereis" these are . A formulation involving only notionsfromtraditional grammarcan distinguish(7) from (9), since inthelatter, whatcomes after the"t hat" is a verb phrase; in the latter, a completesentence;b utone needs an applicableformulation, notjustthe example , ofthe chance thatMoriarty is responsible,butalso all t heothersconsidered intheliterature (the sake of theillustrious client , thewhereabouts ofthemissing treaty,the identity of the younglady'smysterioussuitor,t hemarital status of DoctorWatson).
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Whatevert hecorrectinterpretation of hisearlywritings,Q uinein theend made no claim to be able providea to criterion.In a comparatively l ateretessay of ananthology (Quine 1981) rospective,t heopeningandalmost-title of his writingsfrom the1970's, he devotesa section(§II) to theproblem, and writesthusof thequestionwhere todrawthelinebetweenontologically committedanduncommitted: [T]here is no line todraw. Bodies are assumed, yes; theyare the things,firstand foremost.Beyondthemthereis a successionof dwindlinganalogies . Variousexpressionscome to be used in ways more or lessparallel to theuse oftheterms for bodies, and it is feltthat correspondingobjectsare more or less posited, pari passu; butthere is no purpose in tryingto mark an ontological limit to thedwindling parallelism . My pointis not thatordinarylanguageis slipshod,slipshodthoughit be. We must recognizethis gradingoff forwhatit is, and recognize thata fencedontologyis justnot implicit in ordinarylanguage.T he ideaof aboundarybetweenbeingand nonbeing is a philosophical i dea
For Quine, thedifference between theontologically committedand uncommittedhas become one of degree, not kind. isIta matterof thedegree of resemblancebetweenthepattern of usage ofS expressionsin T andthenorm of usage oft ableandchairexpressionsin everyday or paradigm,thepattern discourse. For Quine, thoughthereis no line to bef ound, yet one can be made, by formalizing (or "regimenting ") thetheory;b utowing totheabsence of asharpdistinctionbetweencommittedand uncommittedin unformalized language,thereis an absenceof asharpdistinctionbetweenappropriatea nd inappropriateformalizations (or "regimentations"). While this view of Quine's is controversial , given astrongresemblance in patternof usage tothenorm orparadigm, even thosetendingtowards compatibilismwilltendto concedethatthereis atleasta strongappeamnce ofontological commitment. Insofaras theburdenof proofis generally on the side maintainingthatrealityis very differentfrom appearance,t heburden of proofwill be onthewould-becompatibilist. This conclusionresembles theprinciplementionedabove, exceptthatit of several. involvesconsideration notjustof onefeatureof usage,butrather Those featuresemphasized in the writingsof Quine, Church,and others include: The usage ofS expressionsas nouns Nominal: The making of assertionsof theform: Quantificational: "Thereis an S that. . ." Identificational:The making of assertionsof theform: "This S is thesame (one) asthatS"
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In thecase oftheuse of natural numberexpressionsin conventional mathematicalandscientificandeverydaydiscourse,theresemblanceis verystrong as regardsallthefeaturesof usagecited. As forthenominalcriterion,we have anelaborate system of numerals: one, two,three,. .. j and besides these, compound phrasesare formed using such expressionsas: thenumberof ... ,
theproductof ....
the sum of .. . ,
The numeralsfunctionsometimes as adjectives,as in: 1001 nightspassed, butoften asnouns,as in: 1001 is composite, whilethecompound phrasesfunctiononly asnouns,and may occuras subject,directobject,or indirectobjectin a sentence . criterion , we haveboththeparticular and the As forthequantificational universal: foreverynumber... ,
thereis a number. . . ,
as well asc omplicatedite rat ions, alternations , and nestings: for everynumberm thereis a numbern .. .. Theseappearnot onl y in theindicativein theorems,butalsotheinterrogative in problemsand the imp erativein algorithms: . is thereany numbern . . .?
takeanynumbern . . .!
As fortheidentificational criterion , we havebothequationsand inequalities: . . . equals. . . ,
... is greater t han. . . ,
... is lessthan. . . ,
as well asassertionsof unicityand multiplicity: thereexistsa uniquenumber.. . , thereexistseveraldistinctnumbers. . .. (And tomentionless philological considerat ions, thereis somet hingexplicit ly called"thetheoryof numbers" cultivated as an importantbranchof mathematicalscience, whereasthereis no "t heory of snags" or "t heory of chances ". Or rather,thereis a theoryof chanc es, thetheoryof probability , butin this theorya chanceor probability is simply a numb er.) It is such featurest hat
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constitute atleasta strongappearance of ontological commitment, a strong prima facie case for the incompatibilistto answer. 3. NOMINALIST PARAPHRASE
Can theavailability of a nominalistparaphrasehelpthecompatibilistanswer this case? Can the availability of a nominalistparaphrasehelp show thatconventional mathematics,including (1), does not imply (2)? One early discussion of"ontological commitment",from aboutthesame period as the earlycontributions of QuineandCarnapand Church,namely, Alston (1958), questionsof ontolseems to suggestthatparaphrasecan never be relevant to ogy. This is surelygoing too far, and it may be well to consider instance an constituting a counterexample to thissuggestion. Considerthemethodofparaphraset hatreplaces"thereis a strongchance that.. ." by "it won'tbe surprisingif it turnsout that. . ." . Applied to somethingthatis ordinaryin the senseindicatedabove, like (7), this method producessomethinglike: (11) It won'tbe surprisingif it turnsoutthatMoriartyis responsible for this outrage . And themethodseems to beordinarily adequate in thesensethatanordinary originaland itsparaphraseseem to be usable more or less interchangeably in thecontextof philosophically unselfconscious discourse. Applied to the extraordinary (8), however, themethodproducessomethinglike: (12) It won't be surprisingif it turnsout which is not aproper English sentence . This outcomemay very well help lead us to view (8) itself as being, notimplication an of (7), but rathera piece of nonsense. The situation seems to bedifferent withthemodal paraphraseof conventionalm athematics,whichturnst heordinary(1) into (3). Again, the method of paraphraseseems ordinarily a dequatein thesense indicatedabove,butin this instance,t hemethod, applied to (2),producessomethingthatseems to be proper English, to be implied by the paraphrase(3) of (1), and to be assertedby modal nominaliststhemselves, namely, (4).If theargumentof (despite compatibilistnominalistsis supposed to bethatthey arewarranted theirprofessednominalism) in asserting(1) because itsparaphrase(3) is assertable , parityof reasoning would suggest thatone is warranted also in asserting(2), since itsparaphrase(4) is assertable.In any case , theoutcome thattheparaphraseof (1) appearsto imply theparaphraseof (2) can only reinforce, not undermine,theappearancethattheoriginal (1) implies the original (2) . This is not to saythatthe modal methodof paraphrasecan never be relevant . Indeed, applied to :
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t han10 (13) Thereis no numbergreater
10 10
buttherecould have been,
themethodproducessomethinglike: 10 10
(14) Therecould not have been numeral a greatert han10 could have could have been.
,
but there
And this outcomemay very well help lead us to view (13) as a piece of nonsense,since (14) seems not to bep roper English. This in turnmight well help lead us to view numbersas appropriately classifiable as "abstract", since theinapplicability of themodaldistinctionbetweenwhatis and what could have been seems to be partof whatis meantby "abstract"(whenthis termis notjustdefined by a list of examples). (Note thatChihara'smethod of paraphraseapplies thedistinctionnot to "abstract"numbers, butrather to "concrete"numerals.) But abstractness is one thing,and nonexistence another. The common featureof theexamplesconsideredso far isthis: If an ordinarilyadequatemethodof paraphraseis applied to abody of assertions T (which mayconsistof a singleassertion,T = {T}) to produceT* (which againmay consistof a singleassertion , T* = {T*}) and to anextraordinary assertiona to producea* , thentheparaphrasewilltendto undermineor reinforcet heappearancethatT implies a accordingas T* does or does not appearto imply a» , The issue ofappropriateformalization may be reconsideredin thelight of the principlejustsuggested. As alreadyindicated,were itagreedthat(1) is to be formalized as somethinglike:
(15) (3x)(Nx /\ ... ) , compatibilismwouldnotbe an issue. For (15)implies: (16) (3x)(Nx) , and (16) may be "deformalized" as somethinglike (2). Butas alreadyindicated,theappropriateness of thisformalization is justwhatis not agreed. Whetherany formalization is uniquely appropriateseems to bedisputedby Quine, who, as indicatedabove, holdsthatthereis no sharpline between . appropriateand inappropriateformalization Buteven if oneacceptsthisnotuncontroversial view ofQuine's , thereis at leasta conventional methodof formalization forconventional mathematics, which would be used by any philosophically unselfconscious logician. This methodis alsoordinarily a dequate,not quitein thesense indicatedabove, butin thesense ofthefollowing claim, sometimes calledHilbert 's Thesis: For ordinary T and (), logicians'judgmentsaboutwhen theformalization ()* of () is implied by or deduciblefrom theformalization T* of T tendto agreewith
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mathematicians ' judgmentsaboutwhen () is deduciblefrom orimplied by (This agreementunderliest heutilityof thelogicians' incompletenessor undeducibility resultsformathematicians, much asChurch'sThesis underlies theutility of incomputabilit y or undecidability results : A resultto theeffect that()* cannotbe deducedfrom T* is supposed bothto explainthefailure of pastefforts to deduce() from T andwarn ofthefutility of suchfutureefforts .) Since this conventional m ethoddoes replace(1) and (2) bysomethinglike (15) and (16), it tends,in accordancew iththeprinciplesuggestedabove, to reinforcet heappearan ce that(1) implies (2). It is noaccidentthatChihara 's methodofparaphrase,likeothersin theliterature, provides(2) with aparaphrasea pparently implied bytheparaphrase providedfor (1). For ind eed, themodalparaphraseis obtainedby way of the conventional formalization , in somethinglikethefollowing sequenceofsteps:
T.
(i) given anoriginallike (1), (ii) formalize it conventionally , obtainingsomethinglike (15); quantifier"3x" withvariablex fornumbers by (iii) replacetheexistential " 0f' withvariable~ fornumerals; a constructibility quantifier (iv) "deformalize"t heresult , obtaininga paraphraselike (3). It is almostinevitable t hata methodof paraphraset hatproceedsin thisway
willproducesomethinglike (4) asparaphrasefor (2). 4. SPECIAL R ELATIONSHIPS
Som etimes thecompatibilistappealto paraphraseis supplementedby a suggestionaboutthecloseness oft herelationship betweenparaphrasea ndoriginal, best expressed in theterminologyof modernlinguist ic theory. In the terminology t hatis p erhapsbestknown tophilosophers , therelationship suggestedwould bethat(3) uncovers the "deep" structure underlying t he"surface" structure of (1). This terminologyderives from anearlyphase of the work of NoamChomsky.' andhas beensupersededin lat er phases; butunder one terminology or another,a distinctionbetweenthe "superficial"analysis of traditional grammarand the "depth" analysisof modern linguisticshas survivedin many latertheories . In anotherterminology , theclaimwould be that(3) uncoversthe "logical form"underlying t he "grammatical form " of (1). (This latter formulation of theclaim suggeststhatQuine'sviews about formalization are beingrejected,a ndtheexistenceof a unique"logical form" and hence auniqueappropriateformalization, is being assumed.) 3Formula te d in Ch ur om sk yit e terminology, the claim tha t "(3) reveals the 'dep t hs ' underlying the 'surfac e ' of (1)" wouldbe the claim that "( 1) and 3 have differentsurface str uctu resbutthe same dept h st rucure t , and this com mon depthst ru ct ures much i more ctur e of (3) than like that of ( 1)" . like the sur facestru
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Therewould beconsiderablehistorical irony innominalistst husappealing to ahypothesisof covert"depth" forms underlying o vert"superficial" charforms ofthekind made familiarby Chomsky. Historically, it has been acteristic of pro-intuitionist anti-nominalists from Dummett(1956) onwards "underto claimthatsentencesaboutwhatnumeralscould beconstructed lie" sentencesa boutsentencesaboutwhatnumbersexist(notso much in a conditions"). Chomskyitesense as inthesense ofprovidingtheir"verification (Dummett derives hisoutlookfrom thesame anti-nominalist sourcefrom which several of Church'sviews derive, namely, Frege .) Originally,it was the nominalists, Goodman and Quine, who were sympatheticwith "structuralist"and "behaviorist"approachesto language; and whenChomsky a languageb eneaththeobserved"surface", theorizeda bouta "deep" level of ", as when aCarnapor aChurchwished tointroducea bstract"propositions they were skepticalandcritical. Irony aside,t hereis a considerable problemof evidence for any claim a bout "depth"analysis. Andperhapsone needs to beespecially carefulaboutevidence when the claim is made, not by a linguist for whom understanding an but ratherby a of thehidden mechanisms of languageis an end in itself, philosopherwith ulterior ontological motives. Evidenceaside, thereis also a considerableproblemof relevance. Suppose therewerethebest possible evidencethat(3) and (4) reveal the "depths" underlyingthe "superficial" (1) and (2). Would not thisstrongclaim aboutthe closeness of the relationfurther reinforcethe ship betweenparaphraseandoriginalserve only to yet appearancethat(1) implies (2), justas (3) implies (4)? At anyrate,it is hardto see howtheclaim that(4) revealsthe"dept hs" underlying underlying the "superficial"(2), and thatthe "depth"structure (2) is quite different from "superficial its " structure, could serve to motivate a denial of (2). Chomsky,afterall, held inishearlyworkthat: (17) Chomsky hasrefutedt hestructuralist Bloomfield and behavioristSkinner, t he"surface" of: revealsw hatgoes on inthe"deeps" underlying (18) The structuralist Bloomfield andbehavioristSkinnerhave beenrefuted by Chomsky; and hencethatthe "depth"structure of (18) is quitedifferent from its "sudid nottakethis to showthat perficial " structure . But he emphatically thestructuralist BloomfieldandbehavioristSkinnerhad not been refutedby Chomsky! Debatesoverontology t akeplaceentirely in "superficial"language:Philosophersare neverheardspeakingnor seenwriting"depth" language,which is an unobservable theoretical posit of technical linguisticscience. The "su/ "depth" distinctiontherefore has no obvious relevance theold to perficial"
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issues of ontology, nominalism,and compatibilism. Butacceptanceof such a distinctionmay make it possible tointroducenew issues, which might be calledbathontology, infranominalism, and infracompatibilism . Call(bothparadigmaticand idiomaticoccurrences of) the"thereis" that occursin audible,legible "superficial" languagetop-quantification, and the counterpart thereofin unobservable , theoretical "depth" languagebottomquantification. Let 8 be some common noun . Ontologywas concernedwith questionsof the kind: Is onewarranted in asserting(6)? In the present terminology t hisbecomes the ques tion: Is one warranted in asserting,"[top8"? Nominalismwas thenegativeanswer to this questionin quantification] thespecialcase of (1) or the like , where 8 is"number" orthelike. Compatibilism was theclaimthatendorsingnominalismis notincompatiblewithmaking certainassertionsofthekind "[top-quantification] an 8 that.. ." because(the occurrenceof top-quantification here beingidiomatic ratherthanparadigmatic) thesedespite appearancesdo not really imply "[top-quantification] 8" . Bathontology would beconcernedwithquestionsof the kind : Is one warrantedin assertingsomething whose depth version is "[bottom-quantification] 8"? Infranominalismwould bethenegativeanswer tothisquestionin the special case where 8 is "number" orthelike. Infracompatibilismwould be making the claim thatendorsinginfranominalismis not incompat ible with 8", because itsdepthversion is the assertion(6) that"[t op-quant ification] 8" . As thehistorical somethingquitedifferent from "[bottom-quantification] remarksabove indicate , in the early d ebateovernominalism,some atleast not the nom inalist,side) took apositionnotutterly (on theanti-nominalist, unlikebathontological infranominalist infracompatibilism (namely, the posiconditions"forexistencetheoremsin mathematics tionthatthe "verification involveconstructions on numeralsd enotingor pluralities instancingnumbers and notdirectlyon thenumbersthemselves). Now all this involvessubstantial a p resupposition , for which evidence would berequired,namely, thepresuppositionthatthereis a "depth"counterpart of the "surface " expression"thereis". The presuppositionhere is a m ost "superficial" occurdoubleone: First,the"depth" versionsunderlying . (Those "superficial" rences of"thereis" must involve someone construction occurrencesof "t here is"thatare distinguishable as idiomaticeven by the coarsecriteria oftraditional grammar, and presumablyalso someoccurrences as notparadigmaticonly by the refined of "thereis" thataredistinguishable criteriaof modernlinguisticsneed not do so,but most must.) Second, most occurrences of this "depth" construction must be normallyexpressible"superficially" using "t here is" . (Some may be normally expressed by some other expression,but most must beexpressibleby this one express ion.) For if th is e is" splits into is not thecase, eitherbecausethe "surface"expression"ther a greatvarietyof different "depth"constructions, or because it issubsumed with a greatvarietyof other"surface"expressionsundersome one "depth"
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thentherewould seem to be no basis fordistinguishingsome construction, one depthconstruction as the analogue or counterpart specifically of the expression "thereis" , and hence no basis forrecognizing a topic ofbathontology underneath the topic of ontology. The questionwhetherthesepresuppositionsdo indeed hold, and with it thestatusof moreparticular claimsaboutbathontological commitmentand thecompatibilityof infranominalism withconventional science, is one I am t heprofessionallinguistsa nd will nota ttemptto settle preparedto leave to here. WhatI would like to emphasizehere isthatthekind ofdistinctionof levels oflanguagerecognized inmodernlinguistictheoryprovides themost (and perhaps theonly) obviousconceptualframeworkin which one might tryto make sense of t hedistinctionassumed, in oneterminology or another, by thosephilosopherswho tryto distinguish"mere" from "ultimate"being or existence,asserting(1) (and forthatmatter(2» in the "mere" sense, and denying (2) (andforthatmatter(1») in the"ultimate"sense. Such positions,which may beconsideredversions orvariantsof compatibilism, have beentaken,forinstance,in theclosingparagraphsof Bonevac 1980, and in theopening paragraphof Hodes 1990 (anotherauthorinfluenced, likeChurchand Dummett,by Frege), where one reads: The answersto questionslike "Are therenumbers?" and "Do sets exist?" are, trivially,"Yes." To not seetheseanswersas trivialities bespeaks a misunderstanding of mathematical discourse. Butto go on and say thatthereis a realm of mathematical o bjectsis to engagein obscurantist h yperbole . Mathematical objectsare second-rate ; they are notamong the"furniture of theuniverse ."
These are authorsof thekind describedearlier,in connectionwith thepasthatthere"are" suchthingsas sage fromChurch,as only meaning to deny numbersin some extraordinary sense of"are" (as contrasted withauthorsof the kind who when theysay, "Thereis a numberthat..." don'treally mean whata philosophically unselfconscious mathematicianwho said this would mean). The distinctionhere between"second-rate"(or "mere") and "first-rate" (or "ultimate")t hingsor objectsdoes notmake any immediatesense. The first, "mere" sense of "existence" makes sense , since it issupposed to be this sense (only)thatoccursin ordinaryscientificdiscussion. In the scientific communitythereis considerableagreementaboutwhich assertionsof existencein thisfirst sense are warranted, suggestingthat"existence" in the first sense is being used in thatcommunityaccordingto fairly definite rules, giving it a fairly definite meaning. The problemis with thesupposed second, "ulphilosophicald ebate. The timate"sense thatoccurs(only) inextraordinary burdenof providingsuch a secondmeaningof "existence"rests,as Churchso long ago said, on those whose writingsor whosephilosophicalviews require it.
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REFERENCES
Alston,W . 1958 Ontological commitments, Philosophical Studies, vol. 9, pp. 8-17. Bonevac, D. A. 1980 Ontological reduction in the abstroct sciences, Hackett . Burgess, J. P. 1983 Why I am nota nominalist,Notre Dame Journal of Formal Logic, vol. 24, pp . 93-105. Carnap, R. 1950 Empiricism, semantics, and ontology,Revue Internationale de Philosophie, vol. 4, pp . 20-40. Chihara,C. 1973 Ontology and the vicious circle principl e, CornellUniversityPress, Ithaca, New York. 1990 Constructibility and mathematical existence, Clarendon , London. Church,A. 1958 Ontological commitment, The Journal of Philosophy, vol. 55, pp . 10081014. Dummett, M. 1956 Nominalism, Philosophical Review, vol. 65, pp . 491-505. Goodman, N. 1956 A world ofindividuals,The problem of universals (1. M. Bochenski, A. Church,a ndN. Goodman,editors), NotreDame UniversityPress, Notre Dame, pp. 155-172. Goodman, N., andW . V. Quine 1947 Steps towarda constructive nominalism, The Journal of Symbolic Logic, vol. 12, pp. 105-122. Hodes, H. 1990 Ontological commitment: Thick and thin, Meaning and method: Essays in honor of Hilary Putnam (G. Boolos, editor),Cambridge University Press, Cambridge,Massachusetts pp . 235-260 . Quine, W . V. 1948 On whatthereis, Revi ew of Metaphysics, vol. 2, pp. 21-38. 1951 On Carnap's view onontology , Philosophical Studies, vol. 2, pp. 65-72. 1981 Theories and things, HarvardUniversityPress, Cambridge, Massachusetts.
MICHAEL DETLEFSEN
PEACE, JUSTICE AND COMPUTATION: LEIBNIZ ' PROGRAM AND THE MORAL AND POLITICAL SIGNIFICANCE OF CHURCH 'S THEOREM-
1. LEIBNIZ ' PROGRAM
Throughout his life, Leibniz gavehimselfto projectshe saw ashavingthe potential to improve theconditionsof humanlife. Among these,none was more importantto him thanthedevelopmentof his ars combinatoria. This ambitiousproject,sketchedin hisearliest writings(cf. De Arte Combinatoria, 1666) and returned to againand againthroughout theremainderof his life (cf letter of January10, 1714 to NicolasRernond"), was originally divided intothreesub-projects:(i) thecalculus ratiocinator (calculusof reasoning) , in which he hoped to codify , in mechanicalform, all caceptableforms of logicalreasoning,(ii) thecharacteristica universalis (or universal c haracteristic) , intendedto serve as a logically perspicuouslanguage fortheexpression of all rational t hought , and (iii) the encyclopedia of human knowledge , intended to catalogthe whol e of received human knowle dge.f Leibniz believedthat , - I wouldlike to thank aud iences at the Un iversity of Notre Dame, the Univers ityof Q ueen sla nd t, he Univer sity o f Sydney, LaTrobe University, Mon ash University and the Unive rsity of Montr eal for usefuldiscussion s of this pape r. Am ong indi viduals, I am to Ala sdairMacIntyre , Graham Priestand the la t eIan Hinckfuss for par ti cul arlindebted y comme nts. thoughtful 1 English tran s lationin Leibn iz 1969, page 654. Ther e Leibniz writ es: ". . . if I had been less distracted , or if I wer e younge r orhad tal e nted young men to help m e , I sho uld stillhop e to creat e akind of universal symbolistic in which alltruth s of reason would be reduced to a kind of ca lculus . . .. this couldbe a kind of univer sallanguage or writ ing . " for the character s and the words them selves wouldgive direct ion sto reas on , and the errors-except those of fact-wouldbe onlymistakes in ca lculat ion." s into two by co mbining the calcu2La te r , L e ibniz consolidated the three sub project lus ratiocinator and the cha rac te rist ica universalisintoa singleprojectca lled' t he gener al scie nce'. Thus, in the "N ew Prop osals" he wrote : "I believetwo things to be necessary for m en to take advantag e of their opportuniti es and to do every t hing th ey couldto contribut e to their own happiness, at le astin the matter of knowled ge .. . These two things are, first an exac t INVENTORY of allthe knowledge disp ers ed and badly arranged(atleast of thatknowledgewhich appears to be most importa nt at the beginning), and second ly, the G ENERAL SCIENCE which sho uldgive us notonly th e m ean s touse knowledge already acqu iredbutalso the m eth odof judging and discovering, in orde rto go furth er andsupp ly what we wa nt . This invent or y I speak of wouldbe very different from sys te ms and dictionaries, and wouldbe com posed onlyof Lists or enumerations, Tabl es , or Progr ession s which wouldserve to keep always befor eus , dur ing some reflecti on ro del iberationof any
445 C. Anthony Anderson and M. Zeleny (eds .), Logic, Meaning and Computation, 445-467.
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takentogether, thesethreedevices wouldprovidethebasis forsignificantimprovementsin humanknowledgeandin theconditionofhumanlifegenerally. He even believedthattheycould be used to serve theends ofpeace and justice.Peacewould beservedby developinga mechanicalmeans ofresolving of epistemic disputes. Justicewould beservedby using themechanization thought to effect a more equitable d istribution ofepistemicgoods, one which wouldincreasethesharesof theepistemicallyleastadvantaged. The mechanizationof thoughtwould servepeace by allowingopposing sides in anepistemicdisputetopeacefully ' calculate' theirway to aresolution . It would servejusticeby enablingthosenotso richly endowed in nativeor 'int uit ive' powers of reasoningto come closer to epistemic paritywiththose more richly endowed with such.addition, In it would maket hesocialbenefits of knowledgeavailableto thoseto whom it could not make the knowledge itselfavailable. We will refer to this generalprogramof promotingpeace and justice or mechanization of reasonas Leibniz ' Program. throughtheformalization In doingso, we do not however insi st thatit agreein everyparticul arwith whatthehistoricalLeibniz himselfmay havehad in mind. It istheideas themselves,r ather t hanperfectaccuracyin attribution thatis central to our concerns.Thatnotwithstanding , we believethatwhatwe heredescribeas Leibniz' Programis a plausibleinterpretation of Leibniz'own views. Ouraim is to assesstheplausibility of Leibniz' Program. The assessment we offer isbasedon Church's celebrated proofoftheundecidability ofclassical first-order validity. It is largely, thoughnot entirely, pessimistic. We argue thatChurch'sTheorem poses a serious theoretical limit on the extentto which programslike Leibniz' Programcan be successful.If we arecorrect , Church'sT heoremwouldappearto havenon-trivial implicationsin therealm of socialand politicalphilosophy. These implicationsseem to have gone largely u nnoticed . Ourassessmentof Leibniz' Programrestsupon theidentification of what we taketo be its key element; namely,condition a we shall ref er to asthe Computability Requirement. This requirementt akes theform of atenetof epistemicsocial policy concerning l egitimate a ttemptsby individualstoshape thebeliefsand actionsof otherswho areco-members with them in a given epistemic community. Specifically, it s tatesthatto be elgitimate[i.e., to be thesortof thingthatproducesan obligationto believethatthecommunityshouldregardas enforceable) , an attemptto shape or conformthe beliefs ofothersin anepistemic communitymust be backed by asupporting sort ,the cat a logue of fact s and circumstancesand the m ost importantassumptionsand maxims which oughtto serveas thebasis of reasoning . . . . [in a disorderlyinventory]o ur very richesmake us poor, somewha t lik e what wouldhappen in a big store which lacked theordernecessaryto findanythingone needed. . . the generalscience helps to make up an orderlyinventory.. . it iswiththegeneralsciencethatwe shallhave tobegin." (English translation from Leibniz 1951, page 581 , em phases Leibniz', square brackets mine).
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'epistemic computation '. By an 'epistemic computation ' of a givenproposition,we mean, roughly , an effectively verifiable formalizeddemonstratiorr' of thepropositionfrom premises locatedin thecommunity's encyclopediaor inventory of acceptedknowledge. t Leibniz advocatedguidanceby such a cond i tionbecausehe believed computational abilityto be a kind ofcognitive'lowestcommon denominator' among humanbeings-a cognitivecapacitythatlies (or atleastcomes close to lying)withintherangeof competenceof allhumanbeings equally. This belief inthecognitiveelementarity or commonalityof computational thinking is whatwe shallrefer tothroughout this paper as Leibniz ' Thesis. Its significance lies in t hefactthattheworkings of such common a capacityare l eastadvantaged to achieve a plausiblyseen asenablingeven thecognitively certainepistemic-politi calobjective-namely,epistemic autonomy,t heconductof theirepistemic affairsaccordingto theirown epistemic lights. Leibniz' Thesis gives thereasonwhy institution of theComputability Requirementis to be seen asrecommendedepistemic policy for acommunity. Given thetruthof Leibniz' Thesis, enforcementof some condition like the ComputabilityRequirementis necessaryfor theadequateprotectionof the epistemic autonomyof thecognitivelyl eastadvantaged m embers of a community. This, atany rate,is trueif thecommunityin questionrepresentsa reasonably diversecross-sectionof humancognitivecapacities. To put thepointsomewhatmore accurately , enforcementof a condition liketheComputabilityRequirementwouldappearto be necessaryto protect whatwe will refer to weak as epistemic autonomy. Weak autonomyconsists and/orto actexceptout in thelimitedrightof anindividualnot to believe of her ownconviction.The ComputabilityRequirementprotectsthe weak autonomyof theepistemicallyleastadvantaged individualsin a community for cla ims calling for their by enforcinga high standardof elementariness epistemic conformity . By enforcingthisstandard,even theconvictionof the epistemicallyleastgiftedshouldbe moved toautonomousconformityand theirweak individualautonomythuspreserved. Enforcementof the ComputabilityRequirementis thus desirableas a means ofsecuringtheepistemic autonomyof the full rangeof citizensin a cognitivelydiversecommunity. Whatwe willargue,however, isthatit alsothreatens anothert ypeof epistemic autonomy-strong epistemic autonomy-forthose who are cognitively more gifted. Strongepistemic autonomyconsistsin thelimitedrightof an individual 3 In ca lling aformaldem onstr ation 'effect ivelyerifiabl v e'. I m ea n to say thatits status well-formed , accord ing to th e community' s st andards of well-f ormedness, is as infere ntially effectively decidabl e. 4The encyclop ed iais somew hatabst rac tlyand ideall yconceived. Spec ifica lly, itis conceived as conta ining not only allact ua l kn owledg e confirma b le yb m eans com mon ly p ossessed and accep t ed within a given community, but, indee d, all kn owledge ,non- ac tu al as well as ac tual, that is confirma ble yb such means.
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to shape thebeliefsand/oractionsof othersin ways she beli eves theyought rationally to be shaped. We willarguethatthereis a limit to which t he interests ofbothweak andstrongepistemicautonomycan be served, and t hat thislimit isplausiblyattributable to Church'sTheorem. We thusclaimthat Church's Theoremposes limitations on theextentto which anenduringstate ofepistemicpeace[i.e., a stateduringwhichepistemicconflicts are peacefully resolvable) can be expected.We will also arguethatChurch'sTheorem(and a family ofrelatedresultshaving to dowiththe 'Decision Problem') pose an equitable limits totheextentto whichepistemic justice-inparticular, distributionof basic epistemic goods-canbe advancedby means broadly thatChurch's similarto thoseproposed by Leibniz. We believe,therefore, Theoremhas notonly social and politicalb utalso moralramifications . 2. PEACE AND EPISTEMIC CONFLICTS
As we willtreatofthem,epistemicconflicts arise when (i) some member of a communitymakes ademandofepistemicconformityon thepartoftherest or some partoftherest oft hecommunity,and (ii) some othermember(s) of the communitydoes not accede to thiscall for epistemic conformity. We assume thatthe community in questionhas a definedand mutuallyagreed-upon understanding of (1) whatconstitutes epistemic conformity Inon-conformity, be called for, (2) whataretheconditionsu nderwhichconformitycan rightly and (3) whatare themeasuresthatmay rightlybe takenin orderto secure conformity.The specifics of theseassumed elementsof agreementneed not concernus here. On ourconception,everyepistemic conflictt husbegins withwhatwe will is, a call to communityto a conform refer to as an 'epistemic claim'-that theirbelief to acertainpropositionor propositions . Ordinarily, these will be propositionsthatthe agentissuing thecall herself believes. This, however, is notessential to the issuing of an epistemicclaim. Whatis essentialis that theyare directedat communitiesand call for complianceas regardscertain mattersof epistemic commitment." 5To avoid confusion , let me offer a coupleof clarificatory remarksatthisjuncture.The a t I takea rather broadand flexibl e understandingof thenotionof community. first is th ar, I am assumingthata given individualmay livein severaldifferentepiste mic In particul communitiesatonce, and thatthese com munit ies may be sm all. Indeed, I am pr epar ed even tocountthep ersonaeof a single divided m ind as com p rising a communityof sorts ing epistem ic agencyshould prov e to be cogent providedthatsuch a sche me ofindividuat , let me say that I am wellawar e thata grea t (which it very w ellm ay not). Secondly many epistemic conflict s do not have so much to do with the truthor falsity of a given belief as with thestrengthof evidence or f or againstit. I am assuming that alls uch disagreementscan, however, be representedas disagreementsregardingthetruth-valu e of som e proposition(e.g., such aproposition s as "The evide nce forp is conclusive" or "The evidence for p is strongert hantheevidencefor q.") . Hence, I believe thatthebasic forms such seeminglymore recondit e of disputeoutlinedherecan serveas a basis for re presenting types of conflictas those centeringon differencesin thest re ngt h w ithwhich th e disputing
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To putit more exactly, an epistemic claim is aquintuple of elements.T he first istheclaimant(the conformant) issuing theclaim. The second isthe targetcommunityto whom the claim addressed.The is thirdis a propositionalelementp giving thepropositional c ontent of theattitude-taking being proposedby theclaimant . The fourthis an attitudinal elementA specifying theparticular epistemic attitude(s) and/oractionalstance(s)towardsp that theclaimantis sayingshouldbe taken. The fifth, finally , is a modalelement . , an obligation In it theclaimantspecifies anormativemodality(specifically or anentitlement of some kind) which shetakesas defining the responsibilities ofthetargetcommunitywithregardto theepistemic attitude-taking featured in her claim. Understoodin thisgeneralway, epistemic claims can be seen as divided into twobroadtypes, dependingupon whetherthenormativeor modalel. We will call claims ement of theclaim is anobligationor anentitlement registeringan obligationto believe (or to actas if one believes)conformal claims. Non-conformal claims, on theotherhand,assertno suchobligation, butmerelyregisteran entitlement to believe and or to act in certain a way. Theythusserveprimarilyeitherto inform those in a given target community facilitate sharing of some featureof theclaimant'sepistemic holdings or to of non-conformal holdingsamongstcommunitymembers for thepurposesof generalepistemic enrichmentand the like . The distinctionbetweenconformaland non-conformal claims isessential to ournotionof epistemic conflict . All conflicts, as we see them, requirethe or actional s tanceA lodging of a conformalclaim (to hold epistemicattitude withrespectto propositionp) to which somemember (the non-conformant) 's targetcommunitydoes notthenconform. of theconformant In orderfor aconformalclaim to belegitimate,theconformant m ust produce agroundcapableof movingthemembers of thetargetcommunityautonomouslyto adopttheposition(epistemic attitude or actional stance)she of aconformal claim to a commuis calling for in her claim . The presentation nitythusincursa debton thepartof theconformant -adebtto provideits members with a means of moving themautonomously to conformity . Indeed, the assertedobligationof conformitypresentedin her conformal claim does . not go into effect untilthis debthas beendischarged This is a key element in o ur conceptionof epistemic conflict. It instituteswhatmight be called rule a of conservat ism in relationships between to thisrule, conformalclaimantsand theirtargetcommunities. Accordingly no change in the epistemic or actionalpositionsof the members of at arget communityis requireduntiltheconformantprovides them with acompensatingwarrant forthatchange--awarrant whosepurposeis to preservetheir epistemic and actionala utonomywhilstsecuringconformity . The rule of conservatismis thusrootedin a concernfor theprotection of theindividual partieshold beliefs .
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from authoritarianism in herepistemic and actionallifeand is designed to of theepistemicautonomyoftheindividualbe insurethatdue consideration includedin anylegitimated emandforepistemic conformity. The issuing of aconformalclaim thusrequirescompensationfrom the conformant.Suchcompensationmust generally taketheform of anargument move the members of the target or groundthatis adequateto rationally communityto autonomousa doptionof theattitude or actional s tancecalled for intheconformalclaim. Properconformantcompensationshouldalso make non-conformity a ctionableby making a properlycompensatednonconformant guiltyof a breachof rationality. The particular features of acommunity'sscheme ofcompensatinggrounds will bepartlydeterminedby how deeply intot he epistemic and actional lives ofindividualsit believest hemechanismsforepistemicconflict-resolution oughtto bepermittedto go. Those who believethatrelatively deeply and strongly held beliefs and actional s tancesshouldbe open toconformant deposmands willrequirethatcompensatorygroundsoffered byconformants sess a highpotencyforautonomously moving rational agentsto conformity. convictionsshouldbe open Those who believethatonly less deeply held to demands of conformitywill be free to adopta scheme ofcompensating groundsof lesserpotency .6 This way ofthinkingof conformant c ompensationputssubstantial weight on theepistemic and actionalautonomyof theindividual. This concern forautonomymay be seen as based, inturn,on a concernforauthenticity. The chief idea isthatan individualwho is calledupon to believe a given p propositionp must be givensomethingthatis capableofmakingher belief in authentic, or anatural rational outflowing of convictionsandinclinations that are, inthetruestsense, her own.I t is onlythroughbeing givensomething capableof inclining her autonomousepistemic selftowards acceptance of p thatbelief inp by someone notpreviouslyinclined by herconvictionsto . Similarlyfor anindividual'sactional stances. believe inp can beauthentic We emphasize this type of autonomy-freedom not to believe and/orto actotherthanout ofone's own convictionsand inclinations -becausewe believe it to be crucialtheissue to of peaceful resolution ofepistemicconflicts. 6It shouldnotbe assumed thatthereis automatically anycorrelation betwccnwhatwe are thinkingof hereas the "depth" of individualbeliefand thestrength of theevidence requiredfor itsenforceable revision. Whata communitycountsas evidenceand, so, what must be acceptedas evidenceamong its individualmembers, must be somethingthatthey can agreeupon. At thesame time, however,it might turnoutthatwhatcan be agreed upon as evidence is not whatany individualmember of thecommunity would herself countas belongingto themost powerfulevidenceather disposal. In suchcases, therewill developan importantdistinctionbetweenwhatmight be thoughtof as publicand what might be thought of as privateevidencewithatleastsome of whatthevariousindividual members of thecommunitycountas among theirmost powerfulevidencebeing beyond thepaleof thatwhich is public. This notwithstanding, theremay stillbe enough(and importantenough)agreementbetweenthemembers of a communityfor theirformation intoa communityto make sense and to have apurpose.
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Threatsto itcanserve aspotentand legitimatemotives fortheuse of force againsta communityby an individual.Demandingof a rational individual , andone thatlegitimately sees herselfas rational , thatshe believesomething thather ownwell-examinedand responsiblydevelopedconvictionsdo not , in animportant incline her to believecalling is u ponher to dosomethingthat sense, she, as arational individual,cannotdo. Such ademandrequirese ither thatshe become anindividualotherthantheone she is orthatshe cease existingas an individualaltogether. Such calls are not always illegitimate. However, whendirectedtowardsindividualswho areunderno obligationto reform or reconfigure theirbeliefs,theyarelegitimate only iftheone issuing thecall also providesa means (i.e., a groundsor reason)capableof making thechangean authentic one. Preservation of a similartype of autonomy - whatwe will refer to here withthatof mere belief) as integrity-in thesphereof action(as contrasted is alsoimportantto the peacefulresolution of epistemic conflicts . Integrity of belief andactionwhich makes the actionsof concernsthatintegration a person an accurateand fullrational expressionof her beli efs. Only by between anindividual'sbeliefsand preservation of this typeof relationship her actionsis she protectedagainstthesortof violationof the actional self thatoccurswhen she isnotat libertyto actin accordanc e with her own convictionsa nd is pressuredto make heractional self a vehicle of expression forthewill or beliefs of another.When done, thisconvertst heactiveself of of another a ndactional one personintoan actional a ppendageor automaton autonomy(integrity)is lost. comprise thetwo fac etsof thattype Together , authenticit y and integrity of autonomythatwe arereferringto here as weak autonomy - that negativ e libertynot to believe and/orto actexceptoutof one's own convictions. Leibniz, we believe, was concernedwith the pres ervationof this type of autonomy. In this, he seems to have followede th lead of hi s Protestant 7 There is a line ofreasoningthatsupportsthe idea predecessor Luther. thattheuse of'computations ' of thesortenvisionedby Leibniz in hi s Ars Combinatoria shouldprovide a means of securingconformityto conformal claims while also preservingautonomy.This reasoningproceedsas follows . By a 'computation'let usunderstand an argumentwhosepremises canbe seen by allc ommunitymembers to belong to t heircommon 'encyclopedia' of acceptedpropositionsand whose validitycan be determinedby a very rudimentary t ypeof character-recognition andcombinatorial judgmentthat is withinthereach of all. Understoodin thisway, eachcommunitymember would be able to verify the premises of acomputationby consulting t hecommunityencyclopedia.Likewise, everycommunitymember would be able to 7This is not int end ed to sugges t, however ,thatLeibniz was ent ire lyLutheranin his er s as well- in p articul arby the theology. He was heavilyinfluen ced by Cat holicthink J esu it FriedrichSp e (d. F . W . C . Lied er, "Fried rich Spe and the Theodi cee of Leibn iz" , Journal of English and Gennanic Philology, 1912: 149-172; 329-354).
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verifythevalidityof thecomputationbecausetheevidenceneededfor such verification is so extremelyelementary -a kind ofrudimentary c apacityto recognize andd istinguisha finitevarietyof basicshapes andto tell w hether theylie some type of elementary o rder.f Relianceon such acapacity,Leibniz said, shouldprovide "a sensibleand, as it were , mechanicaldirectionof themind whicheveryone , even themost stupid, couldrecognize .t' " Hence, it shouldprovecapableof moving themembers of acommunityto autonomous beliefin thevalidityof thecomputation,a t leaston theassumptionthat theirassentto thecommunitystandardsis itselfautonomously t aken.This together w ithautonomously t akenbelief in thepremises (i.e., theencyclopedia) shouldthenprovideforautonomousbelief intheconclusion.Implementationof theComputabilityR equirementshouldthuspreservetheweak autonomy,and socontribute to thepeacefulepistemic existence,of thatsegment of thecommunitymade up of thosewho authentically assentto the propositionsof thecommunity'sencyclopediaa ndwho accepttheinferences s.l" of thecommunity'sagreed-uponstockof 'valid' inference It should,however, also contribute to theepistemic peace of theentire communityin another way-namely,by making thesocialbenefitsof knowledge orauthentic beliefavailable to all.Crucialto thepeacefulresolution of conformald isputesis thatallmembers of thecommunityplay by acommon setof rulesa nd thateach beableto determinefor him- orherselfw hethera given piece ofreasoningis in conformitywiththose rules.T his means that thereshouldbe a universally operablemeans ofdeterminingwhethera given piece ofreasoningis countedby a communityas validand, hence, as reasoning whichthemembers ofthatcommunity areobliged tohonor.Implementation of theComputabilityR equirementshouldprovide forthisby offering a com binatorial or computational means of making such determinations . Hence, even forthosenotableauthentically to assentto the tenetsof theencyclo8 Algebraicthought , bas ed on whatwe are ca llingcharacter recognitio n, was contrast ed edin Leibniz to intuitive knowledge (cf. Leibniz 1961, pages 422-425 , the essay ent it led 'M e et Ideis') . The latter was co ncerned withnotion s directl y, itationes de Cognition e, Veritat while the formerconcerned itselfwiththe signs used to sig nify not ion s .Thus , in intuitive thought , notionsthem selves are the direct objectsof awaren ess and manipulati on, while c thought(which Leibniz also referred toas 'blindthought'),it is symbolicexin algebrai pressions thatplaythese roles . Cf. Leibniz 1965, pages 17-19, for a generaldiscussionof the natur e and utilityof symboli c reasoning. gCf. Lett er toOldenburg(undat e d), in Leibniz 1961, page 14. lOIs idealization to co m m unit ies allof whose (normal) m ember s are tak en to have the abilityto accurately d etermine the validityof the formal rules used in their accepted calcu lus os restrictiv e as to be unint eresting ? We don't thinkso . We may grant thatthe ability of acalculu s of reasoningto preserve weak episte m ic autono my depends upon more than st r ictl y"se ns ible"or "a lgebraic" knowledge withoutther eb y rest ric t ingthe extent to e of those notgifted in int uit ivelogical which it is a usefulm eans of increasingthe knowledg p ower andof bringing withinthe purview of their logicaljudgmentmany cases ofvalid(or n discover beca usethey wouldnever th ink of invalid)reasoningthey wouldot he rwise ever them . This extens ion of the p ower s of the intuitiv ely disadvantag ed would, moreover ,be atleastin significa ntpart due to the "mec hanica l"chara cter of the calculus .
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pedia andtheinferencesof thelogical calculus, therewill betheopportunity to takeadvantageof thebenefitsoffered byconformalclaim-makingwhile atthesame time protecting o neselffrombeing takenadvantage of by other such claim-makers . Key to keepingtheepistemic peace, then, is existenceof a common and commonlyoperablemeans fordeterminingwhether a given piece ofreasoning conformsto communitystandardsof validity. Use of computationsto back -should conformalc laims-the'policy ' ofthe'ComputabilityRequirement' providesuch ameans. Hence, itshouldalsoprovidea means of peacefully securingatleastostensible(as distinctfromauthentic) epistemic conformity . Such, at anyrate,is theconclusionof theLeibnizianargumentwithwhich we areconcerned.As we shall see later,Church'sTheoremhas somethingto say bothaboutthisandabouttheabilityof theComputabilityR equirement to simultaneously preserveweak andstrongautonomy. 3. EPISTEMIC JUSTICE
Theelementary character ofsymboliccalculation is thusthatfeature of Leibniz' calculus ofreasoningwhich issupposed topromoteepistemicpeace. This same featuresuggeststhatimplementationof the ComputabilityRequirementmight also aid in increasingepistemic justice- that is, thedistribution of epistemic goods to thoselying in a widevarietyof differentnatural or non-self-engendered categoriesof cognitivecapacity. Epistemic goods can, of course, be of greatprivatebenefitto anindividual. At the s ame time, theycan also be ofgreatsocial value.T he ability to acquireand publiclyto 'expend' epistemic goods seems clearlyto be an importantd eterminant of humanwelfare . Equallyclearly,thedistribution of suchgoods through geneticand(broadlyspeaking)societalforces isanything butequitable . Birth, upbringingand the useand abuse of personal,social and politicalpower oftenresultin unjustifiedepistemic advantagefor some and unjustifiedepistemic disadvantage forothers. Leibniz wasconcernedwith theseinequitiesand saw theirredressas a demandof justice. He seems to have believed thatdevelopmentof his combinatoricartwouldprovidea means of making suchredress. The key, once again,was itscomputational character. Being ofelementary epistemic character,it could bedrivenby capacitiesthatare moreequitablydistributed acrossthehumancommunitythanare nativecapacitiesof intellect and favorablecircumstancesof upbringing. ofthought wouldrunoff will-power Roughly,t heidea wasthatthecalculus rathert hanintellectual ability . It wouldthus,in effect , reduceintellectual differences to differences in -power will , and such differences could thenbe removed byexercise(albeit,p erhaps,diligentexercise) ofthewill.l l In addi11
Leibniz adm itted,in the end, thatthereareeven some minor inequitiesof will-power
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tion,Leibniz believed,o ne'scomputational proficiency could be so improved throughpracticethatthedullbutdiligentepistemic laborercould, by force of determination , matchandeven exceedtheepistemic holdingsof theintellectually more gifted. Such , atany rate,was his view of howdevelopment of his combinatoricartmight promotegreater j usticein thedistribution of epistemic goods. To understand t hisbetter,let us briefly considerLeibniz' generalconceptionofjustice. He characterized it as"t hecharityof thewise man (caritatem , was defined as"universal benevolence"(ibid.), sapientis)".12 Charity,in turn benevolence as the"habitofloving"(ibid.), loving asthetakingofpleasureor delightin the"happinessof another"(ibid.) (or astheregardingofanother's happiness "as one's own" (ibid.I) , and pleasureor thetakingof delightas a "sense" orrecognition of perfection .P Given theabove, it follows that,since God isthemost perfector beautiful of allbeings.I" we findourgreatesthappiness or pleasurein loving Him.15 One can not , however, love God withoutatthesame time loving her fellow humanbeings. The reasonis thatin loving Godwhatwe loveareHis perfections. Hence, since ourfellowhumanbeings arecapableof growing inthe same perfections,we must desire to seethoseperfectionsr eproducedto the highestpossible extentin them.!" It thusfollows that , to theextentthatwe trulylove God (i.e ., to theextentthatwe truly"sense" His perfection),we mustalsocultivate theincreasein humanbeings ofthoseof God's perfections which can(thoughonly to animperfectextent)be realizedin them.!" For Leibniz it wast husa demand of justicethatwe seek theperfection of our fellowhuman beings. Furthermore,among the ways of perfecting humankind,none wasmore importantthanthe perfectionof knowledge and reason. Allothergifts, said Leibniz, may corrupta humanbeing; reasonand reasonalone isunconditionally wholesome fort hem. I S With improvement of reasoncomes improvementin theabilityto recognize perfectiongenerally and to recognizer ational perfectionin particular . Improvementof reason one'sfellowhuman shouldthusnot onlyincreaselove of God,b utalso love of amongsthuman bein gs. Hence, justice--epistemic justice-demands the encouraging of ed too. Cf. Theodicy, pt. 120. the discourag l2Cr.theprefac eof Codex Juris Gentium Diplomaticv.s (1693) , an Englishtranslation of an excerptfrom which ap pea rs in Leibniz 1951, pages 559-563. Cf. also "Eleme nt s of Natural Law" (1670-1) , pp . 133-4 in Leibniz 1969. 13Cr. Letter to Nicaise 1698, pages 567-568 in Leibniz 1951. l4Cr. Leibniz 1969, page 134, and Leibni z 1951, page 560 . 15This is so, atany rate , providedthatwe areas capableof recognizingor sensingthe s of otherbeings. perfectionsof God as we are ofsensingtheperfection 16And ourselves too, of course. l7Cr. Leibniz 1951, pages 567-570. 18Cr. Leibniz 1951, pages 23-25, and Leibniz 1969, page 224 . Wi energives thedate of this fragmentdescribingtheUniversalCharact e ristic as 1677, Loemker as ca. 1679. Cf. Leibniz 1961, pages 184-189 for theoriginalLatin.
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beings.l? As Leibniz himselfput it: thereligion I follow closely assuresme thatthelove of Godconsistsin an ardentdesireto procurethegeneralwelfare,andreasonteachesme thatthereis nothingwhichcontributes more tothegeneralwelfare of mankindthantheperfectionof reason/"
Leibniz' "computerization'F! of knowledge was thusintendedtodistribute theperfectionsof reasonand knowledge to a gentsof every (ornearlyevery) natural cognitivestripe.22 Thereare, however,limitations on how effective distribution of knowledge by such means can be. In particular, therearelimits on how effectively it mightbe used todistributea uthentic knowledge . The reasonforthisis that thereis more to authenticknowledge orbeliefthanmerely the abilityto c apacityauthentically tojudgethe operatea calculus.It requiresas well the propositionsoftheencyclopedia to betrueandtheinferencesof thecalculus to be valid . Hence, implementation of acalculus of thought can beexpected to distribute a uthentic epistemicgoods onlytothosehavingsuch capacities. We shouldperhaps also notethatLeibniz, likeLutherbeforehim, was deeplyconcernedfor authenticity, since he took it to be anindispensable ingredientof justifyingreligious belief.The basis forthis concernwas the Lutheran c onceptionofjustification before God,thecritical m omentof which 19Cf. Leibniz 1951, pages 24-25. 20Cf. Leibniz 1951, page 17 (from the prefaceto "The UniversalScience"). Another statement to thesame effect, thoughnotoneoccurringin thecontextof a largerd iscussion of the ars combinatoria and its uses isgiven in the Theodicy whereLeibniz writes: . . . thereis no greaterindividualinterestt hantoespouse thatof thecommunity,andonegainssatisfaction foroneselfby takingpleasurein theacquisition of truebenefitsfor men. . . . Som e Christianshave imaginedthattheycould be devoutwithoutlovingtheirneighbor,and pious withoutlovingGod; or t hattheycouldlovetheirneighborwithoutserving elsepeoplehavethought him and couldloveGod withoutknowinghim. Many centurieshave passed withoutrecognitionof thisdefectby thepeopleatlarge;and therearestill greattracesof thereignof darkness. (Leibniz 1966, pages 2-3) 21By 'computerization' , I mean thegiving of a computational m eans of deciding the truth-values of propositions . One importantissue bearingon thequestionof whethere pistemicgoods mightbe redistributed by meansof sucha deviceas Leibniz' combinatorial art is thatconcerningwhethercertainkey ep istem ic goods arelostin the"translation" , so to speak. This wouldhappen if certainepistemicallyvaluablef eaturesof non-combinatorial justifications were lostwhentheywere exchangedfor combinatorial ones, and no compensatinggains were generally to be expected. One might, for example, be concernedabout theloss of(certainkinds of) simplicityand also ofsuch hard-to-get-at t hingsas epistemic authenticity . Moreon thisin a moment. 22Complexityis a legitimatec oncernhere. Theremay be a tendencyfor thecomplexity of justifications to increasein dramaticand troublesomeways when non-combinatorial proofsare replacedby combinatorial ones. This is a well-knownphenomenonin number theory,a nd, indeed,is theprimaryinspirationof thefieldknownas analytic n umbertheory. This is a phenomenonwhich Hilbertdrew upon in motivatinghis Program.
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was "belieffrom theheart"(Glaubenvon Herzen),a moment emphasizedby boththeprophetHebakkuk'Pand St. Paul: "He who and it is one thing through faith is righteousshalllive... for man believeswith his heart andso is justified." 24 Luther'stargetswerethosewho triedto 'mechanize'or automatejustificationby reducingit totheperformanceof certainritualistic actsor, even worse, certainactsof financialexchange . Such practices,he said, pose a threatto truejustification before God. Specifically,t heycan deceivethe practitioner intothinkingthatshe hassatisfiedconditionssufficient for justification when, in fact, she has not. This was preciselythebasis ofLuther's oppositionto such practicesas theselling ofindulgences.PIn general,any attemptto ritualize or mechanizefaith-any a ttempttosubstitute a 'recipe' of practicesfortheessentialelementof belieffrom theheart -was to be rejected. Justification requiresauthenticity of belief;a nythinglessconstitutes a deficiency of involvementw ithGod on thepartof thebeliever. Authenticity of belief(and action)was thusa concernof Leibniz'. It is perhaps curious,then,thathe thoughtto distributebeliefthroughthe operationof a calculusof thought.Such calculation couldperhaps allow one totakeon theappearanceof one whoauthentically believes,but truly authentic belief would requireas wellt hatone begenuinelyconvincedofthe truth of thecalculus' encyclopediaa ndthesoundnessof its logic . Neitherof means. 26 thesecan be obtainedby purelycomputational A calculus of thoughtcan thusbe used todistributea uthentic epistemic goods only tothosewho believe inthe truthof its encyclopediaand the soundnessof its logic . This suggeststhatLeibniz shouldalso haveregarded such belief ascapableof widespreaddistribution a mong humanbeings. Authenticity aside, asignificantpartof whatmakes (atleastmuch) authenticknowledgevaluable is its capacityfor socialexpenditure-that is, its utilityin securingepistemic and actionalconformityand in fending off un23Habakkuk2:4: ". . . he whose soul isnot uprightin him shallfail, but therighteou s shalllive byfaith." 24Romans 1:17: 10:10, emphases mine. Cf. alsoGalatians 3:11, Philipians3:9 and Hebrews 10:37-38. 25This was especiallytrueof theso-called"JubileeIndulgences"declaredby Pope Leo X in 1507 and again in 1513 as partof a plan to raise money to rebuildSt. Peter'sin Rome. One particularly well-knownfund-raiser,a Dominican friarnamed John Tetzel,is represented in documentsprintedin 1521 as havingemployeda jinglegivingtheessentials of themechanism relating t hepurchaseof an indulgenceto thestateof thesoul of adead lovedoneforwhom theindulgencewas purchased : "As soon as thecoin inthecoffer rings, immediatelythesoulto heavensprings." (So baldderGuldenin Beckerklingt/Im hui die Seel im Himmel springt/) 26It is notinevitablet hatthisbe so, however. An encyclopediacouldconsistwhollyof truthsa pparenttoeven theleastpowerfulintellect.Likewise, theinferen ces acceptedas valid/invalid by a given communitycouldbe self-evidently so. We wouldnot, however, expectthata calculus of thought c apableof settling s uchcontentious anddifficultm atters as theexistenceof God and thedivinityof Christ(and, indeed,even much lessconte ntious mattersof ordinaryscienceand everydaylife) toconsistwhollyof such things.
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wantedsuch calls from o thers . In otherwords, a significantpartof what makes knowledgevaluable is thatwhich makes it socially valuable,and that consistsin thecapacityit gives to exercise strongautonomy(i.e., to effectively lodgeconformalclaims) and toprotectweak autonomy(l.e., to effectively ward offunwantedconformalclaims). but For socialpurposes,then,whatis most importantis notauthenticity theabilityto identify those beliefs andinferences to which the communityis that , in the first instance , one prevails committed.This follows from the fact in an epistemic disputenot by showing her claim to be logically implied by o pponent's premises she authentically believes to betrueor by showing her claim not to be implied bypremises she (theopponent)authentically believes. Rather,she prevailse ither(if she is theconformant)by showing her claim to follow from premises and inferences thateven heropponentmust acknowledge as compelling , or (if heropponentis theconformant) by showing heropponent'spositionnot to follow from such. thatcouldbringeffectivemanipulationof the Developmentof a calculus acceptedthoughtof a community withinthe reach of all would thusaid in effecting an equitabled istribution of thesocial benefits of knowledge or . One authentic belief.The question,t hen,is whethert hereis such acalculus , and for reasons similarto those given might thinkthattherecould not be above for thenon-mechanizability of genuine belief or knowledge. Genuine beliefrequiresnot only a calculus , but belief in itssoundness. Likewise, it might be said, social use ofcalculus a requiresnot onlythatcalculusb ut belief in itsacceptanceby thecommunity. This latter , however, is no more securable by means of thecalculus itselft hanis thesoundnessof the calculus. Hence,thereasoningcontinues , the social uses and benefits of knowledge e ar t hanis knowledge itself. no more open tomechanization We believethisreasoningto be faulty. It overlooksimportant the factthat membership in an epistemic communitypresupposes a basic grasp (though not necessarilyperfect a grasp)ofwhatever specificationit isthatthe community gives of its accepted(resp. rejected)propositionsand inferences. Anyone who does not or can not do so canproperly not be said to belong to the community becausethey can not authentically affirm in the given basic way what the communitystandardsare. Such a one not only can not functionas an authentic member of thecommunity (Le., as one whoauthentically accepts thecommunity'spractices),she can not even act as if she were. If thisreasoningis correct , implementation of a calculus thought of might atleastsecurethesocial benefits of knowledge . To theextentthatthis is so it shouldalso be a means of improving epistemic justice. Later,we will considerwhatif any effects Church'sTheoremhas onthis.
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In their1928 book, Grundziige der Theoretischen Logik, Hilbertand Ackermannposed a fundamental problem of logic, to wit:"Is it possible to determinewhetheror not a given s tatementpertainingto a field of knowledge is a consequenceof theaxioms [given forthatfield-MDj?"27 Hilbert and Ackermanndubbed thistheDecision Problem. Hilbert'sinterestin the problem,andhis reasonforregarding it asfundamental, was thatits positive reasoning solution,together withtheprovisionof acomplete systemof logical and acomplete axiomatization of each field of knowledge, would make it in principlepossible to solve all problemsin thatfield in apurelymechanical manner(wherethesolutionto aproblemP pertainingto a given field F is to be understood as consistingin the logical derivationof ananswerto P from the class of acknowledged t ruths of F). The requirement ofaxiomaticcompletenessforF 28 is essential since, without it, it isnot legitimateto identifythe questionof the solvabilityof a problemof F withthequestionof theextraction by purelylogical means of t hatproblemfrom the axioms for F. For similarreasons, the an answer to requirement of completenessforthelogicalapparatus/?(also aproblemthat HilbertandAckermannsucceededin givingdefiniteshape to) isessential.I f the logical apparatusof a system is not complete,then, even supposing its axiomaticbasis to becomplete,non-deducibility of a givenstatementwill not implyunsolvability of thecorrespondingproblem. 27C f. page 10 8 of the English tran s lati on of the seconded it ionof Hilber tand Ackermann (i.e., Principles oj mathematical logic, edi te d by R. E . Luce). 281n ca lling an axi om aticbas is for F complete, we m eanthatan answer to everyproblem of F (i.e., every problem formulabl e in a languag e in which the subject-ma t te r of F ca n be expressed) is obtainable from the axioms of F by purely logical m ean s. This is, of course, an informaldefinition of complete ness ,but it represents what the m or e formal definition s are intend ed to captur e. The usualformaldefinitionof com plete ness or f a e of T , formalaxiomatic syste m T is this: for every se nten ce 8 formulabl ein the languag eit her8 is a theor em of T or -.8 is a theor em of T . This, of course, presumes thatthe problems of F for which one seeks solutionsa reexac t lycoe xte ns ive w i th whatmight be calledtheclassical truth problems of F-that is, theproblems conce rn ing th e classicalt ruth or falsityof a given senten ce or proposition. We call su ch problems classi cal ,b ecause the classicalconcept ion oftruthis presupposed in thedefinition - of every (8 , -.8) pair, it is presupposedthatexac t ly on e will betrue . Thereare, however , ot he rkinds of problems- for instance, classical validity problems. These problems concernthe classical logicalvalidity of a given sentenceor proposition. For such problems, the above formal definitionof comp leteness isinappropriate since it isnotclass icall ythecase thatone member of every (8, -.8) pair is logically v alid. Thus, formaldefinitionsof com plete ness edp end notonly upon thechoice offormalizati on sfor F butalsoupon the kind o] problem whose solva bility is being quer ied. The onlygen er icrequirem entof completeness , itseems, is thatit solve allproblems . " it being understood thatdifferences in the kind of probl em to be solved in differenc es in the formal definitionof solvabilit y. will esult r 29By the com p leteness of a logical appara t us we m e antha t every olgicallyvalidreasoning is rep resented by an acceptable derivationin it. Again, this is an infor ma l edfinition , but it is thatwhich the more formaldefini ti onsof the logica lcom p lete ness are inte nded to capture.
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Godel, of course,demonstrated the existenceof a completeformalappathat ratusforfirst-order classical logical consequence in 1930. He followed up in 1931 with aproofoftheincompletenessofconsistentformal-axiomatic bases forfirst-order theoriesfrom elementary a rithmeticon up. Neitherof thesesettledt heDecision Problem, however . Nor woulddenyingCodel'sincompletenesst heoremandaddingas an assumptionthecompletenessof the axiomaticbases withwhich one might be concerned.P"AxiomaticcompleteF, ness requiresonlythatfor everystatementS pertainingto a given field eitherS or itsdenialis a consequenceof theaxioms for F-andit is one consequenceof the given thingto knowthat eitherS or its denial is a logical axioms, andquiteanother t hingto knowof eitherS or itsdenialthatit is. It is another t hingstillto be able to determineof any given S thate itherit or its denialis a logical consequenceof the axioms for F by applyingsome general effectiveprocedureor algorithm capableof deciding all such questions . t ype of problem-namely,problems concerningeffective It is thislatter -withwhichChurchdealtin his landdetermination by generala lgorithms He gave anegative mark1936 paper "A Note ontheEntscheidungsproblem". solutionforthegeneralproblemof determiningclassicalfirst-order validity . Thatis, he showedthatthereis nogeneral a lgorithm fordeterminingw hether or not a given s tatementpertainingto a given field of knowledge is a classicalfirst-order logicalconsequenceof theaxioms given forthatfield.31 More thatthereis no effective procedur e precisely, and more generally, he showed P such thatfor anarbitrarily selectedset ofsentences~ and an arbitrarily language(i.e., a first - order selectedsentenceS of a non-monadicfirst-order t hatcontainsat leastone predicateofadicitygreater t hanor equal language to 2),32 executionof P determineswhetheror notS is a logical consequence axiomatictheory of E. Assuming thelogicalcompletenessof a given formal F, this impliesthatthereis no effective procedureP forgenerating, for an arbitrarily selectedsentenceS of thelanguageof F, eithera demonstration or arefutation of Sin F. We would now like consider to the significance of this discoveryprojects for of thesortdiscussed in theprecedingsectionsof thispaper; namely, those concerning the use of caomputational device such as Leibniz ' for (i) peacefully resolvingepistemic disputes, and (ii) improving upon theinjusticesof the natural distribution of epistemic goods.
30T he effect iveenumerability of thetheorem-set is also e r quired. 31Actually , ther e was a mistake in Church's originalproof, butit wascorre ctedand the corre ctionpublishedin thesame volume (namely, thefirst) of The Journal of Symbolic Logic as carriedthe original proof. 32The restrictionto non-monadiclanguagesis necessary because the Decision Problem . Accordingto Ackermann, this for th e monadic predicatecalculushas a positive solution was proved for thefirst time by Lowenheim in his 1915 paper "Ube r Moglichkeit en mi Relativkalkiil " (cf. Ackermann, Solvable cases of the decision problem , page 34).
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MICHAEL DETLEFSEN 5. PEACE: WEAK AUTONOMY , STRONG AUTONOMY AND CHUR CH 'S THEOREM
We have suggestedthatthe capacityof a givenmechanism for peaceful conflict-resolution is dependentupon its abilityto preserveweak autonomy. We would now like to considera relatedtype of autonomy , whichwe shall refer to asstrong autonomy. It concernsthe fr eedom of an individualto attemptto conforma community to her beliefs actions or . This freedom is crucialto theabilityof anindividualto realize her goals for herwithina life communitywhileremaininga member of it. Infringements on thistype of freedom, like infringements on weakautonomy, pose a significantthreatto peace, and forsimilarreasons. Persistent refusalto allow a p erson to pursue her goalsthroughthemaking of conformal claims ist antamount to denyingher 'conformalself' a placein the community. This is notto saythattheremight notbe legitimatereasonfor restricting or denyingattemptsatconformalaction.Therevery well m ight be. Thatthis is so, however, does not changethe fact thatsuch restrictionsmight alsoprovidea motive for violence a gainstor defectionfrom the restricting community.P Concernforpeace wouldthusseem to dictatethe need foratleastsome freedom to exercise strongautonomy.At thesame time, however, attempt s toconformthebeliefs andactionsofothersshouldbe enforceableonly tothe extentthatdoingso providesadequatep rotection fortheweak autonomyof thoseupon whom thecall toconformityfalls . The question , then,is how farthefreedom to makeconformalclaimsshouldbe extendedin a system in whichprotection of weakautonomyis to be providedby enforcement of the ComputabilityR equirement . It is atthis point thatChurch 's Theorementersthepicture . It poses a theoretical limiton theextent to which individualslivingundertherule of theComputabilityRequirementare free to make conformalclaims. It does theclaimin questionassert s theinvalidityof some thisspecifically when (a) form ofreasoning,a nd (b) theinvalidityof this form ofr easoningcannot, by Church'sTheorem, be 'computed ' in the given calculus . This limitation to Church's Theorem,a generaleffective wouldnotexistwerethere,c ontrary procedurefordeterminingof any form of inference whetheror notit is valid. By Church'sTheoremthereis thusno generalprotection a gainstsituations in which awould-beconformant is in a positionto wantto make a conformal claimregardinginvalidity(throughher knowing or justifiedlybelievingthat an inferenceis invalid)but notin a positionto be permittedto make it.3 4 33T hro ughoutt his paper when we speak of peac efulresolut ion of conflicts etw b een an individualanda community, we m ean resolut ion s which notonly avoid violencebutwhich also leave the individual' s com m unity mem b ership intact (i.e., which find a way for the conflict to ebresolved h s o rt of"de fect ion" rom f the comm u nity by the ind ividual ). 34 We do not, of course, suppose that authentic belief itself, or any action requiring t ion fr om genu ineconvicti ons , will be elect ive or voluntar y in some t hing lik e direct activa
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Such situationswill , of course, frustrate thedesire forrational epistemic communityon thepartof thewould-beconformant . Should suchfrustration become persistent enoughor,thoughperhapsnotpersistent, should it arise in serious enoughscattered cases, it would have the same potential to disrupt the peace of acommunityas violationsof weakautonomy . Bothalienate an individualfrom hercommunityand thusprovide her with a motive for disruptingits peace. Assuming, then,thatthe ComputabilityRequirement(or some requirement sufficiently like it)necessaryfor is the adequateprotectionof weak autonomyin the face of possible demandsfor epistemic conformity, theredevelops anin-principletensionbetween weak and strongautonomy.The basis of this tension Church is 's Theorem.P 6. JUSTICE: THE EFFECTS OF CHURCH 'S T HEO RE M ON THE DISTRIBUTION OF THE SOCIAL BENEFITS OF KNOWLEDGE
We notedin section3 thattheComputabilityRequirementis not, in andof itself, anadequatemeans ofproducingan equitable distribution ofauthentic knowledge. At the same time , however , we notedthatit faressomewhatbetterwithrespecttodistribution ofthesocial benefits of authentic knowledge . We will now consider what,if any, effect Church'sTheoremhas onthesetwo points. t hedistribution of authenti c knowledge . In this conLet's consider first nection, let us suppose, forthesake ofargument , thatauthentic belief in a community'sencyclopediaand its acceptedlogic can be widely distributed across it . Our questionthenis this: Grantedsuch favorable assumptionsfor thedistribution of authentic knowledge , coulddevelopmentof acalculus of theappropriat esense. 35This does not, of course, m ean thatthereare notsources of tens ion between individuals and communities otherthanChurch's Theorem . There is a parall el type of tension sponsored byGodel's incomplet en esstheorems. This wouldoccurshould onenotice that, thoughthey are not logicallyimplied by the com munity 's encyclopedia, com mit me nt to certainsentences (viz. GOdel sente nces forformalsystem s based on the encycloped ia ) is nonetheless necessarygiven commitmentto it. We believe,however ,thatthetens ion arisy more serious. This is due to the factit applies to ing from Church 's Theorem is generall a broaderrange of senten ces. It wasonlyin 1977 (cf. "A mathem atical incom plet en essin Peano Arithmeti c", in Handbook of mathematical logic, edited by J. Barwi se) thatP ari s and Harrington f oundthe first exa m ple of anundecidableyet m athematicallyinteresting sente nce .Paris and Harringt on lasos howed , however ,thattheundecidabilityof their interestingsentence si im plied by the undecidabilityof GOdel'ssente nce. In this sense, they c tionfails to capt ur eanyt hing refutedany general claim to the effect thatGOdel's constru i te rest . of mathem atical n Another sourceof tens ion, andone thatm ay be of greater practicalimportan ce,is that conce rn ingthe complexi ty/tractabilit y of decidable cases of the Decision Problem . The lengthand / orother com plexity ofthe computationmi ghtm ake it humanlyim possi ble to produce or tocom pre hend it . Indeed, atpresentit appea rs thatthe Decision Problem for classicalconsequen ce si intra ctabl e for allbuta relativ elysm allclass of argume nt -for ms .
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thoughtbe expectedto effect an e quitabled istribution of authentic belief or knowledgeacrossthecommunity? Church'sTheoremsuggeststhattheansweris 'no', at leastto theextent thatcertainlogicalabilitiesare not evenly distributedacrossthecommunity. The specific logical abilitywe have in mind istheabilityjustifiedly to make judgementsof theform'S does not follow logically from I:'. Let us suppose thatcapacityto makesuchjudgementswithoutthehelp of a backing computationis notdistributed evenlyacrossa community. By Church's Theorem,thosewho canjustifiedly make themwill have access to epistemic goods to whichthoserequiringa confirmingcomputationwill not.Supposing (as seems reasonable)thatpossession ofsuchgoods constitutes some kind of privateepistemic perfectionin the onepossessing it, therewill beepistemic perfectionsw ithinthereach oft hoseof higherlogical skill which will not be sharablew iththoseof lower logical skill. Were it notforChurch's Theorem, such asymmetriesof logical knowledge wouldleast at be in-principleremovablethroughtheuse of agenerala lgorithmfordecidingquestionsof validity (invalidity).Church'sTheorem,however,prohibitsthis happening. It thus pointsto atype of epistemic inequitythatcan not be removed t hroughthe developmentof anycalculus of (classical)logicalthought. The inequityjustdescribeddoes not,however, confer any specialsignificance onChurch'sTheoremas a limitationon epistemic justice.The reason is thatsimilarinequitiescan beexpectedto arise fromsourceshavingnothing to do with C hurch's Theorem, forexampleinequitiesbetweencommunity members concerning genuineknowledge of thecommunity'sencyclopedia a nd its logic.Limits on genuinefactual knowledge of any typewill posethesame generalt ypeof limitation on epistemicsharingthatChurch's Theoremposes Thatbeing the case , thelimitativeeffects withrespectto logical knowledge. of Church'sTheorem on justdistributionof genuineknowledge/belief can hardlybe seen asdistinctive. Much thesame must be said of theeffects ofC hurch'sTheoremon the justdistribution of thesocialbenefitsof genuineknowledge/belief.Indeed, Church'sTheorem seems to lack evennon-distinctive effects inthis area. The generalreasonis this: inequitiesconcerningthedistribution of thesocial effects of genuineknowledge/belief arise not from inequitiesof authentic knowledge/belief, butfrom inequitiesin abilityto operateor manipulatet he specified calculus of a communityto suitone's purposes (whetheror not it is authentically possessed by theoperating/manipulating agent). Church's type, however, sincet heyhave to Theoremimplies noinequitiesofthislatter do not withthelinkagebetweena logicalcalculusa nd thelogicalrealityit computes(i.e., theactualvalidityor invalidityof inferences),b utonlywith theoperationof thecalculus itself. All in all, then, theeffects ofC hurch's Theoremon thatpartof Leibniz' Programhavingto do withepistemicjusticeseem to bebothless severeand lessdistinctivet hanthosehavingto dowithepistemic peace.
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7. CONCLUSIONS
One might,of course, say thattheculpritin all of theabove isnotChurch's TheorembuttheComputability Requirement . It is,afterall, anecessarypart of boththeargumentconcerningthetensioninducedby Church'sTheorem between weakand strongautonomy(and, so, of thethreatthatit poses to peacefulepistemic conflict-resolution) and theargumentsconcerningthe inequitable d istributionof authentic epistemic goods. This, togetherwith thefactthatit is a more'negotiable' claim thanChurch's Theorem, might thussuggestthatit rather t hanChurch'sTheoremshouldbe seen as thetrue sourceof the morala nd politicalsignificancethatI haveattributed to the latter . namely, that I have tworesponsestothisclaim. The first is aclarification; I am not presentingChurch'sTheoremas by itself implying limits of any sort. My claims arerathert hat(1) Church'sTheorem in conjunction with a conditionliketheComputabilityRequirementhas suchconsequences,and that(2) this otherconditionby itselfdoes not. This would not, of course , be a veryinteresting claim werethe ComputabilityRequirementan implausiblemeans by which acommunitymight , does notappear seek topromoteepistemicjusticeandjustice. This, however to be thecase, at leastnot inadvanceof Church's Theorem. We are thus led to aperhapsmore surprisingsecond point-namely,thatinstitution of a conditionliketheComputabilityRequirementis not animplausiblemeans of tryingto promoteepistemic peace andjusticein a community. Allow me to elaborate. To begin with,n otethattheComputabilityRequirementis rightin preif cept. It, or some conditionlike it, would have to be enforcedepistemic peace andjusticewere to be achieved in any world in which (i) Leibniz ' Thesis weretrue,(ii) restrictions on weak andstrongautonomytendedto thatnonworkagainstpeace in the ways indicatedabove, (iii) it is allowed computational knowledge of a case of non -consequencemight countas an epistemicgood, and (iv)thereare differences between individualsas regards theirgeneralabilityto graspcases ofnon-consequence non-computationally. I do notthinkthat(i)-(iv) lackplausibility.Withregardto (i), my belief in its plausibility is restricted to communitiesin which theindividualmemof however, bers span a fairly wide range humancognitivecapacities.This, seems to me to be aprettyfair model of w hatthetypicalhumanepistemic communityis like. Even if not all epistemic communitiesareof thistype, it still seemsrightto saythatone cannotaltogether avoid epistemic communities containingpersonsof inconveniently low logical abilitygenerally and still meet her obligations concerningepistemicjustice.As withothersortsof goods, so, too, with epistemicgoods, justicedemandsthatwetryto find ways of rightinginequitiesof birthand upbringingand otheraccidental determinantsof epistemic (mal)distribution. Enteringintocommunitiesinclusive of
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logically challenged individualswouldthusseem to be a likely consequence of seekingepistemicjusticein a world such as ours. More likely, though,misgivings regardingt heComputabilityR equirement will come fromotherquarters-namely, (1) thosewho doubttheabilityof a communityto arriveata satisfactory form ofencyclopedia,(2) those who doubttheexistenceof analgorithm forarrivingatplausiblec andidates forthe logical forms of sentencesframed in anatural language(assumed to bethe, or atleasta, type of languagein termsof whichtheepistemic transactions of thetypicalcommunitywith which we a reconcernedare to beconducted), and (3) thosewho seetheuse ofcomputationto backconformalclaims as standingin need ofaugmentation by certainnon-computational knowledge. It might be thoughtt hattheseproblems are deeper or more seriousthan thosecausedby Church'sTheorem. For reasonsI will nowstate,I do not sharethisview. Let's begin with the questionof theencyclopedia.To avoid misunderstanding,let me say fromthestartthatI sharethebeliefthatit is highly unlikelythatwe, as members of thetotalc ommunityof humanbeings, will ever arriveat a sharedencyclopediathatis at once sufficiently robustto yieldperceptibleresolutions of most matters(even most of themost importantmatters)of moraland religioussignificanceand, atthesame time, a matterof common agreement . This does notmean, however, thatsmallercommunities,communitiescapableofbringingus most of thea dvantages for whichcommunitymembership is wantedin thefirst place, must suffer thesame fate. At any r atethisis so so long astheholdingof such views does not carrywith it ageneralabsence of abilityon thepartof communitymembers to providegoods and services importantto theirgeneralwelfare. It seems doubtfult hatone could saythesame aboutcommunitiesbased on differentials in logical skill. Restricting oneself tocommunitieswhere others do not exceed her logical abilitynor hertheirswould, to be sure , promise to minimize clashesof weakand strongautonomy.At thesame time, however,theremight be connectionsbetweenlogicalabilityand cognitiveability generally thatwoulddictatethatrestricting oneself tomembership in communitiesconsistingof thoseof similarlogical skill would mean a lower level thatthisis so, of availability of goods and services generally. theextent To thereis reason(at least for the non-maximallygifted)to join communities withnon-uniformlevels of logical skill . Church'sTheorem,however, implies thattheseare communitiesat risk for clashes betweenweak andstrongautonomy. Hence, Church'sTheoremwould seem in some ways to bedeeper a and more persistentthreatto communitythanthedifficulty of obtaininga common encyclopedia.Even plausibleutopiancommunities(e.g., communities in whichthereis agreementon such generally controversial mattersas religiousand/orpoliticalbeliefand practicesbut significantdifferences in logicalability)would have to worry aboutits effects.
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Let us nowconsiderthesecondof thethreeabove-mentioned o bjectionsto ourtreatment of theComputabilityRequirement -namely,thatwhich sees theobstacleposed by Church's Theorem as minor in comparison to that posed by thedifficulty of determiningthecorrectlogical forms of conformal claims. The thinkingbehindthis objectionmay be presented as follows. (a) language.(b) To be fit Conformalclaims are to bep resentedin a natural targ ets forcomputation,conformalclaims must be expressed in a logically perspicuouslanguage (i.e., in alanguage which clarifies theirlogical form) . (c) The logical form(s) of natural language claims are often not perspicuous, are generally contextd ependentand, in any event , are notthetype(s)ofthing(s) thedetermination of whichcanbe guidedby analgorithm.Therefore,(d) the ComputabilityRequirementwill beviolatedbefore we ever get to Church's t hepointof having toproducea backing Theorem(i.e., before weever get to computationfor aconformalclaim). Non-algorithmic means will have to be employedin ordertoexpressconformal claims in a logically perspicuousform (i.e., in thelanguageof a first or higherorderpredicatecalculus) . Thus, it is notChurch'sTheorembutthelack of analgorithm fortran s latingnatural language claims intothelanguage of a logical calculus which posesthemore fundamental threat to Leibniz' Program. , the point made in premise (b) I willaccept, for thesake ofargument and focus myattention upon premise (a)-or,better , upon therelationship between(a) and (c). (a) is not, I believe,rightlyregardedas a commitment of Leibniz' Program. At leastnotin thepresenceof (c). Indeed, it seems altogether properthatthe makers ofconformalclaims be requiredto present themin logically perspicuousform; e.g., in thelanguage of a logical calculus (a chcracteristica universalis). Failingthis, a conformantwould not have conformitythatis clearenoughto give presenteda communitywitha call for potential non-conformists a suitablycleartargetagainstwhich to make an in logically effectivecounter-claim. It wouldtherefore seem thatpresentation perspicuoustermsis rightlyviewed as a conditionfor making a conformal claim. To theextentthatthisis so, however,t hepresentobjectionvanishes. Either(c) is correct,in whichcase natural languag e claims are nottypically logically perspicuousand (a) oughtnot to beaccepted, or natural language claims do possess anappropriatelevel of logical perspicuity,(c) is incorrect and thereis no problemof translating from natural languageintologically perspicuousform. Attainment of persipicuityin logical form may be aeal r difficulty . It is, however , somethingthatshouldaccompany a conformant's becomingconvinced of a logical judgment.(How, rationally, could a confor mantbecome convinced of ajudgmentof validityor invalidity w ithoutbeing similarlyconvinced oft he(relevant)logical form of theitem(s) she judgesto be valid or invalid?) Hence , it shouldnot interf ere with anyrational desire to makeconformalclaims.
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Finally,let'sconsiderthethirdof theobjectionsstatedabove-namely, thatconcerningt heneed for acertainbody of non-computational knowledge of anycalculus of thought.W ithoutsuch knowlto confirmthecorrectness edge, theargumentmaintains,thecomputationsp roducedby a calculusof thought would beincapableof shapingthebeliefs ofrational a gentsin the ways thatLeibniz' Program(withits concernfor weakautonomy)requires. The elementary c haracter of thecomputationsof the calculus mtiocinator is notenoughto ensuretheoverall e lementary c haracter of one's knowledge of computedclaims. Inparticular, it seems thatknowledge of thesoundnessof thecalculus would also have be elementary. generalbelief inthesoundnessof a comWe agreethatthereis a need for a munity'scalculuswithinthatcommunity. The questionis whatis required in orderforthatneed to be met. We believe thatit could be metw ithouta graspof (or even an a cquaintance with) aproof ofsoundness.Specifically, we believe it could be based on a rudimentary inductiveacquaintance withthe calculus and itsproducts-an acquaintance thatmight be withinthegrasp of even those of relatively lowcognitiveability. Acapacityfor basicinductive learning must, afterall,e xtendas far (or very nearlyas far) downthescale of cognitionas does thecapacityfor basicsymbolic manipulation.T his being so, rudimentary i nductiveknowledge of thesoundnessof a calculus could be basically as widespreadas thecapacitytooperatethatcalculus . Accordingly, theneed for knowledge thesoundnessof of acalculus would pose noessential obstacleto theexecutionof Leibniz' Program. We havepresentedLeibniz' idea (Leibniz'Program)forbringingepistemic peace andjusticeto humancommunities. We have alsoa rguedthatChurch's Theoremconstitutes a fundamental theoretical limitationon theextentto which it andotherprograms like it can becarriedout. It would be no surprise,however, if more detailedconsideration of sub-casesof theDecision Problemfor classical validityandofrelated issue ofcomputational complexity provedcapableofsharpeningourunderstanding of thesematters.Such work Programby identifying might even point the way to a revival of Leibniz' subdomains of logicalreasoningwithinwhich largea nd importanttractsof humanreasoningmight properlybe seen as being confined, and by showing thatthevalidityof reasoningwithinthosedomainsis decidable,p erhapseven feasibly so. REFERENCES
Leibniz, G. W. 1951 Leibniz selections (P. Wiener,editor), CharlesS cribner 's Sons, New York. 1961 Die Philosophische Schriften von Gottfried Wilhelm Leibniz, vol. VII (C. I. Gerhardt,editor),Georg Olms,Hildesheim. 1965 Monadology and other philosophical essays (P. Schrecker,translator), Bobbs-Merrill,New York.
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Theodicy (abridged)(D. Allen,editor),Bobbs-Merrill,New York. Leibniz : Philosophical papers and letters, second edition(L. Loemker, translator and editor),Reidel,Dordrecht .
M. RANDALL HOLMES
TARSKI'S THEOREM AND NFU
Abstract.T he Tarski paradoxof theundefinability of truthis proved by a diagonalization argumentsimilarto theargumentof Russell'sparadox. In ZFC, Russell 's argumentshows thattheuniversalclass(and largeclass es generally)do notexist. In otherset theories, such as Jensen's va ria ntNFU of Quine's "New Foundations" , largeclasses such as theuniversemay exist; the diagonalization argumentsl eadto somewhatdifferentr estrictions on theexiste nce ofs ets in the presence of differentaxioms. In this paper,we explorethe possibilitythatsemanticsexpressedin NFU may have somewhatdifferentr estrictions imposed on them by thediagonalization argumentof Tarski. A languag e L is definablein NFU, in which thestratified s entencesof thelanguageof NFU ca n be encoded (but, it shouldbe noted, as a propersubclassof L) . Truthforsentencesin L is definablein NFU, andthe reasonthata suitablyadaptedTarskiargumentfails to lead toparadoxis notthattruth forL is undefinable in NFU, butthatquotation becomes a type-raisingop eration,c ausingthepredicat e needed for the "Tarski sentence"to be unstratified .
1. INTRODUCTION
The well-known t heoremof Tarskithattruth of sentencesin any reasonably expressivelanguageL cannotbe definedin thelanguageL itselfis proven argumentsimilarto theargumentinvolvedin Russell's by a diagonalization paradox.The paradoxof Russellshows us thatsome restriction on compre, butit doesnotprescribetherestrichensionaxioms in settheoryis required tion. The traditional approachinvolves"limitation of size" , andis embodied in Zermeloset theoryand its extensions.It is usualto thinkthatRussell's paradoxexcludes"large"sets liketheuniverse,but this is actually notthe case. An alternate solutionto Russell'sparadox(and otherparadoxes)was proposed by Quine (1937) in his system "New Foundations"(NF): comprehensionrestricted to stratified formulae.The consistencyquestionfor this theoryremainsopen (as is generally known);whatis lessgenerally knownis thatthevalidityof thegenerala pproachof Quineto resolvingtheparadoxes has been demonstrated.Jensen (1969) showedthatthetheoryNFU ("New axiom weakenedto allowurelements, Foundations"withtheextensionality but with thesame comprehensionaxiom as "New Foundations"itself),is consistentr elativeto theusualset theoryand remainsconsistentif theaxioms of Infinityand Choiceareadjoined. In theseset theories,"large"sets liketheuniversal s etareprovidedby thecomprehensionaxiom, andtheresparadoxesproceedsalonga differentr oute.An olutionof theset-theoretical all-purpose referenceforsettheoriesof thistype,whichcanserveas a substi469 C. Anthony Anderson and M. Zeleny (eds.), Logic, Meaning and Computation, 469-478.
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tutefor most oft hespecific references citedin thepaper, is Thomas Forster's excellent book Set theory with a universal set (Forster 1992) , although t he emphasis thereis on NF rathert hanNFU. Analogously, we have discovered thatapproachingtheTarski paradoxof thedefinability of truthin a Quine-style set theoryallows a different resolutionof theproblem; theparadoxdoes not preclude the definability of truth any morethanthe Russellparadoxprecludestheexistence of auniversal set. We willpresentan infinitarylanguage(with sentences of finite length butwithinfinitely many primitivepredicates)which does define its own truth predicate,b utavoidstheparadoxofTarskibecause quotation turnso utto be a "type- raisingoperat ion"! This appearsto suggest an alternate approach to semanticparadoxesin general,analogousto thealternate approachto set-theoretical paradoxesembodied in NFU. It shouldbe notedthattheinfinitary n atureof thelanguageused is not accidental.Consideration of theparadoxpackagedsuccinctlyin thephrase "thesmallestnatural number not describablein lessthana billion words " whichexpressesits ownsemanticrelations can be revealsthatany language expectedto haveshortnames for eachand everynatural number, and so infinitely manyatomic names (in a languagewithoutnames, we can say, equivalently, "definite descriptions"in Russell 's sense). 2. TARSKI'S THEOREM
We briefly(andinformally) review theproofofthetheoremof Tarski.1 Suppose thatwe have encodedt heformulae of laanguageL in such a waythat theycan be discussed inL (as numbers, for example). For each formula ¢ withone freevariablex, let"¢" be thecodefor¢. We canthinkof formulas withone free variable as "definable predicates" , holding of an objectif substitution of a name for t hatobjectforthefreeoccurrences of x in the formula . Suppose, moreover,thattruthof encodedsentenc es yields atruesentence of L is a predicatedefinable inL . We thenconsider the following , which should be expressible formally as a x: formula in one free variable The formula encoded by x, when each free occurrence ofthevariable "x" is replaced with t hecode x itself, yields the code of a sentencewhich is nottrue. Less formally , for this to be truefor an encoded formula "¢" in place ofx meansr The formula¢ (as a predicate)does not hold of "¢". 1 For detail s,
see Andrews 1986.
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Calltheformulainformally describedabove 1/1; replacingthevariablex in 1/1 withthecode "1/1" yields asentencewhich assertsits own falsehood! Roughlyspeaking,theresulting s entencesays The predicate1/1 does nothold of"1/1", butfor1/1 to hold of"1/1" means exactly: The predicate1/1 does not hold of "1/1" , so thesentencedenies itself. An obvious requirementfor theargumentto work isthatthenotionof substitution of a specificobjectfor avariablein (thecoded version of) a formulabe definablein L; this holds (forthe usual kind of c oding using numbers)in anytheoryas strongas arithmetic.T he resolution of thecontradiction a pparently mustbe (andindeedmustbe in theusualcontext)t hat thepredicate"is true"of encodedsentencesof L cannotactually be defined in L. 3. PRELIMINARIES IN NFU
NFU is a first-order , one-sortedtheorywithequality,membership, and a unarypredicateofsethood .P The axioms of NFU are as follows : Extensionality: Sets withthesame elementsare thesame. Urelements: Objectswhicharenot sets have no elements. Stratified Comprehension: For eachvariablex and formula
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Note thatthe usualKuratowskidefinitionof the pair «x, y) = {{x}, {x, y} }), which can be used in N FU withoutt heassumptionof Infinity, yields . This a pair with relativet ype two higherthanthetypes of itsprojections is inconvenientb ut not impossible to work with; it has odd effects onthe relative t ypesof functionsa nd theirarguments , forexample. Cardinalnumbers(includingnatural numbers)are defined as equivalence classes of setsu ndertheobvious equivalencerelation ; ordinalnumbers are defined as equivalence classeswell-orderings of undersimilarity . The objects whichoccasiontheparadoxesof Cantorand Burali-Forti in naivesettheory (thecardinality of theuniverseand theordertype of theordinals)a ctually existin NFU butdo not haveq uitetheexpectedproperties. The cardinality of theuniverseis clearly not lesst hanthecardinality ofthe power set oft heuniverse(thesetof allsets), so Cantor'st heoremin itsusual form cannothold. The situationcan be clarified byconsideringtheform of Cantor'stheoremwhichcan be provenin thetheoryof types: there,we of a setA and its powers etP{A} cannoteven askwhethert hecardinalities are thesame, becausethetypesof thesetwo sets are different. Whatcanbe proven, inNFU as in thetheoryof types, is thatthecardinality of PI {A}, theset ofone-elementsubsetsof A, is strictlyless thanthe cardinality of P{A}. In NFU, we candrawthefurther conclusion thatIP I {V}I < IP{V}I < IVI (thereare "fewer " singletonsof objectsthanthereare objects). This shouldnot be toosurprising, since thefunctionwhichtakeseach objectto its singletonhas anunstratified definition. The role oftheset ofone-elementsubsetsof A in theargumentabove inspiresthedefinitionof a parallel operationon cardinals : T{IAI} is defined as IPI{ A} I. It is straightforward to showthatthis operationon cardinals does notdepend on the choice of A; it is unstratified and does not define a (set) function.For eachcardinalIAI , thecardinalexp(jAI) = 21A1 is defined (following Marcel Crabberathert hanSpecker toobtaina slightlystronger definition)as T-I{IP{A}/}; the functionexphasa stratified definition,b ut it is partial(becauseT-I is partial).Observethatexp thusdefined is also thenatural exponentiation functionfor thetheoryof types; theoperations T and T- I can be thoughtof in thecontextof thetheoryof types as projectingcardinalsto "thesame" cardinalsin higher or lower types (from the standpoint of theusual settheory;we have seenthatT{IVI} =I- IVI in NFU, . so thiscannotbe ourpositionfrom theNFU standpoint) The Burali-Forti paradoxdoes not afflict NFU, becauseit dependson the theoremof naive settheoryor Zermelo-style set theorythattheordertype of theordinalsbelowa is equalto a itself.Observethattheordertype of theordinalsbelowa is an objecttwo types higherthana in thetheoryof types; one will thennot besurprisedto findthatthecorresponding t heorem of NFU assertsthattheordertype of theordinalsbelowa in thenatural orderis T 2 { a}, where T {a } is defined astheordertypeobtainedby replacing theobjectsorderedby any orderof type a with theirsingletons.Now the
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-FortiparadoxprovesthatT 2 {n} < n, wheren is the reasoningoftheBurali ordertypeof thenatural orderon theordinals;n proves to begreater t han theordertype of thesegmentbelow itand less thanthelargestordinals. The sequenceof iterated images of n underT can have nosmallestelement, butit is not aset. Such "sequences" (failing to be sets)of iterated images of i mportantrole below . objectsundertype-raisingoperationswill play an Note therole oftype-raisingoperationssuch astheT operationson cardinaland ordinalnumbersin avoidingparadox. The approachto avoiding semanticparadoxhere will allow thedefinitionof notionssuchas truth and synonymy, attheprice oftreatingquotation as a type-raisingoperationof this generalkind and reference as an e xternal relationwith stratification restrictions similarto thoseon membership. 4. SEMANTICS IN NFU
The theoryNFU + Infinity+ Choicecannotdescribeits ownsemantics. This is fortunate, becausetheinconsistencyof thetheorywouldfollow!However, it candescribethesemanticsof an infinitarylanguagewhich capturesthe semanticsof allstmtified sentencesof NFU in a certainguardedsense (if it did thisunqualifiedly, it would stillimply inconsistencyof thetheory!) . The obstruction to NFU introspecting on its ownsemanticsis thatthe relation E is not a set; it s houldbe clearthatfor anyrelation R which is a set, theset {x I '" xRx} is definable , and so theexistenceof theset E would imply theexistenceof theRussell class. differentp urpose" . Observe We use an idea of Grishin(1972) for arather thattherelation of inclusion is a set; it has satratified definitionin which thetworelated o bjectsareof thesame type. Thenobservethatanyformula x E y can be expressedas {x} ~ y, in whichthenon-relation E has been replacedby therelation~. One can thenreplacetheexpression{x} by a to theset ofsingletons PI {V} . variableX restricted In general,a formulain equality,membership, and theprimitive projection relations can betranslated intoa formulain equality a ndtherelations induced by inclusionand projectionrelations on n-foldsingletonsfor eachn in such a waythateachvariableof relative t ype i is replacedby a variablerestricted numberN largeenoughthatN - i is to theset pi'-i{V} for a fixednatural nonegative in each case needed . Notethatany such formula canbe expressed in thelanguage w ithprimitiveunarypredicatescorresponding to each set in theuniverseand binaryrelations correspondingto each setrelation in the universe. The specific unaryand binarypredicateswe need are much more restricted, beingsimply thepredicatesof being an n-fold singleton for eachn, equality,and therelations inducedon n-foldsingletonsby inclusionand the projectionrelations ; so it may seem to be overkill to all usesets andrela tions, 3See also Forster 1992, p ages 64-66.
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butthereis no small set in NFU which containsall oft hesets we need to encodethesepredicates(even thoughtheclass ofsetswe need isexternally countable!). If thesentenceto be translated is not stratified , thetranslation process yields a formula in which some particular variableis replacedby variables representing different i terated singletons of objectsrepresented by thatvariable; sincetherelation betweenthemth and nthiterated singletonsof the same objectfor m i- n is not capturedby any set relationin NFU, the translated sentencewill not be usable in the construction below. We nowintroducean infinitarylanguageL, havinga primitivepredicate "membership in x" for eachobjectx in theuniverseand a primitivebinary predicateR for each set of pairs such thatxRy is to holdexactlywhen (x, y) E R . The logicaloperationsallowed inL are theusualpropositional connectivesa nd quantifiers(L is infinitaryonly inhavinginfinitelymany constants; sentencesof L are finite in length).T he termsof L are variables indexedby thenatural numbers. As we indicatedabove, eachsentenceof NFU can be translated into a sentenceofthelanguage L. Thereare countable sublanguages L; of L capable ofexpressingallsentencesof NFU using norelative t ypesotherthanO-(i-1), butthe"union" of the Li's is not a set,a nd thereis no set ofexactlythe sentencesof L which encodesentencesof NFU. If therewere such a set, paradoxwould ensue. , and indicatehow truth of We developthesemanticsof L in some detail coded sentencesof L is definablein NFU. Use of thetype-levelp air of the previoussectionwill beessentialto allow inductionon structures builtup using pairing. our set theoryas Moreover, we will make an assumptionstrengthening well.ObservethatT{n} = n holds forn = 0,1,2, . .. , where T isthetyperaisingoperationon cardinal n umbersoftheprevioussection. It isimpossible to provetheassertion"T {n} = n for eachnatural n umber n" in NFU + Infinity+ Choice; theattemptto prove it byinductionfailsbecause the conditionon whichinductionis to becarriedoutis unstratified and fails to define a set.Nonetheless, thisassertion,calledRosser'sAxiom ofCounting." is consistentw ith NFU + Infinity+ Choice (andstrengthens it essentially). Atomic termsof L arevariables.T herewill becountably many variablesVi fori rangingthroughthenon-negative integers.T he factthateachsingleton correspondsto apredicatemeans thatwe couldjustas well have name a for . eachconstant in theuniverse, butwe willrefrainfrom thisextravagance of membership in sets and particiAtomic formulasof L are statements pationin binaryrelations.The sentence"Vi E A" is encodedas «i, {A}),O), while thesentence" (Vi, Vj ) E R" is encodedas «(i,j) ,{R}),l). The use therelative t ype of a of thesingletonoperationis to preservestratification; 4Proposedfor NF in Rosser 1953.
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codedsentenceis to bethesame as theintendedrelative t ypeof itsvariables , by predicates,u naryorbinary, areonetypehigher. whilethesetsrepresented Similarly, we define codes for formulas inductively :
""-'q/'= ("q/' ,2); "ifJ /\ 'l/J" = (("ifJ","t/J"),3) j "(V'Vi )(ifJ)" = ((i,"ifJ"),4). It is straightforward to definethepredicate"x is a code for a formula" and the op erationofsubstitution of atermfor avariableby induction;N FU provides morethanenoughset theoretical machineryfor this , butit iscrucial thatthetype-level p air is used to allow s tructural induction on pairs. Notice thata type raisingoperationT on coded sentencescan be deset x which appearsrepresenting a unaryrela tionwith fined: replace each . P l {x} and each relationR with thecorrespondingrelationon singletons Our assumptionof the Axiom of Countingensuresgood behavior of this type-raisingoperation ; in particular, it implies thatcoded sentenceswhich do not involve any constants willactually be sentto themselves,as one would of "ifJ" by an inductively expect! T{"ifJ"} can beobtainedfrom thesingleton is appliedto a(doubly)encodedsendefined setfunction.I f this operation has the effect systematically of lowering types by tence ofNFU, it actually one. One proceedsto define"sat isfact ion" as predicate a of pairs ({ "ifJ" },J) , wheref is a functiontakingeachnatural number i to anintendedvalue for Vi, in the usual way; the singleton o perationis used topreservestratification. The satisfaction p redicate , which isstratified and does define a set , can then theset oftrueclosedsentencesof L in theobvious way . Nobe used to define sentencesof eachL, canactually be encodedas numbers; ticethatthe closed thereare onlycountably many sentencesin each oftheselanguages. 5. THE TARSKI "PARADOX" IN THE LANGUAGE L
The "paradoxical"situationwhich now arises is this. We have defined the ofsentencesof L via stratified construcnotionsof "sentenceof L" and truth expressanystratified sentenceof tions inNFU. Butwe knowthatL itself can NFU. Thus, it appearsthatL may be able tocaptureits owntruth predicate. ofN FU as it To see whatactually happens,it is useful to looktheworld at translated is seen from thestandpointof L . Sentencesof NFU areeffectively intosentencesof a type theory(notinternally describablein L) . The types are inclusion are theiterated images ofV underP l j themembership relations toptype andthetypebelow it) and (ofsingletons in general sets, between the therelations inducedby inclusion on erated it singletons(betweensuccessive lowertypes). Noticethatthisis a "downward"t ypetheoryin whichthereis a top type and nobottomtype; in all otherrespectsit is preciselyanalogous to the usual kind type of theory . A consequence of our assumptionof the Axiom of Countingis thatthetype-raisingoperationT on "pure" translated
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sentencesof NFU (involving noc onstants otherthanthoseused torepresent types) preservestruth value.5 Our construction of thesatisfaction p redicatefor L involvedtherelative typeof codedsentencesofL andtwotypesabove that.In thetranslations of sentencesinvolvingsatisfaction ofthesentence¢ intoL , thedoublesingleton of thecode "¢" willappearin place ofthecode itself. Moreover, thereis a of a further reflection i ntolowertypesinvolved if we considerthetranslation sentenceabout satisfaction intoL; thiswillmentiontherelation ofsatisfaction a ndanyformulasto besatisfiedvia theirquadruple using itsdoublesingleton singletons . Suppose we tryto replicatet heargumentof Tarski. We would usethe predicate: The predicaterepresented by formula¢ is nottrueof "¢" . Butthispredicateis eitherunstratified or senseless(dependingon how it is read). Certainly t hereference to t heformula¢ outsideof quotesin the predicateis, forL , a reference to thedoublesingletonof ¢ . Then it is necessaryto realize t hatthetruth predicateappearingin a sentenceinterpreted ers to a propertynot ofdouble singletonsof formulas inside of L now ref butofdoublesingletons ofdoublesingletons offormulas ; thequestionis then whethert heterm"¢" is to beunderstood as referring to thedoublesingleton understood as a code for a formula (soewhave of ¢, in which case it is not the "senseless " interpretation) or as aquadruple singleton,in which casethe formula isu nstratified and so doesnotdefine aset in NFU or predicatein L. The Tarskiparadoxis blocked bythefactthatquotation of a formula is If we hadtermsreprea type-raisingoperation , preventingdiagonalization. sentingconstants,quotingthem would raisetypes in thesame way. Since thediagonalization is blocked bystratification , it is nota problem thatthe set oftruesentencesof L (as representedby theirdoublesingletons)proves to bedefinablein L by following our developmentabove inNFU. 6. MODELS USING EXTERMA L AUTOMORPHISMS
The same situationcan be modelled in t heusual settheoryusing models of a utomorphisms . initialsegmentsof thecumulativehierarchywith external In fact,thisis how NFU itselfis best modelled. model ofthe usualset theoryZFC (or of We work in anonstandard a utomorphismj and nonstan "enough" axioms of ZFC) withan external dardinfiniteordinala such thatj(a) > a. We considerVa' stagea in the cumulativeh ierarchy . 5See S. Orey 1964 for a discu ssion of the need for the Axiom of Countinghere.
TARSKI'S THEOREM AND NFU
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It is shownelsewhere"thatVa is readilyinterpreted as a model ofNFU. The trickis to observethatVa containsthe muchearlierstageVj(o) of the cumulativehierarchywhich looksexactlylike it. Construeeach elementof Vj(o)+l as a set with its elementsreplacedby the inverse images underj of its actual elements and each elementof Va - Vj(o)+l as anurelement (noticethat therearea lot of urelements!).This is achieved by defining a new membership relation "x En ew Y" as "x E j (y) and j (y) E Vo + 1" . It is straightforward to E n ew is a model ofNFU. provethatVa withmembership relation The semanticswe have been doing in this paper can be clarified in the same context . It is possible to define thesemanticsforthe "fulllanguage" to a (in which each set correspondsto aunarypredicateand each relation binarypredicate)on Va in Vo +2 ' If we replaceelementsof sets with their forNFU andinflatepowersets images underj as in themodel construction to becarriedout withurelements in thesame way(settingup theconstruction in NFU as above), weobservethatthesemanticsforthe"fulllanguage"on Vj2 0 can beexpressedin Va in a way which translates successfully i ntoterms is in factexactlythatof theprevioussections. of En ew . The construction If L is the "real" full language on Va' the language, which we internally see above, is P(L) . We cannotderive theTarski to Va in theconstruction : paradoxhere becausetheneededpredicatewould be
Predicate4> does not hold (in j2(L» of P("4>") , whichcannotbe expressedin our working model set of theorybecausej is theargumentfor external.This is preciselyanalogousto the way in which Tarski'sparadoxfails above . A system resemblingthesystem of this paper in its semanticfeatures, , is disalthough notapparently m otivatedby work inQuine-styl e settheory cussed by Hiller and Zimbarg (1984) . Theirsystem uses additional assumptions whi ch make it farstrongert hanthesystem NFU + Infinity+ Choice . used here , even whenthisis further extendedwiththeAxiom of Counting REFERENCES Andrews,P. 1986 An introduction to mathematical logic and type theory: To truth through proof, Academic Press, New York, pp. 264-265 . Forster,T . 1992 Set theory with a universal set, Oxford LogicGuides, no. 20, Clarendon Press, London. Grishin,V. N. of Quine's NF system to one of itsf ragments(in Rus1972 The equivalence sian), Nauchnotekhnicheskaya lnformatsiya, series 2, vol. 1, pp. 22-24. 6See Forster 1992, pages 67-68.
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Hiller, A. .P, and J. P. Zimbarg 1984 Self-referencewithnegativetypes, The Journal of Symbolic Logic, vol. 49, pp . 754-773. Jensen,R. B. 1969 On theconsistencyof a slight(?)modificationof Quine's NF, Synthese, vol. 19, pp. 250--263. Orey, S. 1964 New Foundationsand theaxiom ofcounting,Duke Mathematical Journal, vol. 31, pp . 655-660. Quine, W. V. 1937 New foundationsfor mathematicallogic, American Mathematical Monthly, vol. 44, pp. 70--80. Rosser, J. B. 1953 Logic for mathematicians, McGraw-Hill,New York; and Chelsea , New York,1978.
GARY MAR
CHURCH'S THEOREM AND RANDOMNESS·
A good work of art can in its entirety be expressed only by itself. -Tolstoy Abstract.AlonzoChurchdemonstratedin a logicallyr igorousway that Leibniz's dream of a logicalcalculust hatcoulddecide alltruthswas not only unfulfilled butunfulfillable . Accordingto Church'sTheorem (1936a) , even firstorderlogic isundecidable . Kleene's (1936) proofof theequivalence of theChurchKleenenotionof .>.-definability a nd theHerbrand-GOdel ( 1931) notionof general recursivenessa nd Turing's(1937) proofof theequivalenceof .>.-definabilityand computabilitywere regardedby Churchas evidencefor Church'sThesis (1936). Church'sThesis is thephilosophicalclaim thata functionof positive integersis c alculable if and onlyif it is X -definable .! Using the notionof effeceffectively tivecalculability, Church(1940) proposed a definitionof randomness . Church's definitioncan be seen as aprecursorto algorithmicdefinitionsof randomness formulated i ndependently by Kolmogorov(1963, 1965) and Chaitin(1966) . In this paper we sketchone way ofrelatingC hurch'sTheorem to randomnessby way of arecentgeneralization of the'paradoxof theLiar (Mar and Grim 1991) thatuses themathematicsof chaos.
1.
The Classical Liar is a sentencethatassertsits own falsity :
IThe boxedsentenceis false.I Accordingto theTarskian(T) schema, asentencestatingthata sentencep is truehas the sametruth-value as p itself. Hence, (1) 'The boxed sentence is false'true is if and only if theboxed sentenceis false. " This paper is a concise presentation of thefoundationof collaborative researchwith PatrickGrim and PaulSt. Denis. See Mar and Grim 1991, and Mar and St . Denis 1994 and 1996. Afterthepaper was submittedforthisvolume,an additional a rticle ( Mar and St . Denis 1999) has been publishedthatgives anelegant c haracterization of theformalsystem used here. 1 Church 1936. The precise epistemological s tatusof Church'sThesis is a subjectof considerablephilosophicaldiscussion. See the Notre Dame Journal of Formal Logic, vol. 28, no. 4, 1987, devotedto thistopic.
479 C. Anthony Anderson and M. Zeleny (eds.), Logic, Meaning and Computat ion, 479-490.
© 2001 Kluwer Academic Publishers. Primed in the Netherlands .
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Butsince it isempiricallytruethat (2) 'The boxed sentenceis false' isidenticalto theboxed sentence, we may infer byLeibniz'sLaw that (3) The boxed sentenceis trueif and only if theboxed sentenceis false. The assumptionthattheboxed sentenceis trueleads totheconclusionthat it is false, and the assumptionthatit is false leads to theconclusionthat it is true . The semanticbehaviorof the Classical Liar can thereforebe true representedas an infiniteoscillation betweentheclassicaltruth-values and false.Suppose we do notrestrict ourselves to theassumptionof bivalence. Whatform couldtheClassicalLiar takewhengeneralized to thecontinuous real-valued case? First, we generaliz e thelogicalconnectivesby adoptinga Lukasiewiczian infinite-valued logic.2 Following Rescher (1969) , we use thenotation ' /p/' to representthevalue oft hesentencep, which may now have as a value any infinite-valued negationrule is : realnumberin the[0, 1] interval.A natural
/rvp/
= 1-
/p/.
Intuitively, the negationof p is trueto theextentthatp is untrue,i.e., to theextentthe value of p differs from the value of 1 completetruth. or T he infinite-valued Lukasiewiczianbiconditional rule is:
/(p ..... q)/ =
1-
I/p/ - /q/I·
(Here thevertical slashesreprese ntthe absolute value of the difference.) Indo not tuitively, the biconditional is trueto theextentthatits constituents . Boththeseinfinite-valued rules arefaithful to clasdiffer in truth-value sical logic : when the values of the sentencesare restricted to the classical tables . truth-values , we obtaintheclassicaltruth Secondly , theTarskian(T) schema can also begeneralized to an infinitevaluedcontext . The (T) schemaimplies thetruth of eachbiconditional whose constituents are thesentenceassertingthatp is trueandthesentenceP itself. by 1 = 1 - I/Trp'/ Using thebiconditional rule, we an c expressthistruth /p/I. Alternatively stated , we have/Trp'/ = 1-lt- /p/I, wheret is the value of 1 orcompletetruth . Replacingtheconstant t with aparameterv (which rangesover the [0 ,1] interval) and replacingthe bivalent truthpredicate'T' ' Vvp' (which is to be read ' v is thetruth-value witha multi-valuedrelation of thesentencep'), we obtainRescher's(1969) schema for hisparametricoperatordevelopmentof many-valuedlogicsr' /Vvp/ = 1 - [v 2A
/(p
--+
/p/I.
Lukasiewiczianinfinit e -valued logic ischaracterized by its rule forthe condition al :
q)/ = MIN{l, 1 - /p/ + /q/} . See Rescher 1969, page 36.
3 Res cher
1969, page 81.
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Intuitively , Rescher'sschemastatesthatthesentenceVvp , namely,thesentencethatv is thetruth-value of thesentencep, is trueto theextentto which the value pofdoes notdiffer from .v? Consideragain, forexample, theClassical LiarA, which assertsits own sentenceassertingthattheLiar istrue , falsity. Tod eterminethevalue of the a calculation using Rescher'sschemayields
/VtA/ = I -If - /A/I, wheret and f are thevaluescompletelytrueand completelyfalse, respectively. How can we representtheself-referential elementin theLiar? One way is to usefunctional iteration: a value for /VU/ can be calculated on value for/VtA/ the basis of anestimatefor /A/, and thenthe calculated can thenbe recycled as a revised estimatefor /A/, and so forth.Formally , 'p' in we representthe process of iteration by replacing the sentenceletter the Vvp schema by 'x n ' and byreplacing'/VU/' by 'Xn+l'. Makingthese replacements and usingthevalue forf, we can nowrepresentt hesemantics for the Classical Liarby thealgorithm:
°
Xn+l =
1-
10 -
Xn
I·
Given aninitialestimatedvalue ofxo, the successiveestimatedvalues for thecontinuous-valued ClassicalLiar will be an a lternating cycle of period 2 Xo and 11 - xol. The singleexceptionis thevalue of1 /2 between the values which is a fixedpoint. Suppose we want to evaluate sentencest hatdo notattribute tothemselves to themselvesa truth-value a particular truth-value v but rathera ttribute expressedas a junction of its previousestimatedvalue." To evaluate such 'v ' in theVvp schemawith afunctional expression sentences,we replace the S(x n ) thatexpressesthevaluethesentenceattributes to itselfin termsof its previouslyestimatedvalueX n :
4The Tarskian (T) schema was stat ed above in terms of sentences; Resch er st a tes hi s Vvp schema in terms of propositions. For the sake of consiste ncy w e have reformulated Rescher's schema in termsof senten ces. For presentpurposes , we setasidethe philosophical cont roversy as to whatshouldproperlybe regardedas the bearers of truth . See Church 1956, page 27, footnote72. e optionsare possibl e. For example, given a seq uencex o, z i , •• • , X n of 5Less restrictiv lengthn + 1, let us edfin e the middle of the sequence m(n) to be n/2 if n is eve n or (n - 1)/2 if n is odd. Then define Xn to be x m(n) if n is even and to be 1 - Xm (n) if n is odd. Then given Xo = 0 and Xl = 1, thesequence {x n} for nEw gen erates the famous Morse-Thue (MT) sequencefrom symbolicdynamics. The MT sequ en ceis of interestbecauseit is aneffect ivelyalculabl c e binarysequence whichis ap eriodic,infinit ely self-sim ila rand intimatelyrelated to the p eriod-doublingroute to chaos (see Schroeder 1991, pages 264ff.). Consider theinfinitelynestedsequenc e of Biconditional Samesayers: f30 : = ""PO; f3l : = PI .... f30, and in general,f3n+J : = (Pn+l +-+ f3o). Then given a standard truth-table assignment to thesentencelett ers, the final oc lumn for each of the nested biconditional s f3n is preciselytheinitialbinaryblock oflength2 n of theMT sequen ce.
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We can illustrate the use of this modified Vvp schema by comparingthe semanticbehaviorof twoLiar-likesentences. Considerfirstthesentence:This sentence is half as true as it is estimated to be true,which we call t he Cautious Truth- Teller. Using the modified Vvp schema, thesuccessivevalues fort heCautiousTruth-Tell er are given by Xn+l = 1 - Ix n / 2 - xn l. Compare this with asecond sentence,This sentence is as trueas the estimated value of the conjunction of itself and its theContradictory Liar. The value of a conjunction, in negation, which we call a Lukasiewiczian logic, is given bytheminimum ofthevalues of its conjuncts: /(P /\ q)/ = MIN {/p/ , /q/}. Hence, thesequence ofe stimatedvalues forthe Contradictory Liaris given by Xn+l = 1 -IMIN{x n , 1 - Xn} - xnl. The differencebetweenthesemanticbehaviorsof thesetwosentencescan be made visuallyperspicuoususing a web diagram. Beginningby plotting a linevertically from (Xo, 0) to (Xn , Xn + d, theweb diagramcontinuest he linehorizontally from (x n , Xn+ d to (X n+ 1, Xn+ d and theniteratetheproClassical cess by usingXn+l forXn. The semanticsforthecontinuous-valued Liar, forexample, appearsin a webdiagramas nestedseriesofsimple squares.
Figure1: Web diagramsfortheCautiousTruth-Teller and theContradictory Liar. The CautiousT ruth-Teller, on theotherhand,yields afixed-point attractor: no matterwhatinitialvalue with which we begin, thesuccessivelyrevised estimatedvaluesare inevitablydrawntowardthefixed point of 2/3. The Contradictory Liar, in contrast , yields afixed-point repellor: for any values otherthanthe fixed point 2/3, the succes sivelyrevisedvalues fort he Contradictory Liar arerepelledaway from2/3 untilthe values settleon the of theClassicalLiar. oscillation between1 and0, characteristic To sum up, thesemanticbehaviorof theClassicalLiar, thoughidentical totheContradictory Liaron classical values , diverges fromtheClassicalLiar on the range of values betweena and 1. Sentenceslike the CautiousTruth-
CHURCH'S THEOREM AND RANDOMNESS
483
Tellerand the Contradictory Liar, on the otherhand, have preciselythe oppositesemanticbehaviorin termsof beingattractors andrepellors around thesame fixed point. Infinite-valued logiccan, therefore , revealintriguing new patternsof paradoxthathave remainedhiddenin a classicalbivalent setting.f 2.
Considernexta more intriguing , infinite-valued generalization of theClassicalLiar,whichassertsof itselfn otthatit issimply false,butthatit istrue to theextentthatit is false:
IThe boxed sentenceis as trueas it isestimatedto be false.I We callthissentencetheChaotic Liar forreasonswhich will s oon become apparent.WhattheChaoticLiar assertsof itselfis thatit is false ; hence, S(x n) = 1 -10 - xnl. Since X n 2': 0, we haveS(x n) = 1 - X n . The algorithm for thevaluetheChaoticLiar attributes to itself,t herefore,is identicalto thealgorithmfor theClassicalLiar. The successivevaluesfor theChaotic Liar will be given byX n +l = 1 - 1(1 - xu) - xnl . The web diagramfor theChaoticLiarmakes it clearthatthecomplexityof thesemanticbehavior
Figure2: The ChaoticLiarhas chaoticsemanticbehavior. 6Considerthe Contingent Liar based an infinite-valued generalization of theparadox due to Kleeneand Rosser (1935) , discussed by Curry(1941,) and also known as Lob 's paradox(see Banuise and Etchemendy 1987, footnote14) . The Conting e nt Liar asserts This sentence is as tnte as the conditional: if this sentence is tnte then q, whereq is some conting e ntsentence . Using the infinit e -valuedrulefor theLukasiewicziancond it io nal, ew have thattheiteratedvalues for the Co nt inge ntLiar is given by the seq ue nce: Xn+1 = 1 -JMIN{1 - Xn + jqj , I} - x nl . The ContingentL iar exhibitsthe fixed-pointsemantic behavior of the Truth- Teller (which asse rts This sentence is tnte) on the interval[0, / q/J andexhibitschaoti c semanticbehavioron theinterval[/ q/ , IJ.
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of theChaoticLiarfarsurpassesthemonotonousregularity of theClassical Liar(see Figure1). The ChaoticLiaris so-calledbecauseits algorithmis chaotic in a precise mathematical sense." Known morefamiliarly as thechaotictentfunction,the algorithmfortheChaoticLiar is paradigmatically chaotic ." The semantic patternsinherentin theChaoticLiar, therefore,exhibitalltheintriguing propertiesof chaoticfunctions.Among thesepropertieswill be: • unpredictability: thesensitivedependenceon initialconditionsof chaoif conditionsare tic functionse ntailst hatpredictionwill fail theinitial not knownwithinfiniteaccuracy; • infinitely many periodic cycles: since the ChaoticLiar has a cycle of period threeit will have cycles of otherperiods all accordingto Sarkovskii'sordering(see Devaney 1989, pages 60-62); • fractal images of the semantics of paradox : fractals, sets witha fractionalHausdorffdimension(see Peitgen and Saupe 1988, pages 28-29), arecharacterized by self-affinity at increasingpowers ofmagnification. Withinthepatternsof paradoxforDualistforms oftheLiar, for excomplex fractal patterns(see Mar and Grim ample, thereareinfinitely 1991) .
The semanticparadoxeshave been atrapforlogicianswho, intheirattemptsto solvetheparadoxes,havetendedto viewthepatterns of paradox as simpler and more predictablethanthey actually are. Even in thesophisticatedwork ofBarwiseand Etchemendyon theLiar (1990), thecyclical regularity ofthesemantical paradoxeshas beenobviousbuttheirincalculable complexityhas remainedhidden. Here, insteadof searchingfor simplepatternsofsemanticstability (as in Gupta 1982 and Herzberger 1982), we have ofsemanticinstability triedtoexhibitinfinitely complexandchaoticpatterns which have gone v irtually unexplored . 7 Devaney 1989, page 50 , notesthatther e arestronger a nd weakerdefinitionsof chaos. Devaney'sdefinitionof chaosis as follows . A functionf : I -+ I is chaotic on a set I if all threeof thefollowing c onditionshold: close (i) f has sensitive dependence on initial conditions: thereexistspointsarbitrarily to x whicheventually separatefrom x by atleast0 underiterationof f, i.e. , 30 > 0 'v'xEI 'v' neighborhoodN of x 3yEN 3n 2: 0 Ir(x) - r(y)1 > 0 (here 'r(x)' representst henthiteration of thefunctionf) ; (ii) f is topologically transitive: f has points which eventually move underiteration from one arbitrarily smallneighborhoodto any other, i.e., 'v' opens sets U, V C I 3k > 0 fk(U) n V "q, ; (iii) theperiodic points are dense on I: thereis a periodicpointbetweenanytwo periodic points in theinterval I , wherea point x is periodic if 3nfn(x) = x, 8The tentfunctionis mentionedin RobertMay's (1976) ground-breaking p aper as a "ma t hemat icacuriosity l ". Here,however ,thetentfunctionappearsas perhapsthesimplest andmost natural generalization of theClassicalLiarthatyieldschaoti c semantic behavior .
CHURCH'S THEOREM AND RANDOMNESS
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Figure3: Infinite-valued dualistforms of theChaoticLiar generatefractal images. 3.
Variationsof theStrengthened Liar, namelythesentence : This sentence t hatthemove is false or neither true nor false, can be used todemonstrate -valuegaps is at best atemporaryexpedientin blocking Liar -like to truth Liar tomotivate paradoxes." Here we use avariationof theStrengthened a limitativetheoremregardingthe calculability of chaos. Combining the Strengthened Liar with the notionofchaoticsemanticbehavior,we arrive at the Strengthened Chaotic Liar: Eithertheboxed sentencehas achaoticsemantic behavioror it is astrue as it isestimated to be false . Does theStrengthened ChaoticLiar havechaoticsemanticbehavioror not? Assuming it does have a chaotic semanticbehavior,it will becompletelyt rue in virtueof its firstdisjunct . However,thesemanticbehaviorof a sentence thatis completelytrue will not qualify chaotic as since it will have theconstantvaluetrueregardless of previousestimates. Assuming theStrengthened ChaoticLiar doesnot have achaoticsemanticbehavior,its truth-value will 9 Mar 1985 (complet ed under the directionof AlonzoChurch) demonstrates this claim ee truth -value gap solut ions,n amely, for Martin 's (1970) CategorySolution , van for thr Fraasse n's (1968) PresuppositionSolution,and Kripke's (1975) SemanticGroundingsolutionto th e Liar.
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CARY MAR
dependentirely on itsseconddisjunct.However,t hesemanticbehaviorofthe seconddisjunctwill mimicthebehaviorof theChaoticLiar, which is known to besemantically chaotic . We havederiveda contradiction in eithercase. This paradoxcan be used (in muchthesame way asCodel(1931) used the Richardparadox, Tarski (1936) used theLiar paradox,and Chaitin(1966) used theBerryparadox)to motivatea limitativetheorem. Non-Calculability ofChaos. Let C betheset ofchaoticfunctionsdefined [0 Assume on theset ofpartialrecursivefunctionsf on therealinterval ,1]. thatAx[I -I(I-x) - x l] is in C butthatAx[l], theconstant functionidentical to 1, is not in .CThen theindex set I(C) = {i : Ii E C} is not effectively calculable. thattheindex set ofchaoticfuncAssume, for aproofby contradiction, tionsis A-definable . We havethatAx[1 - 1(1 - x) - xl] is some h in C and thatAx[l] is some !J notin C. Then we may define thediagonalfunction d(x) to bei . if fxEC, and d(x) to bei otherwise.By thefixedpointlemma, therewill be ak such thatfk = fd(k). Hence, bythedefinitionof d(x), we have fkEC if and only if d(k) = j ; butsince !J(x) is the non-chaotic c onstant functionidenticalto 1, we have fk EC if and only if!k rj.C , which is a contradiction . Therefor e, contrary to ourassumption, I(C) is not A-definable. It follows by C hurch's thesisthat theindex set ofchaoticfunctionsis not effectivelycalculabl e.10 Church'sdefinitionof randomness(1940) for aninfinite series of Os and Is uses the ideathatany effect ively calculable way of choo sing elementsof thesequencewill nota ltert helimit oftheratioof Is inarbitraril y long finite sequencesof theseries. Intuitively, thereis no effectiv elycalculable way of capturingtheinformationcontainedin thebinaryseries. Church's idea can be seen as a precursorto thealgorithmic definitionof randomnessproposed by Solomonoff(1960) and developedindependently by Kolmogorov(1965) andChaitin(1966) . The informational contentof a computerprogramthatcomputes a real numberin thecontinuumbetween0 and 1 can itselfbe representedby a binarystring,which itselfepresents r a realnumberin binarydecimalnotation. The lengthof a stringcan therefore be comparedto thelengthof thestring which it issupposed to compute. The complexity of a number x, K(x) can thenbe defined to bethe lengthof a programipe of minimal lengththat computes x on thebasis of a fixedinput,say o. Thena number x is defined to be random if K(x) ~ x. This definitionof randomnessis notcompletely precise insofaras we have not given a specific definitionof acomputer.However, the problem of defining anidealizedcomputerwas solved byTuring (1936) and Post (1936) in a veryintuitiveway. The complexityof a string lOThis form of the proof makes the resultan applicationof Rice's Theor em (1953) . For a proof using diagonaliz ation and Godel numbering, see Mar and Grim 1991.
CHURCH'S THEOREM AND RANDOMNESS
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is a mathematical conceptthatis notdependenton anyparticular definition of a computer. Chaos andrandomnessare two sides of thesame conceptual coin. Chaotic dynamicalsystems are intuitively r andom. Due tosensitivedependenceon initialconditions , a chaoticdynamicalsystemcan requireinfinite-precision of irreducible theinitialinput; andhenceprogramsforchaoticsystemswill have information. 11 Chaitin'sTheorem (1965) statesthatfor anyconsistentsufficiently expressive formalaxiomaticsystem T includingnumber theorytherewill be numbers whose randomnesscannotbe proved. In particular therewill be some K. such thatT cannotprove thatany specificbinarystringhas complexitygreatert hanK.. Church'stheoremcan be seen as aconsequenceof thistheorem : for any sufficiently expressiveformalaxiomaticsystemT there will bebinarystringswhosecomplexityis greater t hanK. for which Tcannot prove thattheyhavecomplexitygreatert hanK.. Formalsystems are thereforeinformationally limited: eventhoughtheremustbe stringsofunbounded complexity,thereis an upper boundto thecomplexityit is possible to prove any particular stringto have. ChurchtheoremestablishedthatLeibniz's dream of reason was logically unfulfillable , and Church's definition of randomnessultimately held the key to understanding why formalaxiomatictheorieswillinevitablyfallshort. theinformation-theoretic interpretation of thelimitativetheoIn retrospect, rems suggeststhattheseresultsdo notdependon anomalousor pathological singularities . From this perspectiveChurch'stheoremcan be seen as the inevitableoutcomeof incorporating the notionsof chaosand randomness withinformalsystems. llChaose merges natur allyfrom an infinite-va lued v ariati on of Russell 'sp aradoxthat uses a notionof degr eesof randomness. Considerprogramsfor computingbinarystring s together w ithbinaryrepresentat ions of these programs. Ratherthan determining whether the complexityof a stringis simp ly greaterthanor equalto the st ring its elf, we can introdu ce a measure m(x) of how much greater the complexity of the string is. (For example, m(x) couldbe a partially defined functionwhich is thediffer en ceb etw eenthe lengthof theminimal program K(x) for x and x itself.) Then defin e H" = {x : m(x) 2: n & K(x) is defined},and define R W = URn forevery nEw. Note thatfor'VnEw,3N > n, R N f
stringsuchthatn is the leastn such thatxERn, thenwe ca n assigna degr eeof randomness , thatther e si a to x in [0,11 equalto 1 - lin. Now assume, for a proofby contradiction min imal programP, represe ntedby a string#Rw, such thatP output s a st ring en c oding them embersof RW . Is #RwERw or not? If #RwERw, then #Rw is a member of some R" . Then P is a programof finite complexitywhich is generating the numbers of programs of w, then giventhatthe complexity w. If #RwltR arbitrarily h ighcomplexity . Hence, #RwltR of program P is no, we have K(#RW) < no, contradicting thatP is a min imal program for RW; hence #Rw E R W. The Russelli an ont c rad ict ion # Rw E R W iff #RwIt R W ca n be modeled in the infinite-valu ed logic above again yield ingthealgorithmfor the Chaotic Liar. This variationof Russell'sparadoxin termsof degrees of randomnes s suggeststhat infinite-valued generalizations of Rice 's Theoremareworthyof furth er investigation .
488
GARY MAR REFERENCES
Barwise, J., and J. Etchemendy 1987 The Liar, OxfordUniversityPress, New York. Chaitin,G . J . 1966 On thelength of programsforcomputingfinitebinarysequences, Journal of the Association of Computing Machinery, vol. 13, pp. 547-569. 1974 Information-theoretic limitationson formalsystems, Journal of the Association of Computing Machinery, vol. 21, pp . 403-424. Church,A. 1935 An unsolvable p roblem of elementary n umbertheory(abstract),Bulletin of the American Mathematical Society, vol. 41, pp. 332-333. 1936 An unsolvable p roblemof elementary n umbertheory,A merican Journal of Mathematics, vol. 58, pp. 345---363. 1936a A noteon the Entscheidungsproblem, The Journal of Symbolic Logic, vol. 1, pp. 40--41; correction, ibid, pp . 101-102. 1940 On theconceptof randomsequence, Bull etin of the American Mathematical Society , vol. 46, pp. 130--135. 1956 Introduction to mathematical logic, vol. 1, PrincetonUniversityPress, Princeton , New Jersey. Curry , H. B. 1941 The paradoxof Kleeneand Rosser, 1Tansactions of the American Mathematical Society, vol. 50, pp . 454-516. Devaney, R. L. 1989 An introduction to chaotic dynam ical systems, secondedition, AddisonWesley, MenloPark. GOdel, K. 1931 Uber formalunentscheidbare Satzeder PrincipiaMathematica und verwandterS ysteme I, Monatshefte fUr Math emat ik und Physik, vol. 38, pp. 173-198. Gupta, A. 1982 Truthand paradox, Journal of Philosophical Logic, vol. 11, pp. 1-60. Herzberger , H. 1982 Notes on naivesemantics, Journal of Philosoph ical Logic, vol. 11, pp. 61-102. Kleene, S. C. 1936 A-definability a nd recursiveness,Duke Mathematical Journal, vol.2, pp . 340--353. Kleene,S. C., and J. B. Rosser 1935 The inconsis tencyof certainformal logics,Annals of Mathematics, vol. 36, pp . 630--636. Kolmogorov, A. N. 1964 Threeapproaches to thedefinitionof the conceptof 'amountof information', Problemy Peredachi Informatsii, translated as "P roblems in InformationTransmission", vol. 1, 1965, pp . 3-11 ; translated in vol. 1 1965, pp .I-7.
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Kripke, S. 1975 Outlineof a theoryof truth,The Journal of Philosophy., vol. 72, pp. 69(}-716. Mar, G. 1985
Liars, truth-gaps , and truth : A comparisonof formaland philosophical solutionsto thesemanticalparadoxes,Dissertation, Universityof CaliforniaatLos Angeles,UniversityMicrofilms, AnnArbor.
Mar, G., and P. Grim 1991 Patternand chaos: new images inthesemanticsof paradox,Noiis, vol. 25, pp . 659-693 . Mar, G., and P. St. Denis 1994 Chaos incooperation : continuous-valued prisoner'sdilemmas in infinitevalued logic, International Journal of Bifurcation and Chaos, vol. 4, pp. 943-958. 1996 Real life , International Journal of Bifurcation and Chaos, vol. 6, pp. 2077-2086 . 1999 WhattheLiar TaughtAchilles,Journal of Philosophical Logic, vol. 28, pp .29-46. Martin,R . L. 1970 A category s olutionto theLiar, The Liar Paradox (R . L. Martin,e ditor), YaleUniversityPress, New Haven. May, R. 1976
Simple mathematical models withvery complicateddynamics, Nature , vol. 261, pp . 459-467.
Peitgen,Hr-O ., and D. Saupe 1988 (editors), The science of fractal images, Springer-Verlag,New York. Post, E. L. 1936 Finite combinatoryprocesses- formulation1, The Journal of Symbolic Logic, vol. 1, pp. 103-105. Rescher, N. 1969 Multi-valued logic, McGraw-Hill,New York. Rice, H. G. 1953 Classes ofrecursively e numerable set andtheirdecision problems, Transactions of the American Mathematical Society, vol. 74, pp. 358-366. Schroeder,M. 1991 Fractals, chaos, power laws, W . H. Freeman,New York. Solomonov, R. 1964 A formaltheoryof inductiveinference , Information and Control, subsequentlyrenamedto Information and Computation, vol. 7, pp . 1-2. Tarski,A. 1936 Der wahrheitsbegriff in den formalisierten sprachen,Studien Philosophica, vol. I, pp . 261-405.
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Turing,A. 1936 On computablenumberswithan applicationto theEntscheidungsproblem, Proceedings of the London Mathematical Society, vol. 42, pp . 230-265; corre ctionsibid., vol. 43 (1937), pp. 544-546. 1937 Computabilitya nd ~-definability, The Journal of Symboli c Logic, vol. 2, 153-163. Van F'raassen,B. C. 1968 Presupposition,i mplication,and self-referenc e, The Journal of Philosophy, vol. 65, pp . 136-152.
ENRlCO MARTINO
RUSSELLIAN TYPE THEORY AND SEMANTICAL PARADOXES
Abstract.We compare Grelling'sparadoxwith Russell'sparadoxabout propositions,in order to illuminat e their differentnatur es. We crit icizeChurch's ctionof Grellin g 's paradox,accord ing to which it would e an b intenreconstru sional antinomyarisingwithinthesim ple typetheoryandsuitablysolvedwithin the ramified type theory . We claim , on thecontrary,thatthis paradox is not genuinely intensional a nd thatramificationis of no help for its resolut ion. We arguethattheparadoxrestson a highlyproblematicassumption, which can be rejectedwithinthesimple theoryitself, since its allegedevidence has nothing to do withthe lack oframification.In contrast , we show thata reconstruction , similarto the one proposed by Churchfor Grelling'sp aradox, is appropriatefor , which turnso uttobe reallyintensionaland Russell's paradoxaboutpropositions capableof an adequatesolultion by means of ramification .
1. INTRODUCTION
According to C hurch(1976) , Myhill (1979), and Hazen (1983), theRussellian ramifiedtype theorysupplies asolutionto Grelling'sparadox,which arises in the simpletypetheory. In particular, Church(1976) provides aninteresting detailedreconstrucChurchexpresses tion oftheparadox. By usingcertainsemanticalc onstants, theterm"heterological " withinthesimple typetheoryandshows how,within the ramified theory,splitsinto it infinitely many terms "n-heterological " for every level n . He thenshowsthat , while inthesimple theorythequestionof whether"heterological" is heterological leadscontradiction, to in theramifiedtheorytheinfinitely many questionsof whether"n-heterological" is m, heterological are allcapableof preciseand coherentanswers. In this sense Churchconcludes,t heramificationsolvesGrelling'sparadoxof thesimple type theory . ThoughtheRussellian notion of propositional function describedby Church is distinctively intensional, theaxioms he uses are all extensional, so thatthe paradoxin questionseems to havelittle to do withtheintensionality of propositionalfunctions . On theotherhand, thesuperiorityof the ramified logic, comparedwiththesimple one, seems to emergejustin dealing with intensions. In thissituation,it is hardlyplausiblethata paradox,notessentially intensional, be solved by means of ramification.According to this observation, we shall hold that,in the simpletype theory, Grelling 's paradoxrests on anerroneousassumption, quiteextraneous to mattersof intensionality , 491 C. Anthony Anderson and M. Zeleny (eds.), Logic, Meaning and Computati on, 491 -505. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.
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andgetstherefore its appropriatesolutionby means ofthemere rejectionof thatassumptionwithinthesimple theoryitself . Furthermore, we shallshow how, evenwithinthe ramified theory, similarassumptionleads,withthe a help oftheaxiom ofreducibility, to asimilarparadox. In contrast,a genuinelyintensional p aradoxof the simpletype theory , suitablysolved by means oframification,is Russell's paradox about proposiby Russellin appendixB of thePrinciples (1903). It tions, statedinformally exploitsessentially Russell'sconceptionoftheidentityof propositionsandis perhapsthe source of the very idearamification of . An interesting formalizationof thisparadoxis expoundedby Church(1984) himself in which he no longeremploys themethodof semanticalconstants(cf. Church 1976) , but introducesa specialprimitiveconnectivein orderto express anintensional identityof propositions. thatthe methodof semanticalconstants, suitablyreWe shall also show . fined, is adequatefortreating Russell'sparadoxaboutpropositions We believethatsettingup the twoparadoxeswithinone and the same t heirdifferentn atures . formalcontextmay help tobetterunderstand 2 . RAMIFIED TYPE THEORY
We assume acquaintance with Church's(1976) system of ramifiedtype theory, which will be here summarized. Ramified types, shortlyr-types, arerecursively defined :
(a) i is an r-type, (b) if 131, . .. ,13m (m:2: 0) are r-types,so is (131, .. . ,13m)ln (n
> 0).
If k < n, (131, '" ,13m)lk is said to beless than(131 , .. . , 13m)ln. i is ther-typeofindividuals ; (131 , ... ,13m) In is the -rtypeofm-placepropolevel n withargumentsof r-types~ 131, . . . , 13m . sitionalfunctions of Types of the form (i, i,.. . , i) In, with m i's, are indicatedbriefly bymin (Olnis ther-typeof propositionsof leveln) . To everyr-typeis assigned an order: (a) ordi
= 0,
(b) ord(131,. . . , 13m)ln = N
+
n , with N
= max[ord131, ' "
,ord13m]'
For everyr-type,thereare infinitelymany variablesand possibly some constants . F(Xl, "" x m ) , with F being a variable Atomic formulas are theform of Xi variables (orconstants) (orconstant) of type(131, .. . ,13m) In and where the areof the -types r 131, . . . , 13m , respectively . Molecular formulas builtup are in the usual way .
RUSSELLIAN TYPE THEORY
493
We will also use thefollowing definitionof order of a formula (notused by Church, 1976): if h is the greatestorderof freevariablesand constants , and k thegreatest o rderof boundedvariablesin theformulaP, then
ordP = max[h, k + 11. We will usemetavariables p , q forpropositions , F, G forpropositional func. tions,P, Q for formulas(ther-typeswilloftenbe understood) We presupposesome standard s ystemof rules and axioms for logical constants.These will never be explicitlyused, our proofsbeing presentedin a semiformal style . t hecomprehensionaxioms: I A central role is played by AXIOM
2.1. 3p.p == P,
wherep is a propositional variableof r-type Din not free in P, withordP $ n; AXIOM
2.2. 3F. F(XI ' . .. , x m
)
== X\ " ",X m
P,
where F is a variableof type (f31, "" f3m)ln not free inP; Xl, •• . , X m are distinctvariablesof types f3I' . .. ,f3m, and ordF ~ ordP. Finally, we recall the axioms of reducibility, assuringthateverypropositionalfunctionis coextensivewithsome predicative (i.e., of level 1)propositionalfunction :
withG predicative. In virtueof 2.3, identity is definableas indiscernibility with respectto predicativepropositional functions : DEFINITION
2.4.
X =
Y
=def
F(x)
JF
F(y)
wherex , yareof thesame r-typef3, and F is of type (f3)I1. 3. CHURCH 'S FORMULATION OF GRELLING 'S PARADOX
Here we sketchChurch'sreconstruction of Grelling 's antinomy.Given anaturalnumber n ~ 1, every formulaof ordern with a singleindividualfree variable(and without any otherfreevariables)represents(accordingto 2.2) a propositionalfunctionof order$ n . Regardinglinguisticexpressionsas functionsare individuals,such relations betweenformulasand propositional 3-placepropositionalfunctions , which we can express in t he languageby means ofspecialconstants. 1 We followChurch 's convention of usinga bolddotas a left bracket witha corresponds for logic al ing rightbracket as farto theright as possible (while thenorm aldot stand conjunction) .
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In fact, for all n > 0, we introduce t heconstant valn+1 ofr-type(i, i, lln)I1. n+1 val (a, v , F) is intendedto mean thata is an individualvariable,v a formulaof orderx n whose uniquefreevariableis a, and F thepropositional functionexpressedby v. In accordancew iththisinterpret ation, we assume thefollowing axioms: AXIOM 3.1. valm+1(a,v,F) »;.:» , valn+l(b,v,G):>b,G F = G . AXIOM 3.2. 3a3v3F. valn+ 1(a, v, F) . F(x) =x P, with P a formula of order :::; n without free variables except x. AXIOM 3.3. valn+1(a,v,F):> valm+1(a,v,F) , ifm ~ n. From 3.1 and 2.4, it follows that COROLLARY 3.4 . valm+l(a,v, F) :>a ,v,F. vafn+1(b,v, G) :>b,G ' F(x) = xG(x) . of Grelling's By means of such constants,t he predicate"heterological" paradoxis interpreted i ntoan infinitehierarchyof predicates,a ccordingto thedefinitions DEFINITION 3.5. hetn+l(v)= def 3a3F. valn+l(a, v, F) . -, F(v). From 3.3, it follows immediatelythat COROLLARY 3.6. hetn+l(v):> hetm+1(v), if m ~ n . The followingt heoremshold: THEOREM 3.7. (valm+ 2(a,v,G) . G(x) =x hetm+1(x)) :> -,hetn+1(v),ifm ~ n. Proof. Let (1) valm+ 2(a, v, G), and
(2) G(x) =x hetm+1(x). Assuming, for areductio,h etn+l(v),w ithm ~ n, we get, by 3.6, hetm+l(v) and therefore, by 3.5, 3a3F. valm+ 1(a, v , F) . -, F(v) . Using (1) and 3.1, one obtainsthat-'G(v) and hence, by (2),-, hetm+l(v),whichis absurd. -I THEOREM 3.8. (valm+ 2(a, v , G) . G(x) = x hetm+1(x)) :> hetn+l(v)i fm Proof. Let
(1) valm+ 2(a, v, G), and (2) G(x) =x hetm+1(x).
< n.
RUSSELLIAN TYPE THEORY
495
By 3.7 we have -.hetffi+1(v) andhence, by (2),
(3) -.G(v). From theconjunction of (1) and (3) and from.53 if follows thathetffi+2(v) , whence, by 3.6, hetn+1(v), for all n < m. -I Finally, as an instanceof Axiom 3.2, .we have COROLLARY 3.9. 3a3v3G . valffi+2(a, v, G) . G(x)
=x hetffi+I(x).
ffi+1(v) is n+1-hetero3.7,3.8, and 3.9 say, intuitively, thattheformula het logical iff n > m. We believethatall of this is perfectly correct.Whatwe would like to discuss isChurch'sclaim thatsuch resultsc onstitute thesolution,by means of ramification,of Grelling'santinomyof simple type theory . Let's quote Church's(1976, atthe end of §2)argument: If we reduceto simple typetheoryby droppingall level indicators,the infinitelymany constantsvaln+1 coalesceintoa singleconstant,val, whosetype(in thesense ofsimple typetheory)is (i, i, (i)). Theinformal explanation of themeaningof vain+!thenbecomes an explanation of themeaningof val,a nd thepostulates (1) , (3) [our 3.1, 3.2] stillseem to be evidentfrom this intendedmeaning; moreover,(2) [3.4] is still a consequenceof (1) [3.1], and (4) [3.3] becomes tautologous . The proofs oftheorems(5)-(8) [3.7-3.9], stillhold,afterdroppingthelevel indicators,andthelastthreetheoremsthenconstitute a contradiction . This is Grelling'sa ntinomy , as it arises insimple typetheory .
The resolution of theantinomyby ramifiedtype theoryconsistsnot , afterrestoration of thelevelindicators,theomerelyin thefactthat rems (6)-(8) [3.7-3.9] are no longer ac ontradiction , butalso inthat thequestion"Is thepropositionfrom hetm+l(x)autological or heterological?"can now be answered : namelyit is (by. (6)) [3.8] autological atall levels S m+ 1, and it is (by (7)) [3.71 heterological atall levels
> m+1.
4. CRITICISM OF CHURCH 'S ARGUMENT
ThoughChurchpresentsGrelling's antinomyas anintensional paradox,none of the axioms he uses is of an intensional c haracter . In fact , the alleged evidence for these axioms does not seem toundermined, be even if propositionalfunctionsareinterpreted as sets andpropositionsas truth values. This of a makes theargumentin questionsomewhatsuspicious. The resolution paradoxby means oframificationsuggeststhatthe source of thecontradiction consists in a confusion of levels . So, if theparadoxarises also within an extensional context,such aresolution could beadvancedas anargument
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ENRICO MARTINO
e set-theoretical noagainsttheadequacyof simpletypetheoryalso tohandl tions. Thereis certainly a well-known notionof class which could support such a claim ofinadequacy : it is the derivative (sometimes calledlogical) notionof class,accordingto which classes are conceived of asensionsof ext propositional functions,so thata classcannotexist independently of some generating propositional function . This conceptionfinds classes to follow the t heeffectthat,if simple typethehierarchy of propositional functions, with ory is inadequatefor thelatter, so it is inadequatefortheformer. This is not so, however,for the usual and more successful primitive notionof set, accordingto which a set is built up only bymembers, its no otherentitiesbeing involved in its existence . In particular, accordingto thisconception,a set exists quiteindependently of anypropositional functionwe may use to describe it. Simpletypetheoryis certainly a dequateforthisnotion,since the simple thatthe members of a set are hierarchyof sets is justdevised toguarantee pre-existingto the set itself(Zermelo-Fraenkel set theorymay be regardedas a simple type theoryextendedup to transfinite levels) . Thoughthederivative notionis certainly the appropriateone for Russell 's logisticattemptto reduceclasses topropositionalfunctions,nevertheless the primitive notion seems to bequitecompatiblewithRussell'sramifiedtheoryof propositional functions.Moreoverit provides a good ontological tool for satisfactory a jus. Russell observes that,if theexistence tification of theaxiom ofreducibility immediately of classes isassumed from theoutset,t heaxiom ofreducibility function , a coextensivepredicative follows : in fact, ifF is any propositional G is given bythepropertyof belonging to theextensionof F. And, although thataxiom to be anassumptionweakerthanthe Russell seems to believe existenceof classes, he does not offer any otherintuitivesupportfor his axiom buttheintuitionof sets as primitiveentities.?For these reasons, even from thepointof view of ramifiedhierarchyforpropositional functions, the possibilityof arguingagainstthe simplehierarchy ofsetsseems to be highly implausible.Thus,since Church'sderivation ofGrelling 's paradoxis logically unexceptionable, it must rest on some erroneousaxiom, whose fallacy should that be recognizable evenwithintheperspectiveof simple type theory. All Churchsays abouttheplausibility of hisassumptionsis thatthe axioms on constantsvaln + 1 , when passing from ramified to simple theory, seem to be stilljustifiedby theintendedmeaningof the single constant val. On the contrary,we willcontendthat,withintheperspectiveof simple types, schema 3.2, unless it issuitablyrestricted, loses anyplausibility . Whatmakes val awell-determined relation is thepresuppositionthatevery propositional form P(x) [i.e., formula withthesingle freevariablex) representsa well-determined propositional functionF. Besides theintended meaningof logical c onstants and theintendeddomains of quantification ,F 2 An interesting modelfor ramifiedtype theory,wherepredicative functi ons are explicitly groundedon set-theor eticalmembe rship, has been ercentlyconstruct ed by C. A. Anderson (1989 , §V).
RUSSELLIAN TYPE THEORY
497
is determinedby themeaningof non-logical constants(if any) occurringin P(x) . It followst hatvalmust be relative to some understoodassignmentof values (of a ppropriatetypes) to all non-logical constants of the given language E. When introducing a specificconstant"val.c" for thevaluation function relative to E, we get anextendedlanguage£,' = E U {valc}. Grelling'sa ntimony exploitsessentially the factt hatAxiom 3.2, once therestriction on the orderof P is dropped, allowsvalto occurin P. This amountsto assuming val.c= val.c', butno reason or i ntuition seems tosupportsuch identification. Observethat,a fterextendingE to £,' by means of a 3-placer elational symbol R, it would bepatently circularany attemptto assignstipulatively functionrelative to c, the latter being determinedonly to R thevaluation R. Thus, withinsimple type theory,Grelling 's afterhavingfixed a value for paradoxis not anauthentic antinomy . It is rathera proofby contradiction that , however you choose a value R for(so to determineval£:,), thatvalue cannotbe coextensivewith val.c,.3 Churchremarks,in the aboveq uotedpassage, thattheresolution of Grelthefact ling's paradoxby means oframificationdoes not simply consist in that,a fterrestoration of the levels, theparadoxis no longer derivable , but also inthepossibilityto answer precisely thequestionswhetherhetm+l(v)is or notn+1-heterological. But,if ouranalysisis correct,also in simpletype , the intended theory,once theincriminatingassumptionhas beenrejected meaningof valsuppliesa preciseanswerto thequestionwhetherhet(v) is or is not heterological. Sincetherelation valis relative to thegivenlanguage E (withoutc onstant "val"), thepropermeaningof val(a, v , F) is: a is an individualvariable,v is a variableis a, F is thepropositional function formulaof E whose unique free restrictaxiom schema 3.2 to formulasP expressed by v. We must therefore of L, So 3.9 (in its simpletype reformulation) is not aninstanceof 3.2, since het( v) is not inL, and theproofthathet(v) is heterological is blocked. val(a,v, F) , by its verymeaning, can besatisfiedby Furthermore, the formula L, so that3a3F.val(a, v, F) is false when no othervalue ofv buta formula of v is assigned"het(v)". Thus, by Definition3.5, het(v) is notheterological. This conclusioncannotbe formallyderivedfrom the axioms on valcorrespondingto 3.1-3.3. Afterdroppinglevels, .33 becomes superfluous and 3.1, restricted as explained)do not sayanythinga boutval(a,v, F), 3.2 (thelatter when v is not a formula L. of It is easy, however, to reinforce the axioms, so as to deriveresultsq uiteanalogousto those in ramified theory . To that 3These considera tionspresuppose that , in orde r for aanguag l e to be well-d et ermined , all it s non-logicalconstantsshouldbe given. This seems to be in contrast withChurch's ion thatRussell'sformallanguage would be an open language(i.e., (1976, note5) observat withoutany prefixedvocabulary)to which new constants couldbe added step by step. Observe, however ,that adding new constants affe cts the valuationfuncti on . We believe ther efor ethat such an opening (presentor not in Ru ssell)shouldbe banishedfrom the reconstruction in question, on pain of theindeterminacyof val.
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ENRICO MARTINO
end, restate(thesimple typeversions of) .31, 3.2 bothforval.candforval.c', where£' = £ U {vale},and replac e 3.3 by
val.c(a,v,F)
J val.c,(a , v, F) .
The analogueof 3.5, for val.c and vale respectively,supply definitionsof het.c(v),hete(v);and, justas in ramifiedtheory,one provesthathet.c(v)is not£- heterological butis c .heterological. This solutionis essentially thatof Tarskiand leads to hierarchy a of semanticallanguages . ChurchobservesthatalsoRussell'shierarchyof orders may be viewed as aparticular hierarchy of languages,so thatRussell's resoTarski'sbutrather lutionofsemanticalantinomiesis notreally different from a special case of it. Indeed , our restriction on Axiom 3.2, forbiddingoccurrences of val in P, is automatically assuredby theoriginalr estriction on the orderof P. Butthis isexactlythereasonwhy 3.2 is evidentwithinramified theory , while itsunrestricted simple typeversion is not, even withinthesimple type perspective.And whentryingto correctit, we arenaturally led to theTarskianhierarchy oflanguages . In this senseTarski'ssolutionturnsout to be theappropriateone, since itactsonly ontheunique wrongassump. On thecontrary , Russell's tion, leavingthegeneralframeworkunchanged solutionimposes a drasticrevision onthewhole framework, which seems to be unjustified,since thecredibilityof thatframework was not affected atall by theparadox. Since, however, quite independently of semanticalparadoxeslike Grelling's, thereare goodreasonsforrejectingsimple typelogic in favor of ramified logic , as a framework for Russellianpropositionalfunctions (seen ext , afterall, ramified logic provides , as a bysection),one mightthinkthat product,a good alternative to Tarski'sresolution of semanticalantinomies. . We shall show that,evenwithinramified theThis is notthecase, however ory, an assumptionquitesimilarto unrestricted 3.2, butin agreementwith theorderhierarchy, leads , with the help of the axiomreducibility, of to a reconstruction of theparadox. To thisend, define,withinramified theory, DEFINITION
4.1. Q =def 3F. vaI 3(a,v,F) ·G(x)
=x
F(x), withG predicative.
By thecomprehensionand reducibility axioms, thereis a relation R of type (i, i, 111)11 such thatR(a,v,G) =a ,v ,G Q. Introduceinto theobjectlanguage a newconstant val" oftype (i, i, 111)11 for R and define het"(v)=def 3a3G.val"(a,v,G) · -,G(v). Now, assume thatAxiom 3.2 allows val" occurringin P (keeping fixedthe restriction on theorderof P). We havethen,as an instanceof 3.2,
3a3v3F.vaI 3(a , v, F) . F(x)
=xhet*(x).
RUSSELLIAN TYPE THEORY
499
It turnsout het" (v) =11 het3 (v) and theparadoxfollowsfrom thequestionof whetherhet"(v) is or not3-heterological. Here theappropriatesolutionis still the same as in simple type theory : if £0 is the initially given language,valn+lis thevaluation functionrelative £n = £0 u {vall, .. . , vain}, to propositional forms oforder n of language so thatonly formulas P of £n are allowed in.2.3 Butval"is not in£n and theparadoxbecomes a proof bycontradiction thatno constant of £n oftype (i, i , 111)11 can be coextensive with val". In this version of paradox the , the crucialrestriction on 3.2, imposed by theintendedmeaning ofvaln+l, is not automatically respectedin virtueof theordertheory, which t urnsout to be unable to offer any alternative to theTarskiansolution . So thelatter is to be adoptedeven within ramified typetheory . Our considerations seem to be inagreement , to acertainextent,with Myhill's (1979)argumentin defense of the axiom of reducibility. The argument isaddressedto those who broughtagainstthe axiom ofreducibility the charge of allowingreconstruction, a withinthe ramified theory, Grelling's of paradox.Myhill's reply that is such alleged reconstruction of theparadoxis nothingbut a reductio ad absurdum of theunsupportedassumptionthata certainpropositional function (similar to our val") is expressible in the objectlanguage. We certainly agreewiththis conclusion . Nevertheless , Myhill too, like Church , regardsthe simple type version of Grelling's paradoxas an authentic antinomy, of which the ramified theorywould be free, thanksto thathis ownargumentfor theorderhierarchy . He seems not to be aware the axiom ofreducibility immediatelysolves theparadoxeven inthecase of simple types. Whatreally makes thataxiom harmless is not the fact thatit does not allow the reconstruction of theparadox:indeed, as we have seen , it does. Whatmakes it harmless is ratherthe factt hatthe paradox,once reproducedin theramifiedtheory , is quitesatisfactorily resolved in the same way as in the simple theory.
:s
5. RUSSELL'S ANTINOMY ABOUT PROPOSITIONS
Unlike Grelling's paradox, Russell 's antinomyaboutpropositionsis really an intensional antinomyof simpletypetheory : it shows theinadequacyof simple logic in dealing with the Russellian notionofpropositional function and finds its proper solution in ramified logic. From now on we shall, for brevity, refer to this antinomyas Russell's antinomy (not to be confused with the more popularset-theoretical one). The antinomyis presentedinformally by Russell in Appendix B of Principles (1903) and, as Church (1984, note 5) observes, is of historical importanceas Russell's first indicationof the need forsomethingbeyond simple typetheoryto resolve antinomies. Church's (1984) reconstruction of Russell 's antinomyis radically different from thatof GreHing's antinomy(Church 1976): insteadof thesemanticalc onstantval, a newpropositionalconnective';;;;;' expressingintensional identitybetween
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propositionsis used. For abettercomparisonofthetwoantinomies, we shall hereproposean alternative reconstruction of Russell'santinomy , by suitably generalizing themethodof semanticalconstants.C hurch'streatment refers mostlyto Russell'sversion oftheparadoxin §500. Buta simplerversionis s hortpassageat theend of §488: alreadypresentin thefollowing -iffact it be a fact Thatpropositions are a type results from the thatonly propositions can significantly be said to be true or .. . false . But if so, the number of propositions is as great that as of all ob jects "x is identical absolutely, since every object is identical with itself, and . . . . If it is possible, as it seems withx" has a one-one relationx to to be, to form ranges of propositions, there must be more such ranges than there are propositions, although such ranges are only some among objects. Really , thatthenumberof propositionsis as greatas thatof allobjects absolutely is not ameaningfulassertionof simple type theory,since there is no type of allobjects. In any case,the paradoxdoes notexploitthis assertionin its full generality, butonly intherestricted formthatthenumber of propositionsis as greatas thatof allrangesof propositions. This is all thatis needed tocontradict Cantor 's theorem(provablewithinsimple type theory).So theparadoxrestson Russell'sclaim that,by associatingwith everyrangex of propositions,t heproposition"x is identicalw ithx" , we get . Thus, a one-onerelationbetweenrangesof propositionsand propositions in orderto get a formal reconstruction of the paradox,we have toderive thatclaim from somegeneralprinciple,fittingwith Russell'sconceptionof propositional function . A suitableprinciple, which seems tounderlieseveral of Russell'sarguments,is statedby Church(1984, page 518) himself. Let's call itInt (for principle of intensionality): Int. If nandm are namesofdifferent things,b utare ofthesame type,andif
a sentenceis altered by replacingone occurrence of n by m , theoriginal sentenceand thealtered sentencerepresentdifferentpropositions. Churchdoes not formalize , however,thisprinciplein his system: he prefers to derivethe paradoxfrom otherintensional axioms involvingthe logical constants.Yet, his axiomatization, as observed by Anderson(1986), turns outto beincompatiblewith Int (see alsoour§5). Int by means of a natural generalization of the We want to formalize methodof semanticalc onstants. Church (1976, §3) introducessemanticalconstants , analogousto the valn + 1 , for formulasw ith any number of freevariablesof any -rtypes. In fact, for every r -type JL = (.81, .. . , .8m)!n, he introducesa constantvall' of type (i, .. . ,i,JL)11, with m+1 i's. Accordingto theintendedmeaning, vall'(a1,"" am, v, F) says thata1 , " " am are variablesof types .81, .. . ,.8m ,
RUSSELLIAN TYPE THEORY
501
v is a formulawhose variablesa reamong al,. . . , am and F is thepropositionalfunctionobtainedfrom v by abstraction w ithrespectto thevariables
aI, .. . ,am·
So Church's semanticalr elations areonlythoseassociatingwitha formula P thepropositionalfunctionobtainedby abstraction r elativeto asequence of variablesincludingall freevariablesof P. Starting from P, it is possible, however, toobtaina propositionalf unctionalso byabstraction r elativeto some (notnecessarilyalland possibly none) ofthefreevariablesof P and assigningto theothersarbitrary values (ofa ppropriatetypes). It is natural, therefore , to generalize C hurch'srelations as follows . Let L be thelanguageof ramified type theorywithout non-logical constants. For everysequenceof r-types7l"1,••• ,7l"k, It = «(31, .. . , (3m)ln, introducetheconstant val1rl,...,1rk,P oftype(i , . . . ,i,1Tl," " 1Tk ,It)11 (withm+1 i's). The formula val1rl,...,1rk,p (al,' " ,am, v, Ul, .. · ,Uk, W) says thatai, .. . , am aredistinctvariablesof types (31, ... , (3m, v is a formula of I:- whose freevariables,differentfrom al, "" am, are, in orderof first occurrence,bl, . . . , bk ; and W is the propositionalfunctionobtainedfrom v by abstraction w ith respectto al, " " am and assignmentof UI, •.. , Uk respectivelyto bi , . .. , bk. To beginwith,we shallassumefortheseconstants theanalogues ofAxioms 3.1-3.3: AXIOM 5.1. val1r l , . .. , 1r k,p(al , . . . ,arn ,v,uI"",Uk,W)::) . val1r, ,..., 1rk, P ' (al, .. . ,am, v, Ul,. . . ,Uk, w') ::)
W = W' (with It and It' comparable) .
AXIOM 5.2. 3al,
,3am,3vVul , ... ,VUk, 3F . val1rl,...,1rk,p(a l ,. .. ,am,v,uI , . .. , Uk , F)· F(xI, , x rn ) = Xl , . .. , X m P, where Xl, ,X rn is a sequence of variables of types (31, , (3m, ordP ::; orduand Ul, ,Uk are all free variables of P different from Xl,· . . , X rn • AXIOM 5.3. val1r l , • . . , 1r k,p(al , . . . , am ,V,Ul , val1rl,...,1rk,p, (a l ,
,Uk,W)::) , am, v, UI,· .• , Uk, w), if It' 2: u:
We can now formalizeInt by theaxiom AXIOM 5.4. val1r, ,...,1r k,p(v, UI , . • • ,Uk,P) . val1r! , . . . , 1r koJl (v, u~, .. . ,u~, p) ::) Ui = u~ , (i = 1, . . . , k), where It is a propositionalr-type.
If propositional functions,conceivedof as intensional entitiesin thesense of Int, aresupposed to satisfythesimple type hierarchy,t heanaloguesfor simple types of Axioms 5.1-5.4 (obtainedby droppingthelevelindicators) . seem to bequiteplausible . They lead,however, to Russell'santinomy Withinsimple type theory,t hefollowingintensional v ersionof Cantor's theoremis provable,whichassertsthatthenumberof propositional f unctions of anytypeis greatert hanthatof theirarguments:
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ENRICO MARTINO
THEOREM 5.5. For every type u, there is no relation R of type ((fL) ,fL) such that
(a) 'v'F3u. R(F, u) , (b) (R(F,u)' R(C,u))
-:JF ,G ,u
F
= C.
, R be a relation satisfying(a) and(b) . By applying Proof. Let, for areductio thecomprehensionprinciple2.2 to theformula P
=def
3F. R(F, u) . --, F(u),
we get a G oft ype (fL) such that
(1) C(u) ==u 3F .R(F,u) ,--'F(u). By (a) and (b), thereis a v suchthatG is theuniquepropositional function satisfyingtheconditionR(G,v). From (1), it follows thenG(v) == --'C(v), which isabsurd. .., By theway,thefollowing corollary is worthnoting : COROLLARY 5.6. Th ere is no formal language in which allpropositional functions of type (i) can be represented. Proof Suppose, for areductio,t hatthepropositional functionsin question are allrepresentable in one and the same formallanguage. The relation of type ((i),i), which withevery propositionalfunctionassociatesits own representing expression,satisfiesclauses (a)a nd(b) oftheprecedingtheorem, which isabsurd. ..,
toTheorem5.5 is derivable On theotherhand, Russell's counter-example from Axiom 5.4. By applying5.2 to theformulaP =d ef F = F, with F of type (0), we get COROLLARY 5.7. 3v'v'F3p.val(o),o(v ,F,p), and from the comprehensionaxiom, applied to formulaval(o),o(v ,F,p), it followst hat COROLLARY 5.8. 'v'v3R. R(F,p)
==F ,p
val(o),o(v, F,p), with R of type ((0),0).
Thusfrom 5.7 and5.4 we get arelation satisfyingclauses (a)a nd(b) ofTheoof Russell 's antinomyaboutpropositions. rem 5.5. This is ourreconstruction We wantto stressthemain differencebetweenGrelling'sa nd Russell's , thecrucialstep antinomiesemergingfrom ourtreatment.As we have seen in thedeductionofGrelling's paradoxis an application of Axiom 5.2, which is in conflict w iththeintendedmeaningof val: while val is to understood be as relative to theinitiallanguage E (withoutval), it is used as if it were relative
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RUSSELLIAN TYPE THEORY
to theextendedlanguage £.' = E U {val}. Onthecontrary, theuse of Axiom 5.2 in thedeductionof Russell'santinomyis in perfectagreementwiththe intendedmeaningof relations val's: theseare conceived of as relative to the language£. of simple type theorywithout non-logical constants and 5.2 is, accordingly,restricted to formulasP of E: Here, thecrucialsteptowardst he contradiction is theuse of 5.4 for acounter-example to Cantor'stheorem . And sincethelatter is certainly a ppropriateforsimple type theory , while 5.4 expressestheintensionality principleInt, theparadoxshows thatInt is incompatiblewithsimple type theory . In thissense we canconcludethat Russell'santinomy,unlikeGrelling's,is an authentic intensional paradox. Withintheramifiedtheory,we can still derive, means by of theaxiom of reducibility, a theoremanalogousto 5.5, namely THEOREM
5.9. For every r-type J.L, there is no relation R of r-type (J.L) 11, J.L) 11
(a) VF3u. R(F, u),
(b) (R(F,u) . R(G,u))
-:JF ,G,u
F = G.
The counter-example is blocked, however . In fact, ifF is ofr-type(011)11, theorderof theformulaF = F is, by definitionof identity , n + 3. So, by applying5.2, we get
3vVF3p. val(OlnJII,Oln+3(V, F,p) . By comprehension,we canobtaina relation R satisfyingclauses(a) and (b), butits typeturnsoutto be (0In)11,0In+3)ll,so thatit isnotof theform specified in5.9. Thus, Russell'santinomyis happilyresolved bymeans of ramification. 6. SOME FINAL REMARKS
We want to make here some comments on thenotionof intensionality expressed by Axiom 5.4. TakingP =def F(x), withF and x variablesof types (1I")ln and 11" respectively, we have, as an instanceof 5.2, COROLLARY
m .= ordP.
6.1. 3vVFVx3p . val('lI"Jln ,7T,Olm(v,F,x,p) . p ==
F(x) , where
This says, intuitively, thatp is thepropositionexpressedby F(x). As an instanceof 5.4, we have COROLLARY
6.2. val(v,F ,x,p), val(v,G,y,p) -:J F = G· x = y.
Therefollows aprincipleof strong extensionality forpropositional functions:
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ENRICO MARTINO
6.3. Two propositional junctions are identical, provided they have one value in common.
PRINCIPLE
IdentifyingF andG in 6.2, we have, as a corollary of strongextensionality , 6.4. Every propositional junction is injective:it maps different arguments into different values.
COROLLARY
The factthatpropositional functions,precisely invirtueof theirintensional character, do notsatisfya principleof extensionality , evenstronger t hanthe usual one for extensional s et-theoretical functions , might seem an oddity. This is not so, however . In fact,accordingto the notionof propositional functionunderlyingInt, propositionalfunctionsare constituents of (those propositionswhich ar e) theirown values, in saimilarsense as symbols for propositional functionsareconstituents ofthoseformulasin whichtheyoccur. So, for twopropositionalfunctions,to sharea value means to be thesame constituent (in thesame role) of ac ertainproposition.This is whatjustifies strongextensionality . Anderson(1986) has shownthat 6.4 cannotbe consistently a dded to Church's(1984) system. In fact, 6.4 is incompatiblewithChurch'saxiom AXIOM
6.5. 3F. F(XI , ... ,x m )
" ' X l , .••
,X~ P
withXl, . .. , Xm all idstinct,o rdF ;::: ordP, F not fr ee in P, and where' ...' is Church's operatorof intensional identit y between propositions. If Xl, • .. , X m do notoccurfree inP, theF sat isfying 6.5si a constantfunction , so thatit fails to beinjective. The principle6.5, however, does not fit with eth notionof propositional functionjustsketchedabove. In fact , the natural c andidatefor F in 6.5 shouldbe thepropositional function o btainedfromtheproposition(expressed by) P by abstraction on itsarguments.Butsuch anF, because of its ver y genesis, cannotbe aconstituent of P, while,accordingtothatconception,it is a constituent of thepropositionF(XI, . . . , x m ) , so thatthe twopropositions fail to beidentical.Accordingly, inour Axiom 5.2, F(XI, . . " x m ) and P are assertedonly to eb logically equivalent and theyare notexpectedto be strictly identical. The adoptionof 6.5 entailstherejectionof thefeatureof propositional functionsbeing constituents of theirvalues. This is a possible approach and, in fact, it ist heone chosen by Anderson(1989). He bringsoutsom e argumentsin favor of 6.5 a ndproposesa revision ofC hurch'stheory, in which thatprincipleis essentially saved, whileextensionality andinjectivityin their generalform arebanished. Indeed, we believethatmore thanone conce ption of proposition and propositionalfunctionmay be suggestedby Russell'swritings . The one workedoutby Andersonis certainlypraiseworthy in variousrespects. We
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think,however,t hatalso thealternative choice of espousing 5.4 and rejecting 6.5 is coherentand worthinvestigating . ACKNOWLEDGEMENTS
I am grateful to Anthony Anderson and Nino Cocchiarella fortheiruseful commentsto the first draftof thispaper and to Neil Nelson for numberof a stylisticimprovements. REFERENCES
Anderson, C. A. Nous, vol. 20, 1986 Some difficultiesconcerningRusseIIian intensional logic, pp .35-43. 1989 RusseIIian intensionallogic, Themes from Kaplan, Oxford University Press, Oxford. Church,A. 1984 Russell'stheoryof identityof propositions,Philosophia Naturalis, vol. 21, pp . 513-522. lantinomieswiththat 1976 Comparisonof Russell 's resolution of thesemantica of Tarski, The Journal of Symbolic Logic, vol. 41, pp . 747-760; reprinted in Recent essays on truth and the Liar Paradox, (R. L. Martin,e ditor), ClarendonPress, Oxford,pp . 289-306. Hazen, A. 1983 Predicativelogics,Handbook of philosophical logic 1 (D. Gabbayand F . Guenthner , editors),Reidel, Dordrecht,ch. 1.5, pp . 331-347. Myhill,J. 1979 A refutation of an unjustifiedattackon theaxiom ofreducibility , Bertrand Russell memorial volume (G. W . Roberts,editor),Allen and Un- 90. win, Londonand New York, pp. 81 Russell, B. 1903 The principles of mathematics, CambridgeUniversityPress, Cambridge, England .
TERENCE PARSONS
THE LOGIC OF SENSE AND DENOTATION: EXTENSIONS AND APPLICATIONS
Abstract.The purpose of this paper is to exte nd Al on zo Church's (1951 ) .investigat ion of thelogic of sense and den otation in orde r ot address some issues thathave arisen since his clas sic paper on thistopic. Aftersome backgroundmater ial ni secti ons1 and 2, a formulation of the syste m is given in section 3 thattreats em pty terms and function al sen ses, follo wingout some ideas developed by C hurc h(1973, 1974). De re contextsare discussed in section4, where I sugges tthatthese are esse ntially a lreadypresentin the syste m. In section5 the issu e of languag e learnability is addressedin the contextof Frege's infamouslinguistic hierar chy of senses. In section6, I suggestthattheproblematic linguist ic hier ar chyis supe rfluous,b ut thatthis leaves untou ch edthe ontologicahier l ar chy of senses tha tis the subjec t er of Church'swork. m att
1. INTRODUCTION
Accordingto Frege (189 2 andelsewhere) , words havebothsense and denotation.The word 'Socrates ' denotestheman so named, and it expresses a es denotetru t h values , and sense, themeaningof thatword. 1 Wholesentenc express propositions;thesentence'Socrateswas a philo s opher' denotesthe truthvalueTrueandexpresses thepropositionthatSocrateswas a philosopher. In each casethe sense of a worddetermines its denotation.T his is becauseevery sense "presents, " or "is a conceptof' som ething. The sense expressed by'Socrate s' is a conceptof theman, and thesense expressedby 'Socratesis a philosopher ' is a conceptof theTrue.? Being a conceptof is a contingent , non-linguistic relation.i'W hata word orphrasedenotesis whatever t hesense itexpressesis a conceptof (seeFigure1). Fregethought thatcertaincontextscause a shift in these s emanticrelations.In indirect quotation, such as'Agathasaid thatSocratesis a philosopher',t hephrase 'Socratesis a philosopher ' comes to denotethepropositionthatit normally I I use 'de note 'a nd 'refer to' interchangeablyfor Frege 's term ' bedeuten' . 2Here I follow Church's usag e of 'is a conceptof'. Frege , uses 'conce pt' differently,for the denota tionof a predi cate, which is a functionfrom thing s to truth -val ues. 3If the worldhad b ee n differ en t ,the sens e that'Socr ates' ac tually h as might have es been a concept ofsomeone else, or of nob od y atall, and the prop osition that Socrat is a philosopher might have been a conce pt of th e value Fal se. If the lan gu ag ehad been different, the words' Socra tesis a philosopher' mighthave expressed a differentp rop osit ion, butthe proposi ti onthatwe now express by 'Socrates is a philosopher' wouldnot for that reason changewhich truth value it is a conce ptof. 507
C. Anthony Anderson and M. Zeleny (eds.), Logic, Meaning and Computation , 507-543. © 200 1 Kluwer Academic Publishers. Printed in the Netherlands .
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Sense of word
l
is a concept of
Denotation of word Figure1. expresses. Since the phrase has acquireda new denotation , and since its denotation is alwayswhatev er its sense is aconceptof, it must also have acquireda new sense, onethatis a conceptof thepropositionthatSocrates is a philosopher . Thus thesemanticsrequiresconceptsof propositionsin additionto the propositionsthemselves. Frege (1902) also assumed that when such aphrase is reemb edded, as in 'John thinksthatAgathasaid thatSocratesis a philosopher ', thereis anothershift of ednotation , which requir es another shiftin whatsense is expressed. Thuswe also haveconcepts of conceptsof propositions (which areconceptsof truth value s) . Any phrase languag e is thusautomatically associat ed withan infinitenumber of natural of distinctsenses. Frege assumed compositionalityat bothlevels:The sense of a complex phraseis a functionofthesenses of it s parts, and the denotation ofa complex phraseis a functionof thedenotations of itsparts.Thus the semanticshifts ces. As a consequenceof mentionedabove al so apply to the partsof senten thisshifting,every meaningful word orphraseof thelanguage has associated withit an infinite n umberof sensesand denotation s. Frege thoughtof this infinitesemanticalshiftingas a defect ofn atural language,a nd he urged the use ofnotationin which words never shift either sense or d enotation." Instead,when we wish to say ' Agathasaid thatSocratesis a philosopher' i nstead of thesame we would use new words withfixed senseanddenotation wordswithshiftedsense anddenotation; for theordinaryuse of 'Socratesis a philosopher'we wouldwrite: Socrates-is a philosopher., andfor the longer sentencewe would write: Agatha.,said., thatSocrates,is a philosopher, er denoting where'Socrateso'and'Socrates}' aredistinctwords, withthelatt 4 F'rege 1902: "To avoid ambig uity, w e ought eall r y tohave special sig ns in indi rec t i ththe corre sponding signs in direc t sp eech sho u ld eb speech, thoughtheir connec t ions w easy to recognize."
THE LOGIC OF SENSE AND DENOTATION
509
thecustomarysense oftheformer andhavinga new sense of its own. The similarityin wordrootwould let us keep trackof thesimilarsources ofthese language. words innatural Frege nevercarriedout thetaskof designingsuch alanguage , butAlonzo Church(1951) tookup thetaskin his classicpaper "A Formulation of the Logic of SenseandDenotation."Churchdid not develop semantictheory; a the terms 'denote'a nd 'express' do notoccurin thesymbolism." Instead, his taskwas toconstruct and axiomatizethescientificnotationitself.This notation supplies us withordinarylogicalnotationfor the simpletheoryof types together w ithample vocabulary to denotecustomarysenses, concepts of customarysenses, and so on, and it also containsterminology forthenonlinguistic,non-semanticnotionbeing a concept oj. It is a useful scientific notationof thesortFrege envisaged, and it simultaneously is an axiomatic appealedto by any descriptionof theontologyof senses and denotations language(includingan imperfectnatural language).As such, it has wide interest . The goal of thisp aper is to exploreand extendthe usefulness of Church'ssystemconstruedas a generalt heoryof sensesthatis applicableto thesemanticsof natural language,and as a scientific notation for thestudy of Frege'sontology . One ofChurch'sgoals was to say more thanFregeabouttheidentityconditionsfor senses. He explored threealternatives . Alternative 0 requires thephrasesthem(roughly)t hattwophraseshave different senses whenever selves differ(exceptfor the choice of bound variables) , alternative 1 is an intermediate system (whichpermitslambda-conversion withouta lteration of 2 identifies the senses phraseswhenever of they ear sense), and alternative necessarily equivalent . Difficulties have been found with all ofoptions. these thephrasesthatexpress Alternatives 0 and 1 identifysenses so closely with of ofthesemanticparadox es, and themthattheytendto run afoul analogues complicationsneed to beintroducedto guardagainstthis. Alternative 2 is inconsistent w itha principleaboutfunctional senses (discussed below) . My concernsin thispaperare notwitha choiceamongthesealternatives, but ratherw itha numberof issues thatbearon thegeneralusefulness of any of theseo ptionsin doingsemantics. Afterdescribingthe ontology and symbolism (section2) my firstmajorgoal(section3) is to provide acriterion for whenfunctionsof theappropriatetypes can themselvesbe senses. This requiresa discussionof empty concepts. FollowingthatI show (section4) how toextendthesystem to includede re contexts.Next (section 5) I show to avoidcriticismsbased on thepurported how toextendthesystem further unlearnability of thevocabulary.Finally(section6) I discuss reasons for to each word and phraseofnatural language rejecting the viewthata ttributes an infinitehierarchyof senses. I arguefor asimpler view, which is one endorsedby Church. My overall goal is to show thatChurch'sapproachcan 5Church took uptheseissues elsewhere.
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be lessdauntingt hanit sometimes appears,and thatit hasapplicationst hat give it a much wider a ppealthanis oftenassumed. 2. THE ONTOLOGY: TYPES AND FUNCTIONS
Church'sontologyis meantto beFrege's, withtheone exceptionthatthe types of sentence-related entities(truthvalues, propositions,conceptsof entipropositions,. ..) aredistinguishedfrom thetypesof individual-related ties (individuals,conceptsof individuals,. .. ).6 I use mostlythesame types as Church,expressedin a lesscompact buteasier to read n otation . First, thereare non-functional types: Sentence-related types : the type oft ruthvalues is0 ("omicron"), the type of conceptsof truthvalues(thetype of propositions)is 01, the type of conceptsof propositionsis 02, and so on. The basic type 0 is alsowrittenas 00. Name-related types: thetype ofindividualsis t. (alsowritten£0), the type of conceptsof individualsis £1, thetype of conceptsof concepts of individualsis £2 , and so on.
Second,therearefunctional types, thetypesof(total)functionsfromentities of onetypeto entitiesof another t ype: The typeoftotal one-placefunctionsfrom entitiesof typeQ toentities of type /3 is (Q -+ /3). The typesof two-placefunctionsare (Q , /3 -+ 1'), and so on for functionsadditional of places." To illustrate theuse of these types : The name 'Socrates'denotesan individual(of type £) and expresses a sense oftype £1 . (The sense expressedis thusthe righttype to be a conceptof theindividualdenoted .) The predicate'is wise' denotesa functionfrom individualsto truth of values, which is anentityoftype£ -+ 0, and it expresses a sense type £1 -+ 01, thetype of functions from conceptsof individualsto propositions. (Again, the function expressedby thepredicateis a conceptof thefunctiondenotedby it.) 6 Fregeexplicitly i ncludedtruth valuestogether withindividualsin thesame type, which em entcontainsno empty place, and ther efor ewe must regard he calledobjects: "A stat whatit standsforas an object." (Frege 1981 ; translated in Frege 1960.) This is unnatural and inessential to his/ourpurpose. 7This departsslightlyfrom Church, who disallowedmultiplace functions. Instead of a two-placefunct ionfrom pairs of individualsto truthvalues, he uses a one-place functionfrom individuals to one-plac e functions from individualsto truthvalues. Nothingof importancehangson thischoice.
THE LOGIC OF SENSE AND DENOTATION
511
The sentence'Socrates is wise'denotesa truthvalue (oft ype 0) and expresses aproposition(of type od. (The propositionexpressedis a valuedenoted .) conceptof thetruth The truthvalue of'Socratesis wise' is doubly - determined . It is theunique truthvaluethatthe propositionexpressedby thesentenceis a conceptof. This propositionin turnis thevalue ofthefunctionexpressedby 'is wise' fortheargumentthatis theindividualconceptexpressedby 'Socrates'.T he former functionis a conceptof thefunctiondenotedby 'is wise', and the latter conceptis a conceptof theobjectdenotedby 'Socrates'.Finally, the truth value oft hesentenceis determinedatthe level of denotation by being thevalue oft hefunctiondenotedby 'is wise' fortheargumentthatis the individualdenotedby 'Socrates' . These multiplerelations are partof what gives thetheoryits logical complexityandinterest . It is sometimes convenient tospeak of atypethatis a certainnumbern of types this levels"above"another t ypein a hierarchy.For thenon-functional is achieved byraisingthesubscripton thetypesymbol by n . If 0 is thetype symbol f3n, Churchuses Om to abbreviatef3n+m. He introduces this notation forfunctional types too: if 0 is thetype f33 -+ "'(1 then0 2 abbreviatesthe type f3s -+ "'(3 . This is a mere convention,and it is importantto keepthat in mind. Otherwiseone mightassume thatif'o is a functional type, then01 is a typeof entityallmembers of which are senses . In section3 we will see thattheentitiesof functional type 01 alwaysoutrunthe senses oft hattype. We can use theset ypes in discussingordinarylanguage . There,when a sentenceis embedded in an indirectcontext(such as 'Mary believesthat __') thereis a semanticshift, in which the words come to refer to the senses theycustomarilyexpress, andtheyalso acquirenew senses. Frege thoughtt hatin such acontextthename 'Socrates'refers to its customary (of L2) ' Likewise, 'is wise' sense and acquiresa new higher-level sense type acquiresa new sense oft ype (Ll -+ 01}t, thatis, of type L2 -+ Oz. Since the complementsentencedenotesthepropositionthatit customarily expresses s is wise, which is oftype 01) it acquiresa (the propositionthatSocrate new sense oftype 02. As previouslynoted,Frege sawthisshift ofsemantic relations in indirectcontextsas a defect of language,and heproposed that in a scientificnotationwords would never change theirsemantics(noteven but systematically) . Instead,weshoulduse different words thesecontexts, in the different words in a scientific notationthatrepresentdifferent uses of thesame ordinaryword should have some identifiable linkwithone another. Church'snotationmay be used toaccomplishthisneatly, since each of his propersymbols has twoparts; a rootthatresemblestheword oftheprescientificnotation, and atypesymbolthatindicatest hetypeofentitydenoted by thatsymbol in acertaincontext.Thus, if Frege isrightaboutnatural language,8the pre-scientific'Socrates'can be representedby the chainof 8Churchdid not intendhis scientifi c notation to be used for th is purpose. See discussion
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scientificsymbols 'Socrat es.; ",'Socrates,1 " 'Socrates. , " . . . , wherethefirst scientificsymbol mimics thesemanticbehaviorof 'Socrat es' when it is not embedded in anindirectcontext,thesecondmimics thesemanticbehaviorof 'Socrates'when it ise mbedded in a single,indirectcontext,thethirdmimics thesemanticbehaviorof'Socrates' when itembeddedin is anindirectcontext which isitselfin anindirectcontext(as in'AgathabelievesthatHarry thinks thatSocratesis wise') , andso on. I use thesymbol ' ~' for'is a conceptof".? So the following are true: Socrates ,1
~
Socrates. ,
" Socrates.,~ Socrates (Strictly,'~' representsan infinitenumberof two-placepredicatesymbols, each witha type of its own . The firstoccurr ence of '~' above shouldbe '~" " O ~O ' and thesecond shouldbe ' ~'2 ' '' ~O ' ' I omit thetype symbols on '~' becausein such acontexttheycan be uniquelyrecover ed from thoseof its argumentswhen it si used to make asentence(of type 0).) I also follow C hurch in assuming thateveryprimitivesymbol of thelanguageoccurswith aninfinitenumberof differentt ype subscripts ; its basic subscript0, and also 01, 02 , and so on, with each successive occurrence identified by theaxioms as aconceptof thepreviousone. Thus, if we have a primitivenegationsign --, of type 00 -+ 00, we also havet hefollowing signs andaxioms:
ee sets: Church'saxioms come in thr I. There are axioms forthe basic logic ofthe functional calculusw ith identity,includingunrestricted A-abstraction a nd -concretion. I will takeforgrantedthatthereis some adequateset of suchaxioms in the background .
II. Therearethreealternative setsofaxioms forthethreemajoralternative assumptions(Alternativ es 0, 1, and 2)abouttheidentityconditionsfor senses. Most of mydiscussionis intendedto beneutral among these. in section 6. 9! use this notation becau se itis easy to read; I X ~ Y ' is read "X is a conce pt of Y ' . C hurch's nota t ion forthis is t,.Y X ' , whic h ca n be read (in orde r) as' be ing a conce pt of Y holds of X '. I
THE LOGIC OF SENSE AND DENOTATION
513
III. Finally,t hereare some sets of axioms governingthepredicate"~' for being a concept of Theseare myprimaryfocusesthroughout thispaper. One is axiom set 14, which sayst hatany givenconceptis a conceptof at most onething; I takethis forgrantedthroughout my discussion. The controversial ones areaxiom sets 15 and 16, which areexplained in thenextsection. 3. WHICH FUNCTIONS ARE SENSES ?
Churchassumes thatsenses offunctional signs arethemselvesfunctions; e.g., thatthesense ofthesign "F' is a functionthatmaps thesense of's' to the sense of thesentence'F (s )'. Here is areasonwhy. Suppose thatsenses of functional signs arenotfunctions . We still want thesense of acomplexexpressionto bedeterminedby thesenses ofthecomponentexpressionsandthe way in whichtheyareputtogether.Thus therewillbe a functionthatmaps thesense of'F' and thatof 's' to thesense of'F(s)', and this will be systematicfor all sentencesof theform ' F (x )'. So in termsof thiswe can define a functionthatmaps any individualconceptto thepropositionexpressedby 'F(a) ' when 'a' expressesthatconcept.This definedfunctionalreadyexists we arepresuming,andso theontologyis simplerif we idenin theontology tifythesense of'F' withthatfunctionthanif wepresume a completelynew expressions.In addition,we kind ofentityto serve ast hesenses offunctional can exploitthe logical function-argument relation in explaininghow senses combine with oneanother,and thissimplifies thelogic. Thereare several objectionsto identifyingfunctional senses withfunctions of senses(thoughI thinkthatall oft hemcan be overcome) ; in thefollowing fewsectionsI focus ontheone thathistorically led toproblemswithChurch's system. 3.1. Characterization Not all functions from senses to senses can be senses. Suppose thatE and M are twoindividualconceptsthatare conceptsof thesame individuali. For example,E might be thesense of'theeveningstar',M thesense of'the morningstar',a nd i theplanetVenus. Let F be some functionthatmaps E to atruepropositionand M to a falseproposition. Then F cannotbe the sense of anypredicate.T his is becauseF is not "predicational" ; it cannotbe theconceptof apredicatereferencef, which is afunctionfrom individualsto truthvalues. Any such p redicatereference mapst heindividuali to atruth value, andthattruthvalue isthereference of any sentencewhose subject refers toi and whosepredicaterefers tof . If F werethesense of apredicate P, thentherecould be two sentencescontainingt hepredicateP, one whose subjectexpresses E and one whosesubjectexpresses M . These sentences wouldthenhave the same truthvalue while expressingpropositionswith differentt ruth values,t huscontravening theprinciplethatthetruth value of
514
TERENCE PARSONS
a sentenceis thetruth value which t hepropositionit expresses is aconcept of.
Church'ssolutionis to requiret hatsenses ofpredicatesbe predicational, andto requiret hatsenses ofothertypesof functional expressionshavethe analogous property ; he introduces theterm'characterization ' forthegeneral case. A functional sense F characterizes a functionj iff wheneverF's argument A is a conceptof some thinga, F maps A to aconceptofwhatj maps a to: Suppose thatthetypeof F is 01 --+ /31 and thetypeof j is 0 --+ /3. Let A range overentitiesof type 01 and a overentitiesof typeo. Then:
F characterizes j
=df
'v'A'v'a[A
~
a --+ F(A) ~ j(a)].
This is a necessaryconditionfor a function to beconceptof a a function . Whatmore is needed?Church(1951, page 16) supposes thatnothingmore is needed. This assumptionleads to the adoptionof his axiom-sets15 and t heassignmentof any types0 and /3 as describedabove: 16, which hold on 15: If F
~
j, thenF characterizes f.
16: If F characterizes t, thenF
~
f.
Withthe definition eliminated , these say : 15: If F
~
I. then'v'A'v'a[A ~ a --+ F(A)
16: If 'v'A'v'a[A
~
a --+ F(A)
~
~
j(a)] .
j(a)], thenF
~
f.
3.2. The Reduction to Extensionality Thereis a problemwith axiom set 16 . For some types 01 --+ /31 of functions thereare nofunctionsofthattypethatare definedonly for senses. (At least, this is truein theintendedapplications.)Some functionshave adomain 01 thatcontainssome thingsthatare, and somethatarenot,senses. Since all functions are totalon theirdomains, these functions cannotbe defined only for senses. As aresult,t herewill often be several different functions that are equally good atcharacterizing a given function j , different because they differ (only) in how theytreat a rgumentst hatare not senses.This seemingly irrelevant behaviorturnsoutto conflict with axiom set 16thecontext in of alternative 2. In the remainderof this section I show how this happens, andin the next sectionI proposean improvement, similarto thatof Church (1973, 1974). The difficulty takes this form. Church'saxioms forAlternative 2 in his originalpaper permit the derivationof theunpalatable resultthatthereis exactly onetrueproposition.This result is due to theinteraction of axiom set
THE LOGIC OF SENSE AND DENOTATION
515
16 forfunctional senses withtheaxioms thatidentifynecessarilyequivalent propositions.Here is howthathappens.10 In thefollowing, I use theterm'concept ' inconnection witha functionto mean 'concept ofs omething'. The proofexploitsthefactthatin alternative 2 necessarily equivalent conceptsareidentical.Thatis, twoconceptsthatare necessarilyconceptsof thesame thingare identical.T he proofproceeds by defining two c onceptsthatcanbe proved to be con ceptsof thesame function , becausetheycan be provedto characterize thesame functionf j axiom set 16 thusmakes them (provably)conceptsof f. Thus, in alternative 2 theyare identical.However,theyare defined in such a way thattheyare not identical, becausetheydiffer forc ertainargumentst hatare not conceptsof anything (recallthataxiom set 16 ignoreswhatfunctionsdo to non-concepts) . At leasttheydiffer if suchargumentsexist. And suchargumentscan be shown to existwheneverthereis more thanone trueproposition. The onlyescape t hatthereis not morethanone from theinconsistencythenis to conclude trueproposition . of , I use 'T' to name thetruth-value In thefollowing version theproof true , ' N' to name some necessaryproposition,a nd'!'to name an impossible proposition.U We begin by defining afunctionf and thetwo functionsV and W thatwillturnoutto be conceptsof f. For conveniencehere I call thingsoftype -+ unary truth-functions, andthingsof type 01 -+ 01 unary propositional functions.
° °
f
=df the functionof type (00 truthfunctionto T.
-+
00
thatmaps every unary
V =df thefunctionof type (01 propositional functionto N.
-+ 01) -+ 01
thatmaps every unary
W
=df
thefunctionof type (01
00) -+
-+ 01) -+ 0 1
thatbehavesas follows :
If thereis only onetrueproposition,thenW maps every unary propositional functionto N.
otherunarypropositional Otherwise , W maps M to I and maps all functionsto N,whereM is theunarypropositional functionthatmaps N to itselfandallotherpropositionsto 1. thatbothV and W characterize f. Recallthatthis We want to prove first means: F characterizes f == 'v'A'v'a[A • a -+ F(A) • f(a)] . IOThis proof was appare ntl y dis cover ed independentlyby A. F . Bausch (reportedin Church 1973) and by David Kaplan(personal co m munica t ion,) and perhaps by others . Apparentl y, the proofhas never been published. The presentversionis based on a versi on by AlonzoChurch(p ersonalcomm unication). IJ In alternativ e (2) thereis exactl yone necessaryproposition, andexac t lyoneimposs ible one, butthosefactsarenotrelevantto theproof.
516
TERENCE PARSONS
The proofis simple forV, since bothV and f areconstant functions , and the formerm aps everything in its range to N, which isconceptof a whatthe latter maps everything in itsrangeto (namely, T) . So V characterizes f. W alsocharacterizes f, becauseit eithermaps everythingto N, and so it is I) , or else it differs from V only for the identical with V (whichcharacterizes . argumentM in a situationin whichthereis more thanone trueproposition Butin thelatter situation,M is not a concept of anything , and so W still f. characterizes SincebothV and W characterize f , axiom 16 saysthatthey arebothconcepts of f. Now suppose thereare signs'p(o--+o)-+o' and 'Q(o--+o)-+o' whose "senses" are respectivelyV and W; thatis, 'P(01-+ 0,}-+ 01' names V and 'Q(01--+ 0,}-+ 01' names W, and 'p(o-+o)-+o ' and 'Q(o-+ o)-+o' name whatV and Ware conceptsof, namely,f. From the definitions above we have proved p(o--+o)-+o = Q(o-+o)--+o'
Since thissentenceis provable,its sense must be anecessaryproposition.A name of its sense is got by increasingthetypesof allsymbolsin the es ntenc e , thefollowing maythusbe by 1. Using subscriptsto indicatethis raising proved: Nec[Po1 =1
Q 01],
where 'Nec' standsfor 'isnecessary'(of type01 -+ 0). The distinctiveprinciples foralternative 2 (embodied in the seconddisplayedformula on page 21 thatthe termsflanking the sense-ofof Church 1951) thenallow us to say themselvesidentical : identitysign inthejust-givenformula are
Thatis, V sition.
= W. ButV = W
only ifthereis not morethanone truepropo-I
This is only asketchof a proof; inorderto turnit into aproofone would need toactually define'p(o-+ o)--+o' and ' Q (o-+o)--+o ' withinthe formal system, and I havenotbeen able to do this.Indeed, I am not certainthatit can be done. But I agree withChurch(1973) thatit would beundesirableto have senses in the t heorythatcannotbe assigned to any signs on pain of a collapse to extensionality . 3.3; The Problem is In1ependent of Alternative 2
Not everyfunctionoftheappropriatetype can be a sense; necessary a condi. tion isthatit must characterize some function.Buteventhatis not enough withregardto mapping In thelastsectionwe sawthathow functions behave non-conceptsto thingscan disqualifyt hem as senses, inthecontextof alternative 2. One could conclude from thisthatthefault lies with the modal
THE LOGIC OF SENSE AND DENOTATION
517
tive2, whichareindependen tly mplausible.But i identityconditionsofalterna I thinktheproblemis more general.In orderto get att heheartof theissue t in its full generality we need toconsidera relat ed phenomenon: how totrea empty termsand theirsenses. In a scientifi c notat ion we can avoid empty terms,buttheyoccurin nonscientific discourse, andso theirsenses must be in theontology . This was Church's intent(Church 1951, page 15, note16): u • •• tentatively , we shall allow thatin some language , . . , ther e may be names which have a sense b utno denotat ion. Hence we alsoadmit conceptsthatare not conceptsof anything..."
We need to becareful to haveouraxioms treatsuch conceptsappropriately . For it is easy to show, withoutany ofthespecial assumptionsaboutAlternatives0-2, thataxiom set 16 (which saysnothingaboutempty terms) is incompatiblewiththeexistenceof empty termsin any language,atleastif theirsenses areincludedin theontology .P' Here is anillustration . Let s be thesense expressed by 'Superman', and assume thats is not a conceptof anything. , andlet c bethefunctionthatmaps clever Let C be thesense of'is clever' thingsto truthandallotherobjectsto falsehood . Then C is a conceptof c, and C(s) is thepropositionthatSupermanis clever ,which is not aconcept of anything . Let F be thesame functionas C exceptthatF maps s to theproposition thatsnow is white . So F( s) is a concept of truth . Then, by axiom set 16, F is a conceptof c. This is because F coincides with C on allindividualsenses thatare actually concepts ofsomething.So F characterizes c: 'v'A'v'a [A ~ a -+ F(A ~ c(a)J. By axiom set 16this entailsthatF is a conceptof c. But this leads to absurdity if thereexistsany language w ithempty termsanda predicatethat expressesF. Assume thatF is expressed by thepredicate'is such-and-such'. Then 'Superman issuch and such' expresses F(s) , which isthe proposition thatsnow is white, and so the sentenceis true. But 'is such and such' expressesF, which is aconceptof c, sothereference of 'is suchandsuch'is c. But'is clever' also refers to c. So yousubstitute can 'is clever'for'is suchand such' withoutchangingthe reference of thewhole. This yields'Supermanis ' t havechanged, must be true. This clever', which, since thereferencecouldn contradicts thefactthat'Superman is clever ' expressesa propositionthatis not aconceptof anything. My diagnosisis thatthis problemarises notjustbecausesenses of predicatesare functions , but becauseaxiom set 16 makes functionsbe senses withoutlookingatwhattheydo toargumentst hatare not senses.The view 12Church pred ict s th is in Church 1951 , footnote 4, and com me nts on its im po rta nce in Church 1973, p age 25. I thin k I am follo wing out Church's line of thought her e.
518
TERENCE PARSONS
I wish todefendis thata functional sense shouldbe somethinglike apartial function,defined on sensesa ndnot defineda tall for non-senses. This idea is articulated in Church 1973, 1974, andis repeatedin Church 1993, in asom ewhatdifferentform. It is alsoadoptedin Anderson 1984. I willimplement it differently from theway Churchand Andersondo, withina slightmodification ofC hurch's originaltheory,a modificationthatpermits intuitively empty terms. This will alsopermit a simple reformulation of axiom sets 15 and 16. 3.4. Empty Terms and Empty Senses
Frege thought t hatin natural language some termslackreference , hutthat in a good scientific n otationthiswould noth appen. Instead,we wouldtake each termthatlacks reference, and give it one. He presentsthisas an improved way totreata singlenotionof reference,but it is betterto view it as a correctway tocomparetwo different referencenotions.The first is our ordinaryintuitivenotion.I"in thisordinarysense, some words and phrases ee. Instead,thethelack reference , and no theoristcan alterthis by decr oristcan define a econd s ,technicaln otionof referenc e for which reference failure doesn otoccur. The technicalrelationholdsbetweena termand a thingwhenevertheintuitiveone holds; in case theintuitiven otiondoes not notionrelatest heword to a special relatea word toanything , thetechnical "supplied" object.This is the idea I wish toexplore. The "supplied object" idea is similarto Frege's "chosenobject" option, except thatif intuitively e mpty terms are made to refer technically to a specially"chosen" ordinary object, the technicalnotion of referencedocs notencodethedifferenc e between(intuitive)referencefailureandsuccessful (intuitive)referencetothechosenobject. It isbettertoselect for thispurpose a "supplied object," thatis, some entitythatis not alreadyin thedomain of referencefor entitiesof the given type. Forexample, for reference to between terms and individuals , one can extendthis to atechnicalr elation thingsin an expandeddomain consistingof individualsplus somethingelse (e.g., plustheset of all old individuals , assumingthatit is notalready a mong theindividuals).So assume thateach domain of thingsof a giventype is expandedby theadditionof a new thing,called a"zip." We stipulate that technical-reference relates termsofthelanguage to zips whenthetermsreally refer tonothingatall intheintuitivesense of'refe r'. We use thetechnical extensionof 'refer' toemulatereal reference failuretheaccount: by w refers tox
=df
w referst echnicalto x & x is not a zip.
w fails to refer =d f w referst echnicalto a zip. 13! speak her e as ifour ordi na ryint uit ivenotionof den ot at ionncl i ude d th e assumption tha tsente nces d e note tru t hvalues . This is notwhat people nor ma llyassu me. The p oint can be made withoutthis assumpt ion; it is just a bit m orecumberso me.
THE LOGIC OF SENSE AND DENOTATION
519
This gives us all of the neatnessof no-failure-of-reference at thetechnical level, while allowing us totrueto be thefactthatwordssometimes fail to sense. It resolves Frege'stwoapproachesto reference by refer intheintuitive making themmutually complementary a spectsof ananalysisemploying two relations . In more detail : we add to each d omain of non-functional types a special object,a zip, symbolized0", for eachnon-functional type Q . We decreethat no zip is a conceptof anything . Nonfunctional senses of typeQ1 thatare not intendedto beconceptsof anything(in theintuitivesense of'concept of') aredecreedto beconceptsof 0", (in thetechnical sense of'conceptof' symbolizedby'.'). For example, in thetype of truthvaluesthereare now values,truth,falsity, and zip . Any proposition(of type 01) that threetruth in theintuitivesense lackst ruthvalue is now a conceptof 0 0 0 , In thefunctional typesthereare nowautomatically theappropriatefunct herole of zips . We say thatx is the zip of type Q -> (3 iffx tions to play is neatestif maps everyentityof type Q to thezip of type (3. The notation we usethesame symbolforthis,so thatthezip oftype Q -> (3 is denotedby '0"'_13'. (We do notactually need tointroduce newsymbols for the zips; we could use' LX ", (-oX", = X"')' forthezip oftypeQ.) 14 Lastly, we need to takecare in discussingempty concepts, for anentity of a conceptual t ype can beempty in two different senses . In one sense it is empty if it is aconceptof theappropriatezip of lower type. In another sense it isempty if it is not aconceptof anythingatall.The formersorts of entitiesarethethingsthataresuitedto be senses ofsingulart ermsthat, in theintuitivesense, lack reference. The latter are . . . what? In thenext t heaxioms. 15 sectionwe addressthisquestion,and givethefinal form to 3.5. Supercharacterization p ursuedis to gettheeffect of making EarlierI suggestedthatthe idea to be functional senses be partialfunctions defined only on senses. But first we the need to knowwhata sense is. In particular , we need to be able to tell difference between a sense thatis a conceptof nothing,a ndsomethingthat is a conceptof nothingbecause it isnota sense atall. This problemcan now be solved . In the revisedsystem we can define a sense as anythingthat 14This approachbearsstrongparallels to theconstructions in Church 1973, 1974. The principaldifferenceis thatChurchdoes notallowthezips to be members of thedomainsof thingsof a giventype. Thus he need notdistinguishtheintuitivefrom thetechnical s enses r to somethingoutsidethedomain of referring;when a term "refers"to a zip it is eferring of quantification , so we needn'tsay thatit refersto something(implicitly:to somethingin thedomain). The price he pays is a fairlycomplicatedset of replacementsfor axiom sets 15 and 16. 15Distinguishingthesetwo kinds of emptiness is partof theprocess thatKaplan(1973) called"garbagecontrol."I am indebtedto thistalkfor introducing me to thecom plex ity of theseissues.
520
TERENCE PARSONS
is a conceptof somethingin thetechnical sense of concept of Previously this was inappropriatebecausetherecould beempty senses. Butnow an intuitively e mpty sense is a conceptof theappropriatezip, and so we can construenonconcepts( = thingsthatare nottechnically conceptsofanything) as thingsthatarenot sensesatall. of functional senses by saying thatF is a We get the righttreatment functional sense iff itcharacterizes some functionand it maps every nonsense (everythingin thedomain of thefunctionthatis not aconceptof something)to thezip of appropriatetype. I callthis supercharacterization. This notionpermitsus toreplacebothaxiom sets 15 and 16 withwhatis in n otionof "conceptof': effect arecursivedefinitionof thetechnical SUPER-15-16: Suppose thatthetypeof F is a1 ....... /31 and thetypeof f is a ....... /3. Let A rangeoverentitiesof type a1 and a overentitiesof type a. Then:
F ~ f == 'v'A'v'a[A ~ a ....... F(A) ~ f(a)] & 'v'A[-' 3aA ~ a ....... F(A) = 0 i3J This proposalhas thevirtueofsimplicityandof beingbased on anintuitive conceptionof senses. Whetherit escapesthedifficulties of theearliera ccount is to be seen. Withtheintroduction of zips, we now have t ermsin ourscientificnotation that,'intuitively,lack denotation -becausethey (technically) denotezips. Withthisin mind, let usrehearsetheSupermanexampleof thelastsection: Let 8 be thesense expressedby 'Superman', and assume that8 is not intuitively a conceptof anything . So 8 is technically a conceptof 0, . Let C be thesense of 'isclever',and let c bet hefunctionthatmaps cleverthingsto truthand allotherobjectsto falsehood.Then C is a conceptof c, and C(8) is thepropositionthatSupermanis clever, which is notintuitively a conceptof anything , butis now technically a conceptof 0 0' Let F be thesame functionas C exceptthatF maps s totheproposition thatsnow is white. So F(s) is a conceptof truth . Previously,we could showthatF is a conceptof c. This is becauseF coincideswithC on allindividualsenses thatwere(formerly)concepts ofsomething(ignoring8). So previouslyF characterized c, andthenby axiom set16, F was aconceptof f. Butnow 8 is technically a concept c, of something, theindividualzip, 0 ,. In orderfor F to characterize F must map 8 to aconceptof whatc maps 0 , to, which is 00 ' But F maps 8 to a con c eptof truth.So F does notcharacterize c, and the proofbreaksdown.
THE LOGIC OF SENSE AND DENOTATION
521
therevised system escapes thereductionto We shouldalso review how thatplaguedtheoriginalsystem. Here is anattemptedreconextensionality struction of thatproofas givenearlier.We begin withthesame definitions as before:
f =df the functionof type (00 truth-function to T. V =df thefunctionof type (01 propositional functionto N.
--+ 00) --+ 00
thatmaps every unary
--+ 01) --+ 01
thatmaps every unary
We cansee rightawaythatV cannotbe shownto be aconceptof f, because V maps allnon-conceptsto N, not to a zip. And thereare nonconceptsin therangeof V if thereis more thanone trueproposition;any functionthat maps some truepropositionsto trueones and othersto false ones is not a conceptof anything . (We can'tassume atthispoint thatthereis only one trueproposition .) So withSUPER-15-16 in place ofaxiom set 16theproof thatV is a conceptof f breaksdown. Butwe canfix up thispartof the proof.Suppose we replacethedefinition of V withthis: thefunctionof type (01 --+ 01) --+ 01 thatmaps every unary propositionalfunctionF to N ifF is a conceptof something, and to 0 0 1 otherwise.
VO =df
ThenSUPER-15-16 lets us showthatV O is a conceptof f. And so theproof goes on as before . So we have toexaminetheotherfunctional sense, which is: W =df thefunctionof type (01 --+ od --+ 01 thatbehavesas follows: If thereis only on e trueproposition, thenW map s every unary propositional functionto N.
Otherwise,W maps M to I and maps allotherunarypropositional functionthatmaps functionsto N, whereM is theunarypropositional N to itselfa ndallotherpropositionsto f . In theoriginalproof, W characterizes I, andis a conceptof f. WithSUPER15-16 replacingaxiom set 16theproofagainbreaksdown, becauseW maps some nonconceptsto N, not to zip. (At leastit maps them to N ifthereare any nonconceptsof theappropriatetype, and againthereare some ifthere is more thanone trueproposition.) So let usfix up W as we did V. Make W O map non-conceptsto zip, and therest as above . Thenit is easy to show thatW O is a conceptof f, because W O is identical to vo, no matterhow many truepropositionsthereare. So we can continuethe proof. It is easier now, since we can skipthe partthatuses thespecialaxioms of alternative 2-since we have already
522
TERENCE PARSONS
shownwithoutthem thatV O= wo. (This is a reasonwhy I do not see the issue.) specialnessof alternative 2 as being relevant thekey to Butnowthefinalpartoftheproofdoes not go through.W henW Oand VO were thesame on theiroriginaldefinitions, we could prove thatthereis only defeattheconsequence one trueproposition, forthatwas the only way to thattheyotherwisem ap M to different things. Buton therevised definitions, a and to N they do not differ M on; theybothmap it to zip if it is non-concept . if itisn't. So theinconsistencyin theoriginalproofhas not beenduplicated
3.6. Models for the Basic Axioms It would be nice at thispointto have aproofofconsistencyforthisapproach. It is not difficult sketcha to modeltheoryforthesystem withoutthe specialalternative 2 axioms; it isthenpossible topiggybacka possible worlds accounton thatmodel (asin Church 1973, 1974) to securethe consistency of alternative 2. (This sectionmay be skipped withoutloss ofcontinuity.)
3.6.1 Notation Assume thatthenotationis the same asChurch 1951 withtheadditionof names of zips oftheform 0", for everytype a.
3.6.2 Model Theory An arbitrary model issubjectto the following conditions.First,we assume thatthereare disjointdomains for everynonfunctional type. Each domain of non-functional type is to containa singleprimitivezip ofthattype. The domain of thingsof type 0 containsexactlythreethings: truth,falsity, and the zip oftype o. The domain of thingsof type t containsthe zip oftype c along with any n umber (possibly zero) ofo therthings. The domain of tocontainmore thingsthanthe nonfunctional thingsoftype alis stipulated domain of thingsof type a. The domain of type a -+ {3 consistsof alltotal functions from the thingsof type a to thethingsof type {3. The zip of type a -+ {3 is thatfunctionthatmaps everymember oftype a to the zip of type {3.l6
Lambda abstractsareto be interpreted as standingfor (total)functions in the usual way. The universalquantifier of type (a -+ 0) -+ 0 standsforthatfunctionthat maps a functionof type a -+ 0 to T ifthatfunctionmaps everything in its domain to T, to F if itmaps somethingin itsdomain to F, otherwiseto 0 0' l6In theterminologyof Church 1951, every type is now a prefer redtype, and thezips arethedesignat ed memb ers.
THE LOGIC OF SENSE AND DENOTATION
523
The definitedescriptionoperatoriotaof type (Q -+ 0) -+ Q standsfor the functionthatmaps a funct ion f of type Q -+ 0 to x if x is theuniquething of type Q thatf maps to T; otherwiseit maps f to U o ' The conditional is interpreted by thistruthtable : T Uo F
T T T
Uo T T
F T T
Negationis defined byChurchas
·8 =df 8:J F. As a consequence , negation has the following truth table:
Thereis a binaryrelation , being a concept of, defined ondomains. For every nonfunctional type Q, being a concept of is any many-onerelation fromthings of type Q1 onto things of type Q (so everything of type Q is "conceptcd"), type Q1 requiredto be aconceptof the zip of withthezip of nonfunctional type Q . For functional typesof theform Q1 -+ 131 the concept of relation is : extendedby recursion in accord withtheformula F 0 1 -> (31 is a concept of f o->{3 iff VAVa[A is a concept of a
-+
F(A) is a concept of f (a)] &
VA[.3a[A is a conceptof a] -+ F(A) = the zip oftype 13rJ.
The predicate'~' is interpreted as denotingthe relevant concept of relat ion, dependingon itssubscript.
3.6.3 Axioms and rules The goal oft heaxioms and rules is to stateand preservetruth.The logical ones (Rules -V I and Axiom sets -10 1 of Church 1951) are all valid the in models described above exceptfor axiom 8, which needs revision . Axiom 8 says: values . Nowtherearethree.Axiom which presumesthatthereare twotruth 8 can be replaced with thefollowing four axioms:
524
TERENCE PARSONS
So the logi calaxioms and rules are allsatisfiedby any model. Inaddition , we shouldadd more axioms and/orrules totakeaccountof the presenceof newtruth values. This is for completeness,a nddoes notimpact the factt hat theexistinglogicalaxioms andrules are all validated by themodel theory. I do notattempthere tostatetheadditional rules. 17 It is alsoapparentthatthemodel theoryvalidatest heaxioms governing '~' . Axioms 11-13 merelyguarantee t hatwhen thetype subscript a on a primitivesymbol is replacedby al thenthenew symbol names aconceptof whattheold one names. Since e verythingin any model isconcepted,and sincethereare nospecialaxioms for theseelevations oftheprimitivesymbols, thismay be stipulated.Axiom set 14 saysthat"conceptof' is many-one,and thisis guaranteed by thecharact erization of our models . And finally , ournew axiom Super-15-16justputsintowordsthemodel theor eticcharacter ization of theconcept of relation forfunctional types. I concludethatthemodel theor y validates the logical rules andaxioms as well as those governing' ~ '. 3.6.4 Possible Worlds Model Th eory We can extendthe model theorygiven above toconform toalternative 2 by thepossible worldstechniqueemployed inChurch 1973, 1974, which is a bit easierfor us becauseof ourlogicaltreatment of zips asmembers of their dornains.l''This lets us construct models foralternative 2 thatdo not ent ail thatthereis only onetrueproposition.Detailsare notpursuedhere. It is alsoopen to us not to do this, and to leavethe theory as stated, withoutcommitment to any formal or mechanicalanalysisof the identity conditionsof senses. (This is my preferredapproach.)!? 17This may notbe possibl ewiththe given notation . The connec ti ves as ewlyint n erpreted ap pear not to be truth-fun cti on all yomplete c e.g., ( ther e appe a rs to be no way to define a conju nction thatis true iff bothconj unctsaretrue, fals eiffeit he rconj unc tis false, and othe rw ise zip). Thus, normal ways of fillingoutthe rules to "thr ee- valued" completen ess al developm ent do not necessarily work in the usual way. If this is the case, the natur wouldbe to add a primitive conjunct ion or disjunction with its own semantical cla use in y in principl e of com p let ingthe rules the m od eltheory;ther e the n seems to be no difficult and axioms . 18Church (1973, 1974) p resumed tha t the zip of a given type is not a m ember of any dom a in; this requiredspec ia ltreatmentof them in the m odel theo ry, and required complicated replacementsfor axio m sets 15 and 16. 19If m odaliti es are present in the language, then there si a difficulty wi thaxiom 16 in
THE LOGIC OF SENSE AND DENOTATION
525
4. DE RE CONTEXTS
Frege seems to have disavowed de re constructions . When Russellsuggested thatthe propositionthatMont Blanc is over 1000 meters high has Mont t heidea thata concrete Blanc as one of its constituents, Frege balked at objectcould bepart of a sense. He apparently concludedfrom thisthatno de re construction makes sense. I say"apparently" becausehe probablydid not thinkin theseterms, and to saythisattributes to him ageneralview thatgoes beyond anyparticular one thathe had.2o Churchfollows Frege in this; his discussion ofconceptsis framedentirely thatcapturesde dicto readings,butnot de in termsof a scientificn otation re ones.Butthisis a matterof focus, and is not requiredby theontology at hand. Indeed, theontology a tChurch'sdisposalcontainsalltheingredients de re constructions.This is whatI one would need to givesemanticsfor a arguehere. Considertheambiguityin 'Agathathinkssomeone is aspy'. The de dicto readinglinksAgathawith thepropositionthatsomeone is a spy. The de re readingsays thatthereis someone whomAgathabelieves to be a spy . Let us . The wholesentence consider thetypesof thesymbols in thede re reading value,andis of theform: denotesa truth
3x[Person(x) & Believes(Agatha, Spy(x) )1. This is trueiffthefollowing is trueon some assignmentto 'a': Person(a) & Believes(A gatha,Spy( a)). Butwhatis assignedto a in the first conjunctmust be an individual,and the objectof 'believes'denotesa proposition. So we have aconstituent of the form : Spy(a) which, in thisc ontextrefers to paroposition(oftype ad when'a' refers to an individual.T his can happenonly if 'Spy' refers, in thiscontext,tosomething oftype £ ---. 01, thatis, to a function from individualsto propositions . In the de dicto reading,on theotherhand,'Spy' refers to faunctionoftype£1 ---. 01, from conceptsof individualsto propositions. So to mirrora de re indirect addition to the one discussed in thispaper. Kaplan(196.i) p oin ts outthat in order for a functionto be a concept of anotherit shouldnecessaril ycharac teri zeit. In terms of the : presentproposal ,one needs to add to theright-handside of SUPER-15-16 the condition \fAV'BNec[A
=1
B
-+1
F(A)
=1
F(B )]
where 'N ee' m ap s necessary proposition s to truth.I am indebted to David Kaplan for pointingthis outto me. 20T he Mont Blanc dis cussion is in F'rege 1904 ; there is a sim ilar examp le usin g Mt. Etnain F'rege 1914 ? The gen er alrejecti onof de re const ruc t ionss cons i iste ntwithall of s response to Russell on the p aradox about sets of Frege's state d views; see especially hi proposition s in F'rege 1902.
526
TERENCE PARSONS
contextin a scientificnotationwe need to use a different symbol from the one thatwe need to use to mirrora de dicta indirectcontext . This should be truthvalues, sosomething no surprise; thetwocontextscan have different must havechanged. Maybe thisis it. This hypothesismeshes nicely withChurch'sontological development, language which allows us to bifurcate senses of words . Predicatesof natural must express twa senses, a de re one and ade dicta one.21 Whena predicate ofnatural language is embedded in whatwe normally call ade dicta indirect de dicta sense, and when it isembedded in context , it comes to refer to its whatwe normallycall ade re indirectcontextit comes to refer to its de re sense. In a scientificnotation thereis no reference shifting,and so we need additional symbols to denotethese senses.In Church'soriginala pproachwe language: needed aninfinitechain of symbols for each predicateof natural one symbol for the predicatein directcontexts , one for thet ermin a singly embedded de dicta context,one for thet ermin a doublyembedded de dicta context , and so on. When de re indirectcontextsare takeninto account, language needsatleasta double chain: one symbol eachpredicateofnatural t ermin a singlyembedded for thepredicatein directcontexts,two for the the de re), two forthetermin context(one forthe de dicta use and one for a doublyembeddedcontext(one for thede dicta, andone for thede re), and so on. (Actually, as we see below, it will belittle a more complicatedthan this, toaccommodate,e.g., de dicta beliefsaboutde re beliefs.) The multiplicationofsymbols becomes a bit more rococo, butthereis no difference in principlethanwith thestraightde dicta approach:witheach different contextin whichthenatural language symbol appearswe get adifferent symbol . This is thestandardtradeoff ; one avoids the refin thescientificnotation languagein favor ofmultiplesymbols in the erential ambiguityof natural scientificnotation . Thereis no worryaboutfindingthesensesthemselves , fortheyarealready . Thatis, therearealready functionsof theappropriate presentin theontology types, functionsfrom individualsto propositions. Whatis new is toappeal to some ofthemas senses of symbols. tothesenewentities.T here We can easilye xtendthe"conceptof' relation es is no dangerof confusing the use of this relation in de dicta cases with us in de re cases, for thet ypesare different. The original"conceptof' relation introducedby Churchrelatesa function of type 01 -> {31 to a function of type 0 -> {3. Callthese"Fregean " instancesof theconcept of relation.The which I call "Russellian,"are those otherinstancesof theconcept of relation, thatrelatea functionof type 0 -> {31 to afunctionof type 0 -> {3. For easy readability I use 't>' for theRussellianinstancesof the concept of relation, 21T his means thatsimple senten ces of natural languag e, such as ' Holmes is clever', are ambiguous. I thinkon independentgroundsthatthisis correct;t heambiguity si essent ia lly thatof the"referential/attribut ive" distinction .
THE LOGIC OF SENSE AND DENOTATION
527
to calla ttention to thedifference between these instancesand theFregean ones, thoughit isjustas correctto use'~ ' as the root symboland leave it to thetypesubscriptsto distinguishthem. The Fregean and Russellian instancesareidentical fornonfunctional types. For functional types theRussellian ones can be defined by a simple version ofsuper-characterization: RUSSELL-15-16: Fa -
f31 [> f a-f3
iffFa -
f31
Russell-characterizes fa-f3,
where"Russell-characterize " means:
Va[F(a) ~ f(a)].22 The two versions of Super-15-16 now applyautomatically to multiplyembedded contexts,includedmixed de ref de dicto contexts . As an illustration of how this should work , here are somesample sentences andtheirtypes whenrepresentedin scientificnotation,(in which for simplicityI omit notation for thesubjectsof 'believe'). Agathais a spy. SpY t_o(Agatha.) . Mary believes(de dicto) thatAgathais a spy. B 01-0 (SPYtl-Ol (Agatha t1))·
Sue believes(de dicto) thatMary believes(de dicto) thatAgathais a spy.
BOI-0(B02-01 (SpY t2_02(Agatha t2))).
Mary believes(de re) thatAgathais a spy. BOI-O(SPYHOI (Agatha.j]. Sue believes(de re) thatMary believes(de re) thatAgathais a spy. B 01- 0 (B 02-01 (Spyt- 02 (Agatha.) )).
Sue believes(de dicto) thatMary believes (de re) thatAgathais a spy. B OI-0(B02-0, (SPYq - 02 (Agatha t1))) ·
The relations among thevariousrepresentations of 'Spy' arethen: SPYt2-02~ SPYq_Ol~ SPYt_O,
SPYt_02 c- SPYt_Ol
[>
SPYt_O'
SPYtl-02 ~ SPYt_Ol
[>
SPYt_o'
220ne can ask about using 't>' in definingRussell-charact erizationinsteadof '~' . If the type f3 is nonfunction al ,this is m erelya changein notation . Otherwi se the question arisesof ther e being a multiplicityof types of instan ces of concept of relations , not just theFregeanand Russellianones I am discussing. I haven't explored this.
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TERENCE PARSONS
Notice thaton this accounta conceptcan be aconceptof more thanone thing. Butthisdoes not really violatetheidea thatconcept of is functional, a never be aconceptof two becausethe types are different; conceptcan differentthingsin thesame sense of'conceptof'. A simple illustration of this is thatthe de dicto sense of anordinarypredicate(say 'is a giraffe ') , which is afunction is a Frege-conceptof thedenotation of thatpredicate t ruthand from nongiraffes to falsehood (and perhapsfrom from giraffes to the de re sense ofanother(possible) zip to zip). Butit is alsoautomatically predicate , one whosedenotation wouldmap concepts of giraffes to truth and conceptsof nongiraffes to falsehood (andperhapstheconceptof a zip to zip) . Thus it is a Russell -conceptof thatdenotation.So it is aFrege-conceptof one thing, and aRussell-concept of another. 5. LEARN ABILITY AND THE HIERARCHY
5.1. Learnability
Frege'ssemanticswith its infinite hierarchiesof senses has often been used to illustrate a problemaboutlearnability of language(cf. Davidson 1965). If people learnlanguage by learning themeaningsof words, and if senses are meanings, and if each word has an infinite numberof different senses, then languageis apparently unlearnable. Having learnedthe meaningof 'giraffe' for'Sue is a giraffe' , it wouldappearthatwe would not yet know meaning the ', let alone 'She didn't realize of 'giraffe ' in 'GeorgethinksthatSue is a giraffe thathe thoughtthatSue is a giraffe'.But this seems to fly in the face of thedata;multipleembeddings do notappearto cause difficulty for the understanding of thesimple wordstherein. This difficulty is reflected Church in 's developmentby the fact thatto every word ofnatural languagetherecorrespondsan infinitenumberof distinct primitivewords. Church'slanguageis (apparently)unlearn able because it has an infiniten umber of primitives, and this capturesthe ideathatits t hat,were ittrue,would make Fregeansource is atheoryofnatural language natural languageunlearnable too. This is theissue to be discussed inthis section. The issue arises from the most usual interpretation of Fregeansemantics thatis alreadyin an indirect accordingto which when a word phrase or contextis embedded in an additional i ndirectcontext,it comes to refer to thesense thatit expressed beforereembedding, andit thuscomes toexpress a new sense;this is a consequence of the ubiquitousprinciplethata word or phrasealways refers (in any given context)to whateverits sense (inthat context)is a conceptof. Church 's developmentof Frege's scientific notation is independentof thishypothesis(as I willarguein the next section),and the hypothesisitselfis subjectto doubt(as I will also arguein thenextsection). Buttheissue is worth discussing in any event. Sopurposesof for discussion
THE LOGIC OF SENSE AND DENOTATION
529
here Iadopttwohypotheses . The first isthatnatural language obeys Frege's principlethatreembeddingsin indirectcontextscause additionalshiftsin sense andreference . Thesecond isthatit isdesirabletostudya scientific notationthatreflects thisprocess byassumingthatwheneverwe havethesame rootsymboloccurring in thescientificnotation withvaryingtypesubscripts, thesecombinationsrepresenttheuses of a single word of natural language in variousdegrees ofembeddings. Then an examplewould bethestringof symbols: (For simplicityhere I ignorede re senses.) The firstdenoteswhatan unembedded use of 'is a spy' denotes,thesecond denotesthesense thatan unembeddeduse of'is a spy' expresses(which iswhata singlyembedded use of 'is a spy' denotes),t hethirddenotesthesense thata singlyembedded use of 'is a spy' expresses (which is whata doublyembedded use of'is a spy' denotes),andso on. This linkagebetweenthescientificnotationand naturallanguageis not inherentin thescientificnotationitself,but it is worth exploringas a possibleapplication . The claimthatnatural language is unlearnable accordingto Frege's theory of natural languageis easilyundercutby presuming thathumans have an innateabilityto predictthereembeddedsense of a word of natural language from itsunreembeddeduse. The factthata sense of a newly embedded sign is itselfnew totheheareris no more aproblem thanis thefactthatthe sense of a novel combinationof words is new tot hehearer;in bothcases we have a(partially) innateabilityto getthenew sense fromthealready familiarinputs. Languageunderstanding could workt hisway,andif it does, thereis no specialproblemof learnability; thereis justtheusualsituation thatspeakersneed to be able to projectthemeaningsof novelconstructions from familiarones. Interestt henshiftsto thisinnatenesshypothesis, which hasconsequences thatare worthstudying . One consequenceseems to be thatif this innate abilityexists, it must beindependentof theparticular human languagein question.So let usconsidera "contextfree" form of t hathypothesis: Given any sense what soever,there is a unique ascendent atsense, of th such that if any word in any language expresses thatsense in any context, thenthat if context is reembedded one more time , thatword in thatlanguage in that reembedded context expresses thatascendent .
If this is so, thereexists afunctionthatmaps senses totheirascendents , whichobey thishypothesizedpattern.We can representthis functionin the scientificnotation , and askwhatare thelogicalconstraints on it. Suppose we introducea symbol '1l" for this function.Suppose the (de dicta) sense expressedby 'is a giraffe' in o rdinary c ontexts is something,g , oftype£1 --+ 01 . Then when 'is a giraffe' embedded, is its sensemust be 1l'(g),which will be
530
TERENCE PARSONS
of type £2 --+ 02. And when it isreembedded,its sense becomesit(1't(g)), of type £3 --+ 03 . And so on. With it in thescientificnotation we no longer need multipleoccurrences of any other symbol with the same rootand different t ype subscriptsto representprimitivenon-logical predicates.Exceptforthelowestsubscriptof theform 01, all symbols with higher subscriptscan be replaced by occurrences ofthesign withthelowest-sense subscriptprecededby applications of it. For example,we can replace the infinite sequence of scientific symbols: SpYL_O ' SPYL1-ol'SPYL2-02' SPYL3-03" "
by:
Spy L_o, SPYL1-o1, it(SPYL1_0.),it(it(SPY L1- 0 l
)), • ••• 23
So the onlyremainingsource ofinfinitudein the primitive vocabulary would betheinfinitenumberof distinctoccurrences of 'it' itself, withtheir distinctt ypesubscripts . And these can all reduced be to one if'it' is construed as whatChurchcalls an"improper" symbol,thatis, as untyped. This would be justifiedon thegroundsthat'it' representsan innatecombinatoryability, and is notthenormalsortof item thatmirrorstheuses of items ofn atural languagevocabulary . Whataboutcomplex signs; can these too reduced?Yes, be if we assume thatthe it functionsatisfies twoprinciplesof compositionality, one for functionalapplicationandone forabstracts.Suppose we define: x is a sense
=df
3y[x ~ y] .
Thentheconditionswe want are : 1. If ! Oil-131 is a sense andaO I is a sensethen
it(fOl-131(aol)) = it(fOl-131)( it(aol)) · 2. If ..\X01[M131] is a sensethen it(..\XO I [M131J) = ..\X02[it(M131)] .24 23The intent of the hypothesisis clearlyt hat1t(s) is alway s a conceptof s. 24T he re is acomplicationhere: the variablefollowing'>.' on the left handside is 'X" I ' andthaton the rightis ' X" 2' , which apparently does not bind the 'X"I ' thatremainsinside 'M131 '. For example, an instance of (2) is:
1t(>.x" [SPY" - 01 (x" )]) = >.x'2[1t(Spy'1 -01 (XLI»], which, after applyingcond it ion (1) tothe inside yield s:
1t(>'x" [SpY'I_ 0 1 (x,,)]) = >'X'2 [1t(Spy"-01) 1t(x" )1 · Here the rightmost'X'I ' is apparentlyu nbound. This oddity cannotbe removed merely by changingthe 'x,, ' to ' X' 2', for thenthe resulting'1t(X' 2)' is the wrongtype to be an argumentof '1t(Spy"-01 )'. There are a numberof ways ofhandlingthis issu e. The simplestis the m ostartificial ;t hisis to change 'X" I ' to ' X"2 ' on the right,a nddecree that when theargumentof ' '0' is a varia bletheresultis the same as if ' 0' ' was notthere; '1t(X' 2)' is automa t ically j ustthe same as 'X'2'. This ad hoc move seems to workadequat ely. (This approachrequiresa slightmodificationof cond it ion (1).)
THE LOGIC OF SENSE AND DENOTATION
531
A good applicationof (1) tells us, e.g ., thattheascendentof thecustomary sense of'Agathais a spy' is a function of theascendentsof thecustomary senses of itsparts: it(SpY'l-ol(Agatha,J)= it(SpYq_oJ(it(Agatha,J). RecallChurch'spolicy thatuniformly raising thetype subscriptson any primitivesymbol by 1producesa name ofthe(de dicto) sense of the original symbol. If this is combined withtheassumptionsabove, we can derive that if F is a closedformulathenwe have: whereF 1 is whatyou get by raising every typesubscripton every symbol in F by 1, provided that everyprimitivesymbol in F denotesa sense; otherwise this is nottrue. The indirectsense of'whatJohnsaid' is thustheascendentof its customary sense.Butthis doesnot mean thattheascendentofthecustomarysense of 'whatJohnsaid' is represented by placing'it' in front of 'whatJohnsaid': it((£(ol-ol-ox Ol)[Said(j,X Ol)])· statedforcondition(1) above.) Sup(This does notsatisfytherestrictions pose thatwhatJohnsaid is thatAgathais a spy, which is alsowhatMary said. Thentheascendentof whatJohnsaid is theascendentof the propositionthatAgathais a spy, which isit(SpY'l_Ol(Agatha)): theascendentof whatJohnsaid = theascendentof whatMarysaid = it(thepropositionthatAgathais a spy), Buttheascendentof thesense of 'whatJohnsaid': it(£(o2-od-OlxolISaid(j,X Ol)]), Anotherapproachis bothmore natural and more complicated . We suppose thatvariables come in sequences, distinguishedwithin the sequence by theirtype subscripts. So for anytype a we considernotjusta variable'x", ' of thattype, but thesequence x"'O,X"'l'X"'2,X"'3' '' ' of variableswith thecommon variable-root ' x ' . Then we do not make assignmentsto variables,we make assignmentsto sequencesof variables . An assignmentto thesequenceX"'O' X'" 1 , X"'2 ' X"'3' ••• assigns a chainof thingsto thesequence, assigningto each 'X"'i+l' a conceptof whateveris assignedto x",, , Further,it must assign to eachX""+l the uniqueconceptwhich is 11-(the thingassigned to x",.) for alli ~ O. Then when we givetheconditionsfor quantification or abstraction we do notspeak of assignmentsthatdifferfrom a given assignmentat most in whatthey assign to 'X"'l', we speak of assignmentsthatdiffer from a given a ssignmentat most in whattheyassign to thewhole 'x' sequenceof variables . (We must takecarein writingformulasthento in distinctabstractsso as to avoid an extendedkind of clash use distinctvariable-roots of bound variables.)Then we leavecondition(2) exactlyas is, since thevariable' X "' 2 ' followingthe 'A' in effectbinds thevariable'X"'l' inside theabstract,and thecondition yieldstheresultswe want. (If thisapproachis adoptedin detail,it willprobablyrequire therefinedtypesdiscussedin section5.2 below.)
532
TERENCE PARSONS
is somethingelse entirely. The ascendentof the sense of 'whatJohnsaid' i= theascendentof thesense of'whatMarysaid'. The aboveconsiderations cannaturally be duplicated for de re ascendents of sensesalongsideof theirde dicta versions. This leaves eachfunctional de dicta and de re, withtherest projected word having two basic senses, from these, and one wonders aboutreducingthis pair to one.This is easily accomplishedif one does notexpect an informativeanalysis. One could simply hypothesizethatthe de dicta meaning of a worddeterminesits de re meaning, andintroducea "sidewaysprojection"functionsymbol for this. Thenall oft hevarioususes could bereducedto one. 5.2. Refining the Type Hierarchy
The aboveconsiderations suggesta more refinedstructure for thetypehierarchy. Compare thesenses expressedby ' snow is white' and by ' what John examples: said' in these two Mary believes that snow is white, Mary believes what John said.
Bothare oftype 02, but the former is anindirectsense and thelatter a customarysense. Thatis, the former ist hesense of anembedded phrase, andthelatter is thesense of a non -embedded phrase. In addition,t helatter is not theindirectsense of anyphrasein anylanguage . This can be shown as follows. Assume thatwhatJohnsaid wasthatsnow is white. It is apparentthat the de dicta senses of Mary believesw hatJohnsaid, and Mary believest hatsnow is white are different. The matricesof thesesentencesare the same, and so the customarysense of 'whatJohnsaid' must be different from theindirectsense of 'snow is white'. Representthe directuse of thesentence'snow is white'by So, which refers to truth a value. Its singlyindirectreference (its customary t he propositionthatsnow is sense) is representedby SOl ' which refers to white, and itsdoublyindirectsense is representedby S02 ' which refers to . Representthedirectuse a conceptof thepropositionthatsnow is white of 'whatJohnsaid' by W O I ' because thephraserefers to a proposition. (In thepropositionthatsnow is white, soW O I = SOl') Its singly fact, it refers to indirectreference is denotedby W 0 2 ' and wearguedabove thatW 0 2 i= S0 2 ' Now suppose for reductio thatthecustomarysense of'whatJohnsaid' is the indirectsense of somephraseX, so thatW0 2 = X 0 2 ' Since theindirect sense of aphraseis theresultofoperatingon itsdirectsense byit, thisentails
THE LOGIC OF SENSE AND DENOTATION
533
thatX 0 2 = 'fj(XO , ) . However,X 0 1 = 8 0 1 • (This is becauseX 0 1 = whatever X 0 2 is a conceptof, which is whateverW0 2 is a conceptof, which is WO " i.e., 8 0 1 . ) So, substituting 8 0 1 for Xo p we concludethatX 0 2 = 'fj(80 1 ) . But 11"(80 1 ) = 8 0 2 • So we provedthatW 0 2 = 80 2 , contrary tothedatafrom which -j we started. If thisargumentis generalizable (or even if itsconclusionis consistently generalizable by hypothesis)thisleads na t urally to theidea thatsenses are not justcategorizedintotypes by Church's type symbols, they also come stratifiedwith in these types by thelinguisticlevel ofsymbols thatcould expressthem. E.g., a given sense might be inherently a kind ofthingthatis language e mbedded n times, and expressibleby a word orphraseof natural notexpressibleby any word or p hraseembedded m times, withm I- n. This thensuggestsa different t reatment of the"conceptof' relation forfunctional types. A functional sense oftype 0:1 -+ {31 shouldnot beconsideredto be a partialfunctiondefined forconceptsof type 0:1, it shouldbe defined for senses thatareconceptsoftype 0:1 and of linguisticleveln . This suggestsan interesting complicationof thetheorywhich could very well lead to fruitful results . I don'tpursueit herepartlyforreasonsofspace, andpartlybecauseI suspectthatthereis a bettera ndsimplerway toproceed.This is articulated in thelastsection. 6. A SIMPLER ACCO UNT
The idea thatthereis a hierarchyof linguisticlevels ofense s corresponding to thelevels ofe mbeddingin indirectcontextsp redictsdistinctionst hatare ed for. not readil y apparentin the data. Theirabsenceneeds to beaccount I suggestthatthebestaccountof natural language is one thatis farsimpler thantheone contemplated in thelastsection. t hatleads in thisdirection , Here is one smallb utimportantconsideration discussed in Bealer 1982, pages 38-41. 25 The following a rgumentseems to a ndvalid: be well-formed J ohn thinksshe believes, Mary believeseverything Johnthinksshe believesthatsnow is white, :. Marybelievesthatsnow is white. The types of thesymbols in thefirstpremise are constrainedby its form, which seems to be : V'x[jThinks(mBelieves(x))
-+
mBelieves(x)].
25Bealerargues tha t th e argume ntis notrepresentabl e in C hurch 'ssystem , but this is the ver sion of the syste m as used by Church; in the previ ou s sectio n I argu e that Church 's " de re const ruc t ions , whic h si what Beal erurges should be system p ermits "R ussellian used.
534
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The firstoccurrence of x is in a doublyembedded context,and thesecond is in a singlyembeddedcontext . So how canthesentencebe well-formed? T he answer must bethattheembedded 'believe' refers to the de re sense of its customaryuse. The typeofx is 01, andthe types of the predicatesare x[jThinks(mBelieves(x))- > mBelieves(x)] The second premise hast heform: jThinks(mBelieves(Snow iswhite)) and theargumentis valid. -clauses. The logical However, thisexampleleads to aquestionaboutthat symbolism seems to permit a construal of the second premise whereby it is well-formed and the argumentis not valid. Yet this does not seem correctfor the original argumentin English.The invalidreadingis easilyillustrated by substituting for 'snow iswhite'the phrase'whatSam said'. The argument is then Mary believeseverything J ohnthinksshe believes, Johnthinksshe believeswhatSam said, :. Mary believesw hatSam said. that theargument It is now possible to read the second premise in such a way is not valid.T he theoryyields thisreadingby construing'believes' in the second premisede dicta : jThinks(mBelieves(w hatSam said)) Thefascinating questionposed by Bealer is why suchreading a is not available ofthesecond when we use athat-clause . Thatis, why istherenot a construal premise with thesetypes: jThinks(mBelieves(Snow iswhite)), which yields a form accordingto which the a rgumentis not valid? Thereare twonatural answers to this question. Onethat is theinvalid construal is somehow equivalent to theearlierone thatmakes theargument valid. The otheris thattheinvalidconstrual does not exist in t helanguage. I consider these in t urn.
THE LOGIC OF SENSE AND DENOTATION
535
6.1. The Equivalence Hypothesis
Suppose thattheEnglish sentencedoes permita doublyshiftedreading , but one thatis equivalent to the singly shiftedreading.The assumptionis that theseareequivalent : jThinks(mBelieves(Snow is white))
If thepatternis general,it must be becausethe de dicta sense of' believes' maps thedoublyembedded sense of 'Snow is white'to thesame thingthat thede re sense of'believes'maps thesinglyembeddedsense of 'Snow isw hite' to. In symbols:
B o I -+ o I (SOl) = B 02-+ 01(S02) If theexampleis further complicatedby an additional embedding,theresult is thattheseareequivalent :
aRegretsjThinks(mBelieves(Snow is white))
Again, if thepattern is general i,t must bebecause the de dicta sense of 'thinks'm aps whatthede dicta sense of' believes'm aps the doublyembedded sense of'Snow iswhite' to to th e same thing thatthe de re sense of 'thinks' maps whatthe de re sense of 'believes'maps the singlyembedded sense of 'Snow iswhite'to to. Insymbols:
T 01 -+01 (BoI o, (SOl)) = T 02 -+0 1 (B03 02(S03)) -+
-+
It turnsoutthatalthough 'Snow iswhite'has an infinit e numberof distinct de dicta senses, and 'believes'and 'thinks' do too, theircombinationyields exactlythesame resultas applicationof thesinglyembedded de re senses of 'thinks'a nd 'believes'to thesinglyembedded de dicta sense of 'Snow is
white'.The infinitecomplications,aftertakingus once eacharoundthebarn, end up cancelling one anothero ut. This strongly suggests thatthesimpler hypothesisis on therighttrack. 6.2. The Simpler Hypothesis
The simpler hypothesisdenies thatwhen words orp hrasesof ordinarylanguage arereembedded in an indirectcontext they undergo an additional semanticshift. One shiftmakes words come todenotetheir senses, and additionalshiftshave nofurth er effect. This hypothesis requir es one minor
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alteration of Frege's principlesfor words innatural language . Insteadof: A word orphrasein anycontextd enotestheuniquethingthatits sense (in thatcontext)is a conceptof, we need: A word orphrasein a directcontextdenotestheuniquethingthatit s sense is a conceptof; a word orphrasein an indirectcontextdenotes its sense.26 In termsoftheexamplessurveyedabove, theproblematicreadingssimply do for whytheyare equivanotexist,andso thereis no problemof accounting lent tot heotherreadings.I thinkthatthisis therightanswer;the additional semanticshiftrequiredof reembeddedtermsis entirelyu nmotivatedby any dataof natural l anguage.FFurther,t hetheoryis simpler withoutit, becauseit requires thatwords have only customarysense and reference. This alsobypasses thelearnability issue entirely , since thereare no new senses to understand. This may be Church'sview as well; he has expressedtheopinion(personal communication) : , a denotation and a sense. ". . . a name has justtwo kinds of meaning Indeed the sense of a given name may be denoted by another , name and thislatter name may be expected to have a se sen of its own . And so on ad infinitum. But this is not the same as saying thatone name may havethree kinds of meaning (such ,as e.g., a denotation, and a sense, and after thatalso anindirect sense, . , .)." I agreeentirely . Church'smotivationfor hisaxiomaticstudydoes notcome from thepurported"reembeddingcausesreshifting"view ofwhathappensin natural language . This view is notarticulated in Church'swritings.Instead, he simply says thingslikethis (Church 1951, page 12, note 13): "The hierarchy of conceptssuccessively of higher orders arises as soon as wesupposethata concept, like anythinge whichcan els be discussed at all, is capable of having a name given to it. For a sense of a name of a concept is a concept of the next higher , and order so on ." So Church'saxiomaticstudy,withits ontological hierarchy oftypesis needed even ifnatural languagelacksthelinguistichierarchypopularlyattributed to Frege.28 26This changeis requiredfor thetheoryof natur al language . No changeis requir ed for thescientificnotation , which containsno indirectconte xts. 271 arguethis from diffe rentperspectivesin Parsons 1981 and 1996. 2BI thus take exceptionto Geach's (1980, page 95) claim that "Withthis view, Church's hierarchyof senses colla pses lik e a house of ca rds." ("This view" = the view thatan unambiguous termhasa single sens e, andits refer enc edepends on thissense together w ith thekind of propositional cont ext in whi ch it is embedded .)
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7. AFTERWORDS
The precedingpaper was completedsometime before 1995 . Afterit was typeseta numberof issuesarosefrom conve rsationswith DavidKaplanthat requiresome developmentsin section3 regardingthequestionof when a functionon conceptsis itselfa concept. Ratherthanpatchingthepaper,29 it seems more useful to describetheproblemsthatneed addressing , since it is not clearwhethercertainchanges are necessary , and sincetheyturnon issues thatwill have to be faced anyone by working inthevicinityof Church's framework . Threeissues arediscussedhere: thereductionto extensionality, thequestionofmodalandepistemicuniformity,andthetreatment ofcomplex names havingempty names as parts.
7.1. The Reduction to Extensionality When I wrotethe paper I did not know how tocarryoutthe proofof a reduction to extensionality withintheobjectlanguage of Church 1951, alter native2, withoutmaking an assumptionthatevery sense can be assignedto a symbolof some language . In fact, aproofcanbe given, thoughalongquite differentlinesthantheone discussedthere . The proofis short.If thereareonly twotruth-valu es, thenthereare only and theyall havet his property : thatif you fourone-placetruth-functions, applyanyoneof themthreetimes, it's thesame as applyingit only once. So thefunctionthatmaps each monadictruth-function totheresultofapplying it to athingonce isthevery same functionthatmaps each monadic truthfunctionto theresultof applyingit tothatsame thingthreetimes. So: Letthevariablef be ofthetype0 -+ 0, thetypeoffunctionswhichmap truthvalues totruth-values . Let To be aconstant name of type OJ forconvenience we use theexistingname oftruth(thatis, To is a particular logically t rue sentence).T henwe can prove :
2, thesenses of aprovableidentityare themselvesidentical, In alternative andin Church 1951 you producethename of a sense of a formula raising by all of its typesubscriptsby 1. So thefollowing m ust be truewhereF is of the type 01 -+ 01, thetypeof functionst hatmap propositionsto propositions:
AFF(ToJ = AFF(F(F(T o.») . Butthisis plainlyfalse if thereare morethantwopropositions. This proofdiffers fromtheone discussedin thepaper becauseit does not 29 1 did in factmake som e changes inthe p aper, including correcting a crucialerro rabout zips . The read eris indebted to David Kaplanforsparing him or er h som e confusion whil e r ains. readingthe paper. I am responsible for whateverconfusion em
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even appealto thetroublesomeaxiom 16. AlthoughChurch(1973, 1974) does notindicatethatany problemsariseindependentof axiom 16,theyare allelegantly solved intheway proposedin his paper, which is toadd a type subscriptto '>" itself . The basic '>.' of type zero obeyslambda-conversion; thetype-raisedversions of'>.' do not-they thusrequirespecial treatment. Withtypesubscriptsin place, thep roofgiven above has a different result ; it yields: The effect of t hesubscripton '>.' is as if itrestricts theresulting functionto natural domain; it is defined only for those onethatis not defined on its whole was thingsin its domain thatare conceptsof something. This restriction accomplishedin Church(1973, 1974) by designing the axioms in sucha way thatlambda-conversion is obeyed by thesubscriptedlambdaforarguments thatare concepts,and not for others. In thespirit of the ip z formulation in my paper, thesubscriptedlambdacan beintroduced by definitionas the functionwhich behaves as expectedif itsargumentis a conceptofsomething, andotherwiseyieldstheappropriatezip. Thatis, it can be defined in context in thisway: >'dot) ¢ /31 = /,F01-+/31'v'/ol[[--,390/01 ~ 9/3 & F 01-+/31 (fol)= 0/311 V F 01-+/31(fol)= ¢ /3J
It isthenmost efficient to follow Church'slead, andproducethismodified axiom: SUPER-15-16: F 1 ~ 1 == 'v'A'v'a[A ~ a ---> F(A) ~ I(a)] where F1 is defined to be>'lxF(x). 7.2. Kaplan's Argument for Necessary Uniformity Kaplan(1964) indicatesa nother problemwith axiom 16.T hataxiom requires thatfor a function to beconceptofsomethingit a mustactually characterize thatthing- but the axiom does notrequirethatthis benecessarilytrue. 30 which showsthatthis lack Becauseof this, an argumentcan be given c ertainnatural of whatKaplancalls"necessaryuniformity"conflicts with principlesaboutmodality(providedthatsuch anonnecessarily uniformsense thelanguage) . This problemdiffers from is actually assigned to some sign of the one discussed above that in (1) it is not solved by the device discussed above, and (2) it arises independently of thespecialprinciplesof alternative 2, and so itextendsto a widerangeof phenomena. t hatallthatis requiredby In supportof point (2) is theconsideration theargumentis thatprovablesentencesbe subjectto necessitation, andthat necessitation be achieved byproducinga name ofthesense oftheproved formula, which can be formed in some regular way (e.g., by type-raising).T hen 30T he argumentwas personally communicate d to me by aDvid Kaplan.
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you need a couple of very plausiblemodal principles. Nowhere does the arguequivalent senses. For ment appealto aprinciplethatidentifies necessarily this reason,t heargumentcan be applied even when wedistinguishnecessarilyequivalent senses. And we want to distinguishsuch senses inorder to accommodatethewell-known phenomenonthatcertainnecessarytruths , known by e.g., certaincomplex truthsof arithmetic,are notautomatically virtueof knowing simple ones. similarto themodalexamHere is how aproblemaboutknowledge arises, ple discussed byKaplan.Suppose thatwe have asystemof axioms for sense a nd thatwe havealreadypatchedaxiom 16 in accordance and denotation gamma thatmaps each with themodalconstraint.Then define a function necessarypropositionto thesense of atrivialtautology , maps each impossible propositionto thesense of atrivialimpossibility,and maps contingent , propositionsto themselves . For example,let'To' be some simple tautology andassume that'T0/ denotesits sense. Let gamma map necessarytruths to o,, and all o therproposiitonsto themselves . T o" necessary falsehoods toT -'1 as Kaplan.) (Noticethatgamma is alreadynecessarily uniform, proposedby Let G expressthisfunctiongamma as its sense, sothat'G I' denotesgamma, and 'G' denotestheuniquefunctionthatgamma is a conceptof (which is in to itself,thoughthatdetailis fact thefunctionthatmaps eachtruth-value notrelevant to theargument).Let 'So' be anysentence,a ndlet'K ' standfor 'you know' (so'K ' denotesa functionthatmaps the propositionsthatyou , and therest to falsehood). know totruth instanceof Leibniz'sLaw: Now noticethatthefollowing is an
Reflecting on t hisfact, andunderstand ing the sens es of theingredients , you should know the propositionexpressed.So this should be true: K[So, =1 T o, -+1 GI(So,) =1 GI(T o,)]. Butthefollowing schemaseems to be aplausibleprincipleof epistemic logic (atleastit holds if you reflect on your knowledge thiscase): for K[A 1 -+1 Bil -+ -,K[-'IAtl-+1-,K[-'IBd·
This schematogether with modus ponens appliedto ourresultabove yields :
This argumentholds good nomatterwhatsentence'So' is. So suppose that 'So' expresses somenecessarilyfalsepropositionwhich you donotknow to e-thatis, you don'tknow itsnegation;it is one ofthose recondite be fals necessaryfalsehoods of arithmetict hatonly afficionados are aware . Then of , theantecedent ofwhatwe have proved istrue.So by modus ponens we infer:
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TERENCE PARSONS
However, byhypothesis,'G 1 ' denotesgamma, andso G1 (SoJ = G 1 (T 01) = T 01' Substituting, we get:
'I
T 01 and
,K[,d'I T ol =1 T olll · Butthisis factually wrong; you arenot ignorantof thetruth : ' I ['I T 0 1 =1 T ol]. So somethinghas gonewrongsomewhere. One mightthinkthatwe need torepairthisby addingan epistemic conditionto axiom 16 in a wayparallel to Kaplan'ssuggestionaboutthemodal case. We need not j ustnecesssaryuniformity , butalsoepistimic uniformity . The conditionwe add would have to betheform: of
Vao IVbo I K [ a = 1 b-+ 1 F oI-+f3I(a ) =1 F oI-+f3I(b)]. The problem with this strategyis thatthereappear to be hosts of other p atchesto exampleswaitingin thewings, each oft hemrequiringadditional axiom 16. We will needadditional constraintsfordeonticlogic, forthelogic of locutionsinvolvingdesirability,perhapsalso forprobability.This worry motivatesone toconsideralternative solutions . As an alternative, suppose thatwe do notexpectourintensional logic to ruleouttheexistenceof conceptslikegamma. Thereis anotherplace to cut offthereasoning . In themodalcase, perhapswe were toohastyin adopting unrestricted necessitation . The factthata sentenceis provablemeans thatwe are surethatit expressesa truth,butthis need not bebecauseit expresses a necessary truth . In fact, in the modal argumentgiven by Kaplan,the provablesentenceexpressesa propositionthatis not necessary , as thefollow up argumentd emonstrates . In theepistemiccasesomethingsimilarhappens; there , even thoughyou knowthat thesentencein questionexpresses a true proposition,you do nottherebyknowthepropositionthatit expresses. This is possiblebecauseyou do not know which propositionthesentenceexpresses. thepropositionit Knowingthatthesentenceis truedoes not foster knowing expressesif whichpropositionit expressesis notapparentfrom thewording of thesentenceitself . This argument,however,a ppearsto cutbothways. The reasonthatyou cannotinfer knowledge of the expressed propositionfrom knowledget hat thesentenceexpressesa truthis thata natural apparency conditionis not satisfied : you do not know which propositionis expressed. And thisis so even thoughyou knowthemeaningsof thepartsof thesentence . The problem withthepredicateG is not thatyou do not know its meaning; theproblem is thatknowingits meaningdoes notput you in apositionto knowwhat getsexpressedwhen youcombine it withotherphraseswhose meaningyou also know . So it can be used to construct sentenceswhose meaningsare not apparentfrom themeaningsof theirparts. Butdoesn'tthatexplainwhy we wantto ruleoutthemeaningof G as a posssible sense?Perhaps. Butwe don'tknow how to developtheory a oftheontology of sensesthatwill rule out all oft hesenses thatgenerateproblemslike this . Instead,we might permit
THE LOGIC OF SENSE AND DENOTATION
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themin theontology a ndthenbe carefulto taketheirpathological behavior o urrules of inference. For example, modal intoaccountwhen formulating necessitation shouldapplytosentencest hatconsistonly of signsw ithmodally uniformsenses, epistemic ascentshouldapplyto sentencest hatconsistonly of signs thatsatisfytheappropriateepistemic condition,a ndso on. This is as easy toimplementas it is tointroducesuch constraints on theaxiom for theexistenceof concepts. The advantagein approachingtheproblem in this way is thatwe face theparticular problems exactlywhen theynaturally arise. The epistemic problemarises when we a ddtooutaxioms a specialinference rule epistemic of , we need to askwhetherit is valid,and ascent. When we add such a rule underwhatconditions.And here may bethebest place to face anddeal with specialconstraints . 7.3. Zip-ity Doo-Dah
The problemof empty names is raisedby Frege's discussionof "Odysseus was set ashoreat Ithacawhilesound asleep". Frege arguestherethatif a name lacks reference, so must a complex sign containingthatname; so thefailure of reference'Odysseus'explainswhy of thesentencecontaining it lacks anactual t ruth-value. This has consequencesfor the senses assignedto complexnames. Church(1973, 1974) putsit succinctly in his footnote 5: "if thesense-valueassignedto one ofthevariablesis not aconceptof anything ... thenthesense-valueof theentireformulamust correspondingly not be a conceptof anything..." I suggestedaddressingthisproblemby havingintuitivefailure of reference be emulatedby technical-reference to a zip. However, I did not end up witha symbolism in which anintuitively empty name (onedenotinga zip) forces a complex namecontainingit also to bee mpty (to denotea zip). A simple counterexample is negation,which producesa truesentencefrom one denotinga truth-value zip. A choice must be made here . If you insist thatthe languageobey the principlethatconceptsof zips beget concepts of zips, thenyou end upwitha working logic thatis so impoverishedit is hardto formulate s tronggeneralprincipleswithinit. I didn'tsee how to do this withoutdepartingsubstantially from Church'ssystem of axioms, and so I introduceda three-valued logicthatdepartsminimallyfrom his, and is stronganduseful intheway thatit handleszips. Butthenit doesnotobey theprinciplethatzips beget zips.(It does not obeytheprinciplethata name must denotea conceptof a zip if apartof thatname denotesa conceptof a zip.) A language t hatemulatesEnglishshouldthenbe constrained by being t hatobeys the"zips beget zips" principle. At least , it limitedto vocabulary shouldbe limitedin this way if Frege isright. (And thatis whatI had in mind in theSupermanexample.) Frege'sinsightshouldbe dividedintotwoprecepts. One is thatall failure
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TERENCE PARSONS
of reference (as opposed to obliquecontexts)shouldbe treated alike, sothat e.g., if onesentencethatlackstruth -valuehas atruenegation , theyshould allbehave this way (if negationdoes notcreateobliquity). The otheris thatreference failure shouldpropogatefrom simple namesto complexnames withoutthesecond. This containingt hem. I implementedthefirst of these is not clear from my paper becauseI was not clear whenwroteit. I ACKNOWLEDGEMENTS
I am indebtedto AlonzoChurchfor hisgraciousreplies to some naive questionsabouthis systems, and to David Kaplanfor variousdiscussions about sense and reference over theyears. I alsothanktheUCI L&M group: Peter Woodruff,P ennyMaddy, andJeff Barrett forcriticismsandsuggestions. REFERENCES
Anderson, C. A. 1984 Generalintensionallogic, Handbook of philosophical logic II (D. Gabbay and F . Guenthner , editors),Reidel, Dordrecht,pp . 355-385. Bealer , G. 1982 Quality and concept, OxfordUniversityPress, Oxford. Church,A. , Structure, method, 1951 A formulation of thelogic ofsense and denotation and meaning: Essays in honor of Henry M. Sheffer (PaulHenle ,H. M. Kallen , and S. K. Langer,ed itors),LiberalArtsPress, New York, pp . 3-24. 1973 Outline of arevisedformulation of thelogic of senseanddenotation ( Part I) , Nous , vol. 7, pp. 24-33. s ( Part 1974 Outlineof arevisedformulation of thelogic of enseanddenotation II) , Nous, vol. 8, pp. 135-156. e and denotation , Alternativ e 1993 A revisedformulation of thelogic of sens (I) , Nous, vol. 27, 141-157. Davidson,D. languages,Proceedings of the 1964 1965 Theoriesof meaning and learnable international congress for logic, methodology and philosophy of science (Y. Bar-Hillel , editor),North-Holland , Amsterdam. Frege, G. 1891 Funktionand Begriff(Functionand Concept); translated in Frege 1960. in Frege 1892 UberSinn und Bedeutung(On SenseandReference) ;t ranslated 1960. D 1902, in Frege 1980, pp . 152-154. 1902 Letterto Russell, 28 ecember , 13 November 1904, in Frege 1980, pp. 160-166. 1904 Letterto Russell , undated , in Frege 1980, pp. 78-80. 1914? Letterto Jourdain 1960 Tronslations from the philosophical writings of Gottlob Frege (P. Geach and M. Black,editors), Blackwell, Oxford.
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Gottlob Frege, Philosophical and mathematical correspondence, (G. Gabriel, H. Hermes, F. Kambartel,C . Thiel, and A.V eraart,editors;H. Kaal,translator), Universityof Chicago Press, Chicago .
Geach, P. 1980 Some problemsaboutthesense and reference propernames, of New essays in philosophy of language, (F. J. Pelletier and C. G. Normore, editors),CanadianAssociationforPublishingin Philosophy,Guelph,Ontario,pp . 83-96. Kaplan,D. 1964 Foundationsof intensionallogic, Dissertation, UniversityMicrofilms, Ann Arbor. 1973 The Churchreformation, a talkgiven totheWesternDivision APA. Parsons,T. 1981 Frege'shierarchies of indirectsense and theparadoxof analysis , Midwest Studies in Philosophy, vol. VI, pp . 37-57. a ndmeaning, F'rege: Importance and legacy (M. 1996 Fregeantheoriesoftruth Schirn ,editor),W alterde Gruyter,New York and Berlin . TP Theoryof meaning for natural language , forthcomingin a volume on Reference (A . Grayling,e ditor),OxfordUniversityPress, Oxford.
MARK RICHARD
ANALYSIS, SYNONYMY AND SENSE
Abstract.The paradoxof analysisis this: some analysesoughtbe informative; butsince analysands and analysand urn must be synonymous,a correct analysiswill betrivialand thusnon-informative . This paper begins by distinguishingtwo sortsofsynonymy, phrasalandstru ctural.T he distinctionresolves theparadox:onlysubstitution of structural synonyms is guaranteed to preserve whatis said; onlyphrasalsynonymyis requiredin analysis . An accountof analysis is sketched,a ndthesolution defendedagainstobjections. Church'swell-known Fregeansolutionis comparedand twoobjectionsarebroughtagainstit. A problem, which theparadoxpresentsfor Fregeanism, is discussed. The Fregeancan avoid theproblem if he is willingto abandontheclaim thatsense is transparent (viz., the competentspeaker knows on reflection , of terms with the same sense, thattheymean thesame). An argumentagainstthisclaim is developed. An appendix compares thenotionof structural synonymy withChurch'snotion of synonymousisomorphism, and discusses theexchangeof Church, White,and Blackon theparadox.
o. Moore'sconcisestatement of theparadoxof analysisasks us toconsiderthe statement one would makew ith (1) To be a brotheris to be a male sibling, a statementhe supposes to be a faire xampleof ananalysis . Moore (1942, page 665) observes The paradoxarises from thefactthat,i f thisstatementis true , then it seems as if itmust be thecase thatyou would bemaking exactly thesame statement if you said: (2) To be a brotheris to be abrother. Butit is obvious thatthesetwostatements are not thesame; andobvious alsothatnobodywould saythatby asserting"To be a brotheris to be abrother"you were giving an analysisof theconcept"brother". 1
The problemseems to rest on four assumptions: Identity: An analysisis whatis said by a sentenceof theform To be an
A is to be a B (or by one oft heforms: Being an A is being a B, The 1 For ease of futurereference,I have takenthelibertyof displayingand numberingthe sentence'tobe a brotheris to be abrother ' in thecitation;M oorequotedit.
545 C. Anthony Anderson and M. Zeleny (eds.), Logic , Meaning and Computation, 545-571 . © 2001 Kluwer Academic Publishers. Printed in the Netherlands .
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MARK RICHARD concept A is the concept B, To V is to V', or To have an A is to have a B} . In suchsentences'is' names therelation of identity .
Synonymy: When such asentenceexpressesa correctanalysis,A and B (and thus'being an A' and 'being a B', etc.) aresynonymous. Compositionality : Substitution of synonymsforsynonyms,atleastin a positionlike thatof 'to be a B' in theabove, doesnotaffectwhata sentencesays. Triviality : Genuineand correcta nalysisis nottrivial,b utwhatis said by a sentenceof theform To be an A is to be anA, if 'is' thereinnames a relation of identity , is alwaystrivial .f These are inconsistent withtheclaimthattherearesome genuine,correct analyses . I willbegin by suggestingthatwe have twonotionsof synonymy, phrasal and structural synonymy. If I am correcta boutthis,theparadoxof analysis has a simple solution,since itturnsoutthatif by 'synonymous' we mean is false; if ' phrasallysynonymous', Synonymy is truebut Compos itionality we mean 'structurally synonymous', Compositionality is true,Synonymy false . Section 1 developsthis response. Section 2 responds to some objections and discusses relations betweenanalysisandsynonymy. Section3 criticizes a Fregeanaccountof theparadoxdue to Church. Section4 arguesthata adoptthesolutionto theparadoxofanalysiss ketched Fregeancancoherently in section1. It also poses aproblem forFregeansinvolvingtheparadox. I arguethata Fregeanbestsolves theproblemby abandoningsuch principles as Transparency : It is impossible for a(normal,rational) personto under standexpressionswhich haveidenticalsenses but not be awarethat theyhaveidentical senses (or besuchthathe wouldimmediatelycome to knowthisif he were to reflect it). on and thatit in factadds to theplausibility I arguethatTransparency is false, ofFregeanismif it isexplicitly r ejected . A corollary tothisis thatconstraints such as: Frege's Constraint : It is impossible for arational p ersonto believean objectto be F undera sense ormode of presentation m whilesimulta neouslydisbelievingit to be sounderm,3 2This formulation of the paradoxassumes thatit is propositionsor thoughts-th e claims made by sentenceutt e rances-whichare trivialor nottrivial. 3This is a simplificationof a principleStephenSchiffercalls' Frege's Constraint ' . For discussion,see Schiffer 1990.
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andshould,be rejectedby theFregean. can coherently, 1.
Synonymy, atleastin the Frege-Russell t radition,is closely tied to whatis said, withexpressionssynonymousif theyinevitablymake thesame contributionto whatis said by sentencesin whichtheyappear. Here, 'inevitably' indicatestwo degrees of variation,as an expressionmay appearin different sentencesand may be used indifferentspeech situations(contexts, as they are sometimes called) . Such accountsof synonymypresupposethatthereis a relation-call it thesemanticvaluerelation -whichrelatesa contribution to whatis said, anexpression,and a contextof use. Semanticvalue echoes syntactickind totheextentthattheclass ofsemanticvalues of each s yntactic kind forms a (more or less) natural "semant ickind". For example, the semanticvalues ofnounsare allproperties,where apropertyis (minimally) somethingwhich(relative to a waythingsmightbe) determinesa set of individuals.Syntactic form andsemanticvalues are s upposed todeterminewhat is said in a broadlycompositionalfashion. At theleast,sentencest hatdiffer only inwhatlexicalitems theycontainwill saythesame thing,if thelexical items on whichtheydiffer havet hesame semanticvalues. Given allthis, a standard a ccountofsynonymyhas itthatexpressionsaresynonymousif and only iftheyhave in all contextst hesame semanticvalue.' This accountof synonymyseems to me atbest incomplete . In orderto explainwhatis missing and to introducet hecontrast betweenphrasaland structural synonymy, let us for themoment accepta broadlyRussellianview of proper names, on whichwhata name contributes to whatis said-the semanticvalue of aname-is identifiedwithits bearer . Consider, now, the complex propositional name: (3) thatall menarecreatedequal. This contributes a certainproposition - "Jefferson 's Doctrine",we mightcall it5 -towhatis said. Forexample, (4) Scaliadoubtsthatall men are c reatedequal, ; it says this, inpart, seems to saythatScaliadoubtsJefferson'sDoctrine because(3), as itoccursin (4), contributes Jefferson'sDoctrineto what(4) says. 41 will h enceforthignore conte xtsensitivityso far as possible. For exa mple, I will suppress referenceto conte xt of use, when incorporatingit wouldbe a routin e matter . I am alsoignoring for themost partissu es which have to do with thepossibilityof a shift in semanticvalues acrossdifferentpositions withinsenten ces. I am , of cour se ,using 'semant icvalue' in a manner quite differen tfrom thatin which it is used by many Fregeans (Dumm ett,for exa m ple). . e no semantically 51 am using this as a name, nota description,and assume it to hav signific antstructure.
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Now considerthepropername (5) Jefferson'sDoctrine. Whatdoes this contribute to whatis said? On a broadlyRussellian(or "direct reference " or "naive")view, itcontributes its bearer , Jefferson 's Doctrine . So (5) and (3) makethesame contribution to whatis said. So they are synonymous.f One mightfindthispuzzling , for it seemsthatsentence4 and (6) ScaliadoubtsJefferson 's Doctrine are not synonymous. As ScottSoames points out, a person might be acthatJefferson's quaintedwiththeproposition"by name" (she has been told Doctrinewas originally p ropoundedby Locke, thatit is one ofthecornerofIndependence,etc.), butshe might not recognize stonesoftheDeclaration the propositionwhen it isexpressedcanonically. Suchpersonmight a untheirtruth."This atleast derstand(4) and (6) buttakedifferent views on suggeststhatthesentencesa renotsynonymous. Butif (4) and (6) are not synonymsforsynonymsdoes synonymouswhile (5)and(3) are,substituting .f notpreservesynonyrny One reactionto theargumentis thatit simply draws yet anotherintolerableconsequencefrom broadlyRussellianassumptions; butwe alreadyknew thattheidea thatnames contribute (only)theirbearersto whatis said leads tointolerable consequences . Such aresponsemisses an inter estingfactabout (3) and(5). For (3) and (5) in someimportantsense havethesame semantic function : theyare devices for introducingt hepropositionthatall men are createdequal intow hatis said. And so, giventheperspectiveon synonymy outlinedabove, thereoughtto be a sense in which they are synonymous. However, (3) seems to"contribute" variousthingstowhat(4) says (equality, forexample)which (5) does notc ontribute to what(6) says. So whilethere is a sense in which (3) and (5) are responsiblefor the samecontribution to whatis said, thereis a sense in whichtheyare not;t hus,thereis a sense in whichtheyaresynonymousand a sense in which t heyare not. It will be helpful introduce to some terminology.I will saythat(3), as it to whatis said which occurs in (4), isresponsible forcertaincontributions it does notproperly make. For example, (3) is responsibleforcontributing it doesnotproperlymake the contribution equality towhat(4) says, although 6Sp eaking st rict ly, synonymy followso nlyif thecontribution s arethesame in allcontexts. 7 See Soames 1989. 8Note, we arenotsubstituting within an intensi onacontext l ere. h In anycas e, thesame pointca n apparentlybe made with Jefferson's Doctrineis true Thatall men are creat edequalis true .
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in question . (Rather , 'are equal'properlymakes thatcontribution.)An expressionwill beresponsibleforthecontributions properlymade by all of itssubexpressions,thoughit itselfp roperlymakes buta singlecontribut ion to whatis said. In general,thepropercontribution of anexpression is whatwas called above its semanticvalue. So a noun's propercontribution is a propert y, a name's contribution (in broadlyRussellianframework) is itsb earer,and so on. In theexample above, I suggest, both (3) and (5) make thesame propercontribution to whatis said (namely, Jefferson 's Doctrine),b ut(3) is responsible , as itoccursin (4), forcontributions towhatis said for which (5), in (6), is not. It isthis thatexplainshow (4) and (6) come to saydifferent things. Introducing a last piec e ofterminology, let us saythatexpressionsare phrasally synonymous if theyalwaysproperlymake thesame contribution to whatis said: structurally synonymous if theyare alwaysresponsibleforthe same contributions." It may soundas if theclaim, x to whatS says, (a) e (as itoccursin S) is responsibleforcontributing
simply comes tothis: (b) e (as itoccursin S) has apartwhosesemanticvalue isx . It's notthatsimple. Before explainingwhy, let me observet hatbothresponsibilityand propercontribu t ionare relationsbetweenan occurrenceof an expression in a sentenceandan occurrence of aconstituent inproposition a . Thus, thedistinctionbetween them can be drawnonly if oneacceptsa view like Russell 's, on which wha t a sentencesays is astruct ured e ntity , withoccurrencesof meaningfulexpressions in a sentencecontribut ing constituents to thepropositionexpressed. Suppose we acceptsuch a view ofpropositions.As we haveanalyzedthe 'Jefferson'sD octrine ' example, thename (5) Jefferson'sD octrine has no (semantically significant)syntacticstructure.It makes the same propercontribution to whatis said as thesyntactically complex c reatedequal, (3) thatall men are but the twoexpressions,(3) and (5), are responsiblefordistinctcontributions. Isuspect thatit is rarefor asyntactically simple expression to be responsiblefor preciselythesame contributions as a syntactically complex one. But one might holdthatit is atleastpossible tointroducea simple expression-apredicateconstant , for instanc e -as meaning the same as a 9Some of the them es of the bal an ce fo the section, as well as fo the appe ndix, are developed in Richard 1993. That articl e does not , however , use the ter ms 'phras a lly u llysy no ny mous'. syno nymous' and 's t r uctra
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complex expression(such as apredicateabstract) , so thatinterchange of thesimple expressionfortheotherwouldpreserveidentityof whatis said. Churchoffersjustthispossibilityin criticismof theidea thatsentencessay thesame thingonly if onecan be turnedintotheotherby alphabeticvariationand exchangeof constantsforsynonymousones.l? (Thatis, Church offersthispossibilityas anobjectionto theclaim thatintensionally isomorphic sentencessay the same thing.)I see noargumentt hatwhatChurchsays is possible is not. synonymy is in some sense "completesynonymy" , subSince structural stitution of structural synonymsshouldpreservewhatis said, even ifthose synonymsarenotsyntactically isomorphic. It is nothardto design accounts, of the relationsof sentencesto whatis said, which allow for this. I will themout,the sketchone, withthe hope thatby showingone way of fleshing notionof responsibilityand thedistinctionbetween phrasaland structural synonymy, will beclearer. Suppose, then,thatpropositionsare structuredlikethesentenceswhich express them.11 In factsuppose, for simplicity, thata propositionis an "interpreted sentencestructure" -whatresultsfrom replacinglexicalitems in asentencewithreferents, propert ies, andso forth , andannota tingphrases with theirsemanticvalues. Forexample, suppose thepropositionexpressed '-that is, let ussuppose, the sentence by thesentence'Tom loves Jill
S NP
----------VP
V'
'Tom'
f
'loves'
NP
I
'Jill'
FigureA entity is thestructured lOSeeChurch 1954 . Page refer en ces rae to the reprintedversi on. The criticism ascribed to Churc happears atpage 161. 11 So far as Ican tell, it is not p oss ible to char ac teriz ethedistinction between structurala nd phrasalsy no ny my withoutm aking some mildly contentiousassum pt ionsabou t prop ositio nalstru ctur e.
551
ANALYSIS, SYNONYMY AND SENSE S NP
Tom
thestateof affairs of Tom loving Jill VP lovingJill
-------------i
V' loving V Tom
i
loving
loving
NP
Jill
1
Jill
FigureB On thispictureof propositional s tructure , a propositionis somethinglike a treeeach of whose nodes annotated is with aconst it uent . I assume we can give rules which generatet hepropositiona sentenceexbottomup" , giveninformation a boutthe "propopresses, working "from the sitionalcontributions" of lexicalitems and thesentence'ss tructure. Let me mentiona few suchrules, enoughso thatwe can discuss thesortsof con'brot her' and ' malesibling', tributionst hatsimple and complex nouns, like may make towhata sentencesays. I will givet herules as rules which map labeledbracketingsto labeledbracketings(or, equivalently , annotated t ree example, structures to annotated treestructures). Some such rules-for (R1) PC([NP 'Tom']) = [NP Tom Dector] (R2) PC([N 'sibling'])= [N being a sibling] (R3) PC([A 'male']) = [A being a male] -associatepropositional c ontributions directlywith a lexical item. For example, thethirdrule maps thestructure
A
i
'male' FigureC to thestructure
A
i
being male FigureD
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Otherrulestakepartially interpreted structures, such as
N'
T
N
T
being asibling FigureE andinterpret theirroots. Forexample,we will have rules which "pass values thestructure in FigureE to up thetree".One suchrule will map N'
T
being a sibling
N
T
being asibling FigureF Anotherrulepassingvalues up the t reewillconspirewith (R3) to map AP
T
A
T
'male' FigureG to AP being male
T
A
T
being male FigureH
ANALYSIS, SYNONYMY AND SENSE
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And therewill be rules which interpret adjectival modification . One sucha rule will anointtherootof
-----N'
AP
N'
A
N
a
f3
i i
i
T
FigureI withtheresultofapplyingan operation - letus callit conjunction-to what annotates therootsoftheAP andthelower N'. Such a rule m aps thephrase 'malesibling' to of beinga N' theconjunction being asibling
~and
AP being male
i
A
N' being asibling
i i
N
T
being male
being a sibling FigureJ
My preferredway of using'semanticvalue' identifiesthesemanticvalue of a phrasewithwhatends upannotating thephrase's rootwhen weapplysuch rules. So thesemanticvalue oft heNP 'Tom ' (given the above rules) is Tom Deeter;thesemanticvalue oft henoun'sibling' isthepropertyof being a sibling;thesemanticvalue oft henoun'male sibling'is thepropertyof being ' in thisway, we identify a male sibling.If we use theterm 'semant ic value whata phraseproperlycontributes to whatis said with itssemanticvalue. by And wecandefineexpression e is responsible for constituent c inductively, describingdirectlytheresponsibilitiesof lexicalitems ('male', forexample, , andby sayingthata phrase is responsibleforthepropertyof being a male) of theform
(F) [x [vI···1 [V2 "
.J...[V" ···Il
MARK RICHARD
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is responsiblefor itssemanticvalueandforwhatever t heconstituent phrases 1 through Yn are responsible . 12 I trustit isclearthatwithinthisframeworkthereareatleasttwo ways in which apropertylikethatofbeinga male sibling can come to beconstituent a of of a proposition. Firstof all , it can be introducedvia theintroduction some properties -say, being maleand being a sibling -and an operation , such as conjunction,which maps the propertiesto the propertyof being a male sibling . Such is theway thatbeing a male siblingentersintothe propositionthatTom loves a male sibling. It is alsoperfectly possibleforthe propertyof being a malesiblingto be aconstituent of apropositionwithout thepropertiesof being a male or being a sibling to beconstituents . If we introducet henoun'goy' withthestipulation: y
considertheproperty,beinga male sibling.. . to be a goy is to be that, thisplausiblyhas theeffect ofgiving thenoun goythepropertyof being a male siblingas its semanticvalue;thestipulation does not, however,result of theproposition,that in thepropertyof being a malebeing a constituent Biff is a goy . The frameworkjustsketchedallows for thirdpossibility. a Sincetheseof lexical i tems areassignedto them manticvaluesand propercontributions directly,t hereis no reasonwe mightnot haveexicalitems l which were responsible for anumberof constituents . We could inprinciplehave anexpression 'yog' whichhadthesemanticeffect of t hephrase'malesibling'. Thatis, what 'yog' contributes to apropositionis exactlywhat'malesibling'does: PC([N 'yog']) = PC([N' [A male][N' sibling])), which is to saythatits propositional c ontribution is picturedin FigureJ. If suchis thesemanticsof 'yog' and'goy', thenthepropositionthatTom loves a male sibling is the propositionthatTom loves a yog . Buttheproposition thatTom loves a goy is not t hepropositionthatTom loves a yog , even given thatthepropertyof being a goyis thepropertyof being a yog. Thus we accommodatethepossibilityenvisionedby Church.13 Let usreturnto theparadoxof analysis . A simple line ofresponserests synonymy. Continuing upon thedistinctionbetweenphrasalandstructural forthemoment to workwithina broadlyRussellianaccount , theresponse may be developedas follows . We areassumingthatallnouns,whethersyntactically simple like (7) brother, or complexlike 12For furthe r discussion, see the appendix. d iscussionsee append ix. 13Again, forfurther
ANALYSIS, SYNONYMY AND SENSE
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(8) male sibling, havesemanticvalues oft hesame sort,properties. It is reasonable to assume thatassociatedwitheach syntactico perationwhich can be used to form a nounis an operationwhich takesthesemanticvalues of itso perandsto a property.For example,associatedwiththesyntactico perationwhich forms (8) from 'male'and'sibling'is an operation("conjunction ") whichmaps the properties,being maleand being a sibling,to thepropertyof being a male sibling. When a nounis complex like (8), it will be theresultof applyingsome syntactic o peration0 to some of itsc onstituents . The contribution thenoun (properly)makes towhatis said will bethepropertyresulting from applying thesemanticoperationassociatedwith0 to thecontributions made by O's operands.Buta complexnounwill beresponsible forcontributions beyond theone it makes. The contribution (8) properlymakes towhatis said by (9) I am a malesibling is thepropertyof being a male sibling; (8) isresponsibleforthis,as well as forcontributions made by 'male' and by 'sibling'. Suppose thatit is truethatbeing a brotheris being a male sibling. (7) contributes being a brotherto whatis said; (8) contributes being a male sibling towhatis said. So, giventheidentityof theproperties , thereis a sense in whichthetwo nounsare synonymous. Precisely : (7) and (8) are phrasally synonymous. Buttheyarenotstructurally synonymous. At least I can see noreasonto saythatare, anda compellingreasonto saytheyare not. Fortheclaimthatallbrothersare brothersis different from t heclaim thatallbrothersare male siblings.B uttheclaims would bet hesame if (7) and (8) werestructural synonyms, forthentherewould be no difference in theconstituency or structure of theclaim made by asentencecontaining(7) and thatmade by replacingoccurrences of (7) inthesentencewith(8). The applicationof allthis to theparadoxof analysisis as follows.T he Paradoxis thatthefourclaims, Identity , Triviality,Compositionality, andSynonymy, areinconsistent w iththeclaim thattherearegenuineanalyses.The solutionis thatthereis an equivocationin the notionof synonymy. We may acceptIdentity. And we may a cceptTriviality in t heform: Analysisis not trivial , but whatis said by a sentenceX is Y is trueif X and Yare phrasally synonymousandtrivialif theyarestructurally synonymous.l" We may accept SynonymyWhen being an A is being a B expresses a correctanalysis, A and B (and thusbeing an A and being a B, etc.) aresynonymous, 14 X and Y here ca nnot e b quantifierphrases like ' the man in the corner ', butmu st be singularterms, withthe notionof term const rue dbroadlyeno ughso that(for exam ple) count asterms. infinitivesin subjectposition(as in 'to be happy is to be a Libertarian')
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if by 'synonymous ' wemean 'phrasally s ynonymous', since in a correcta nalysis, thepropertyassignedto thetermwhichgives theanalysandum will be thesame as thatassigned to theterm giving theanalysans,a nd thusthe twotermswillmake thesame propercontribution . Butif we so understand 'synonymous', we must reject Compositionality:Substitution of synonyms forsynonyms, atleastin a positionlike thatof 'to be a B' in theabove, does notaffect whata sentencesays, since onlyexchangeof structural synonyms willpreservewhatis said. We may acceptCompositionality b utnotSynonymy, if by 'synonymous' wemean structurally synonymous,since, as justargued,t heterms in a sentencegiving synonyrns. P a correcta nalysiswillnotbe structural If we respond in this way tothe paradoxof analysis , we can maintain, withMoore,thatwhenone gives ananal ysis , one is nottalkinga boutwords, butaboutwhatwordsmean, or present,or designate.On this account,an y ; in givingan analysis,one analysisis an accountof thenatureof a propert is in no waytalkingaboutwordsor thesentencesin whichthey occur. And theaccountis in accordwiththe factthatif thereareanyanal yses , they arc not trivial(and is consist entwiththeclaim thatthere aresom e analyses). I am notsuggestingthatthedistinctionbetween phrasaland structural synonymy resolvesevery puzzleor problem aboutanalysi s. For example, nothingI have said providesan accountof properties,or ofoperations , as, whichmap propertiesto properties.Thus, I havesaid sociatedwithsyntax nothingwhich might answersomeone who wasskepticalaboutthetruthof Identity . My concernin thissectionhas been onlywiththe broadlysemantic questionpos ed by theparadoxof analysis , namely: How can sentence s differingonlyby synonyms say differentt hings? 15The account in th e text need s tobe generalized, for we wou ld wantto be able to apply er is to be a male sib ling' or ' being it to ana lyses igven by sente nces lik e 'to be a broth a brother is being a m ale sibling'. I sha ll ot n do thather e, since I do not thin k thatso gen er ali zing, in and of itself,is much m or e enlighteing n tha n whatI have thus far said. (What is said in the text also needs to be gen er aliz edto be applied to the ana lys isof propositions; forsi mp licity I pretend through outthe discuss ion thatonly prop erties and relationsare theobjects of analys es.) Why notjustsay that'bro t he r'a nd 'm ale sib ling' ar e notsy no ny mo us,or atleastthat they do not make the same contribu t ionto whatis said , and solve the paradox of anal y sis thatway? This "solut ion" ign or es the factthatana lysisis closely relate d otdefiniti on. Furthermor e, it seems thatthe semantic va lue of a noun- wh a t it co nt ributesto what is said-is a er; prop erty. If so, then the semant icvalue of ' brother' is the prop erty of being a broth the sem antic value of 'm alesiblin g' is the prope rtyof being a male sibling . These are presumabl y the sam e p roperty. But the proposed "solut ion " is one on which we sim p ly say thatthe semantic values ofthe two nouns are different. T he foll owingsect ion dis cusses furth er some of the relations between ana lysisand synonymy.
ANALYSIS, SYNONYMY AND SENSE
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2.
The skepticalcriticsays: YouacceptSynonymy. Youthendistinguishtwo sortsof synonymy, phrasaland structural.But obviouslyenough, structuralsynonymyis real synonymy. Attheleast,t hesynonymyof structural synonyms is greater thanthatof phrasalsynonyms, since onlystructural synonymscan be substituted for oneanotherw itha guarantee thatwhatis said is preserved.So, since 'brot he r" is "more synonymous"with itselfthan with 'malesibling',shouldn't one say, onyourview, thattheclaim (Bl) To be a brotheris to be abrother, providesa better analysisof theconceptbrothert handoes theclaim (B2) To be a brotheris to be a male sibling? Suchcriticismcanonly beansweredby sayingsomethingaboutthenature ofanalysis.As I see it, thereare twosomewhatdifferent reasonsforwanting an analysis,a ndthustwosomewhatdifferenta ccountsone mightoffer oft he purposeor pointof analysis , and thusof itsnature. In some situations,an analysisis soughtfor explanatory purposes. We are "in touch"witha propertyor relation-that is, we are able to refer to and thinkaboutit, and able, in favorable situations,to ascertainor detect its presence.Butthoughit is clearthatwe are intouchwith theproperty, it is notclearhow we come to be in touch with it. In some such cases, a proper analysiscan providethematerialsfor explainingthis. Acorrect analysisprovides a collection of propertiesand relations(and "a mode of compositionthereof') , which are(underthemode ofcomposition)identical with the analysandum, and withall of which we are touch. in Given such, we may be in apositionto answerthequestion (Q) How is itthatwe are intouchwiththeanalysandum?
A second reasonto seek ananalysisis classificatory:One wantsan analysis becauseone has apropertyor relation which one believes is (in some sense) reducibleto othersa ndsome mode ofcombinationthereof.Giving an analysis,of course, is a way of specifying sucha reduction. This proto-analysis of analysisanswersthe critic'squestionaboutsynonymy. Suppose an analysisis soughtfor explanatory purposes. Then a necessaryconditionon ananalysisis thatit providethematerialsfor an acceptableanswerto (Q). (Bl) and its ilkcannotprovidesuch; (B2) and its ilk, ceteris paribus, can. Suppose an analysisis soughtin orderto providea reduction.( Bl) cannotprovidea reduction , (B2), ceteris paribus, can. I do notclaim-andpractitioners of analysislike Moore did not claim thatwhenevera questionlike (Q) arises, analysisis possible. Likewise, I do , thereis an not claimthatwheneverone propertyis reducibleto another
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analysiswhichstatest hereduction.Besides therequirements justsuggested , an analysism ustexhibita "decomposition", of apropertyor arelation , which was in someimportantsense "conceptually accessibleto us" beforetheanalysis was given. Wemighthave anabilityto refer t oa propertywithouthaving whichcouldcountas decomposiconceptualized it- t hat ,iswithouta nything tion ofthepropertybeing in anyplausiblesense conceptuall y accessibleto us. Natural kinds generate examplesof this. It seems possiblethatin some such is a "complexproperty"- eventhoughwe cases apropertyhas a"nature"-it do notconceptualiz e constituentsinto whichthepropertycan be "resolved". If so, theproperty 's naturewill not be conceptually accessibleto US. 16 Thus we havethepossibilityof examplesin which aquestionlike (Q) may arise, or in whichreductionis possible, butin which(Q)'s answeror thereduction is not to beobtainedanalytically. Summarizing: Analysesarecertaintrueclaims ofpropertyidentity . They differ from"trivial"p ropertyidentitiesin termsofstructure andconstituency. An analysishas constituents which can "becombined" to yieldtheanalysandum (usingthe"mode of composition"encodedby thesyntaxof a sentence accommodatingsuch constituency ;a expressingan analysis)a nda structure "trivialidentity " does not.Whethera claim whichsatisfiestheabove conditions gives ananalysisis a relativ e affair: A claims ian analysisfor a person x, or for a population P, onlyif it satisfiestheabove conditionsa ndinvolves accessible tox or a decompositionof theanalysandum which isconceptually to P. If allthis is correct , it explainswhy "complete"- t hat is,s tructural synonymyis not erquiredof theforms occurringin a sentencewhich gives an analysis.And so, if allthisis correct,we have a response to the original objection.Butone mightatthispointwonderaboutthestatusof Synonymy: Why shouldwe suppose thatsynonymyof forms inany sense is requiredin a sentencegiving ananalysis? Historically, I suspect,Synonymy wasmotivatedby theidea thatan analysis must reflectconceptually accessible structure, along with t hepresuppositionthatsuchstructure would always (andonly) bereflected by synonymous forms. The idea, put prettyroughly, ist hatanalysisis "conceptually accessible" only if obtainablea priori ; butif anidentityis knowablea priori, the terms in any sentenceexpressingit must be synonymous,or at leastanalytically equivalent.(And if it be asked, why does analysishave to reveal conceptually accessiblestructure, theanswerwill be in erms t of one ofthe purposes-theexplanatory -whichmight inspireone to seek ananalysis.) The criticworriedaboutSynonymy will not be satisfiedwith suchhistorical defenses of it. imagine I her posingthequestion: If allthis is correct aboutthe"point" of analysis -ifthis is really whatthosewho wantedphilonot simply reject Synonymy? sophicalanalysesof conceptswereafter-why 16W hich is not tosay thatit couldnot come to be so accessi ble.
ANALYSIS, SYNONYMY AND SENSE
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Suppose we wantan explanation.All onereallyw ants, or needs at any rate, to answer(Q), is a collection of propertiesand relations(and a mode of t hat(a) theproperties(underthemode of comcomposition)which are such position)areidentical w iththeanalysandum, and(b) we are"in touch"with theseproperties(in theappropriateway). A sentence
(K) To K is to K' , mightmake a claim whichprovidedsuch acollection withoutK and K' being synonymous. Indeed, considerthe paradigmexampleof a conceptwhich is a candidateforanalysis,knowledge.P resumablythereare non-trivial necessaryandsufficientconditionsfor knowingsomethingto which ourpractices of ascribingknowledge are sensitive. Shouldwe solvetheGettierproblem, we will do so by w ritingdown asentenceof theform of(K) statingsuch conthatwhen ditionswhich gives ananalysisof knowledge. It quiteunlikely is we do this, K and K' will besynonymousin any sense whatsoever.T hus, analysisdoes notrequirethatanalysans and analysandum be expressibleby synonymousexpressions. Much thesame, thecriticcontinues , could besaid aboutthereductiveaccountofanalysis:Given thata propertyis somethingwhich is to be specified independently of how it might be conceptualized (so thata propertyis, or correspondsto, somethinglike afunction-in-extension from worlds to individuals), thereseems no reasonwhy illumination on thenatureof aproperty mightnotbe castby theclaim made by(K), thoughits componentinfinitives werenotsynonymous. In so arguing , ourcriticseems to assume thatmeaningis in some interestingsense transparent, so thatexpressionswhich havethesame meaning must, if weunderstand t hem, be such thatwe knowthem to havethesame meaning. For thereasonshe gives forrejectingSynonymy is this:something could dowhatan analysisis supposed to do(and hence be an analysis): but while weantecedently understood t hetermsin which it was given , we found if underit surprising . This is an objectionto Synonymy only antecedent standingofsynonymsmust suffice forrecognizingt heirsynonymy, or atl east theirco-extensiveness. Thatmeaningis thustransparent is a familiarview. It is associatedwith Fregeanviews, on whichmeaningis (intimatelyr elated to) sense;andsense, since it is a way ofthinking,must be transparent.I willarguedirectly againstsuch views in section4. At themoment, I observe onlythatthere are coherentviews ofmeaningon which: (a)expressionsA and B may be synonymousin languageL; (b) personX may be a competentspeakerof L, and therefore understand sentencesin which Aand B occur;but(c) X does not, even onreflection, know thatA and B are synonymous, or (supposing them to be predicates)thattheyapply to the same things. Almost any view ofmeaningwhich seesthemeaningof apredicat e as beingdetermined , ctors, will allow for this. Consider, forexample, in part, by broadlysocial fa
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TylerBurge's (1986) view, on whichthemeaningof some termsis determined reflection to bestexplainthe by whata communityofspeakerswould find on application oftheterm. Suppose this is whatmakes 'sofa' mean ' upholst ered piece offurniture m eantforseatingseveralindividuals' . Then a competent sp eakerwho understands'sofa' need not know, even on reflection , that'sofa' and the longerterm are synonymous,since the reflection need not reveal whatthereflection of thecommunitywould decideaboutthetermY It is not truethatto respondto theparadoxof analysi s as I have is, in effect , to give up Synonymy (and, therefore , a bett er responsethanmine is simply to admit that synonymyand analysishave nothingto dowitheach other.)So long as it is allowed thatmeaningmay be opaque, thesolutionof section1 is in principleopen to us . 3.
For aFregean,meaning,tied as it would seem to be to sense, issupposed not to be opaque. How does the Fr egean respondto theparadox? In a wellknown review of articlesby Max Blackand MortonWhite, Church(1946, suggestion. 18 page 133) made thefollowing The paradoxof analysish asan obviousanalogywithFrega's puzzle. . . as to how anequation , say 'a = b' , caneverbe informativ e-because, it seems, iftheequationis truethen'b' is replac e ableby 'a' , andhence 'a = b' is thesame in meaningas 'a = a', ... theparadoxof analys is is a specialcase ofFrege'spuzzleand is to be solved inthesame way .... . . . by thedistinctionof sense and denotation sis thata brotheris a malesiblingis notaboutthe classes . . . theanaly brother , male, and siblin9 butaboutthecorre spondingconcepts (which are the senses of the class names ' brot her',etc.). Hence a correct expression of the analysismust employ names of theseconcep ts. In theexpression'b = ms', thenames 'b', 'm', and 's' must denote the concepts'brother ', 'male', and 'sibling' respectiv e ly-notthe classes. Now like anyothername, a name whichdenotes a conceptmust have, besides its denotation,a sense. From the truthof theequation'b = ms', it followsthatthenames 'b' and 'ms' have hesame denotation , henceone may replacetheotherin a sentenc e withoutchangingthe denotation(truthvalue). But if theanalysisexpressed by 'b = ms' is non-trivial, the names 'b' and 'ms' have differentsenses, hence the replacementof one bytheother in a sentence may very well h c ange thesense (proposition)expressed . 17Burge has, in fac t ,argued thatone can act ively o d ubtthe syn onym y withoutbein g inco m pe te nt. s , such as that 18Limitations of space preventm e from discussing other Fregean account in Anderson 1993.
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I assume thatChurchwouldacceptthe forms mentionedin Identityas expressingan analysis . If so, Churchapparently a cceptsIdentity, ComposiandTriviality,19as well as a modified versionof Synonymy: tionality, Synonymy*If being an A is being a B expressesa correctanalysis,A and B are synonymous. WhatChurchrejectsis inferencessuchas The nouns A and B aresynonymous, So being an A and being a B are synonymous. Apparently,C hurchsees the forms lastmentionedas creatingan oblique context,in whichtheembedded nounsname theirordinarysense andexpress conceptsofthesesenses; when ananalysisis non-trivial, thesenses expressed are distinct .e" It seems to me thatthereare twoproblemswiththisresponse. Given the claimmade by asentenceof theform (10) To V is to V*,
let usspeak of its corresponding biconditional, meaning therebytheclaim made by asentenceof (roughly)t heform (11') WhateverVs V*s, andvice versa. And let us also speak of thetriviality corresponding to (10), whichis simply theclaim made by asentenceof (roughly)t heform (11") WhateverVs Vs, and vice versa. For example,thebiconditional corresponding to theclaim expressedby
(12) To know apropositionis to havejustifiedtruebelieftherein , is theclaim made by (13') Whoeverknows apropositionhas justifiedtruebelieftherein , andvice versa,
whilethecorresponding t riviality is thatexpressedby (13") Whoeverknows apropositionknows it, and vice versa. 19 Actually , Church's wordingsuggests thathe think s thatin some cases an anal ysis might be trivial ; note the condition al i' f an ana lysis si non-trivi al'. I willignore this com plication, since it is orthog onalto whatI have to say aboutC hurc h' s esponse, r given the admission thatnotallanalysis si such. 20 Which is notto say thatthey express their indirect sen ses. As NathanSalmon p ointed outtom e, Churchrejects Frege's notion of indirect sense. For discussion, see Salmon 1993, especia llynote10.
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One objectionto Churchis thaton his account , whoeverknows thetrivialitycorrespondingto ananalysisknows thebiconditional correspondingto it-butthisis simply nottrue . I can know, thatall who know proposit a ion know it, withoutknowingthatall who know proposition a havejustifiedtrue beliefplus X therein,where X iswhateveryou add to justifiedtruebelief to get ananalysisof knowledge.I!Analogously,Churchmust allowthateitherI cannotbe surprisedby B-whereB is thebiconditional corresponding to acorrectanalysis -orI can be surprisedby T, where T isthetriviality corresponding to theanalysis.For asurprisinganalysis,n eitherhornof this dilemma lookspromising. . A second problem with Church'saccountis this. Let us forthenonce pretendthatknowledge isanalyzed as justifiedtruebelief.Churchwishes to respecttheintuitionthattheclaim (A) theconceptknows proposition p is theconceptknows proposition p, is trivial , thoughtheclaim (B) theconceptknows proposition p is theconcepthas justified true belief in p, 22 which gives ananalysisof knowingthatp, is not trivial. Butif (B) is not trivial,neitheris theclaim
(C) takewhateveryou like x : x fallsu ndertheconceptknows p iff x falls undertheconcepthas justified true belief in p. If knowing (A)and othertrivialities does not suffice (given normallogical
prowess) for knowing (B), shouldnot it suffice for knowing (C). However, it would seemtheclaims (P) Whoeverknowspropositionp fallsu ndertheconceptknows proposition p, andvice versa, (Q) Whoeverhas justifiedtruebelief in p falls undertheconcepthas justified true belief in p, and vice versa, as (A). On Church'saccount , (Q) is identical w iththeclaim arejustas trivial
(R) Whoeverknows p fallsundertheconcepthas justified true belief in p, andvice versa. Butsurelya Fregeansuch as Churchholdsthatanyone who knows (P)and (R) will, iftheyareat allr ational and reflect for justa moment, know (C). trivialities suffices for So Churchis committedto sayingthatknowledge of 21This pointis made by TerenceParsons (1981) . 22Here, p issupposed to besome fixed proposition.
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knowledge of (C) . And this seems to imply t hatknowledge of trivialities suffices for knowledge of (B). And this seems to imply that(B) itselfis trivial-a claim whichChurchseems committedto denying. 4.
I would urge Fregeans suchChurchto as adopttheaccountof theparadox of analysissketched insection1. Of course, the account comes ata price. A relatively small costtheaccount of is the pictureof thoughtsas structuredcollections of senses . Once this isaccepted,phrasalsynonymy may be identified with (invariable)identityof senses ofphrasesas wholes;s tructuralsynonymy with(invariable)identityof sensescontributed to thought expressed. Acceding to this , a Fregeancan saythat'brother'is phrasally b ut notstructurally synonymousto 'malesibling'; thus,the claimthatbeing a brotheris being a male sibling distinct is fromtheclaimthatbeing abrother claim, hasthesenses of is being a brother . The former,butnot thelatter 'male' and'sibling'as constituents. Entertaining the former , one might say, requiresinter alia thinkingof brothersas males and siblings ; entertaining does notrequire that. thelatter A somewhatgreater cost may be paid, if one acknowledges thepossibility of surprisinganalysis. For on a Fregean version thepresentaccount, of in a correctanalysis theexpressionsgiving analysans and analysandum have thesame sense. Thatis, thesense of thosephrasesas wholes -theirFregean semanticvalues-are t hesame. Fregeanshavegenerally heldthatwhen expressions havethesame sense, a speakerwho understands bothwill know thatthey aresynonymous,atleastif she minimally reflects on . Buta it surbutin prising analysis would be one giventermsantecedently in understood, termswhich we need not recognize, even afterminimal reflection, apply to to the same things. I do notthinkthatwhattheFregean ought to do here is reject section l'saccountoftheparadoxof analysis;q uiteindependently ofwhetheror not it is accepted,theFregeanis going to have to abandonthe ideathatwhen expressions havethesame sense, theunderstanding speakercan recognize this. To make a case for this, I being pointingout by thatanalysispresentsa quitegeneralproblemfor aFregean,whetheror not sheacceptsmy account of itsnature . To see why, let us returnto thepretense (A) Theclaim,thatknowledge is justifiedtruebelief, is ananalysisof knowl sentence edge, and this claim the is claim expressed by the English 'knowledge isjustifiedtruebelief'. It seems possiblethattherebe someone-Freddie,callhim-suchthat (B) Freddie does not know thatwhateverhas knowledge has justifiedtrue belief, and vice versa,
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but (C) Freddie is anormal, competentspeakerof English, with reasonable logical prowess , whounderstands 'knowledge'and'justifiedtruebelief', and who has refle ctedupon thesentence'whateverhas knowledge has justifiedtruebelief,andvice versa'. If (A) through(C) are jointlypossible, thena Fregeanview which acexpressionswiththesame sense (inthecontextofthe ceptsthatexchange of exchange)preservessense, must rejecteither
Synonymy; If being an A is being a B expresses acorrectanalysis,A and B (and thus'being an A' and 'being a B', etc.) make thesame contribution to whatis said, or
Transparency., It impossible is for a(normal,rational)person to understandcontextinsensitivepredicate s A and B which makethesame contribution to what'ssaid butnot knowwhatis said by Whatever A s Bs and vice versa (or be suchthathe would come to know t hisif he were to reflect on it).23
I takeit thatrejectingTransparency., requires rejectingTransparency, the to this discussion . constraint mentionedin theintroduction surprisedby an analysis,even Surely itis possible thatsomeone should be thoughshe antecedently understoodt hetermsin which it was given . If so, thensomethinglike (A)through(C) arejointlypossible.F" If we thinkthata sufficientconditionforunderstanding an expressionis satisfyingconventional interesting analysis standards for itsuse,25 andweacceptthatphilosophically is possible, thejointpossibilityof somethinglike (A) through(C) borders on the obvious. It follows ata th Fregean must reject eitherSynonymy; or Transparency.This is quite independent of whethershe acceptstheaccount of analysisproffered insectionl. WhatshouldtheFregeando? Church'sway was torejectthe first claim . I 's solution wasunfortunate . I offer adirectargument havearguedthatChurch againstTransparency.Suppose thatI introducea predicateA by defining it thus x is an A =df X is an BCD, andthatI introducea predicateZ by defining itthus x is a Z =qf x is a YXW. 23
1 pretendthat'knowled ge'a nd'j ust ifi ed truebelief 'are contextinsensitive forpresent purposes. 24S p eaking strictly : The sortof sta te ofaffairsfor which (A) thro ugh (C) are repres of a propertyin terms which are antecede ntl y sentat ives (in which ther e is an analysi understood , but which is surpris ing) is pos sible. (Giv en thatknowledg e is not justified truebelief, it is presumablyimpossibl e that (A) through(C) them selv es be true.) 25 1 borrowthephrasefrom Soames , op. cit .
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The terms in thedefiniens maythemselvesbe definedterms, all of which and all of which I have definedterms in of terms which are I understand, thatby replacing defined themselvesdefinedterms. Surely it could be the case terms with definedterms I might eventually transformthedefinitionof A thesame expression,thoughevenafterreflecting and thedefinition of Z into on the claim,t hatbeing an A is being a Z, I do not know thatit is true. (Someone who saysotherwiseshouldtakea course inanalysisor topology.) Shouldn't we saythatin defining AandZ as I did, Isetthe sense of 'x is an A' to bethatof 'x is aBCD', andanalogously for 'x is aZ'? Afterall, Iwas giving a definition. If thisis so, thesame may be trueof theotherdefinitions involved. So long substitution as ofexpressionswiththesame sense preserves .26 It shouldbe stressedthatthe sense, it followsthatTransparency is false argumentdoes notdepend upon the ideathatthemeaning (and thusthe . The definitions sense) of one'stermsmightbe set, inpart,by socialfactors may have been given by solitary a m athematician. If sense is nottransparent, one canunderstand sentenceswiththesame sensebutnot realize thattheyhave thesame sense. And so, ifunderstanding premiss a in (relevant) a sentence,assentingto it,andbeing willing to use it as reasoningsuffices foro ne's believingwhatit saystrue--aclaim which seems quitetemptingwhen made aboutthesentencesof deductivesciences like mathematics-it seems to follow t hata rational personmay believe the sense , of a sentenceS and believethesense of itsnegation.So Frege's Constraint mentionedin theintroduction, is false. Some will saythatthedenialof Frege's Constraintabsurd is ; only someone who does notunderstand w hata sense issupposed to be could bet empted p erson might beto deny it.Butit is not absurdto holdthata rational given thatsenses are thesortsof thingswhich lieve and disbelieve a sense, (inter alia) can be and sometimes are introducedor articulated by definitions. Suchdefinitions,to steala phrasefrom TylerBurge(1979), articulate conceptual structure, by spellingoutinterrelations amongconcepts.Conceptualstructure, simply becauseit may be complex, need not be transparent s tructure is transitive;the obvious to a thinker:The identity of conceptual identity of conceptual structure is not. Sense, qua cognitively accessiblestructure which may becapturedor im26The substitutions are, we may suppose, substitutions of structuml synonyms forstructural sy nonyms. One might respondthatalltheFregeanis committedto is thata speakerwho understandsexpressions can (perhaps underidealization w hich rarelyobtains)recognizetheir synonymy. If so, thereis no bar to theFregeanacceptingthesolutionof section1. Of clearthat that position will course,if this is theFregean's position, it is notaltogether supportanythinglike Frege's Constraint , since in non-idealcircumstancesa speaker may understandingly accepta sentenceS and thenegationof a sentenceT (T and S withthe same sense), therebyacquiringcontradictory beliefs. And if theFregeanpositionwillnot supportanythinglike Frege's Constraint, it is notaltogether clearthatFrege's puzzlecan . be solved(simply) by distinguishingb etweensense anddenotation
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posed viadefinition,need not betransparent . One may apprehendthesame sense twice over -may thinkaboutan objectin thesame way twice over withoutrealizingthis. (Again, I stressthatthis point has nothingto do withthefamiliarpointthatmeaning, since it is socially constructed, is not thesame sense transparent .) Someone may believe and disbelieve one and because, forexample, theconceptual structure which isthatsense is accessithat ble to one in different ways which do not allow one to it seeis the same thingwhich is accessible to each case . The questionnaturally arises, Shouldn ' t we thenidentifysense with the "modes of access" we have to conceptual s tructure, therebypreservingthe unF'regean such transparency of sense? It shouldbe pointedout howutterly a responseis. The modes of access inquestionare(atleastin many cases) vehicles which we manipulate in thinking. somethingliketherepresentational Thatis to say, thesemodes of access to sense are reallywhathave senses, not the senses themselves . They are thesortof thingswhich Frege has in mind when hecautionshis readersnot tomistakesenses for ideas . Whatgives a word, in anatural languageor a "language of thought" , its apparatus . Butnot every sense, are the connections it has to our conceptual connectionto ourconceptual a pparatusis relevant to sense. For example, differences in spelling,p ronunciation, and time ofacquisitionareirrelevant to extremelyrelevant to identityof sense. Butthesesortsof differences may be ourabilityto "process" asentence.In particular , such "orthographic" differences may block our abilityto seethattwosentencess tandin one oranother . This relationof equivalence,even though we understandb othsentences WhatI am urging(andwhatis somewhat much should benon-controversial. o rthography " (in some controversial) is thatwe recognizet hat"differences in suitablybroadsense) may inhibitourabilityto seethatexpressionsare synonymous, eventhoughsuch differences do not inhibitour understanding . At incisiveness- our leastsometimes, it isourfinitudeandlack ofc omputational inabilityto perfectlymanipulatethe symbols which realize our thoughts whichaccountsfor our failure to identities see of sense. At least some of the time, thatwe fail to see identityof meaningis nottheresultof our failing to t heexpressionswhosemeaningsareidentical. understand Let me sum up whatI have triedto do inthis discussion. I began by introducing a distinctionbetween two s ortsof synonymy, phrasalandstruc tural.P hrasesare phrasally synonymousif "as phrases" theyintroducethe same constituent into whata sentencesays; they arestructurally synonycontributions" to whata sentencesays. I mous if they makethesame "total paradoxof analysisin the arguedthatthedistinctionallows us to solve the version I cited from Moore, since the ambiguityin thenotionof synonymy Synonymy andCompositionality, on induces anambiguityin two of the claims, whichtheparadoxrests. I defended thissolutionagainsttheobjectionthat it amountedtoabandoningtheclaim, thata sentencewhich gives an analysis
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is an identitywhose flankingtermsare "really"s ynonymous. I arguedthat thestandard F regeansolutionto theparadox, Church'ssolution , is defective, and thatit is open to aFregeanto adoptthesolutionpresentedhere. Finally,I arguedthatsince it seems we can be surprisedby analysis, thoughwe understand theterms in which it is given , theFregeanwhoaccepts thatanalysisis possible has to give upeitherthe idea thatthesentence which expresses an analysisis an identityflankedwithsynonymousforms, or must abandontheview thatone whounderstands synonymousforms will recognizetheirsynonymy (and thusmust abandonclaims such as Frege's Constraint).I argueddirectlyagainstthelatter . Whetherthis argument , if successful , underminesthe motivationfor preferringa broadlyFregean accountof meaningand thoughtto abroadlyRussellianone is, perhaps, an issue betteraddressedon another o ccasion.F APPENDIX
1. For thosewho likehair, I give statementsof therulesfor constru cti ng propositionsalludedto insection1. In statingsuch rules,it is most convenientto understand l abeledbracketings so thatwhenYI in (i) [x YI Y2 ... Yn] is not itselfof theform
[z WI W2 . . . Wn], thenYI in (i) is itselfa "label", as is 'X' (and botharelabelsof thesame node). This means, forexample,that [N' being a malesibling[AP being male [A being male]] [N' being a sibling[N being a siblingJJ], names thestructure illustrated in FigureJ above. ,Furt her , when proposia tionalcontribution of theform C = [x Y*1 Y*2 . .. Y* n] is assignedto the phraseP = [x Y1 Y2 .. . Y n], we say thatY" 1 is thesemanticvalue of P; we also call Y*1 t hesemanticvalue of C . Thensome rulessimply pass semantic values uptheproposition: PC([N' [N oJ]) = [N' SV([N 0] PC([N 0])], PC([AP [A oJ]) = [A P SV([A 0] PC[A oJ]. Othersinterpret a structure [x 01,02,"" On), given interpretations forthe a's: 2 7 Jody Azzouni , David Braun, Graeme Forbes , and Nathan Salmon made helpfulcom m entson various draftsof this paper , for which I thankthem. I also thank thePhilosophy Departmentof theUniversityof Rochesterfor helpfulcomm ents on an oralpresentati on.
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MARK RICHARD PC([N' [AP a][N' ,8]]) = [N' CONJ( SV([AP aD, SV([N' ,8])), PC([AP aD, PC([N' ,8])].
Here, CONJ is an operationwhich maps a pair of propertiesto aproperty; thevalue ofC ONJ(P,Q), ofcourse, is to be"theconjunction" of P andQ (so thatCONJ(beinga male, being asibling)is being a malesibling). It is clear, I hope, thattheserules suffice to assign thestructure picturedin FigureJ to thephrase'malesibling'. 2. ThosefamiliarwithChurch'snotionofsynonymousisomorphism will recognizethenotionof structural synonymy,as I have fleshed out,as it being kindredto Church'snotion . Church(1954, page 161) proposed thelatter as therelationin whichsentencesstandwhen theyexpress the same belief. Speakingsomewhatroughly,expressionsare synonymouslyisomorphic providedone can beobtainedfrom theotherby a sequenceof operations involving: alphabeticvariation;exchangeof eitherindividualconstantsof descriptionsfor either synonymousindividualconstants or descriptions;exfor either synonychangeof eitherpredicateconstants or predicateabstracts mous predicateconstants or abstracts . (Church'scriterionforsynonymy, in thisdefinition , is identityof sense.) Justas structural synonymydoes not requiresyntactic isomorphism, synonymouslyisomorphicexpressionsneed not be syntactically isomorphic,since apredicateconstant may be synonymously isomorphicwithan expressionwithsemantically significantstructure. Church,however,cannotdraw adistinction,analogousto thatbetween synonymy, betweensynonymousexpressions(thatis, phrasalandstructural ones withthesame sense) and those which are synonymouslyisomorphic. He is, forexample, committedto sayingthatallpredicateswhich have the same sense aresynonymouslyisomorphic. For suppose thatpredicatesF and G have thesame sense, but are notsynonymouslyisomorphic. If we suppose (as, in thiscontext , is fairenough)thateverypredicateis eithera predicateconstant or apredicateabstract,it follows thatF and G are both predicateabstracts,since a predicateconstantis synonymouslyisomorphic withany predicate(constant or abstract) withwhich it issynonymous. "But ...", Church(1954, page 161)counsels"nothingpreventsus fromintroducing (say) a predicatorc onstantR as synonymouswiththeabstraction expression (Ax)[ . . . X •••J, and takingR == (Ax)[ . . . X •••] as an axiom. And ift his is done, thenR must be interchangeable with (Ax)[ . . . x ] in allcontexts, includingbeliefcontexts,being synonymouswith (Ax)[ x . . . ] . . . ." Thus, F' and G' which weresynonymouslyisomorphic wecouldintroduce c onstants withF and G respectively.Being constants withthesame sense, theywould be synonymouslyisomorphicwith eachother . Butsynonymousisomorphism expresses the sam e belief as, is supposed to capturetheequivalencer elation and is presumablytransitive.Thus in a situationin which we were to make such introductions, F and G would besynonymouslyisomorphic. Since making theintroductions would not affect themeaningsof F and G, F and G
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are in fact synonymously isomorphic. To avoid this result, it is sufficient (and, I believe,necessary)to allow thatpropositionshave aconstituent structure, and thattheirconstituents structure as well.I f we allowthis, thenthefollowing may have aconstituent possibilitypresentsitself:F is thepredicate(Ax)[A x & Bx], Gis (AX) [CX & Dx] . Thesenses of thepredicatesas wholes are the same, since the sense&of maps the senses of Ax and Ex to thesense to which it maps the sensesCx of and Dx. This makes F and G phrasally synonymous. So long asthesense pairsof argumentsto thesame value, and the senses of & can map different of F's syntacticc onstituents are constituents of whatis said by asentence occurrences of F and G in whichF occurs,sentenceswhich vary simply by need not express the same thought.For inthiscase, thepredicatesF and G will beresponsibleforcontributing distinctsenses towhatis said, and will not bestructurally synonymous. 3. Finally , I notein p assing that , given thatthe notionsof structural and phrasalsynonymyare fleshed out in the way indicated,t hesolutionoffered here to the paradoxofanalysisdoes not run afoul ofobjection an tosolutions likethatof Max Black.T his objectionis due toMortonWhiteand extended by Churchin a well known reviewpapers of by WhiteandMax Black (1946). According to White, Blackassumed that"no twosentencescan express the same propositionif one expresslymentionsa relation which isdistinctfrom the relation expressly mentionedby theother". Call thisBlack's Principle. White(1945) offers 21 is 3 times 7, 21 is thrice7, as a counterexample; Churchsuggeststhatif B has thesense of 'AUAXAy(U = xy)' ,
sentencesrelated as are b=ms, B(b,m,s) ,
constitute a counterexample. The accountof theParadoxof AnalysisI am proposingis not commi tt ed to Black'sPrinciple . An analysisis a claim which is anidentityand so is expressible by a sentenceof theform To be an A is to be a ,B where 'is' functions to pick out identity an relation and B possesses syntactic complexitybeyondthatof A. The constituents oftheanalysisare not merely b utvariouspropertieswhich "go to make it up" , thepropertybeing analyzed
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by anoperation associatedwith propertieswhich aremapped totheanalysans thesyntactic s tructure (of B) in somesentenceexpressingtheanalysis . This outtheexistenceof nouns A* and B* which does not in and of itself rule arestructurally (and thusphrasally) synonymous, butare suchthatB* has, and A * does nothave, semantically significantsyntactic s tructure . One can t heexpression'thrice'via thedefinition allowthatsomeone whointroduced
x is thriceY =df x is threetimes y , insuredthatWhite'spair of sentencesmade the same claim. On the view I am proposing, an "atomic" expressioncould be responsiblefor anumberof contributions to what issaid by sentencesin which it occurs. Of course, if 'thrice'were sointroduced,thesentence'to bethricey is to bethreetimes y' would not express an analysis,since it would make the same trivialclaim as 'forx to bethreetimes y is forx to bethreetimes y' .28 REFERENCES
Anderson,C. A. 1993 Analyzinganalysis,Philosophical Studies, vol. 72, pp. 199-222 . Burge,T . 1979 SinningagainstFrege, The Philosophical Review. 1986 Intellectual norms and foundations of mind, The Journal of Philosophy, vol. 83, pp. 697-720. Black, M. 1945 The "Paradox ofanalysis"again: A reply, Mind , vol. 54, pp . 272-273. 280ne might reject White's origin al t" hrice " -objecti on toBlack. For, theresponse goes, the claim made by For x tobe thrice y is for x to be threetimes y providesan analy sisof the relati on , x is thrice y . Butif White's pair of senten ces say the same thing, then theclaim made by this lastsenten ce is in some sense trivial,s ince it is the verysame claim as thatmade by For x to be three times y is for x to be thr ee times y . I have some sympathywith this objection, since I have some sym pa t hy with the idea thatsemanticstru ctur e must be art i culated(so thatif x is a constitu entof claim p, andS expresses p , ther e is a constituent of S whosepropercontribution to whatS says is x) . But I thinkone may cogentlydefendWhite thus: If White's pairof sentenc es aresynonymous, then' to be thric e y is to be 3 times y' does not quit e express an ana lysis. But it does expresssomething(thatfor x to be threetimes y is for x to be threetimes y) from which p who knows this last claim can an analysisca n be quiteeas ily "de rived". For a erson reasonthus: "T he re si a relation,x is threetimes y . Let us int rod uce'todrice' as a verb thatindicatesthis relation(in the way that'knows' indicates knowledg e). So, for x to drice y is for x tobe thr ee times y" . Someone who reasonsin this way has reasoned her way toan analysisof the drice-s-i.e., thethric e -relation .
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Church,A . 1946 Review of Black and W hite(1945) , The Journal of Symbolic Logic, vol. 11, pp. 132-133. isomorphism andidentityof belief , Philosophical Studies, vol. 1954 Intensional 5, pp. 65-73; reprintedin Propositions and attitudes (N. Salmonand S. Soames, editors),OxfordUniversityPress, Oxford, 1988, pp. 159--168. Moore, G. E. 194~ A replyto mycritics, Th e philosophy of G. E. Moore (P. Schilpp, editor), Open Court,La Salle , p . 665. Parsons, T. 1981 Frege's hierarchi es of indirectsense and the paradoxof analysis,Midwest studies in philosophy, vol. VI (Frenchet al., editors),Universityof . MinnesotaPress, Minneapolis Richard,M. 1993 Articulated Terms, Philosophical Perspectiv es, vol. 7 (J . Tomberlin,editor),NorthridgeP ublishing,A tascadero. Salmon,N. 1993 A problem in the Frege-Churchtheoryof sense and denotation,Nous, vol.27, pp. 158-166, Schiffer , S. 1990 The mode-of-prese ntation p roblem , Propositional attitudes (C. Anderson and J. Owens, editor s), CSLI , Stanford . Soames, S. 1989 Semanticsand semant iccompetence, Philosophi cal Perspectives, vol. 3 (J. Tomberlin,editor), Northridge P ublis hing, Atascadero , California . White ,M. 1945 A noteon the"Par adox ofanalysis, " Mind , vol. 54, pp . 71-72.
NATHAN SALMON
THE VERY POSSIBILITY OF LANGUAGE A Sermon on theConsequencesofMissing Church"
1.
An Englishspeakerin uttering t hewords, (0) 'The earthis round',says, or asserts,t hesame thingas a Frenchspeakeruttering t hewords, (0') 'La terre est ronde',' The thingassertedis a proposition,thepropositionthatthe earthis round. Thatthereare propositions,as distinctfrom thesentences thatexpressthem, is a commitmentof psychologyandotherhumansciences, which ascribe beliefs and otherpropositionala ttitudes . The existenceof propositionsis an integral p artof ourordinaryconceptionsof consciousness andcognition,a nd therewith of ourordinaryconceptionof whatit is to be a person. The evidence forthis commitment to propositions,qua extra-linguistic entitiesexpressedby intra-linguistic entities,is compelling . No one has done more to bring thatevidenceto the attention of philosophersthanAlonzo Church,and nowheredoes he do sowith more forcethanin his elegant and farsightedpaper "On Carnap'sAnalysis ofS tatements of Assertionand Belief."2 Thatdeceptivelybriefnotepresentsa sharpand tellingcriticism of one possibleattemptto do away withpropositions(in the sense that Churchintends)in favor oft hecognitivedispositionsof speakers vis a vis particular sentencesof alanguage("semanticalsystem"). The argumenthas
*As a student,I had theprivilegeand distinctgood fortuneto takenumerouscourses thinkersof this century . My own from AlonzoChurch,one of thetrulygreatanalytical elgeof thisexintellectual developmenthas benefitedenormouslyfrom themasterfultut traordinary logicianandphilosopher,in whose honorthisessay was written, shortlyb efore his death. I am grateful to my audiencesat UCLA in 1994 and at UC Berkeleyin 1996 fortheircomments, and to C. AnthonyAndersonand SaulKripke fordiscussion. lThesesentencesareto be takenin theusualsense throughout, as expressingthatthe earthis spherical. 2 Analysis, 10, 5 (1950), pp. 97-99; reprintedin L. Linsky, ed., Reference and Modality (OxfordUniversityPress, 1971) , pp . 168-170. See alsoChurch's"A Formulation of the Logic of Sense and Denotation,"in Henle, Kallen,a nd Langer,eds., Structure, Method, and Meaning : Essays in Honor of Henry M. Sheffer (New York: LiberalArtsPress , 1988), pp . 3-24 at5-6n; "Int ensionaIsomorphism l and Identityof Belief," Philosophical Studies, vol. 5 (1954), pp . 65-73, reprintedin N. Salmonand S. Soames, cds., Propositions and Attitudes (OxfordUniversityPress, 1988), pp . 159-168; and Introduction to Mathematical Logic I (PrincetonUniversityPress, 1956), p . 62n. 573 C. Anthony Anderson and M. Zeleny (eds.), Logic, Meaning and Computation, 573-595. © 2001 Nathan Salmon. Printed by Kluwer Academic Publishers, The Netherlands.
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a broad,generic sweep; it is equally applicableto any of a wide range of sentencesor otherintra-linguistic items theoriesthatposit a reference to especiallyattributions in thesemanticcontentof a varietyof constructions, of propositional a ttitude, even includingsuch theoriesas may be proposed thatdo this whilesimultaneously embracingpropositions. (The particular theoryChurchcriticizes,as Carnapintendedit, arguablydoes just this.) The argument,which invokes thet ranslation of varioussentencesbetween the Church Translation Argument. 3 languages,has come to be called Some philosophersof language,many ofthemin pursuitof a propositionless world , haveattemptedto rebuttheTranslation Argument ." Numerous or underwriterst odaydismiss theargumentas weak, fallacious, otherwise whelming, ortheysimply ignore it . WhereChurchwastedfew words, some of his critics have minced even fewer . MichaelDummett,who does not himtreatthis self rejectpropositions,nevertheless said that"it is difficult to argument"frivolously objectionvery seriously.t'"PeterGeach labelled the bad.?" In thejudgmentof thepresentauthor,t hecurrentdismissive attitudetowardtheChurchTranslation Argumentconstitutes a quantumleap studyof semantics, backward,a nd is, from thepointof view of the formal 3In all of his writingson thesubject,Churchreminds thereaderthatthebasic insight as a testfor determiningwhetheran expressionis being used or of employingtranslation mentionedis due to C. H. Langford . See alsofootnote12 below. 4 The following is a partiallist: TylerBurge, "Self-Reference a nd Translation ," in F. Guenthnera nd M . Guenthner-Reutter, eds., Translation and Meaning (London: Duckworth , 1977) , pp . 137-153, and "Beliefand Synonymy," Journal of Philosophy, vol. 75, no. 3 (March1978), pp . 119-138; RudolfCarnap,"On Belief-Sentences : Reply to Alonzo Church,"in Carnap's Meaning and Necessity (Universityof ChicagoPress, 1947, 1956) , pp . 230-232; DonaldDavidson, "The Methodof Extensionand Intension,"in P. A. Schilpp, ed., The Philosophy of Rudolf Carnap (La Salle,111., 1963) , pp . 331-349, at 344-346; MichaelDummett, Prege: Philosophy of Language (Cambridge, Mass.: HarvardUniversity Press, 1973, 1981), at pp . 372-373, and The Interpretation of Frege's Philosophy (Cambridge, Mass.: HarvardUniversityPress, 1981) , at pp . 90-94; PeterGeach, Mental Acts (London: Routledge,1957), at pp. 89--92, and "The Identityof Propositions ," in Geach's Logic Matters (Oxford: Blackwell ,1972) , pp. 166-174; StevenLeeds, "Church's Translation Argument ," Canadian Journal of Philosophy, vol. 9, no. 1 (March1979) , pp. 43-51 (I thankMarkRichardforprovidingthisreference);BrianLoar, Mind and Meaning (CambridgeUniversityPress, 1981), pp. 29--30, 152; HilaryPutnam, "Sy nonymy,and the Analysisof BeliefSentences,"Analysis, vol. 14, no. 5 (April 1954), pp . 114-122, reprinted in N. Salmonand S. Soames, cds., Propositions and Attitudes (OxfordUniversityPress, 1988), pp . 149-158; W . V. Quine, "Quantifiersa nd PropositionalAttitudes,"reprint ed in L. Linsky, ed., Reference and Modality (OxfordUniversityPress, 1971) , pp . 101-111, at 110; Mark Richard, Propositional Attitudes (Cambridge UniversityPress, 1990), atp. 1561£; IsraelScheffler," An Inscriptional A pproach to IndirectQuotation, " Analysis, vol. 14, no. 4 (March1954), pp. 83-90, and "On SynonymyandIndirectDis course," Philosophy of Science, vol. 22, no. 1 (January1955), pp . 39-44, at43-44n. Whileno attemptis made hereto respondadequately to each of thesecritiques , much of whatwill besaid hereis applicabletoa numberof them. 5 Frege, Philosophy of Language, p . 372 . 6"The Identityof Propositions ," p . 167.
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regrettable in theextreme." The pointat issue is by nomeans narrowly confined tothequestionofwhethera ttributions of propositional a ttitude involve Argument a commitmentto propositions. Those who rejecttheTranslation have typicallyfailed tocomprehendthemore generalpoint on which itis based, and have therebyfailed toappreciateone ofthemost fundamental factsconcerningthephenomenonof understanding. This failure is especially dramatic in thecase ofDummett. As I shallargue, Dummett'sfailure to grasp thelargerimportof Church'sargumenthas led himand his followers to defend aseriouslydistortedversion of Fr ege's theory , one thathas the consequence(clearlyunintended)t hatlanguage,as a mode ofcommunicationanda vehicle of cognition,is altogether impossible. I shallendeavorhere A rgument,especially to explainthemain thrustof theChurchTranslation in regardto thoseaspectsof theargumentthatare misunderstoodby its detractors . I shalldo so by reference toparticular a lacunain Frege's philosophy ofsemantics,showinghow Church'svaluable insightshelp to resolve a longstanding controversy concerningFrege'snotionof indirectsense. Generalizingslightlyon Church'spresentation, considerthetheorythat thefirst ofthefollowing two Englishsentencesis analyzabl e (definablewith fullpreservation of meaning)by means of thesecond:" (1) Chris believesthattheearthis round. (2) Chrisaccepts'The eart h isround'. The analysans(2) may be expandedif necessaryto focus on aparticular meaningforthesentenceit mentions, perhapsby addinga phraselike'as a sentenceof English', 'as a sentenceof thesame languag e as this very sentence', or 'as I, thepresentspeakerof this verysentence,would mean it '.9 It 7Moregenerall y, it is my con sideredview thatthe att e mpt to avoidan onto logy ofxtra e linguisticabstractent it ies yb an appe a l to intra-lingui stic subs t itu tes is philosophically mi sguided. The reas ons forthis judgmen tarecom p lex. As an excess ively bri ef sum ma ry, I m entionthatthe onto logyof everyday discourse is repl etewith abst rac tent it iesother tha nproposition s . The philosophicalsecuritythatis supposed to be afford ed by replacing proposit ion swithsentences is lar gelyan illu s ion, since senten ces ,no less than proposition s, are abs tract entiti es. Manysentences-infinite lymany, in fact- ar etoo lon g to be writt en or utt e red by any conceivable crea tu re. Moreover,p rinciples conce rn ing th e identification of sente ncesarenotnearlyas "exte ns iona l" as sometimes is supposed. This lastdifficulty oftenmanifestsitselfin the need to re sortnotm erelyto sente ncesas such, butto sentences as sen te n ces of a particular language, or to sentences as meaning that such-and-such, etc. •8For many applicat ions, the requirem entof full pr eservation of meaning(whereby, as a conse que nce, on e b elieveswhatis expressed by th e analy sa ndum if and only ifone also believeswhatis expressed by the analy sans) may be weakened to m ere logical equ iva lence , and in some application s , to such wider equivalencerelat ions as m od al eq uiva lenceor a priori equival en ce, withlittl e effect on the overallfor ceof the argume nt. t require 9C hurch 's conceptionof linguistic expressions and their semantics is such as o tha tsuc h sema nticatt rib utesas sense, denotat ion, and truthvaluealways be relativized to a particularlanguag e. O thers favor a co ncept ion cacording to which relat ivizat iono t a languageis unnecess ar y or even ina ppropr iate. C f. Peter Geach's protests in Mental
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should eb noted th atany possible expansion of(2) thatapparentl y mentions the propositionthat the earth is round, or otherwi se apparentlylogically entails the exist enceof a proposition, would not eb suit able asp artof an att e mpt to relievet he authorof (1) of his or her apparentcommitmentto proposit ions,although it may be appropriate for other purposes.I? The particular word'accepts' is used schematically; an y of a varietyoftransi tive ver b phrases may be substitute d. As an example, one may replace 'accepts' witha carefullyformula t edphrase like 'is disposed , on reflectio n, when sincereand nonret icent t, o ass ent to some sufficient ly und erstoodtran s lat ion or other of'. Alternati vely, one m ay subst it uteQuine's word' believes-t ru e'I! or even Acts, pp . 88-89; and David Kapla n, "Words," Proceeding s of the Aristotelian Society , vol. 64 (1990) , pp . 93- 116. Although Church's present at io n fothe Tran sla t ion rgument A assumes his view on this iss ue, the argument can also be presented withthe opposite view al ity(as it is her e). presu p posed, or withneutr lOO ne addend um thatwould obvio us ly be inadm issi ble for the purpose of eliminating comm itmentto proposition s is the p hrase' as exp ressi ng the pro posit ionthatthe earth is round' , or mor e simp ly'as meaning thatthe earth is round' . (Cf. footnote 7 above.) C hu rc h y ent a ils his t inadmi ss ibleone, explicitlyconsiders a p oss ibl eexpans ion of (2) thatlogicall adm itting that suc h an expa nded version may yield (1) as a logical consequence. He does not note thatthe expansion in question would not be suitab le for the elim ina t ion of propositions. (It may even leadtocircularity,if thep ro posed analys is is extend ed to su ch constructions as ' Sentence S m eans in languageL thatthe earth is round' in additionto ar ex pa nsio n of (2) he considers does (1).) Instead he notes, correct ly,tha t the pa r t ic ul notpreserve the mea ning of (1). Again he util izes tran s lationto crys t alize the poi nt. In addition, he cons ide rsem bed ded constructio ns like ' Jones belie ves tha t Ch ris believes that the earth is round', arg ui ngthat alleged analysesof this and of its tra nslat ion may even differ intruthvalue intheir respective languages. (C f. footnote 8 above. ) Carnap says in resp on se (op. cit ., p . 230 , in a highl ycomp ressed paragrap h) that he intended precisely such an expansion of (2), whileconce d ingthat C hurch's objectionis correct. Carnap'soverall espo r nse toC hurch is unclear and puzzling. In Meaning and Necessit y, p p . 63---{)4, scarce lya page afte rpresenting his ana lysi s of stat eme nts of belief, Carnap says of analysis in gen eralthataltho ugh analysand um and analysans mu st be logicall y i as Carnapalso puts it, the equiva lent , hey t need not be int en sionall ysomorphic--or ctur e . Davidson analysis m ust preserve intension but need no t preser ve intensional stru (op. cit. ) and Put na m (op. cit .) argue, in effect, that Carnap's analysis of state m ents lar,therefore, is not intended to capture their mea ning of assertion and belie f in particu but only somethi ng lo gically equivalent - th us mak ing Church's objections ina ppli ca bl e. , Carnap conce des not only C hurch's objection , but fu ther mo re that In sharp contrast (1) is str ictly not even a logica l conseq uence of the intended version of (2) (p p , 230--231). C ur ious ly , Carnap alsoendorses Putna m 's respon se (p. 230), while p roffer ing na alternat ive y equivalent ) or f (1 ) ver sion of (2) as a scientific replac ement (presumably now logicall (pp . 23 1-232). The textua levide nce suggests,however, thatCarna p co nfused Putnam's respo nse on this point wit h Putnam's respon se to a separat e objection by Benson Ma tes conce rn ingembedded constr uc tions. In "Intens ional Is omorphism and Ident ity fo Bel ei f," C hurch extends the Transla ti onArgum entagainst Putn a m's res pon se to Mates . See also the introduct ionto N . Salmon and S. Soames, eds., Proposi tions and Attitudes, (Oxford UniversityPress, 1988) , pp . 13- 14n lO. T he issues here are numerous and quite comp lex. Cf. foot not e28 below. Insofar as i timately rela t ed tossiues concerning em bedded constructionsare invo lved, the dispute is n F'rege 's no tio nof indi rectsense. See footnote 25 below. llQ ui ne, op, cit., atpp . 109--110. The expression ' belie ves-tru e' is, however, signi fica ntly m isleading. For discu ssion , seem y "Rela tional Belief," in P. Leonard i and M . Santambro-
THE VERY POSSIBILITY OF LANGUAGE
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thephrase'believesthepropositionexpressedby'. The latter cannotbe used if one'sobjectiveis to avoidcommitmentto propositions , butmight be used, forexample, by one (such asDummett)who supposes thata 't hat '-clause is to beunderstoodby referenc e to itscontainedsentence.Moreover, thepair of words,'believes'a nd'accepts',may be replacedby otherpairsof suitably ', etc. related t erms, e.g., 'asserted ' -'uttered','disbelieves'-'rejects We demonstratet hatthe proposed analysisfails asfollows:Translating bothanalysandum and analysansinto French, we obtain,
(I') Chris eroit que la terre est ronde. (2') Chris aceepte 'The earthis round'.
(The word ' aecepte' is used hereschematically fortheliteral Frenchtranslationof whateverEnglishphrase replaces'accepts', subjectto thesame possible variationsmentionedearlier .) We pause tonotethatthe proper translation for (2) isnot (3') Chris aceepte 'La terre est ronde '.
This sentencementionsa particular Frenchsentencenot mentionedin (2), while lacking any mentionof theEnglishsentencementionedin (2). It is thus (2') rathert han(3') thatcapturest heliteral meaningof (2).12 Likewise, (1) and (I') are literal t ranslations of oneanother . But it is evidentthatthe Frenchsentences(I ') and (2') are not synonymous. For they "would obviously independentp ropositions-to a convey different meanings"-indeed, logically (I') conveys Frenchspeakerhavingno knowledge of English. Of the two only the contentof whatChris allegedly believes . Sentences(1) and (2) must therefore differ inmeaningin English, contrary to theproposedanalysis. 2.
As notedabove, thescope oftheargumentis wide. It is equallyapplicable, forexample, againstthe redundancy or disquotational theory of truth, according to which the Englishsentence''The earthis round'is truein English' thatthe simply reduces to'The earthis round'.The argumentdemonstrates twosentencesa rein fact logically independent.It is alsodirectlyapplicable to alongstanding controversy in orthodoxFregean theory . Frege heldthat theEnglishsentence(0) ordinarily denotesits truthvalue-in this case,t he valuetruth ("theTrue" )-butthat , whenoccurringin anindirector oblique (ungerade) contextin English,as in (1), itinsteadhas itsindirectdenotation, denotingwhatis ordinarily its Englishsense, thepropositionthatthe gio, eds., On Quine (Cambridge UniversityPress, 1995). 12Cf. G. E. Moore, "Russell's 'Theory of Descriptions ' ," in his Philosophical Papers (New York: CollierB ooks, 1959), pp . 149-192, at 156-157, wherehe ant icipat esL angford 's translat ion test.
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earthis round.P Frege heldfurthert hatwhen (0) occursin any English indirectcontext,it takes on its indirectsense, which is aconceptof its indirectdenotation, or ordinarysense. Butwhich ofthemyriadconceptsof the propositionthattheearthis roundis therebyexpressed?Carnapcomplained that"Frege nowhere explainsin more ordinarytermswhatthisthirdentity is." 14 The matterremainscontroversial. Dummettarguesthat,for Frege, the indirectsense inEnglishof (0) is theordinarysense inEnglishof thephrase (4) theordinarysense inEnglishof 'The earthis round'.15 Dummettdoes notsupportthisexegetical thesisby citingany passagefrom Frege'swritings.Insteadhe providesquitegeneralconsiderations in favor of theinterpretation : Thereis nothingin whatFrege saysaboutdirect[i.e., ordinary)a nd indirectsense andreference [denotation) to ruleo utthepossibilitythat, although d istinctexpressions, in thesame or differentlanguages,may have thesame sense, no sensecan be given to us save as thesense of some particular expression; such athesiswould fit very well whatFrege sometimes says totheeffectthatwe can graspthoughts only via words or othersymbols. But,if so, then,if an expressionstandsfor a sense, and does so invirtueof its sense, thatsense must involve areference , overtor covert,to someexpression-the same or different-whose sense its referent is (op. cit., p . 91) . It isperfectly c onsistent tocombinethethesisthattheobjectofbeliefis a thought [proposition)withthethesisthatwe canapprehenda thought only asthesense of asentence(in a verbalor symbolic language) . It is therefore e quallyconsistentto combine thethesisthata clausein omtio obliqua [indirectdiscourse),or anexpressionwithinthatclause, standsfor itsordinarysense withthethesisthat,in understanding it as referringto thatsense, weapprehendthatsense as beingthesense of thatclauseor expression. . .. Frege [adheres) to b oththesetheses (p. 94). 13 "Ube r Sinn and Bedeutung",t ranslated as "On Sense and Reference ," in R. M. Harnish, ed., Basic Topics in the Philosophy of Language (EnglewoodCliffs, N.J. : Prentice Hall, 1994), pp. 142-160, at 144. 1 followChurchin translating Frege's use of ' Bedeurathert han'meaning'or 'reference',and Frege's use of'Gedanke' as tung' as 'denotation' 'proposition'r athert han'thought ' . I alsouse theword 'concept'in Church'ssense, which is verydifferentfrom Frege'suse of 'Begriff' . 4 1 Meaning and Necessity, §30, pp . 129-133. 15 The Interpretation of F'rege's Philosophy, atpp . 89-100. This representsa turnabout forDummett. In his earlierwork, F'rege : Philosophy of Language, he consideredthethesis thattheindirectsense of (0) in Englishis thecustomarysense of (4), onlyto dismiss it as "rather i mplausible " (p. 267) . Boththisthesisconcerningindirectsense, andtheexegetical thesisthatFrege'stheoryimplies theformerthesis,aredefendedin GaryKemp, "Salmon on FregeanApproachesto theParadoxof Analysis," Philosophical Studies, vol. 78, no. 2 (May 1995), pp. 153-162, whereinspecific referencesto Frege's writingsareprovided(at p . 160) in supportof Dummett's interpretation .
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Dummettadds further on thatFregethought t hat"we cangraspa thought only asexpressedin some way" (p, 98), andthatthethesisthattheindirect sense in Englishof (0) is theordinarysense of (4) "doesappearto follow from thetwotheses, takentogether : thethesisthata sensecanbe given only as thesense of someexpression,and thethesisthatan expressionin omtio obliqua standsfor itsordinarysense" (p, 95). 16 He continues,saying that "ourgraspof the sense of theexpression.. . , on Frege's account , ... leaves us withno access tothatsense save asthesense of someexpression" (p . 97).17 EchoingCarnap, DummettarguesagainstFregethatit ispreferable not to confer on an expressiona conceptof itsordinarysense to serve as thesense expressedin indirectcontexts.J'' Despite his misgivings aboutthe notion of indirectsense, Dummettexplicitlyendorsestheidea that,as he puts it, "a sense can be given to us only the as sense carriedby some particular expression." He explainsthatto saythis"implies thatthemost directmeans by which we can refer Englishto in thesense expressedby, say, 'themoon' is by using thephrase'thesense of'themoon"" (p. 95). Dummettsays moreoverthat"we cannotrefer tothe sense of mostexpressionssave by explicitallusionto the expressions" (p. 90) . And hejoins withFrege, as Dummetthas interpreted him, in holdingthat we apprehendtheindirectreferent as being thesense oftheexpression which weperceiveas occurringin the O1utio obliqua clause. . .. our way ofgraspingwhatthesense of an expressionis, rendersus incapableof detachingthesense from everyactualor hypothetical means of expressingit. . . . when we taketheexpressionin oratio obliqua as standingfor its ownsense, we areconceivingof thatsense as thesense of thatvery expression(pp. 97-98).
AlthoughDummettdoes notshareFrege'sbelief intheneed forindirect senses, it is clear from theforegoingpassagesthatDummettmaintainsthe following thesis(which he cites in s upportof his interpretation of Frege on indirectsense): In orderfor us to conceive of any sense at all orderto (in form a beliefa boutit, or tospeculatea boutit, or even merely entertain to a 16Dummettpointsoutthatthe inferencein questionis actually i nvalid,but he allows ed by the additionof a thirdthesis (which Frege heldand (pp. 98-99) thatit is validat which Dumm ettrejects)as premiss , to wit, that"thesense of an expression is theway in which its [denotation] is given to us." 17Bishop Berkeley argued, againstLocke's doctrineof "abstracti on," thatone cannot conceive of, forexample, a colorwithoutalso conceivingof some shape or otherof that color. Accordingto Dummett,Frege did nothold,in the sam e spiritas Berkel ey, merely thatone cannotconceiveof a sense in any way withoutalsoconceiving of it as thesense of some expressionor other. It is not thatany conceptionof a sense we have , must be accompanied in cognitionby a conceptionof it asthesens e of som e express ion. The point, rather , is thatin conceiving of a sense, we conceiveof it onlyas the sense of e, for some expression e orother , since we have no conceptionof the senseother thanas the sens e of e, forsome expressione, forsuc h a conceptionto accompany. (T his will bemade more precise to C ha rlesChiharafor bringingthisdistinction to my att e ntion .) below. I am grateful 18Carnap,op, cit., atp . 129; Dumm ett, Frege: Philosophy and Language , pp . 266-269.
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thoughta boutit, etc.), theremust be some expressione (of some language l) such thatwe conceiveof thesense inquestion(or as Dummettalsoputs it, thesense is "given tous") as theordinarysense of e (in l) j we areunable to conceive of saense in any wayotherthanas theordinarysense of e (in l), for someexpressione (of some languagel). 19 Let us callthis Dummett's Thesis . Theconsiderations citedby Dummettin favor of his interpretation of Frege areunconvincing , especiallyin lightofFrege'sexplicitpronouncement (which bothCarnapand Dummettmay haveoverlooked)t hattheindirectsense in Englishof (0) is rather t heordinarysense inEnglishof thealternative phrase (5) thepropositionthattheearthis round.F'' One may wonderwhethert hephrases(4) and (5) arenot themselvessynonymousin English. In fact, Geachexplicitly a rguedthaton Fregeantheory, suchphrasesarecompletelyinterchangeable (op. cit., p. 168-169). And Dummettevidentlyt reats suchphrasesas synonyms.F! Furthermore, Dummett's interpretation depictsFregeas havingheldthattheordinarysense of (5) in Englishpresentsthepropositionin questionas theordinarysense of some particular linguisticexpression. The only plausiblecandidatefor thatexpressionis theEnglishsentence(0).22 This wouldmake (5) a mere English paraphraseof (4)-evenif it is a"lessdirectmeans" of denotingthe propositiondenotedby each. If (4) and (5) are indeedsynonymousin English,then Dummett'sformulation of Frege'sthesisconcerningindirectsense is simply another way ofsayingthesame as Frege. Any suspicion that(4) and (5) aresynonymousmay be dispelled,however, bytheTranslation Argument . Of course,(4) and (5) denotethesame propositionin English, but theirliteral Frenchtranslations - ' le sens ordinaire en anglais de 'The earthis round" and 'la proposition que la terre est ronde', respeCtively -clearly c arrydifferentm eaningsfor th e Frenchspeaker who knows noEnglish.Hence, (4) and (5) arenotordinarily s ynonymousin English,i.e., theydifferin ordinaryEnglishsense. Furthermore,t he Translation Argumentdemonstratest hatthe thesis Dummetterroneously a ttributes to Frege is inany case incorrect.For just 19Dumm ettdoes notexplicit ly sp eak in a relat ivized manner of the sense of an expreseticalreferen ces to the languageI are includedher e for sion in a language. The parenth the benefitof thosewhose view of linguistic express ions and their sema nt ics requires for propriety the relativiz ation to a languag e. See footnot e 9 above. 20 " Uber Sinn und Bedeutung," at p . 149 of Harnish. 21 In The Interpretation of Frege 's Philosophy, at p . 89, ther e occurs an otherwise inexplica bleswitc h ,wherein the discussion, suddenlyand withoutnotice, changes its focus cationof the indirectsense of a es ntenc e like (0) withthe ordinarysense of from an identifi the correspondinganalogu e of (5), to anident ifi cat ion ofthe indirectsense of ' Aristo t le' insteadwiththecustomary sense of 'thesense of'Aristotle ' '. . See also Kemp, 22Cf. the argumentgiven by Dummett op. cit ., p . 95, top paragraph op. cit.
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as (0) expresses its Englishindirectsense in (1), (0') expresses its French indirectsense in(I'). Yet unlike(2'), (I') evidentlymakes nomentionof (0). It is probable,furthermore, thattheparticular linguisticitem creatingthe '.23 For indirectcontextin (1) is nottheEnglish word'believes'but 'that termsdenotingthesame propositionare typicallyinterchangeable, with no affect ontruthvalue, following occurrences of 'believes','asserted' 'doubts' , , etc. Considerforexamplethe following inferences :
CarnapbelievedFrege'scentral doctrinein thephilosophyofmathematics . Frege's central doctrinein the philosophy of mathematicswas logicism. Therefore,Carnapbelieved logicism. Churchdoubtslogicism. e to logic . Logicism is thedoctrinethatmathematicsis reducibl Therefore,Churchdoubtsthatmathematicsis reducibleto logic. By contrast , theword 'that'is equally involved substitution in failures thatdo not involve belief 'Bertasserted ( that . .. ') and even insubstitution of any kind, for examplein contexts failurest hatdo notconcern attitudes concerningmodality('It is a necessarytruththat . . .'). The word 'that ' is which whenattached to an plausiblyregardedas a device ofsense-quotation, Englishname for the propoEnglishdeclarative sentenceforms thestandard sitionordinarily expressedby thatsentence(in a manneranalogousto that in whichsyntacticq uotation marks form thestandardname of the expression enclosedwithin). As such, 't hat' would be aparadigmaticungerade device ("obliqueoperator")of English , with thephrase't heproposition' ocas a grammatical appositiveto the't hat-clause.v' curringin (5) functioning This hypothesison Dummett'sproposalimplies that,since (0) isinducedby the 'that'prefix toexpress its Englishindirectsense in (1), (1) itself may be rewritten w ithoutalteration of meaning in a formdispensing with any ungerade device (beyondordinaryquotation marks), simply by substit ut ing synonym (5) withinit: (4) fortheshortenedversion of its alleged (6) Chrisbelievestheordinarysense inEnglishof 'T he earthis round'. Since this is asentenceof preciselythe form of(2), theTranslation Argument, as originally given, disprovesthe theorythatit is anEnglishparaphraseof 23It must be admittedthatthe construction'Ch ris believestheeart h isround',without theword' t hat', is p erfe ctlygrammaticalEnglish, and (0) occursnonextensionallytherein . This phenomenon, howeve r, does not immediatelyyield the resultthat' be lieves' is a . It is arguabl e that(0) is induced to shift o t the nonextensional (ungerade) op erator indirect mode her e by other (p erhaps pragmatic)factors . (Indeed, linguists commonly refer to the phenomenonas ". that'deletion .") 24Cf. my Freqe's Puzzle (Atasc adero, Ca.: Ridgeview , 1986 , 1991), atpp . 5-6; and "Refs ," in D. Gabbay and F. Guen ere nceand InformationContent: Names and Description thner , eds., Handbook of Philosophical Logic IV : Topics in the Philosophy of Language (Boston: D. Reidel, 1989) , pp . 409-462, at440--441.
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(1). Again by contrast,the hypothesisthatthe Englishword 'that'is a device forsense-quotation strongly s upportsFrege'sactualc laimconcerning theindirectsense of asentencelike(0).25 3.
A fewyearsafterChurch'spaper appeared, Geachprotested, Veryoften,w hatwe countas correctt ranslation willincludetranslation ofquotedexpressions;a translator of Quo Vadis wouldnotfeel obliged to leave all theconversationsin theoriginalPolish, and we should countit as perverselywrong, not pedantically correct,if he did SO.26
Othershave sinceechoedGeach'scomplaint,n otablyTylerBurgeandDummett.27 It must be observedthateach ofBurge,Dummett,andGeachoffers a replydifferentin variousrespectsfrom theothertwo-andit is notobt heyshare vious thatany ofthethreeagreesentirelyw ithany other-but an emphasis on divergencesbetweentranslation in actualpracticeand the special,sense-preserving t ranslation proposedby Church . Translators would very likelyproffer(3') ratherthan(2'), theyargue,as a correctrendering of the English(2) into French. No one deniesthat(3') fails topreserve (2)'s mentionof (0); (3') mentionsdifferentwords of adifferentlanguage altogether . Insofaras preservation of meaning,or sense, requir es thepreservationof denotation, renderingthe English(2) intoFrenchby means of (3') does notpreservemeaning.P ButBurge, Dummett,and Geacheach deny 25T he Translati on Argumentis not supportiveof allaspects of Fregeantheory . As is shown in my articl e "A Problemin theFrege-ChurchTheoryof Sense and Denotation," Noiis, vol. 27, no. 2 (1993) , pp. 158-166, whiletheargumentsupportsFrege'sthesisconcerningindirectsense, it therebyleadsto aninconsistencywhencombined witha Fregean solutionto theParadoxof Analysis,of a sortadvocatedby Churchin his reviewof the Black/White exchangeconcerningt heParadoxof Analysis,The Journal of Symbolic Logic, vol. 11 (1946), pp . 132-133. I arguein thearticlet hatrelinquishing t heFregeansolution to theParadoxof Analysisthreatens theFrege-Churchtheory,by collapsingFrege's and Church'soriginalargument for thepivotaldistinctionbetweenthesense and denotation of an expression. (Whereasthis difficultye xposes a potentially seriousweakenessin the Frege-Churchtheoryof sense and denotation , it does nothingto weakentheforce ofthe Translation A rgument .) 26 Mental Acts, pp . 91-92. (I thankSaulKripke for providingthisreference .) and Translation,"e speciallypp . 141-144; Dummett, loco cit . 27Burge, "Self-reference Burge'sresponseto theT ra nslat ion A rgumentis alsoendorsedby James Higgenbotham , in "LinguisticTheoryand Davidson's Programin Semantics,"in E. Lepore,ed., Truth and Interpretation : Perspectives on the Philosophy of Donald Davidson (Oxford: Basil Blackwell, 1986), pp . 29-48, at 39n. (I thankHigginbothamfor providingthe lastreference .) 28This point is explicitlyacknowledgedby Dummett (Frege : Philosophy of Language, p. 372; The Interpretation of Frege's Philosophy, p . 90) . It is more or less acknowledged by Burge(op . cit., p . 141), although his overallr esponse to the Translation A rgumentis explicitlypartof a programto avoid commitmentto propositions(pp . 152-153). Geach espousesan extremeskepticism regardingt henotionof synonymy. He arguesthat(3') is "of thesame force as"(his versionof) (2), and that(3') and (2) are "reasona blyequivalent ,"
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thatcorrectt ranslation must preservemeaning-insofar as thepreservation of meaningrequiresthepreservation of denotation . Thus Dummettwrites, "Thereis no groundfor the presumptionthatthepracticalcanonsof apt translation alwaysrequirestrictsynonymy. Onthecontrary , translations of fictionand, equally, of historicaln arrative (includingthe Gospels) always translate evendirectlyq uoteddialogue .,,29 Amplifyingtheargument,B urge says, "translation of foreignquotedmaterialaims at conveyingthe'point' of thepassagethatcontainsit" (op. cit., p . 145). And accordingto Burge, whatis crucialto thepointof sentenceslike(2) is not part of the grammar or semantics of the sentences selves them .
It is betterseen as involvedin a convention presupposed in the use
and understanding of such sentences . ... What isinvolvedin rightly construing the expressions thatare mentioned in problem sentences like [(2)] is .. . the ability to understand those expressions as they would be intended if they were used by the person who s the userelevant token of the sentence in which they are mentioned 146).(p.
Burgearguesthattheoperativenonsemantic , pragmaticconventionin connectionwith (his version of)the English (2) directsone tointerprett he sentenceit mentionsin thespecified manner, and that thisyieldstheresult, contrary to Church,that(3') is a bettertranslation of (2) intoFrenchthan is (2') . This generalline ofcriticismwas cogentlyrefutedindependently by HerbertHeidelberger , Casimir Lewy, and LeonardLinsky.i''' It is completely A rgumentwhether(2') is deemed a correct inessentialto the Translation but spec ifica llystops s hort ofdeclaringthatthey are eit herthe sa me , or different, in meaning. By thesame token, Geach does not propose (2) as a meaning-pr eserving ana lysis e artherefor e why he does notconcede thatanyauthorof (1) is com mit te d of (1). It is uncl to theexistence oftheprop ositionthatthe ea rt h isround(this being whatthe Translati on Argumentis aimed at demonstrating) , and le t this be his reasonfor recommending that theauthorsubstitut e (2), which lacks any suchcomm itment. (Cf. Quine, op . cit., and m y "Relat iona lelief B " for discussion.) In any event, Dumm ett's reply has the advant age over Gea ch 's (and Quine's) that it does notdepend (at leastnot to the same extent ) onany implausibleor otherwisecontroversial skeptical theses concerningsynonymy. Burgeclaims thatthesentenc e he proposes for (2) involves self-denotation,in such a way thatthe best tran s lati on willpreserve thisfeatureatthe expense of denotation . The detail s of Burge 's argument willnot be pursued her e. One m inor correction should be noted, however . Contraryto Burge's initialclaim (which he cred its to W. D . Hart)that translati on of a self-de not ingenten s ce eit he r pr eserves self-denotation or denotationbut notboth,translations may be given thatpreserve neither one. (For whatever it is worth, it is even probablethatsuch translations have been given in act ua lpractice.) Burge's argumentrequires onlytheweakerclaim thatno tran slationof a self-de not ingenten s ce preservesbothdenotation a nd self-denotation. 29 F'rege : Philosophy of Language, p . 372 . But see also The Interpretation of F'rege '5 Philosophy, p . 90. 30 Heidelberg er, Review of Dumm ett 's F'rege : Philosophy of Language, in Metaphilosy 1975), pp . 35-43, at 42-43; Lewy, Meaning and Modality ophy, vol. 6, no. 1 (Januar (C ambridgeUniversityPress , 1976) , atpp. 64-66; Linsky, Oblique Contexts (Universityof C hicagoPress, 1983) , at p. 8. (I thankC . Anthony Andersonfor providing the lasttwo
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translation intoFrenchof theEnglish(2). The claimthatan actualt ranslatoror interpreter wouldoffer(3') insteadis similarlyirrelevant. The Transw iththesemanticsof (1) and (2), lationArgumentis concernedexclusively and notwithanypragmatic"conventionsp resupposedin theiruse" or with anyresulting " pract ical c anonsof apttranslation " of textscontaining e ither . " 8' is a translation into As Heidelbergerc orrectly n otes, theconstruction L' of the L expression8" may even be replaceduniformlythroughout the argumentby "8' hasthesame meaningin L' that8 has in L" , therebyeliminatingany allusionto translation. Any controversy c oncerningtranslation (in particular , whetheraptor correcttranslation must preservemeaning) is thusseen tobe entirely i rrelevant. Dummettappearsto haveconcededtheirrelevance of theissue ofwhether theEnglish(2) wouldbe translated into Frenchin actualpracticeby (2') ratherthan(3'). Yet (I') is clearlynotsynonymousin Frenchwith (2'), whichexplicitlymentionstheEnglishsentence(0). Once it isconcededthat (2') has thesame meaningin Frenchthat(2) has inEnglish , the criticof the Translation Argumenthasno alt e rnativeb utto chall enge theremainingpreF renchtransla tionof the English(1). In miss, that(I') is a sense-preserving response to unspecifiedcritics(presumablyincludingat leastHeidelberger, ."! And indeed, Dummett's Lewy, and/orLinsky), Dumm ettdoes justthat interpretation of Fregeimplies thatFregeshouldhaverejectedthispremiss. ation, Frege is supposed to have held t hat For accordingto thatinterpret referen ces .) See also Saul Kripke, "A PuzzleaboutBelief," reprint edin N. Salmo n and S. Soames , eds., Proposit ions and Attitudes (O xfordUnivers ity Press, 1988), pp . 102-148, at 1421125. In his discussion in Th e Interpretation of Frege's Ph ilosophy, Dumm ettattrib utes bo th the thesis thatthe indi rect sense of (0) in English is the ordinary sense of (4) , and the exeget ica lthesis tha t Frege held the former thes is, to Heid elberger. Dumm ettalso the re ger of inconsis tentl y conjo ining th ese theses with an endo rse me ntof the accuses Heidelber TranslationArgument(pp . 9 1, 94). These at t ributi ons aredubious. Alt ho ug h eidelb H erger defends the Translati on Argumentagainst Dumm ett's crit icism, he explicit ly eclin d es to endo rse th e argument and instead expresses sympathy for the alt e rnativ e crit icisms of it by Davidson and Putnam(p . 43n). Moreover , he does not straight forward ly propose es to Frege the thesis that eit he r ofthetheses in question . Instead he corre ctlyattribut the indirect sense of (0) in Englishis the ord ina rysense of (5) , while m is-identifyingthis thesis withthe alt e rnat ive thesis which Dumm etthad branded "rather implau sibl e"(p . 37; see footnote15 above). As noted ea rl ierD , ummettalso fails ot distingui sh betwccn these fails todistingui sh prop erlybetween two theses concerningindirect sense, and therefore the correspond ingexeget ica l th eses concern ing Fr ege on indrect sense. (It sho uld al so be noted thatwhereasHeid elb erger xp e licit lyattribut es to Frege the thesis thatthe indirect sense of (0) in English is the ord ina ry ens s e of (5), he also attribut es to Frege a fallacious argu me nt for t hatthesis which Frege does not give, and which is in factinconsistentwith his views.) 31 The Interpretation of Frege's Philosophy, pp . 90--91. The prem iss is explicit lyreject ed by Kemp , op. cit., and Ste phe n L eed s, op , cit. Curiously, Dumm et tulit mat ely n e dorses the premi ss ( p . 94) , on the ground that (1) has the same "convention al isgnifica nce" in Englishthat(1) has in Fren ch (p . 99). It is unclearwhy Dumm ettinsists never th eless that it is illegitimate for C hurch to assert this prem iss , on the same or very similar ground, in s is. the course of his argument agains t the p roposed analy
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(1) is synonymousin Englishwith(6), whichmentionsthevery expression (0), whereasthe French(1') insteadmentions(0') . In thegeneralcase, the proponentof (2) as anEnglishanalysisof (1) willcontendthat(1') is synonymous in Frenchwith (3'), which explicitlymentions (0'), and therefore does not have t hesame meaningin Frenchthat(1) has in English. Such a atthis point, since proponentmight evenenlistthesupportof translation thesense-preservingt ranslation of (3') into Englishbearslittle resemblance to (1).32 Thoughtheargument'sa ppealtosense-preserving t ranslation may be thus thwarted by controversy , Church'sgeneralpointmight still be made without appealingto anycontestedtranslation . Churchsupportsthe premiss that (1') and (2') are not synonymousin French byobservingthatthey"obviously convey different meanings" to aFrenchspeakerwho knows noEnglish,with t he contentof whatChris is supposed to believe . In only (1') conveying a similarvein, one might objectto theproposalthat(1) is synonymousin Englishwith (2) on theg roundthatdifferentinformationis conveyed to an speakerby (2')-which, Englishspeakerby (1) thanis conveyed to a French it has now beenconceded, is a sense-preservingFrenchtranslation of (2). In response tothispossibleobjection , StephenLeeds objectsthat whatinformationa sentenceconveys to a hearerdepends not only on whatthesentencemeans buton whatbackgroundinformationthe hearerhas. The mere abilityto understanda languagecan constitutesuch backgroundinformation;forexample, ' Luther sprach: 'Hier someone who speaks German thanits steh ich " will convey more to stricttranslation intoEnglish: ' Lut her said : ' Hier steh ich' , will to a monolingual s peakerof English(op. cit., p . 46).
This lastpointmight also beillustrated by (2) and (2'). The two have the same meaningin theirrespectivelanguages,yet an English speakerobtains more informationon the basis oft heformerthana Frenchspeakerobtains . Churchhimself noted, as partof his argument , on the basis oft helatter thatthe knowledge of what(0) means in English would enableone to infer to Chris. Leedscontendsthat from (2') thecontentof thebeliefattributed it is arguablythis ancillary knowledge,r ather t hanthecontentsemantically expressedby (1), thatlikewiseaccountsfor theinformationconveyed to an speakerby (2') . Englishspeaker by (1) thatis not conveyed to a French Indeed, as we have seen , Burgearguesthatthe pragmaticsinvolved in the paraphrase(1» directs use of (2) (andtherefore also intheuse of its alleged t hementionedsentence(0). one tointerpret It is herethatthisgeneralresponseto theChurchTranslation Argument point of theargument -orperhaps I betraysa failure to grasp thecentral 32For more on thi s sortof issue, esp ecially asit relat esto Burge's response to the Translation A rgument , see my "A Problem in the Fr ege-ChurchTheoryof Sense and Denotation, " atp. 166n15.
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shouldsay, atleastone ofthecentral pointsof theargument -andin a sense, one ofthemain pointsalso ofFrege'sphilosophyof language. 4.
Just as the "practicalcanonsof apt translation" are entirelyirrelevant to theChurchTranslation Argument,so also istheinformation o btainedon the basis of anutterance.To use aterminologyI introducedin previouswork, theinformation o btainedfrom anutterance caninvolve not only semantically encoded information, butalsopragmatically imparted information. 33 (What countsas apt translation, forthatmatter , may also beconcernedto some extentwithinformation of bothkinds.) The Translation Argument , by contrast,is concernedexclusively with theformer. One who knowstheEnglish language is ableto inferadditional i nformationfrom anutterance of (2) not semantically c ontainedwithin(2) itself. Whilet hisadditionalinformation is therebypragmatically imparted(atleastindirectly) to anEnglishspeaker by theutterance , it is notdirectly"conveyed" by (2) inthesense relevant Argument. Rather,it is inferredfrom thesemantically to theTranslation containedinformationtakentogetherwiththeancillary knowledge of w hat (0) means in English. WhenChurcharguesthat(I') and(2') conveydifferent information to aFrenchspeakerwho knows noEnglish, he isspeakingof the informationsemantically encodedin each, thepropositionsthattheFrench French sentencesconvey totheFrenchspeakersolely invirtueof theirliteral meanings. He explicitly c ontrasts thiswithinformation thatmay be inferred from thistogether withknowledge of English.And indeed,thevery conclu sion oftheargument(thedenial of which Leeds seeks to defend) that is (1) and (2) semantically containdifferentinformation,despitethe factthatthe contentof (1) is easilyinferredfrom anutterance of (2) given knowledg e of theEnglishmeaningof (0). Translation between languagesis invoked merely as a device facilitate to our seeing the difference insemanticcontentthatexists between (1) and (2)-oras a specialcase, between(1) and (6)-despite theease withwhich one is inferredfrom theother,relying on our knowledgeEnglish of . The merely auxiliaryrole oftranslation in theTranslation A rgumentwas first notedexplicitlyby Churchhimself: The existenceof morethanone language is notusually to bethought of as a fundamental groundfortheconclusionsr eachedby thismethod. Its role isratheras a useful device to separatethosefeaturesof a statementwhich areessentialto its meaning from thosewhich are language,theformer merelyaccidental to itsexpressionin a particular butnot the latter being invariantu ndertranslation. And distinctions 33Frege 's Puzzle, loco cit . (footnote24 above), pp . 58-60 and elsewhere especiallypp ( . 78-79,84-85, 100, 114-115) .
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(e.g., of use and mention) which areestablishedby this method it shouldbe possiblealso to see more directly.i"
In theapplicationoftheTranslation Argumentto thequestionofindirect directlyis a spesense, thedistinctionthatone should be able to see more cialdistinctionbetweenthe meaningsof (4) and (5). This same distinction accountsfor the differencemeaningbetween(1) in and (2). Whatexactlyis thisspecialdistinction,and how arewe supposed to see The answerto this iswhatI taketo be it withoutresorting to translation? thecrucialpoint,and the very p ointthatChurch'sdetractors have failed to appreciate . Churchputs this pointby notingthattheproposed analysisof (1) must be rejectedon thegroundthat[(2)] does notconveythesame informationas (1). Thus (1) conveysthecontentof what[Chris believes]. . . it is not even possible to infer (1) as consequence a of [(2»), on logicalgroundsalone-butonly bymaking use of [an]item of factual information,notcontainedin [(2), that(0) means in Englishthatthe earthis round]("On Carnap'sAnalysis ofS tatements of Assertionand Belief," pp. 168-169 of Linsky).
Consider(6), as a special case of (2), and thepurelysemanticdifferences thatChurchis limningbetween it and (1).Church'smain pointis that(1) attributed to Chris in a specialmanner,a gives the contentof the belief mannerin which (6) does not. In s hort,(1) identifies thatcontent.To be sure, both(1) and(6) specify thecontent,but(6) does so only bydescribing it inthemannerof (4), as whatever sense (0) ordinarily expresses in English. This is indeed a way of conceiving of the Englishordinarysense of(O)- -it is a conceptof theproposition,in Church'sterminology-but it is aconcept of thatsense thateven one who has no understanding of (0), as anEnglish expression,may possess withperfectmastery,providedhe or she knows only that(0) is a meaningfulEnglishexpression. For such apersonis in aposition expresses inEnglishits ownEnglishordinary sense. to infert hat(0) ordinarily This inferredknowledge istrivial;it doesnothingtofurther thequesttolearn what,by contrast, one whounderstands (0), as asentenceof English,t hereby knows, viz. that(0) specifically meansthis propositionin English: thatthe earthis round. This is essentially a special case of Russell'smore generaldistinctionbetweenknowledge by acquaintance and knowledge by description.35 The lat34 "Intensiona Ilsomorphism and Identifyof Belief," in Salmonand Soames, p . 168n22. In a similarvein, Churchhad writtenin "A Formulation of theLogic ofSense and Denotation,"at p . 5n: "This device is not esse ntialto theexpla nat ion, but is helpfulin order to dispel any remnantsof an illusionthatthereis somethingin some way necessaryor transparent abouttheconnectionbetweena wordor asentenceand its meaning,whereas, of course,thisconnectionis entir ely artificial a nd arbitrary ." 35 Russell , "Knowledg e by Acquantanc e and Knowledgeby Description ," reprintedin N. Salmonand S. Soames, eds., Propositions and Attitudes (OxfordUniversityPress, 1988), pp .I6-32.
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teris a distinctionbetween two kinds of knowledgethings,as of opposed to knowledge of facts (in French, two kinds connaissance of as opposed to savoir). Knowledge of tahingby acquaintance might beexplainedas a conceptionofthethingqua "thatF," perhapsperceptually ostendedor otherwise demonstratively selected,for aparticular sortal ' F' (which might even be the universal sortal,'thing').Knowledge of tahingby description,by contrast, is uniquely satisfies e," invoka correctconceptionofthethingqua "whatever ing some purelydescriptiveconditione. In Church'sterminology, knowledge of a thingby descriptionis exactlythe conceiving of thatthingthroughthe apprehensionof aconceptof it.36 36In calling a condition"purelydescriptive,"I mean to preclud e itsbeingsuch acondition as mightbe expressedin theform ' t heconditionof being [identicalw ith]thatvery F ' . It is assumed herethatChurch's notionof a concept likewiseexcludessuchconditions.Cf. my Reference and Essence (PrincetonUniversityPress and Basil Blackwell, 1981), pp . 14-23 (wheretheterm'descriptional' is employed insteadof 'descriptive') . a cquaintance in a verystrictsensewhichexcludedthepossiblityof acRussellconstrued quaintance w ithparticulars otherthanoneselfor mentalitems directlycontainedin one's consciousness . The distinctionitselfcan be drawnindependently of thissevererestriction, however,and is clearlylegitimatewith regardto more familiarnotionsof acquaintance . One such notionis thatof havingperceptual , or othernatural or "re a l,"cognitivecontact witha particular personor object- thesortof connectionthatis sufficienttoenableone to form beliefsor otherattitudes about theobject(in an ordinarysense). A som ewhat stricter n otionimposes thefurther cond it ion ofknowing who or what theobjectis, in an ordinarysense. (Some philosophers,h avingevidently o verlookedthepossibilityof perceiving anobjectwithoutknowingwhoor whattheobjectis, have confusedthesetwo broader notionsof acquantance , See my "How toMeasuretheStandardMetre," Proceedings of the Aristotelian Society, New Series, vol. 38 (1987/88), pp . 193-217, especiallyat 2~201n, 213ff.) Throughout 1 use the term' ident ificat ion' fornotionof a acquaintance implyingknowledgeof who orwhattheobjectis. Some contemporary n eo-Russellian t heoriesof meaning deny thatone who knowsthecontent of (1) automatically therebyknowswhatproposition it is thatChris believes(i.e., automatically therebyknows at leastone propositionthat Chris believes). Such theoriesmay holdinsteadthatone who knows thecontentof (1) is therebyacquaintedwiththebelievedpropositionin some less familiarway, and may not know exactlywhatpropositionis in question . Knowing whatF so-and-sois (as special cases, knowingwhatpropositionChrisis herebyheldtobelieve,or knowingwhatpersonor who-e-so-and-sois) may be a matterknowingof so-and-so, de re, thatso-and-sois it (him/her),believingthisfactaboutso-and-sowhileconceivingof it in aspecial, identifying way. (See my Frege's Puzzle , pp . 103-118, on thenotionof believinga propositionwhile takingit a particular way; see alsofootnote43 below.) Even such theories , however, will generally r ecognizean importantepistemologicald ifferencebetweenthecontentsof (1) and (6), such thatknowingthefactdescribed by (6) falls w ellshortof knowingthatfact describedby (1). It may be acknowledged, forexample, thatone who knowsthecontent of (I)-by contrast withonewho merelyknowsthecontent of (6)-ipso facto kows something de re aboutthepropositionthattheearthis round,namely,thatit is somethingChris believes(even, perhaps, withoutknowingexactlywhatpropositionis in question). This is sufficientfor my primarypurposein thediscussionto follow . For related d iscussion,see Mark Richard,"Articulated Terms," in J . Tomberlin, ed., Philosophical Perspectives, 7 : Language and Logic (Atascadero,Ca.: Ridgeview, 1993) , pp. 207 -230. It is possiblethatthereis a gradation of notionsof acquaintance . This wouldnot make Russell'sdistinctionbetweenknowledgeby acquaintance a nd knowledgeby description untenable ; on thecontr ary, it wouldmake for amultiplicity of legitimateRusselliandis-
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Russellthoughtof thetwo kinds ofknowledgeas mutually exclusive. To be sure,theverysame thingmay be simultaneously known in each of thetwo ways, buton Russell'sconceptionof thedistinction,no knowledge of thing a by descriptionis also acquaintance with thatsame thing. Otherphilosophersembracea strictrepresentational epistemologyaccordingto which all t heapprehensionofconcepts(in knowledge of thingsis achievedonlythrough Church'ssense), so thatall knowledge thingsis of ultimately knowledge by description(in Russell'ssense). Anyknowledgethatcan be called'acquaintance',on such a view, is merely peculiar a kind of knowledgeby description, one in whichtherelevant descriptiveconditionor conceptis of aspecialsort. It is arguablet hatFrege inparticular held astrictrepresentational epistemology. Andindeed,thisepistemological stancemay liebehindtheprincipal divergencebetweenbroadlyFregeanand Russelliansemantictheories.F Thereis an infinite-regress argumentagainstthetenability of thisstrict representational sortof epistemology:If conceivingof a thinginvariablyinparticular thing vokes aconceptofthatthing,t henin orderto conceive of any x, one must, in thatveryactofconceiving,a pprehendsome antecedently understoodconceptCI which is aconceptof z. But by thesame token,in orderto knowCI, one would have to apprehendsome furthera ntecedently understood c onceptC2 which is aconceptof CI, andso on. It seems to follow thatin orderto conceiveof anythingat all, one would have apprehend to each of aninfinitesequenceof antecedently understood c oncepts. One is invitedto concludefrom thethreat of this infiniteregressthatall knowledge of thingsultimately restson "direct"knowledgeof things,knowledgethatis not mediatedthroughconceptsof thosethings.i'" Textualevidencesuggeststhat,in a sense, Frege insteadembracedthe infiniteregress,via his infinitehierarchies of indirectsenses (ordinarysense, indirectsense, doublyindirectsense, and so on).39 It is a possibilityalso tinctions.The distinctionon which I rely inthediscussionto follow need n ot be made completelyprecise, and will besufficiently o bvious tosupporttheconclusionsr eachedby its means. epistemol37As explainedin thepreviousnote,Russellhimselfhelda respresentational otherthanoneselfand the mentalcontentsof one's ogy with regardto allparticulars consciousness . ContemporaryRussellianshave typicallyfavoreda less restrictive e piste"directly,"i.e., withouta ppealing mologyon whichone knowsvariousconcreteparticulars to individuating q ualitative conceptsof thoseparticulars . argumentin Russell, op. cit ., at pp. 28-29 of Salmon and 38Cf. the infinite-regress Soames. 390ne may notstraightforwardly concludefrom theinfinite-regress argumentthataccordingto thestrictrepresentational sortof epistemologyin question,in orderto conceive of a thingx, one must in that very act of conceiving apprehend each of infinitelym any concepts. For althoughtheepistemologyrequiresthattheactof conceivingof x necessarilyinvolvesan actof apprehandinga conceptCI of x, it does notrequirethattheact of apprehendingCl necessarilyinvolvesconceivingof CI. One might even labeltheactof apprehending a conceptwithoutconceivingof it akind of "directacquaintance " withthe concept. One may noteven concludefrom theinfinite-regress argumentthataccordingto strict
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thatChurchholds asimilarepistemological view.40 Even so,bothFregeand Churchdoubtlessly recognizedthatknowledge of at hingby acquaintance has a very different cognitiveflavor from mere knowledgedescription by (i.e., from knowledge of thethingnotby acquaintance) . The distinctionis made evidentin thefamiliarcontrast found inpairs of ,.41 The former identifiesthe colortermslike'white' and 't he color of snow color inquestionin a waythatthelatter does not. One who, becauseof limited experience, is ignorantof boththecolor of snowand the color of Church's hair, may still knowthatChurch'shairis thecolor of snow, i.e. , thatChurch'shairhas thesame colorthatsnow has, whatevercolorthat may be. If such apersonmay be said to knowwhatcolorChurch's hairis, thenhe or she knows it merely description by . Justas he or she knows that Church's hairis the color of snow, he or she also knows thatsnow is the color of snow.Whatsuch apersonlacks istheknowledge of which color that is, theidentifyinginformationnormallytakento becontainedin thewords: 'Snow is white'. A personwho hasbeen deprivedof sight sincebirthmay be incapableof knowing any color exceptmerely bydescription(e.g., as the property,visuall y manifestedin some manneror other, of reflecting light of representational episte mo logy, inorde r ot concei veof anyt hi ng on e mustalreadyapprehend each of infinit ely m an y concept s. What is required on Frege's doctrine of the hierar chy is that one be able to acqu ire anappreh ension o f anyone of the infinit ely m any concepts on s en ce thatembeds an expres sionwithin dem and, so to speak, in orde r tocom pre he nd a ent a sufficient lyl argenumber of ungernde ope ra tors. Strict representation al ep iste mology ess yieldsa possibleanswer to Russell'sfamous "no backwardroad" observat ion neverthel and the difficultyit is suppos ed to rai se for Frege's doctrin e . It is true thaton Frege' s theory , for anyt hingx ther eare countl ess concepts of x , no one of which cap be singledou t as privil egedor as the "des igna te "d or "standard"con cept of x. Butaccord ing ot st rict al epis te mo logy, inconce ivingof a con cept C1 thro ugh one's acq ua inta nce representation withit, one is thereby appre hen d ing aspec ia lide nt ify ingconcept C2 of cr . By atte n di ng ou gh one's acquaint an cewith it, to what one is appreh end ing, one ca n con ceive of C2 thr and hence throughone's apprehe nsio nof a specia l ident ify ing conce pt C3 of C2 , and so on. Thus it seems thatone need onlyatt e nd to what one is app rehe ndi ngto gene ra te aproper y starti ng froma sing leconcept. Even if ther e is no "backwa rd ro ad" Fregean hierarch (privilegedbranch) from a thingthatis not a concept to a conce ptof the thing, it seems there may be a backwardroadfrom aconce p t to a con cept of the conce pt. (Iam ind ebted her e to remarksmade in a seminarby SaulKripke andto later discussion with C. Anthon y Anderson.) Cf. my "Re ferenceand Information Content: Names and Descriptions," loco cit., and "A Problem in the Frege-Chur ch Theoryof Sens e and Denotation ," at p. 163. 40It must be noted, however ,thatC hurch eject r sFrege's notionof indirect sense. See "A Probl em in theFrege-Church Theory of Sense and Denotation ," pp, 164-165nlO. 41A number of philosoph ers have noted differences of mean ingbeween such terms . For e, sec D. M. Armstron g , "Ma t eria lism,Prop erties and a sa m ple of th e relevantliteratur Pred icates ," Monist, vol. 56, no. 2 (April 1972), pp. 163-176, at 174; J acgwon Kim, " O n the P sycho-Physical Identity Theory," American Philo soph ical Quarterly, vol. 3, no. 3 (July1966), pp. 227-235, and "Eventsand TheirDescr ipt ions,"in N. Rescher , ed ., Essays in Honor of Carl G. Hempel (Dordre cht: D. Reid el, 1969) , pp . 198-21 5, at 205- 206 ; Bernard Linsk y, "Ge ne ra l eTrms as Designator s," Pacific Philosophical Quarterly, vol. 65 (1984) , pp . 259--276; N. L. Wilson, "T he Troubl e withMea nings," Dialogue, vol. 3, no. 1 (June 1964), pp . 52- 64. (The lastis evidentlythe ancestorof the other discussions.)
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such-and-such wavelengths.) The legitimacyof thisdistinctionbetweentwo different ways of knowing the color white(or of knowing the colorChurch's of thatit will bereadilyrecognized even by hair, etc.) seems sufficiently obvious knowledge thosewho holdthattheknowledget hatsnow iswhiteis ultimately totheeffectt hatsnow haswhatever color it is thatuniquelysatisfies acertain, specialvisualcondition(perhapsa phenomenological condition) . The analogybetween thepair of contrasting colorterms and (4)- (5) is striking . In particular, knowing theEnglishmeaningof (0) by thedescription (4) is not a way of understanding (0), in anyordinarysense. As notedabove, someone whospeaks no English , upon learning onlythat(0) is a meaningful expressionof English , is therebygiven itsmeaningby thisdescription.The personstilllacksinformationspecifyingwhatthatmeaningis-towit, the identifyinginformation t hat(0) expresses inEnglishthepropositionthatthe earthis round. (Compare againknowingwhatcolor snow is only by the description'thecolor of snow' vs. knowing thatsnow is this color: white.) ArgumentagainstDummett'sapparent Earlierwe appliedtheTranslation hypothesisthat(4) and (5) areordinarily synonymousin English. A more directapplicationoftheargumentis possible. The translations ofthefollowing metalinguistic sentencesreveal afundamental difference in meaning: (7) 'T heearthis round'ordinarily expressesin Englishthepropositionthat theearthis round. (8) 'The earthis round'ordinarily expressesin Englishtheordinarysense in Englishof 'The earthis round'. The meta-Englishsentence(8) is tautologous, or virtually so. By contrast , (7)-ormore simply ' (0) means inEnglishthattheearthis round'-"conveys the content ," in Church'sterminology , of (0) in English; it identifies the Englishmeaningof (0), semantically specifyingit in a waythat(8) does not even come close to doing. Translation into anotherlanguageof both(7) and (8), and likewise the translation of both (1) and (2), is merely apedagogicalaid which more clearly reveals theirdivergent semanticpropert ies. We havealready seenthat the que s tionof whetherthestrictsortof translation thatis invoked inThe Translation Argumentconforms with the practiceof actualt ranslators and interpreters is quiteirrelevant to thepurposefor whichtranslation is pressed be into service.Whatis essentialfor thatpurpose is thatthe translation differences whatChurchcalls'literal' , i.e., sense-preserving .42 The relevant in meaning between the English (4) and (5)-which is theprincipalpoint of theargument -canbe seen independently of translation, forexampleby appealingto thestriking analogywith 'white'and 'thecolor of snow' . And as Churchsays, itshouldbe possible also to see the pointdirectly. 42 "Inte nsional I somorphism and Identityof Belief,"footnote25 (p . 168 of Salmonand Soames).
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NATHAN SALMON 5.
RecallLeed's contention t hatit is arguablynotthesemanticcontentof (1) in Englishbut theunderstanding of what(0) means (which, accordingto Burge, ispragmatically requiredby (1)), thataccountsfor theadditional informationthatan Englishspeakerobtainsfrom (1) buta French speaker does notobtainfrom thesense-preserving t ranslation of (6). This observation is meanttoisolatetheclaimthat,semantically , (1) specifiesthecontent of the beliefattributed toChris(as does (6)) in precisely t hemannerof (4). Butthis claim flies intheface ofChurch's main point,that(1) semantically "conveys thecontent " of Chris's alleged belief, identifyingthatcontentin a special mannerexhibitedby neither(6) nor itsFrenchtranslation . The contentof Chris's belief is not merely pragmatically impartedto anEnglishspeaker by an utterance of (1) via an inference. Contraryto Leeds, the content of Chris's belief issemantically encodedby (1). Indeed, Leed's rejoinder is nearlyenoughself-defeating.For itimplicitlyrecognizesthatthereis a specialway ofthinkingof thepropositionthattheearthis round-away of being acquaintedw ithit-such thatone who knowsthat(0) in English expresses thatpropositionconceived inthatway, knowswhat(0) means in English, whereasone who knows only that(0) in Englishexpressesthe Englishordinarysense of (0) (conceived in t hatway, by thatdescription) does not knowwhat(0) means. This specialidentifying"way ofthinking"of thepropositionseems to becarriedin Englishby (5), and it would seem to be preciselythisthatso markedlydistinguishes(5) from (4). It isnatural to assume, in fact,t hatthespecialway ofthinkingof thepropositionis nothing lessthantheconceptsemantically expressedin Englishby (5). It is difficult to imagine how else onemightexpressit.43 The foregoingconsiderations revealthereasonsforthefailure of thetheory thatDummettmistakenlyimputes to Frege. Fort hehypothesisthattheindirectsense of (0) inEnglishis theordinarysense of (4) yieldst heerroneous conclusionthatone whounderstands no Englishbeyond thewords 'Chris believesthat',a ndwho is informedthat(0) is a meaningfulEnglishsentence withoutbeing given itsactualmeaning, may therebyknow what(1) means in English.In fact, it would seem thatwhathe or she still needs orderto in knowthemeaningof (1) is to be giventhemeaningin Englishof (0) qua the propositionthattheearthis round,i.e., via theconceptthatis ordinarily expressedin Englishby (5) rathert hanthatordinarily expressedby (4). It is thuspossible to seedirectly,w ithoutresortingto translation, thesupe43presumably understanding(0) , as a sentenceof English, ent a ilsknowing what(0) means in English, but it is arguabl e that under s tanding , in a strictsense, requiresmore thanthis (p erhaps acquiringthe knowledgeof the meaning of (0) in a spec ia lcom putational m anner). Ther e are delicat e issues, on which the presentdiscussion is neutral , concerningtheextentto whichtheEnglish m eaningof (5) ca pt ur es th e way ofconceiving thepropositionin questionthatis pos sessed by one who correc t lyunderstands(0) as a sentenceof English.See footnote36 above .
THE VERY POSSIBILITY OF LANGUAGE
593
riorityof Frege's thesisconcerningindirectsense overthethesisDummett erroneously attributes to him. The foregoing also dramatically exposes a fatale rrorin Dummett'sThesis thatthe only way in which we can conceive of any sense thesense is as of some particular linguisticexpression. Using Church'sterminology (in combination withsome of Dummett's), thethesis may bestatedmore precisely thus: For everyconceptci , if someone conceivesof ci , thenthereis some conceptC2 of ci suchthat that very actof conceivingof ci consistsin foreveryconceptC2 of ci thatwe apprehendingC2; and furthermore, can apprehend,thereis some expressione (of some possible language : (i) Cl is theordinarysenseof e (in I);and (ii) C2 presents I) such that ci as theordinarysense of e (in I), i.e., to apprehendC2 is exactlyto conceive of ci ast heordinarysense of e (in1).44
Although D ummettattributes thisthesisto Frege, it is in fact deeplyout of syncwiththefundamental character andstructure ofFrege'sphilosophyof language.Frege does saythatwe apprehendpropositionsthroughwords.V' And it seemsclearly t ruethatwe do thiswithat leasta greatmany propositions. It by no means follows , however, thatwe areunableto conceive of thepropositionexceptas whateversense it may bethatthosewords ordinarilyexpress. Frege wouldcertainly haveinsisted,againstDummett, that thereare infinitelymany ways todenotethepropositionthattheearthis roundwithoutmentioningparticular linguisticexpressions, eitherexplicitly or implicitly : 'thepropositionthatColumbusset out to prove bysailing westwardto India', 'thepropositionthatAristotleproved onthe basis of theshape of theearth'ss hadowon themoon', 'Chris'sfavoriteproposition', 'thefirstthingColumbusassertedupon sightingthemainlandof America', etc. These unremarkable phrases,and many others,express graspableconcepts that"give us" thepropositionin waysotherthanas thesense of an expression.t" More important,t heveryapprehensionof a proposition, even by means ofthewordsthroughwhich weapprehendit, crucially involves an acquaintance identifyingconceptionof theproposition, a knowledge of it by and not merely as whatever it isthatthese wordso rdinarily express. This is 44See footnot es 19 and 36 above. The firstconj unctexpresses strict represen tationalism wit h rega rd toconce pts. On a Fregean theory the word 'as ' occurrin g in Dumm ett's ct ions ' . . . is given to us as - ' and 'conce ives of. . . as - ' mu st be itselfregarded constru as an ungemde operator . Dumm ett ' s Thesis, as formula ted hereandas given by Dumm ett himself,thusinvolv e s quant ification intoa nonextensionalconte xt. (I am not claiming thatDummettwouldaccept my sp ecific formulation of his thesis , onlythathe explici t ly endor ses thethesisitself, which 1 formulat e thus,and attribut es thesa me thesis to Frege . The textualevide nce forthese claims is clear.) 45For example, in "On the Scientifi c Justifica tionof a ConceptualNotation" (reproduced in his Conceptual Notation and Related Articles, T . W . Bynum, ed., (Oxford University Press, 1972), pp . 83-89) , Frege says that "we thinkin words ... , and if notin words,then or othersymbols" (p . 84). See footnote15 above. in mathematical 461n lightof this, Frege undoubtedl y rej ected even theweaker, Berkeley anthesis mentioned in note 17 above.
594
NATHAN SALMON
tru e in fact of understanding in general ; thereis no comprehensionwithout identification. Dummett'sThesis precludesthepossibilityof a languaget hatworksthe way Fregethought alllanguageworked,and even theverypossibilityof languageitself. OnDummett'sview, intaking(0) toexpresssuch-and-such, one must, in thatvery actof cognition,conceiveof theconceptin questionas whatever sense is ordinarily expressed bye, forsome particular linguisticexpressione- t herebeing no otherway for us to conceiveof anyconcept.This theorymakes all ofl anguageunintelligible for us inprinciple. Dummett's Thesis leavesourgraspof language in a stateof ignorancee xactly .analogous tosomeonewho, from birth,sees theworldonly inblackandwhiteandshades of grey. A personwho hasalwaysbeencompletelycolor-blind may knowthat physicalobjectsarecolored,a nd thatcertainof them have thesame color as certainothers(snow and Church'shair,grassandemeralds, etc.), butin some sensehas no way ofl earning w hatcoloranythingis. Dumm ett'sThesis reducesus all tothestateof theinternational traveler who knowsthatthe wordshe or she seesand hearshave meaning, butis completelyignorantof whatthose meaningsare. Indeed, thetheoryrendersus considerably worse off thanthetraveler . A touristcan identifytheforeignmeanings by consultinga phrasebook. Dummett'stheoryhas theconsequencethatwe are unableeven tounderstand e xpressionsby translating them intoournative language.For theonlyknowledgewe have ofthemeaningsof our own words is by thedescription : whateverit is thatthosewordsmean. On Dummett's theory,t hereis no identification of meanings, no knowledgeof meaningby acquaintance, onlyknowledgeby description.This makes theveil ofsyntax utterly opaque,and thewall ofu nintelligibility impenetrableeven inprinciple. Since therecanbe no comprehensionwithoutidentification , Dummett's theoryhas theunint e ndedconsequencethatwe comprehendnothing. V The fundamental flaw inDummett's theorycanalso be seen throughthe theoryof definition , by means of a variationof theinfinite-regr ess argument consideredabove. Accordingto Dummett'sThesis, in knowingthattheordinarysense of anexpressione l (in a languageIt> is such-and-such , one must conceiveof thatsense as theordinarysense of some expressione2 (of a lan guagel2). This is tantamount to theclaim thateverymeaningfullinguistic expressionthatthespeakerunderstands is understood by means of a kind of verbaldefinition:T he ordinarysense of el is thesame as theordinarysense 47It is arguabl e thata p er son co mp lete ly ol c or-blindfrom birthmay neverth e less learn of the colorgreen (d e re) that grass and eme ra lds ar e that color, by learning that they reflectlig htof such-and-such wav elength s under visua lly ormalconditi n on s. Even so, ther e st illseems to be some knowledge thatthose of us who see gree n(as gree n) have, and that thecom plete lycolor -blind erson p lack s, conce rn ingthe color o f grass. In some sense , the comp letely color-blind erson p stilldoes not know how grass looks, phen om enolo gically , with regard to color. By cont ra st, if Dumm ett'stheor y ';"ere correct, non e of us could know of the propositi on that theearth is round(de re), that(0) expresses it in English (letalon e could weunderstandwhat(0) m eansin Englis h-see footnot es 36 and 43 above).
THE VERY POSSIBILITY OF LANGUAGE
595
, el =def e2.48 The definiens e2 would of e2, or as it is sometimes written likewise beunderstood by meansof ananalogousdefinition , e2 = d ef e3, and so on. It is well known , however, thatnot all expressions of a languagemay be understood by means of ve rbaldefinition.T herehas to be a st ock of primitives, whose sense is learnedin some wayotherthanby verb aldefinition(by ostensive definition,for example). Dummett's Thesis leadsultimately to a vicious circularity among thedefinitions . And thismakes allunderstanding impossible evenin principle. The rootproblemwithDummett's Thesis is thatit restrictsall knowledge of meaningto knowledge by d escription , precludingeven thepossibilityof ouridentifyingtheordinarysense of anyexpression. Since ther e is no comprehensionwithoutidentification , Dummett's Thesis makes comprehension utterly unattainable. And this,I submit, excludestheverypossibilityof the phenomenonwe calllanguage. If I am correct,t hemain point of theChurchTranslation A rgumentis se in Englishexexactlytheantithesisof Dummett'sThesis. The 'that'-clau pressionslike (1)and (7) carrieswithit aspecialway of conceptualizing the contentof thesentencefollowing'that',an identifyingway ofthinkingof thepropositionwhichconstitutes acquaintance rathert hanmere knowledge by description.Church's appealto translation serves to illuminat e thisfundamentalpoint. Frege's explanation of theindirect sense of (0) in English as theordinarysense of (5)rath er thanthatof (4) almostcertainly refl ect s his owngraspof thissame fundamental point, and with it aepudiation r of Dummett's Thesis. Insofaras understanding requiresacquaintanc e, whatis it thatwe are requiredto beacquainted with? If thereis no comprehensionwithoutidentification , whatis it thatwe are identifying?Notjustsentences in alanguage , as we have seen . Butif notjustsentencesin a language , then whatelse? The Translation Argumenthas beenwithus a long tim e -longer eventhan I have. Yet manylanguage t heoriststodayhave missed its principalpoint. If nothingelse, thereis an importantlesson to bel earn ed fromthe failure of eth argument'scriticsto appreciatethatpoint,andfrom thedecisive collapseof at leastone theorythathas been proffered in defianc e of theargument. NathanSalmon Departmentof Philosophy Universityof California SantaBarbara,California 93106 USA Email: [email protected]. edu 48This notationis normall y used for defin it ion swith in a sing le an l gu age . For present purposes we m ay thinkof the union of allof the separat e languag es spo ken by a particular speaker as cons t itu ting a sing lecom pre he ns ive angu l age. (Any resul t ingexical l ambig uities m ay be resolv edby m ean s of disamb iguatingsubscripts.} Alternat ively, the notat ion s , for examp le by writ ing: is easily extendi b le toaccommodate inter-language definiti on [e j , II] =def
[e2, bJ.
INDEX lA, see -X-calculus, first-order
224, 225, 226, 229, 230, 237, 244 Hilbert,238 sets, 245 Skolem,233 signature,280, 281 term,280 Allwein, G., 80 , 82 Alston, W., 437 analysis Church'sTranslation Argument,575 classificatory , 557 decomposition,558 explanatory , 557 identity,545-546,555, 561 conceptual structure, 565 Frege'sConstraint , 546, 565, 567 and property,s tructure constituency , 558 sense, 563-564: modes of access, 566 transparency, 546, 566 truthof, 556 inconsistency , 546, 565 natureof, 557 paradoxof, 545-546, 554-556,560,563,566, 569,582f surprising,562, 563, 564 synonymyof forms, 558, 567 Anderson,A. R., 83f, 97f, 191£, 193, 195, 197f, 198f, 200, 208, 211, 213, 214, 560f R~, 194 weakimplication , viii Anderson,C . A., 395-427,500, 504, 518 Andrews, P., 470 Andrews'RecursionTheorem, 16
2A, see -X-calculus, second-order
E-determinacy , 118 E-induction,129, 133 A-critical , 203, 209, 210 a-frame, 87, 91 assertional, 87 associative , 93: classical, 91 canonical , 91 pre-relation-algebra , 88 protoDe Morganmonoid, 89 abstraction setand -X-, , 290 Russell'sparadox,303 adjointfunctors,181-184 Ackermann,W., 211, 229, 230, 405f, 417 decisionproblem,458 w-order logic, 239f R, 3088 Ackermann constants in, 202f R~, 194 substitution rule, viii Aczel, P., 304 Adams, P. W., 279 algebra Boolean, 79, 82, 83, 94 , 101 atomic,185-189 framed, 90, 102 pre relation , 93 relation, 91, 95, 102 Heyting,101 Lindenbaum,185, 186, 187, 188, 199, 200 pre relation, 84, 85, 88, 93, 95 77-108,84,85,90, relation, 91,95,103 algebraic school(algebraist), 222, 223, 597
598
INDEX
Angelelli, I., 40lf antinomy , see paradox Apostel,T. M., 408f Apostoli, P.,3-59 arithmetic analysis,248 axiomatization and G6del's incompletenesstheorem, 241 first-order, 248, 313 notcategorical, 235f sentences/propositions of, 6 incompletenesstheorem,248 Peano(PA), 314 provably recursive function, 320, 322, 327 realizability, 361 -363 second-order,314 Armstrong,D. M., 590f Asperti,A., 353 assertional a-frame,86-87 associativity, 264, 265, 266, 267, 269 Austin,J . L., 418 automorphism Boolean, 157 invarianceu nder(AIness), 157, 159, 160, 161, 162, 163, 165, 167, 169, 170, 171,173 axioms, 240 accessibility, 348 Ackermann's,303 choice (AC), 116, 126, 232, 239f, 469, 471, 473, 474, 477 realizability, 358 Church'saxioms, 139~140 intensional, 500 comprehension,469, 471, 493, 498, 502, 503 low, scheme of, 115 , 121, 136
stratified, 471 continuity, second-order,228, 237 completeness , geometry, Hilbert's,235, 236, 237 counting,475, 477 descriptions,139 domain closure,366-367, 368, 384 extensionality, 139, 144, 225-226,395,396,398, 471 extremal,Hilbert's , 228, 229f first-order logic, 229 Skolem, 241 foundation,113, 121, 126, 239f,304 infinity, 27, 43, 139, 146 , 469, 473,474,477 completeness,251 lowreplacement,134 map theory, 290, 291, 299-303,309-310 abstraction, 300 application,299, 300 equality, 299, 301 existence, 302 extensionality , 301 minimality, 301 monotonicity,301 quantification, 303 QuartumNon Datur (QND), 289,300,301 renaming, 300 selection, 301 substitutivity, 299 transitivity, 299 truth ,299 Markovprinciple, 358, 362-363 occur,equalitytheory , 367, 370-371 pairing, 115
INDEX axioms (continued) Peano's axioms, 140, 230f powerset,134 reducibility , 228, 229, 492, 493,496,498-499,503 replacement,235, 239f restriction, Fraenk el's,235, 236, 237 second-order,Zermelo, 241 logic, sense anddenotation 512-513 separation,231, 232 sumset, 116, 134 TertiumNon Datur,289 urelements, 471 axiomatization analysis , 228 arithmeticand Oodel's incompletenesstheorem, 241 first-order, 227, 233 geometry,H ilbert , 228, 229 logic,Hilbert,229 mathematical school, 223 set as non -logical,235 set theory , 225, 231, 232, 238 Bachmann, F., 236f Bar-Hillel , Y. , 236f Barendregt , H., 18If, 275-285 Barwise, J ., 78, 96, 162, 221, 246, 314f, 483f, 484 BCK,206f Bealer, G., 405f, 41Of , 534 Beghelli, .F, 161 Bell, J . L., 3, 186f, 213 Belnap,N. D., 83f, 91, 97f, 19If, 193, 195, 197f, 198f, 200, 207, 208, 211, 214 4-valuedinterpretation , 17 R~ , 194 truththeory , 16 weakimplication , viii Benacerraf, P., 234 Berarducci , A., 278
599 Bergmann, G., 95 Berkeley, B., sense conception, 579f Berline, C., 288, 292, 297, 301, 304, 305, 306 Bernays, P., 227, 230 Churchon Platonism , 247f language(logic) first-order , 246 higher-order , 227 relativism,243 .a-convertibility, 276 Bialynicki-Birula, A., 89f, 93f Bibliography of Symbolic Logic, ix Birkhoff, G. , 91f, 407 bivalence, 231 Black, M., 402f, 545, 560, 569 Boffa, M., 118f , 132 Bohm, C., 275, 278, 280, 281, 282 Bonevac,D. A., 442 Boole, G., 185, 222, 224 Boolean algebra , 79, 82, 83, 83, 94, 101, 222 atomic, 185-189 automorphism, 157 constant , 164, 166 De Morgan monoid, 104 function , 166 lattice,159 logicprogramming, connectivesand quantifi ers, 388 negation , 83, 103, 104, 213, 214f protoDe Morgan monoid, 83, 84 relation, 166 algebra,94 Boolos, G., 3, 243f, 249 Bradley, .FH., 95f Brouwer ordinal , 343, 345 Brown, C., 80 Bunder, M., 105f
600
INDEX
Burali-Forti's paradox,287, 292, 304,472-473 Burge, T ., xiii, 560, 565, 574f, 582, 583, 585, 592 Burgess,J. P., 429-444 CO-structure, see Church-Oswald structure calculus (calculi), 222 classicalpropositional,139 extended,Hilbert's,229 formal,228 Lambek,103 relations, 78 weakimplicational, 19lf Cantor,G., 240 paradox,472 set theory, 225, 231 theorem,472, 500, 501, 503 Carboni,353, 359 cardinal numbers,472 quantification, 155 quantifiers, 154 Carnap,R., xi, 236f, 250f, 400-401,404, 406~ 429, 433, 440, 574f, 576f, 578 Cartesianclosed category, 181 categoricity (categorical) first-order, 250 vs. second-order context , 234 natural number,Dedekind, 240 Peanopostulates, 230f set-theory Skolem, 235 Von Neumann,237 statements, 229 category sets, 182 theory, 353 Cayley'stheorem,271 Chaitin, G. J ., 479, 486, 487 Chan,D., 365,366,373-374,382
Chang,C. C., 297 chaos,non-calculability, 486 randomness,487 Chellas, .BF., 340 Chihara , C., 431-432, 434 Chomsky, N., 439-440 Chuaqui,R., 150 Church, A ., vii-ix, xi-xiii, 214 Alternative (0), (1), (1 *), (2), 421-423,514,516-518 analysisparadox,545, 560-561,563, 582f triviality, 562-563 anti-nominalist , 440 axioms, 139-140 intensional, 500 boundvariableabstraction, 4 connectives elimination,414f semanticalrules, 414f -415f definition,contextual, 408-409 existence, 442 functional abstraction, type-free, 6 Grelling'sparadox,491, 493-499 higher-order (second-order) logic(language),243f, 244, 246, 249, 250 identity criterion,396f, 401: counterexample to Carnap's, 401; failure of substitution , 403; senses, alternatives , 509; synonymous isomorphism, 403, 404, 568 formalizedintensional logics, 395 intensional for propositions,492, 499-500
INDEX
Church,A . (continued) implication,converse, 407f indirectsense, against Frege'snotionof, 590f intensional axioms, 500 isomorophism, 550 A -calculus,105, 181, 182, 275,287,418: typed terms,331 -definablefunctions,313 notation in relation algebra,101 logic belief,401 intensional, 421 sense anddenotation, 395, 404,421,507,509.510, 537-538,560: axioms, 512-513, 523-524; collapse to extensionality , 516; contexts,de re and de dicta, 526; models, 522: possible worlds, 524; predicates and functions,514: partial,518; terms withoutd enotation, 517, 519f, 541; type hierarchy, 536 map theory, 288 MR~, 194f model,125 monoid (c-monoid), 83f, 191, 198, 199, 200, 201, 203, 205, 208, 209 natural numbersas iterators, 146 negation,420 inferential definition,195 non-effective consequence, 247 numerals,313, 321
601 objectlanguage , defined notation, 400f ontological commitment, 432 Platonism,247f propositional function , 504 presuppositionof all, 219 propositions, 433, 57 ramifiedtypetheory,492 randomness,479, 486 relevant logic, 206, 211 rules, 139 Russell antinomyof propositional functions,499-505 paradox, 492 semantical constants, 491-492, 500-501 relations , 501 set theorywith auniversal set, 109, 130, 136 comprehension,135 lowreplacement,116 well-founded set, 133 simplicityof meaning,402f symbols, properand improper, 409 translation argument,407f, 574-575,587 truthbearers,481£ type-freefunctional abstraction, 6 typetheory , 139-140, 145, 491 simple, 509 use criterion,201 weaktheoryof implication , 77, 106, 191, 192, 193 Church-Oswald construction, 125, 134, 135 model, 113-134 structure, 114, 128 Church -RosserTheorem,vii, 276 Church's Theorem,vii
602
INDEX
Church's Theorem(continued) Leibniz' program, 445-466,479: ars combinatoria, 44 randomness, 479-487 Church'sThesis, vii-viii, 479 chaos, 486 constructiv e nonconstructive distinction , 42~ realizability , 357-358, 362 undecidability , 439 Chwistek, L., viii Clark,K. L. completion,365, 368 equalitytheory , 367 soundnessresult, 368 class logic, 224, 229 map theory , 290 propositional, 227, 496 second-order,228 set theorymodel, 237 Skolem, 233 Von Neumann,236 classicalvaluation, 185 compactness, logic programming, 370 Zermelo, 242 complete theory,366 completeness , 250f 3-valued, 367 axiom of infinity, 251 canonicalcompleteness theoremforR_&, 199 expressive, 155 firstcompletenesst heorem forR _ &, 199 Ccdel'stheorem, 241, 246 logicprogramming,366, 383-384 propositional logic, 230 second-order a rithm eti c, deductiv e completeness , 230f
complexity,c omputational, 486 consequence first-order, 249 non-effective n otion, 247 second-order, 249, 251 set-theoretic , 251 consistency G 2 ,40 G+,x , 38 NaDSetI, 39 proof, mathematics,226, 230 constants logical, 163 -166 semantical,491-492, 500-501 construction, 241 Cooper, R., 162 Corcoran , J. , 61-75 , 149, 227f, 249 CR, 213, 214 CRVp, 214 Cresswell, M.J., 401£ cumulativehierarchy , 239, 476-477 Curry , H. B., 4, 105, 106, 192, 483f Davidson , D., 402f, 528, 574f, 575f Davis, M., 333 Dawson, J ., 242f De Bruin,P., 275,277,278 De Morgan, A., 81 Boolean algebra , 185 De Morganmonoid, 104 lattice , 83 monoid, 78, 79, 82, 83, 87, 88,89 negation,83, 103, 104, 420f protoDe Morganmonoid, 93, 103 decidability LR,108 R_&,205 relevant , 195
INDEX Dedekind, R., 234f, 240 continuity axioms, secondorder,228 finitenotion, 237 mathematical school, 223 natural numbers, set theoretic, 233 realnumberconstruction , 241 second-order a rithmetic , 73, 248-249 DeductionTheorem,81 definable(definability) A logicalrelation, 340 -provability,325 uniformly, 170 definit
Fraenkel , 233 Skolemand Zermelo ,239 Zermelo, 231, 232 notion, 238 definition,implicit, 240, 241 Demopoulos,W. G. , 3 De re cont ext s,see sense and denotation Detlefsen,M., 445-467 Devaney, R. L., 484 Dickson, L. E ., 197 Dickson's Fact,203 Dickson's Lemma, 197, 202, 203 Dilworth,R. P., 81 Dosen, K., 80, 81 Dummett,M., 440, 574, 575, 582 Church'sTranslation Argument Dummett's Thesis, 580-581,592-594 quotation t ranslation , 582-585 directand indirectsensei denotation , 578-579 Dunn, J . M., 77-108, 198f, 199f, 200,208,211£, 213
603 4-valuedinterpretation, 17 Edwards,D. J. , 279 effectively e numerable deductiveconsequence, 220 Enderton,H. B. , 155 entailment, viii, 199f equalityt heory, see logic programming Etchemendy , J ., 149, 156,222, 483f,484 Euclid,222 excludedmiddle, 231 ExclusionLemma, 91, 92 existence idiomaticvs. paradigmatic use, 433 theorems, 429 ExtensionLemma, 91 extensionality, 114 axiom, 139 propositional function,504 strong , 503-504 theorem, 100 Feferman, S. , 221, 246, 288, 290, 299 leastfixedpoint,21 Feigner, U., 303 Feys, R., 192 Field, H., 245£ filter maximal,94 prime, 94 Fine, H. B., 470 Fine, K. , 97, 198f finite modelproperty,191, 200 LR,206 semilattice s emantics, 208 finite-variable fragment , 187-189 finitelyground ed , 196 finitism, 227 first-order analysis,248
604 first-order (continued) arithmetic , 248, 313 notcategorical , 235f standard model, 248 axioms, Skolem, 241 categoricity, 250 .cornpleteness, 241, 246 formula, 230 , 231 foundations, 251 inadequate,246, 247-248 inductionschemaforN, 46 language,4-5, 38, 227, 228, 229, 232, 242, 244 Codel'scompleteness theorem,241 set-theory, Skolem, 233, 235, 239 logic, 221, 228, 229, 230, 231, 249-250 decision procedure,none for, vii, viii foundations , 251 map theory, 302 second-order,as against, 245, 251 type-freeabstraction ,3 Markovprinciple,358 ontology, 251 realizability, 353-363 second-order , as against,249, 250 semantics,227, 234, 244 prooftheory,243f set theory,251 variable,224, 227, 239, 249 quantification, 233, 239 Fitch,F. B., 4 Fitting,M . R., 387, 391 4-valuedinterpretation , 17 truththeory, 16, 19 fixed-point,7 greatest,115 map theory , 290, 301, 303 Tarski-Knast er, 135
INDEX
theorem,278 Flagg,R. C., 290, 299 formal deduction,Codel, 243 language,222 syntaxandsemantics,234 formalism(formalization), 227, 430 conventional, 438 Forster,T. E., 109-138,470 Fortune,S., 313, 314, 315f foundations, 251, 251£, 252 logic, 238 first-order, 251 second-order,234-235, 251 map theory,292 mathematics,220, 229, 232f, 233, 238 Skolem'srejectionof set theoretic , 234 Quine, 245 relativism , 240 set theoryas, 238 Fraenkel,A., 225, 233, 234, 235, 236, 238 frame, 85, 87 articulated , 85 associativeassertional , 95 classical , 94 concreterelation a lgebra , 102,103 Meyer-monoid,90 models oftheA-calculus, 104 refinedrelation algebra,97, 98,99 framed Meyermonoid, 90, 94 Peirce groupoid, 86, 87 monoid, 93 semi-group, 92 pre relation a lgebra , 88, 93 protoDe Morganmonoid, 89,93
INDEX relation a lgebra,90, 91, 95 Frege, G. , 3-4,40, 402f, 418f, 440, 442, 574f Church 's Translation Argument,574f, 575 de re contexts,525 directand indirectspeech, 508f identityon objects,228 indirect denotation, 577 sense, 575, 576 languageas unlearnable , 529 logicistschool, 223, 224 natural numbers, 39 Russell'sparadox,226 second-order variables , 227 , viii-Ix, sense anddenotation 225, 507 infinitesense semantic hierarchy,528 ontology , 509, 510 shiftscausedby indirect contextreembeddings, 529 termswithoutreference , 518 thinkingin symbols, 593 Friedman, H., 326 Freyd, P., 353, 359 functionalcalculus , see first-order logic fusion, 194 G 2 , 52 gaggletheory,79, 96f Gandy,R. 0. , 139-147,220 Geach, P., 536f, 574, 575f, 580, 582, 583f Gellert , W ., 417 Giirdenfors , P., 158 Geach,P. , 402f, 406f Gentzen , G. calculus , 6, 17, 31 LK,lO
605 freeobjectvariables,8 sequents , 81 structural rules, 103 trees, 11 Gilmore, P., 39, 221, 244 boundvariableabstraction, 4 first-order Gentzen-style sequentcalculus (NaDSetI), 6, 12, 13, 16, 17, 19 leastfixed point, 21 LK,3 numerals , 34, 38 revisionoperator,20 Russell's paradox,39 semanticgroundingof primitive-recursive queries,14 Girard, J.-Y. , 343, 347 monoid,82 second-orderA-calculus(2..\) , 313, 315, 322 Givant, S., 78, 102f Godel, K. , 241--244 completeness of first-order logic, 230, 241, 246 Dialecticat heorem,321 effectiv ely enumerable consequenc e, 220 first-order vs. second-order logic, 227, 231 incompleteness arithmetic , 241 theorem,vii-viii, 73, 248, 459,461f logic asaboutclassesand relations, 224 recurs ion general , 479 primitive, 319 Richardparadox, 486 Russell'sparadox, iterative concept of set, 226 Zermelo,242
606 Goldfarb , W., 223, 232f Gonseth, F. , 243 Goodman, N., 429-432, 440 Grattan -Guinness, 1., 232, 242f Grim, P., 395f, 479, 484 Grishin, V. N., 473 group, 269-271 theory , 261, 270, 271 subgroup, 271 Grue, K., 287-311 Grundlagen, 3 Guerrini,S., 275, 278, 280, 281, 282 Gupta,A. K., 16, 17,484 Gurr,D., 80 Gyssens, M., 78 Handley , W. G., 313 Hart,T . P., 279 Hazen, A., 244, 491 HEAP, 202, 203, 204 Heidelberger , H., 583 Hellwich , M., 417 Henkin, L., 221 ion, 222 interpr etat Henson, C. W. , 109, 110, 111 Herbrand,J ., 479 Herzberger, H., 484 Higgenbotham,J. 582f higher - order concepts,finitudeand well-foundedn ess, 251 intuitionistic logic, 362 realizability, 353-363, 361, 363 Hilbert,D ., 22lf,231,417 algebraicschool, 238 axiomaticmethodapplied to geometryand logic , 229 completeness , Hilbert's program, 230 Decision Problem, 458 s-operator,map theory, 302 geometryaxiomatization as second-ord er, 228
INDEX languag es first-ord er, 246 higher-order, proponent , 227 mathema tical chool, s 223 w-order logic, 239f primitiverecursion,319f program, 238, 243f, 25lf Russell's paradoxandsets, 226 substitution rule, viii Hiller , A. P., 477 Hochberg,H., 419 Hodes, H., 442 Holmes, M. R., 469-478 Howard, W. A., 343 Curry-Howard Isomorphism, 106 Hume, D., 3 Huntington , E., 223, 228, 229f Hyland,353, 358, 359 identity indiscernibili ty, 493 intensional , 499-500 propositional functions,504 identity , criterionof, 269 associativ e m eanings, 416-418,422 axiom ofextensionality, 395, 396, 398 commutativemeanings, 411-415,421 commutativity, 415-416 definitions,411, 414 contextual, 408-411 explicit, 408, 410-411 stipulative, 409, 410-411 intensional entities, 395-423 sema ntics, 421, 422 intrinsically simple m eanings, 402 isomorophism intensional , 401-402
INDEX synonymous,400-404, 410, 420, 422: alleged counterexamples , 405-408: commutation : conversionof nonsymrnetricals, 405-406, 407; r:elations and connectives:conjunction,412-413,415; disjunction,414, 415; materialequivalence, 414; truth -tables , 412-414; symmetricals, 405,406,407; stipulation, 403-404,407 A-conversion, 405, 411f, 421 metal object-language, 400f-401£, 402 one-level , 395 ontological, 395, 397, 398, 399,401,422,423 propositions,395, 400 composition,422 redundant meanings, 418-421 identityfunction , 418 semantical , 395,396,401 , 402,403,415,421,423 observational consequence , 398 senses, Church'salternatives, 509 symbols proper,409 improper, syncategorematic, 409 totality condition,397, 399 true-ofor satisfiedby, 396, 397, 398 two-level, 396 , 402f impredicative,38 definition , 232f natural numbers, 42, 43 property,222
607 inaccessible cardinal , 239f ordinal , map theory,305, 306 inclusionrelation as a set, 473 incomplete(incompleteness) SLDNF-resolution , 369 mathematics , 439 theorem,G6del's,vii-viii, 73, 248, 459, 461£ independence , 223 universe,171 induction first-order, 43 schema for N, 46 pre-induction, 51, 52 infinite formulas , 229, 230 languages , Lowenheim-Skolem theorem, 227, 230 logic, Zermelo, 242 setsand powerset,250 infinitedivision principle(lOP), 191, 196, 197, 202, 203 instrumentalist, 430 intensional (intensionality) axioms, Church, 500 Cantor 's theorem,501-502 entiti es, criterionof identity , 395-423 logic, viii-ix, 395 principle,500 propositional function,504 Russell'santinomy(propositionalfunctions) , 503 Introduction to Mathematical Logic, viii, ix interpretation canonical , 199 first- vs. second-order , 249 formallanguage , 222 hereditary ( h-interpretation) , 199, 201, 203, 204, 208, 209
608 interpretation (continued) higher-order, 221 lambda,280, 281 logical , 164 constants objects,150 logicist , 224 Peirce, 227 propositional , 250f pure relevant logic, 198 quantifier, 229 Schroder,227 settheory, Skolem, 233 relativism,240 subclassesof a fixeddomain, 227 intuitionistic (intuitionism),238, 429-430 revisionism, 430 anti-nominalist, 440 invariance,150 automorphisms(AIness), 149, 157, 159, 160, 161, 165, 167, 169, 170, 171, 173 isomorphism, 171 permutation (PIness), 149, 150, 151, 152, 154, 156, 157,170, 172, 173 inverse, 269 isomorphism, 266, 267 -270 Boolean, 267 group, 271 inversion, 269-270 map theory, 297 theorem,270, 271 Jeffrey,R ., 243f Jensen, R. B., 109, 469 Johansson,I., 193, 194f Jonsson,B., 79, 80, 84, 92 Journal of Symbolic Logic, ix Jung, A., 339 Kamke, E., 408 Kaplan, D., xi-xii, 404f, 405f,
INDEX
422f, 519f, 525f, 538-541 , 576f x-Scottdomain, 288 Kastner,H., 417 Kaye, R., 125 Keenan,E. L., 149-180 Keisler, K. J., 297 Kemp , G., 578f, 580f, 584f Kim, J. , 590f King, J., 405f, 406f, 423f Kleene, S. C., 275, 276, 277 3-valuedinterpretation, 16, 18,19
x
-calculus,182 -definablefunctions, 313, 479 liarparadox, contingent, 483f recursivefunctions and realizability, 357 T-predicate,30, 357-358 truth-tables, 390 Klein, F. A., 149 Kolmogorov, A. N. , 479, 486 Korner, F. , 132 Kraegeloh , K.-D., 78 Kreisel,G., 247, 248, 236 Kripke, S., 584 arithmeticmodel restrictions , 13 decidable LR,205 R_,195-196 R_&,205 frame, 340-341 infinitedivision principle (IDP) . 191 leastfixedpoint,21 liarparadox,semantic grounding,485f logicrelation, 339-341 R~ as conservative extensionof R_ , 194 semanticgroundingof
INDEX
primitive-recursive queries, 14 Kripke's Lemma, 197,202,203 Kunen,K., 365, 366, 367, 384, 387, 388, 389, 391 Kiistner,H., 417 LR containment theorem,207, 208 decidable,195, 205, 209 finite modeltheorem,207 finitesemilattice corollary, 211 LR+,195 Lakoff, G., 149 Lagrange 's theorem,271 >. (lambda) -abstraction, 139, 289, 290 -calculus , vii-viii, 101, 105-106, 181-184,275 ,a-convertibility , 276 ,a-equivalent, 300 definable functions , 279: 2A, 320, 322, 327; computability , 322-327 discriminator,275, 282 first-order type (IA) , 313 frame models, 104 foundations, 287, 290 lists, 279 map theory , 287-311 second-order t ype (2A), 313,314,315-317,318, 319,321 ,322,323,324 self-interpreter , 275, 277, 281, 282 simply typed,313 substitution, 284, 285: variable , 276 successor, 284 term: dosed, 332; simply typed, 331, 332: valuation, 332 type: -free, 278 ; first-order,
609 315, 326: rank(rnk), 315, 316; functional, 315; Peano, 326-327: strong normalization theorem, 327; universal, 315 untyped,324 -conversion , 139 -definability (definable) , vii-viii, 275, 313-314 decidable model , 339 logicalrelation , 339-341 numeric functions, 314 problem: decidability conjecture, 333; absolute , 332; relative, 332 provability undecidability, 331-342: theorem,338-339 word, 334-335 identity,criterion of, 405 interpretation, 280, 281 negation , 419-420 term,275-285 closed, 276, 282 logicalrelation, 339 map theory , 287, 288 preword-term,335-337 subterm,318 strongly normalizable, 318, 324 type-free , 275 untyped,322, 323 Lambek, J ., 80, 182 calculus, 103 Langford, C . H., 574f language finitary(finite), 221, 366, 475 first-order , 227, 228, 229, 232, 235, 242 controversy , 243 semantics,234, 244 set theory , Skolem, 233 formal, 221, 222
610
INDEX
language(continued) hierarchy,Tarski, 498 higher-order , 249 semantics, 234, 250 infinite(infinitary),365,366, 367, 470, 473, 474 learnability, 528-532 natural sense and referenceshifts causedby indirect contextreembeddings, 529 infinitehierarchyof senses, 509 semantics,509 unlearnable and innat eness hypothesis,529 object -meta, 236 proof, 243 semantics,236f possibilityof (Church's Translation Argument), 573-595 second-order , 227, 244, 250 Lassez, J.-L. , 375 lattice Boolean , 157, 159 extensionsof R~ -ordered groupoid, 80 monoid,81 semi-group,80 Lauchli,H., 169 Lawvere, F. W., 182 Leeds, S., 574f, 584f, 585, 592 Leibniz, G. W., vii, ix Leibniz' program applicabilit y and distribution , 454-455 ars combinatoria, 445, 451 Church'sTheorem,458-461 , 487
axiomaticcompleteness , 458 distribution of social benefitsof knowledge , 461-462 epistemic inequit y, 462 incompleteness,G6del, 459,461£ computability requirement, 446-447,451-453,460, 461, 462 abilitydifferences,464 advantages, 463-464 objections,464-465: encyclopedia,obtaining, 464; natural vs. logical language , 465; soundness, 466 weak vs. strongautonomy , 460-461,463,464 Decision Problem, 458 encyclopedia,c ommunity, 445, 447, 451, 452, 464 epistemic authen ticity,451, 455-456 autonom y, 447-451 : integrit y, 451; strong, 447-448,457,460-461 ; weak, 447-448, 451, 457, 460-461 claim, 448-449: conformal, 449-450, 453; conformant compensation,450; non-conformal, 449 computation , 447 conflict, 448: rule of conserv a tism, 449-450 evidence, 450-451: belief, 450f; sufficiency , 451 justice, 453-459, 462, 463: socialutilitya nd capability , 456- 457
INDEX peace, 452, 453-457,462, 463 mechanization of thought/reason, 446 symbolic vs, intuitive reasoning , 452f thesis,447 Leivant,Do, 313-329 lemma Exclusion , 91, 92 Extension,91 Squeeze, 91, 92, 93 Levin, M. I., 279 Levy, Ao, 236f Lewis, Do, 149,229 Lewy, Co, 583 Lindstrom, Po, 245 Lindstrom'stheorem,245f, 246 Linsky, B., 590f Linsky, t., 407f, 423, 583 Lloyd,J. W. , 365, 385 Loader,R., 331-342 Loar, B., 574f Lockemann, P. C., 78 logic 2-valued, 387 3-valued, 387 Aristotle , 74 assemblies,realizability, 355 basic, 66-67 belief,Church, 401 Booleanconnectivesand quantifiers, 389 class, 229 classical A-calculus compared,289 Lukasiewiczian infinitevalued,480 propositional,139, 185-189,301 compactness,Zermelo, 242 constants,156, 163-166 countable, 69
611 elements, 166 finite, 67 first-order , 221, 222, 227, 228, 229, 230, 231, 238f basic propositions generalized , 67 completeness , 69, 241, 246 emphasized,G6del-Skolem, 243 many-sorted , 221 second-order,as against, 251 formal, 227 (see higher-order second-order) , 221, 222, 228, 229, 230, 244, 249f classicalmathematics,249 construction , 241 intuitionistic logic, 362 Quine, 244-245 infinite(infinitary),189, 224, 228, 229 Lukasiewiczian , 480 valued, 480 , 483 inten sional ,423 intuitionist ie, 339 higher-order , 353 propositional,189, 194 Lukasiewiczian ,480, 482 many valued, 480 mathematical, 238 metalogic, 61 modal,340 84, 188-189 85, 189 objects,149-180 w-order, 239f programming,see logic programming proposition, 229 ramified, see ramifiedtype theory relevant m inimal, 214
612
INDEX
logic(continued) pure 191-217 relation, 61-62, 229, 261 A-definability, 339 -341 second-order(see higher-order), 221, 225, 227, 244 arithmetic,324 categoricity , 234 first-order,as against,251 G6del's incompleteness theorem,241 minimal, 349 relativism,240 variables,234 Quine, 244-245 sense anddenotation, 507-542 settheory, asa gainst , 245 simple type, see simple type theory substructural, 80, 81, 103-105 uncountable, 72 Logic of Sense andDenotation, viii, 395, 404, 507-542, see sense anddenotation logicprogramming(and equality theory) , 365-391 2-valued mod el,387, 390 3-valued completeness,367 logic, 387 semantics, 365, 367, 387-391 soundness,367 Clark completion,365, 368, 387 equalitytheory, 367 soundnessresult,368 compactness, 370 completetheory (completeness) , 366, 383-384
completeddefinition,368 constructive negation,365, 366, 373, 383, 385 domain closure axiom, 366-367, 368, 368 equalityt heoryformulas algorithm : reducing formula ton ormalform, 376, 384; replacing : everyequalityformula, 377; negation,377; simple equalityformula, 378 basic, 383, 385 inequations,381-382 indirect,379 normalform, 373-387,388 strictly : normal,374, 377, 380, 382, 383, 384; simple, 374, 376, 377, 382, 384 variable : irrelevant, 379, 381-382, 383; relevant , 380, 381; solved, 376, 380 graphcolorings,381-382 Herbrandmodel, 365-367, 384, 388-391 hierarchicprogram,386, 387 language finite, 221, 366 infinite, 365, 366, 367 occur axiom, equalityt heory , 367, 370-371 Prolog,365 SLD-resolution , 384 SLDNFrefutation , 369, 387 resolution , 365, 369, 387, 391 tree, 369, 387 logical vs . non-logical, 221 logicism (logicist) , 223, 224, 228 boundvariableabstraction ,4
INDEX numbertheoretic, 3 sets, 245 school, 223, 225, 238, 244 Skolem,233 Zermelo, 238 Longo, G., 353 lowcomprehension(theorem ), 115, 127, 128, 133 Lowenheim, L., 229, 230, 231 Lowenheim-Skolemtheorem,227 categoricity, 234, 235, 237 compactness,245 countable model, 233 first-order logic, 246 set theory , 251 Lowenheim'searlyversion, 230 relativism,239, 242, 243 upwardanddownward,228f Lyndon,R C., 78, 79, 95 Mac Lane, S., 181-184,407 MacQueen, D. B., 315f Maddux,R , 78, 79, 81, 84, 102f Maher, M. J ., 366, 375, 382, 383, 384, 385 Malcev, A., 382, 383, 384 Martin,R L., 485f map theory, 287 -311 adaptivestructure , 305-306 axioms andrules ofinference, 290, 291, 299-303 abstraction, 300 application,299, 300 equality,299, 301 existence,302 extensionality, 301 minimality , 301 monotonicity,301 quantification , existential , simple, 302 QuartumNon Datur (QND), 289, 300, 301 renaming, 300
613 selection,301 substitutivity, 299 transitivity , 299 truth , 299 classicalmaps , 291, 302 equality , 299-300 label index ed, 293-294 monotonic,301 partially ordered set, 304 occurrences continuous , 290, 299 discontinuous,296, 298, 299, 301 semantics,287,291,292 infinitelooping(..1) , 289 one non-function (rT), 289 truthandfalsehood,289 termsandwell-formed formulas, 290 tree, 292-293 truth , 299-300 well-foundedn ess, 291, 303, 304 not, 299 with classical map (MTC), 288, 289, 290, 291, 292-297 associativity, 307 axioms and inference rules, 309-310 classicalmap, 298 definitionsused inaxioms, 307-309 x-chain,295-297,304 x-compact.297, 304, 305, 306 x-continuity, 295-297,300 K-cpo,304 x-premodel, 297, 301 x-prime, 305 x-Scottdomain, 288, 291, 297, 304: x-prlme algebraic , 304 x-small, 297, 304
614 map theory(continued) model, 294, 298, 304-306: x-denotational semantics , 304 partially ordered sets, 295 priority , 307 proof,299 Russell'sparadox,303 semantics,288, 292: finite sets, 292 syntax,298-299,307 upper bound,295-296 ZFC, 302 m ap withwell-founded (MTW), 288,289,290, 291, 292, 305, 306 Mar, G. , 479-490 Marriott, K., 375 Marshall, M. V. , 150 Martin-Lof,P., 343-351 Martino , E. , 491-505 Massey, G. J. , 185-189 Mates, B. , 576f mathematical induction , 38, 72 school, 223, 225, 226, 229, 231 May, R. , 484f McCarthy,J ., 279 McLarty , C., 353-363 McRobbie, M. A., 81, 195f, 209 Meaning and Necessity, xi Mellor, D. H . , 410f, 419 membership, 224, 471, 473, 475, 477 non-logical, 225 , Mendelson, E ., 186, 300 meta -language,222 'first-order, 230 , 242 object-distinction , 236 second-order,250 -logic, 61 -mathernatical, 229
INDEX
Meyer, R. K. , 77, 78, 81, 83, 95f, 96, 98f, 105, 191-217 Meyer monoid, 83, 84, 85, 89, 94, 95,103,104 frame, 90 framed,94 Mitchell,E., 134, 136 MLR+,195f MLR--.&v, 195 MLR~&v, 195 model, 150 3-valued, 16, 18, 19, 388-390 4-valued , 17 Boolean-valued, 83 Church -Oswald, 113-134 Church's,125 countable logicprogramming,370 set theory,Skolem, 233, 241 first-order , arithmetic ,248 Herbrand,365-367, 384, 388-391 infinite, 154, 229f A-cal culus, 331 logicist, 224 map theory,293, 294, 304-306 Russell'sparadox, 303 NF 2 , 116 . NFO, 118, 121 NFU, 476-477 non-standard , 251, 476 ordinarysettheory,237 Oswald,118 permutation,128-129, 130 presheaves(Kripke models), 363 Rieger-Bernays permutation , 109, 110-113, 114 standard , 246, 248 theory , 222, 223 prooftheory , distinction from, 234
INDEX set theory , 236, 241 Von Neumannset theory, 236f Moerdijk, I., 182 Mogensen, T . lE., 275, 278, 281, 282 monoid Church(c-monoid), 191, 198, 199, 200, 201, 205, 208, 209 commutative(c-monoid), 197 free, 200, 202 partially ordered (c-po-monoid), 198 De Morgan,87f, Montague,R., xi, 248 Moore, G ., 226f, 227, 228f, 229, 230, 231, 238f, 241, 243, 545, 556, 577f Moss, 1., 150, 158, 161 Mostowski,A., 149 MR~ , 193, 194f, 206, 207 MR~&, 195,206,207 Myhill,J., 246, 290, 299, 396f, 404,423,491, 499 NaDSetII , 39 natural number,6-7, 25, 39, 105, 232f 2~, canonical r epresentation , 321 analysis,248 assemblies, realizability, 353 cardinaln umber,472 categoricity, Dedekind,240 Churchnumerals,276 classes of , Frege, 228 HEAP, 202 intuitiven otionofset, 241 iterators , 146 realizability, 356, 361, 363 powerset,360 second-order language , 227 set theoretic , 233 species theory,345
615 typetheory,140 negation Boolean, 191, 213, 214f, constructive , 365, 366, 373, 383, 385 De Morgan,191 minimal, 193-194 Newman, M. H. A., 140 NF 2 , 113, 132, 135 consistency , 137 model, 116, 132 NFO (New Foundations,Open), 118 and lowcomprehension,121 nominalist(nominalism) anti-nominalist and pro-intuitionist , 440 bathontology, 441-442 Chihara 's paraphras e method,438-439 compatibilist (compatibilism), 429-432, 435,437,441 existence , mere vs. ultimate , 441-442 formalization , conventional , 439, 439 forms, depthvs. surface , 440-441 evidence lack, 440 idiomaticvs. paradigmatic, 441 incompatibilist (incompatibilism), 431-432 infracompatibilsm , 441 infranominalism,441, 442 modalparaphrase, 431, 437-438 natural numbers, appliedto, 43 ordinarilyadequate , 437 paraphrase , 429-442 "thereis", 441-442
616 non-logical, 225 membership, 251 predicaterelation, 228 terminology,228, 247 nonconstructive proofs , 429 numbertheory,3 O'Donnell , M., 313, 314, 315f Ono, H., 206f ontological commitment, ix, 429-442 ontology logical,objects,159, 160, 164, 166, 167. 171, 172 mathematical, Quine, 245 second-order,249 sense anddenotation Frege, 509 types and functions, 510-513 set theory,first-order , 251 operations associativity , 264, 265, 267, 269 cancellation , 264-271 closed, 264, 266, 267, 269, 270-271 rightsoluble , 264, 268-269 ordinal Brouwer, 343,345 notations , 343 map theory,297 regular, strongly inaccessible ,map theory,297 Orlowska, E., 95f Oswald, U., 109, 113 'P-embedding, 129, 130 'P-extension,121, 125, 134 paradox analysis,545, 583f Berry, 486 Burali-Forti, 287, 292, 304, 472-473 Contingent (Lob's), 483f Cantor,472
INDEX
Grelling , 491-492, 502 Church 's solution criticized , 493-499 intensional, 491 , 495, 503 Liar, 479, 487f Chaotic, 483-484: properties, 484; semantic behavior, 486; Strengthen ed, 485 Classical , 480, 481, 483, 484 contradictory, 482-483: fixed-p ointrepellor, 482 Dualist , 484,485 functionaliteration, 481 TruthTellerCautious, 482-483: fixed-point attractor , 482 Richard, 486 Russell,39, 225, 226, 287, 292, 303, 469, 470, 487f, 491 antinom y of propositional functions, 499-505 semantical, 470, 473, 498 Tarski, 469, 470, 475-476, 477 Parsons, T ., 507-543 partition , 271 set 263-264, 270 Peano,G. arithmetic (PA), 3-59, 41, 43, 314 G+·x , 46 provablyrecursive function,320, 322, 327 axioms, 13, 73, 140 in A , 46, 47 realizability, 361-362 A-calculus , 313-329 mathematical school, 223 quantifi ers, 224 types, 313, 314, 315-317, 318 closed (cPeano) ,316-317
INDEX variablesover classes of natural numbers, 228 Peirce, C. S., 78 algebraicschool, 222 framedPeircegroupoid,86, 87, 91 framedPeircesemi-group,92 groupoid,81, 85, 86, 91 monoid, 81, 87f, 93 semi-group, 86, 92 system interpretation, 227 Peitgen,H.-O. , 484 Petry,A., 132 permutation,157, 267 invariance(PIness), 149, 150, 151, 152, 154, 156, 157, 170,172,173 Piperno, A., 275, 278, 280, 281, 282 Platonism , 227 Plotkin,G. D., 150, 169, 331, 339, 340 -Statrnan conjecture , 331, 333, 335 Plumwood, V., 198f Poincare, H., 232 positivism, 220 Post,E. L., viii, 486 Powers, L., 105f Pratt , V., 78 pre relation algebra,84, 95 predecessorrelat ion , see relation , predecessor primitiverecursion,see recursion, primitive Prior, A. N., 419 programminglanguage s emantics, 3 projectionrelation, 473 Prolog,365 prooftheory , 3, 10-12, 230f finitary, 238 first-order andsemantics, 243f,246
617 model theory,d istinction from, 234 proposition(propositional), 62-63, 593 attitude, 575 basic, 65 commitmentto, 575-577, 583f function,228, 239, 240, 241, 497 class, 496 intensional , 501, 504 Russellian,491 nominalism,440 Russellian,228,491-492, 496, 498 sentences , distinctfrom, 573, 575f used extensivelyin (see) sense anddenotation , logic of protoDe Morganmonoid, 81, 82, 93,94,103 a-frame, 89 fullframed, 89 Przymusinski, T . C. , 365, 366, 373-374,382 Putnam,H., 149, 151, 574f, 575f quantification (quantifiers) , 222-224, 232 agents,183 cardinal , 155 first-ord er , 239 map theory , 302 interpretation , 229 prenexuniversal,231 propositional , 192, 211-212 second-order , 249 Skolem, 233 Quine, W. V. 0 ., 118f, 574f, 576, 583f antifoundationalist , 251 nominalist 4, 40
618
INDEX
Quine, W . V. O. (continued) criterion of identity, 403, 404f objection-to for propositions,395 ontological , 398 two-level, 396formalization, 438-439 higher-ordersystems 244-245 New Foundations (emphNF)settheory, 469 nominalism, 429-432 ontological commitment,ix, 432,434-435 relativism,246 second-order ontology, 249 quotation semanticrelation shift, Frege, 507 (typeraising),470, 473 R, 191, 193, 195, 196, 198, 199, 200, 209, 213 R-+ , 191-193, 195, 196, 204, 206, 208, 212, 214 R-+&, 195, 197, 198, 199, 200, 202, 206, 209, 213 decidable , 205 finite modeltheorem,205, 208,211 finitizing lemma, 204 numericalcompleteness theorem,201, 203 secondcompleteness theorem , 200 R~, 193-194,206,212,213 RVp, 214 R~, 192, 193,212,213,214 Rvp -+&,213 R~P, 212, 213 RM,214 Ramsey, F. P., viii, 229, 410f, 419 randomness, 479-487 chaos, 487
definition, 486 Rang, B., 226 Rasiowa, H., 89f, 93f, 185f rationalism,see foundations realnumbers map theory , 296-297 realizability, 358-359 realism, 220 realizability, 353-363 assembly, 353-356 arithmetic , 357 arrow, 353 carrier,353 Cartesianclosed, 354 caucus, 353 logic,355 modest, 358 modulus,353 product,353 quotient , 356 subassembly, 354: almost negative , 356; double negation(d.n.) closed, 356 axiom of choice , 358 Church'sThesis, 357-359, 362 complemented,356 decidable,356 effective topos (EJJ), 359-362 arithmetic,361 finite limits , 339 equality , d.n. closed, 357 equivalence, 356 -357 EjJ, 359-360
higher-order, 363 logic,361 integers,358 Kleene'sT-predicate, 357-358 Markov principle, 358, 362-363 natural number,356, 361, 363
INDEX
powerset,360 negation,356-357 Peanoaxioms, 361-362 powerobject, 361 presheaves(Kripke models), 363 recursivefunctions , 357 rules, 354 weakly classify, 360 Recw , 314, 320, 321 recurrence, simultaneous, 319-320 A-definability in simple type, 321, type-uniform,320 recursion partial , 353 primitive, 14, 22, 29-30, 319, 324 A-definability, 314 , 327 realizability, 357 simple types, 320 species theory,351 theorem,15,47 theory ,3 refinedrelation a lgebraframe, 99 relevance(relevant) decidability , 195 divisibility,210 division, 200 implication , viii, 191 infinite divisionprinciple (IDP) , 202, 203 infinitedivisor,202, 203 logic, 77, 83, 96f minimal, 214 pure, 191-217 numerals,208 relation algebra,77-108, 84, 94, 95, 103 concreteframe, 102, 103 frame, 90, 91, 97 refinedframe, 98, 99
619 equivalence,262-264, 270, 271 identity , 262 inner,264-267, 270 singulary , 267, 271 multiplication of, 261, 266, 269-270, 271 relat ive product,261 reflexive , 262-264 ,270, 271 symmetric,262-265, 270 transitive, 262-264, 270 triangular , 262-263 relational language,153 relativism , 227, 243 first-order, 252 axioms, 24 Zermelo, 241 representation function, 209 Rescher, N., 480 schema, 480-481 residuation theor em, 193 Resnik, Mo, 226 restrict ed word, 122 Reynolds,J oCo, 313, 315f Rice, a. G. , 486f, 487f Richard, M., 406f, 545-572, 574f, 588f Riche, J., 197f, 206 Rosser ,J on., 182, 483f Church-Ro sser Theorem, vii Routely,R., 77, 78, 83, 85, 86, 88, 89f, 91, 93, 96, 198f, 208 Routely,v., 89f RoutleyMeyer frames, 77-108 semantics,96f, 100 o, 232 Russell, B algebraicschool,compared, 229 antinomyof propositional functions, 499-505 intensionality principle, 500 semanticalconstant , 492, 500-501
620
INDEX
Russell, B.(continued) class, 115, 471, 473 definit as vague, 232 higher-order variable , 228 identity,intensional, 406f, 492 intensional logic, ix knowledge byacquaintance vs. by description, 587-591 logicism, 4, 223 paradox,39, 225, 226, 287, 292, 303, 469, 470, 487f, 491, 492 propositions,499-505 propositional function , 228,491-492 , 498: class, 496, quantifiers,192 quantifier range, 224 ramifiedtypetheory,38, 227, 491-492,496,498 Salmon, N., 561£, 573-595 Sato, To , 367, 384 Saupe, Do, 484 Saxton, r, v., 78 Scedrov,s., 353,359 Scheffler, I., 574f Schiffer,So, 546f Schoenfiies,s.. 232f Schonfinkel,Mo, 230 school algebraic , 222, 223, 224, 225, 226, 229, 230, 237, 244 logicist, 223, 225, 238, 244 mathematical , 223, 225, 226, 229, 231 Schroder,Eo , 226, 230 algebraicschool, 222 calculus of relations, 78 system interpretation, 227 weaktheoryof types, ix SchroderBernsteinTheorem, 135-136
Schroder-Heister , Po, 80, 81 Schroeder , Mo, 481£ Schiitte , K. , 4 Schwichtenberg, n., 313 Scott, D. So, 16, 105, 111 cardinal,126 x-Scottdomain, 288 Scott,P or., 182 second-ord er arithmetic(SOA), 324, 325 2A,314 arithmetical formula, 325 second-order completeness, 230f standardmodel, 248, 326 attacked , 245f axiom of continu ity,228 class, 228 consequence,243, 251 first-order , as against,249, 250, 251 A-calculus(2A), 315-317, 318, 319, 321, 323 language , 227, 236, 250 logic , 4, 61-75,227 arit hmetic, 324 Codel's incompleteness theorem, 241 relativism, 240 Quine, 244 logical cons equence,235f minimal logic, 349 quantification , 43, 249 set theory,231 Skolem, 235 terminology,228 relativi sm, 240 variable,227, 228, 239, 249 semanticalp aradoxes, viii semantic(semantics) 3-valued , 365, 367, 387, 388 4-valued , 16 Churchmonoid, 209 ClassicalLiar paradox,481
INDEX continuous valued, 482 complexity ,9 consequencerelation , 220 constants , Church, 491-492, 500-501 denotational , 277 finite model,s emilattic e property , 208 theorem , 211 first-order, 221, 227, 244 prooftheory,243f, 246 relativism , 252 formal,250 Frege, infinitesense hierarchy, 528 G 2,52 grounding,6 higher-order, 234 terminology,245 independent,97 Kripke-style , 78 language , 221 formal,222 logicalconstants , 164 map theory,289, 291, 292 model theoretic,149, 221, 222, 236 NFU, 473-475, 477 non-standard , 221, 228, 240 operational semanticsfor R-+& ,197 paradox,470, 473,484 primitive,13 prooftheory,230f proposition(Translation Argument), 574 realizability, 353-363 relational, 95f relations, Frege,quotation , 507 shift,sense anddenotat ion , 508 Routley-Meyer, 77, 95 second-order,249
621 sense anddenotation, Church's alternatives, 509 set theoretic , 250 syntax, distinctionfrom, 234 Urquhart, 209 UrquhartRoutely semilattice , 200 semi- Thue system, 333 sense, 545-570 sense anddenotation, 582f alternative 2, 514, 516-518 axiom set 16, 514-515, 517, 520 necessaryuniformity argument,538-541 SUPER-15-16,520-521, 524, 538 axioms, 512-513,523-524 models (truth tables), 522-524: conditional , 523; negation , 523; possible worlds , 524 erizat ion, 513-516 charact Church, 507, 509, 512 compositionali ty,508 conceptof, 508, 509, 510, 512, 513, 514, 515-516, 517,519,520,521 ,523, 524,528 Fregeanvs. Russellian 526-527 functional types, 529 consistency,522 contexts de dicto, 525-526, 528, 531-532, 534, 535 de re 507, 509, 525-528, 532, 534, 535 directand indirect , 508f, 511-512,528,579 indirectreembeddings: it, 529-531: compositionality, 530;
622
INDEX
sense anddenotation (continued) shiftsin senseand reference , 529 mixed,527-528 denotation, 508 directand indirect,578-579, 581 empty concepts,519 termsandsenses, 517, 518-519, 520: compounded,541-542 extensionality reduction, 514-516, 521 infinitehierarchy of senses, 509 learnability of language,528, 529 innatenesshypothesis, 529 simplerhypothesis,536 logic of, 507 - 542 one trueproposition,514--516 ontology, 525, 526 typesand functions , 510-513: functional : type to type, 510; name-related types, 510; sentence-related types, 510 paradoxof analysis,560 proposition,510, 511 propositional function,511 semantic(s) alternatives , Church's, 509 shift,508-509, 511 sense, 508, 511; 526 collapse(reduction)to extensionality , 516: proof, 537-538 function(functional) , 513-525: partial,518 predicate , 513-514 supercharacterization, 519-521
terms intuitive , 518 truthvalue, 511, 513 conceptof, 511 denotation, 511 type (semantic)hierarchy, 511 raising,516 refining, 532 -533 simpleraccount,533-536: equivalencehypothesis, 535: simplerhypothesis, 535-536 symbolism, 512 unary propositionalfunction, 515, 521 truth -function,515, 521 zips (suppliedobjectof reference),518-519,520, 521 sequentcalculus, 6 set algebraica ndlogicist,244 class aspropositiona l function,496 coherent,291 complement, 267 finite, map theory,292 hereditarily low, 115, 132 infinite, 238 intuitivenotion,241 iterative concept,225, 226, 236, 249 logicalconception , 225, 234, 235, 244, 249, 251 low, 114-116,123-124,126 memb ership, map theory,302 non-well -founded,235, 287 partially ordered,m ap theory, 295 , 304 power, 116 primitivenotionof, 496 regular , 131
INDEX singletons,472, 473-476 sumset,116 symmetric,269 union, 268 universal set, 109-138,469 well-founded , 116, 128, 130, 131, 132-133,225 set theory AFA,304 antinomies,244 axiomatization, 225, 231 categorical, 237 coset,27Q--271 criterion of identity,397-398 development,history,227, 233 first-order, 232, 251 ontology, 251 higher-order , 243 KF, 134, 135 logic, as against,245 map theory , 287, 288 Mitchell , 136 model theory , 241 naive, 236, 250, 472 NF, 135, 469, 470 NF 2,135 NFO ,135 NFU, 469-477 infinityandchoice, 473, 474,477 satisfaction, 475, 476 semantics, 473-475 stratified ( stratification) , 473, 474 typeraising, 469, 470, 473, 474,475,476 New Foundations , see NF Quine, 134 second-order logic, 245 relativism,235, 239 Skolem, 233, 234, 235 set abstraction , 289, 290 typetheory , 496
623 universalset, viii, 109-138 Von Neumann, 236f Z, 134 ZF, 109,110 Shapiro, So, 219-252 Shaw-Kwei,Mo, 19lf Shepherdson , r. Co, 365-392 Sher, Go, 150, 165 signature , 280 algebraic,280, 281 binarytree, 280 term,280 Sikorski, R., 185f Silva,J. S., 149 Skolem, To definit, 239 first-order and axiomatizations Godel, 241 proponent , 227, 231 foundational goals, 251 functions , 230 higher-order concepts, 251 infinitary language avoidance, 230 paradox,231 propositional functions , 240, 241 totality of, 219 Quine, 245 relativism,234, 243, 246, 250 set theory,225, 233, 238 categoricity , 235, 237 natural number,24lf standardinterpretation, 246 typetheory,244 variables,higher-order, 240 Slaney,r . x. , 202f, 213f SLD-resolution , soundness, 384 SLDNFrefutation, 369, 387 resolution , 365, 369, 387, 391 incomplete,369 tree, 369, 387
624 Slomson, A., 186f Smullyan , R., 261-271 Soames, S., 548f, 564f Solomonov,R. , 486 soundness 3-valued, 367 Clark'ssoundnessresult,368 Species Theory,343-352 accessibility, 346, 351 unprovability, 348 computability, 346, 347 consequences,347 constants, 345 equalityr elation , 346 language axioms and rules of inference , 348 constants,function, predicate,logical , 348 natural numbers, 345 pairing, 344, 351 predecessorrela tion,343, 346 primitiverecursive,351 realizability definition,349 predicate,349-350 rules ofc ontraction , 344 term,344 type, 344, 349 wellordering , 348, 351 SqueezeLemma, 91, 92, 93 stability of reference(theorem), 152 Stabler,E. P., 149 Statman,R., 313, 331, 333 Stavi, J., 150, 158, 161, 165 Stone,M., 79, 92, 94, 101 Stone'sRepresentation Theorem, 185, 186, 187, 188 stratification, llO,III synonymy, 545-570 analysis,561 communityof speakers, 559-560
INDEX
compositionality, 546, 549, 555, 556, 561, 566 defined, 546 Dummett's Thesis, 580-581 explained/justified, 559 form, 558 objections , 559 meaningas transparent, 559 opposing senses, 548 phrasal , 546, 549, 555-556, 557, 563, 566, 568, 569, 570 rejectionof, 558-559 semantic operation , 555 value, 547, 549 , 553-555, 563, 567 560 sense anddenotation, structural (structure) , 546, 549, 555, 556, 557, 563, 566, 568, 569, 570 complete, 550, 558 conceptual accessible, 558 rules, 550 - 553 synonymousisomorphism, 568 synonymyrule, 561 , 564, 566 transparency, 546, 564-565, 566 triviality, 546, 555, 561-563 syntax(syntactic) complexity, 9 equality,A-calculus, 276 map theory,291, 298-299 semantics, distinctionfrom, 234 Tait, W. W., 346, 347 Tamaki, H., 367, 384 Tarski, A., 102f, 227f Booleanalgebra,79, 80, 84f, 92 first-order language,4-5, 38 languagehierarchy , 498
INDEX logicalpropertiesas invariant underallpermutations, 149-150 paradox, 469, 470, 475-476, 477 Liar, 486 solution,viii, 498-499 predicationstratification, 13 relation a lgebra postulates, 78 relativism,243£ truth(T) schema, 9, 479, 481£ infinitevalued,480 Tarski'stheorem,469-477 tautology, 188 Ter Meulen, .A , 158 termmodels G 2 ,52 Tharp,L., 245, 246 Thiel,C., 230 Thistlewaite, P. B., 105, 206 Thomas, W ., 226 Tiuryn,J ., 339 Translation Argument,ix, 407f, 574-595 Dummett'sThesis, 580-581 indirectsense, 587, 592-593 knowledge byacquaintance vs. by description , 587-591, 592-595 meaning(denotation) preservation, 582-583 quotation t ranslation, 582-585 pragmatically imported information,586 semantically encoded information,586 True(the), Frege, 507 truth conditions,201 functions,157, 160 map theory,299-300
625 redundancy or disquotational theoryof, 577 reference,151 theory,3 , 19 truth in an L-structure, 54 undefinabiIity,see Tarski's theorem TUring, A. M., 140, 145, 146, 182, 479, 486 Church's Thesis, viii type application,167 equality,167 -free, 6 lowering , 475 membership, 167 raising, 469, 470, 473, 474, 475 simple, 319 theory,225, 244, 472, 473 downward,475 intuitionistic , 343f ramified, viii, 227, 229, 240,241 ,491,492-493, 503: Grelling 's paradox, 497- 499; orderof a formula , 493 simple, viii, 229,471 : Cantor 's theorem, 501-502; Church, 509; Grelling 's paradox, 491-492, 495- 499; Russell 's antinomy (propositional functions) , 499-501 weak, ix, undecidable(undecidability) Church'sThesis, 439 haltingproblem, 357 A-definability, 331-342 theorem, 338-339 word problem,333 universal TUringMachine, 333
626 urelement, 235, 239, 345, 469, 471,477 Urquhart , A., 97f, 195£, 198f semilattice , 208, 209 universe, 209 validity first-order, 69 Van Benthem,J., 78, 150, 158, 162, 165, 167, 169, 171, 172,173 Van der Does , J. , 158 Van Eijck, J., 158 Van Fraassen,B., 149, 150, 485f Van Heijenoort,J ., 223 Van Gucht, D., 78 Van Oosten,J ., 363 variable firs~order, 224, 227, 239, 249 quantification , 233, 239 higher-order, 232 range, 222 second-order , 227, 228, 234, 239, 249 Veblen, 0. , 220f, 223, 228, 229f Venn, J., 222 Von Neumann, J. , 192f, 230, 236 axiom of replacement , 233f categoricity , set theory , 237-238, 250 relativism , 250 set theorymodels, 237 Wagner, S., 220f, 251£ Wang, H., 242, 246 Ward,M., 81 Warner,S., 408 Wells, R., 421 Westerstahl, D., 150, 157, 158, 165, 172 Weston, T. , 250 Weyl, H., 149, 232, 233, 234 intuitionism,232, 247 set theory , 238 White,M. G., 545, 560, 569 Whitehead , A. N., 229
INDEX Williamson,T ., 395£, 396f Wilson,N . L., 590f Wittgenstein , L., 223, 224 Wolfram, D. A., 331 word derivable,337-338 encoding,333-335, 339 A-definabl e, 334-335 preword-term , 335-337 problem, 333-335 encoding,333 undecidable,333 rank,124 restricted,124 uniquerestricted , 124 White, M., 545, 560, 569, 570 Whitehead , A. N., 118f Woodruff, P. 95 Wright,C., 3 Xeroxlemma, 209, 211 ZF, 113, 134, 135 andfoundation , 121 axiom of foundation independentof, 113 non-well-found ed sets, 287 relativism , Skolem, 235 second-order,239 type theory,496 ZFC first-order, 227, 231, 236f, 239 map theory , 288, 290, 291, 300, 302, 306 wellfoundedne ss , 304 Russell'sparadox,469 second-order , 236f, 239 Zermelo, E. consequence , second-order, 243 definit , 233, 239 effectiv elyenumerable consequenc e, 220 languages first-order 232, 246
INDEX
higher-order proponent, 227
Codel'sincompleteness theorem,241 infinitary logic,242-243 logicist school, 238 non-standard model and NFU, 476 Russell'sparadox, 226 set theory, 225 intuitivenotionsdefined in, 234 Skolem, 240 well-ordering theorem,231, 232
Zimbarg, J. P., 477
627
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J. M. Bochenski, APrecis ofMathematical Logic. Translated from French and German. by O Bird. 1959 ISBN 90-277-0073-7 P. Guiraud,Problemes et methodes de la statistique linguistique. 1959 ISBN 90-277-0025-7 H. Freudenthal (ed .), The Concept and the Role ofthe Model in Mathematics and Natural and ISBN 90-277-0017-6 Social Sciences. 1961 E. W. Beth,Formal Methods. An Introduction to Symbolic Logic and to the Study of Effective Operations in Arithmetic and Logic . 1962 ISBN 90-277-0069-9 B. H. Kazemier and D . Vuysje (eds.), Logic and Language . Studies dedicated to Professor Rudolf Carnap on the Occasion of His 70th Birthday. 1962 ISBN 90-277-0019-2 M. W. Wartofsky (ed .), Proceedings ofthe Boston Colloquium for the Philosophy of Science , 1961-1962. [Boston Studies in the Philosophy of Science, Vol.1]1963 ISBN 90-277-0021-4 A. A. Zinov'ev, Philosophical Problems ofMany-valued Logic . A revised edition, edited and translated (from Russian) by G. Kung and .D. Corney. D 1963 ISBN 90-277-0091-5 G. Gurvitch, The Spectrum ofSocial TIme. Translated from French and edited. Korenbaum by M and P. Bosserman. 1964 ISBN 90-277-0006-0 F.J 1965 P. Lorenzen,Formal Logic. Translated from German by. Crosson. ISBN 90-277-OO80-X R. S. Cohen and M. W . Wartofsky (eds .), Proceedings ofthe Boston Colloquium for the Philo. [Boston Studies in the Philosophy sophy of Science, 1962-1964. In Honor of Philipp Frank of Science,Vol.11]1965 ISBN 90-277-9004-0 E. W. Beth,Mathematical Thought. An Introduction to the Philosophy of Mathematics. 1965 ISBN 90-277-0070-2 E. W. Beth and.JPiaget,Mathematical Epistemology and Psychology. Translated from French by W. Mays. 1966 ISBN 90-277-0071-0 G. Kung, Ontology and the Logistic Analysis ofLanguage. An Enquiry into the Contemporary Viewson Universals. Revised .,edtranslated from German . 1967 ISBN 90-277-0028-1 R. S. Cohen and M. W. Wartofsky (eds.), Proceedings of the Boston Colloquium for the . [Boston Studies Philosophy ofSciences , 1964-1966. In Memory of Norwood Russell Hanson in the Philosophy of Science, Vol. III] 1967 ISBN 90-277-0013-3 C. D. Broad, Induction, Probability, and Causation. Selected Papers . 1968 ISBN 90-277-0012-5 of A of the G. Patzig,Aristotle's Theory ofthe Syllogism. A Logical-philosophical StudyBook Prior Analytics. Translated from German by J. Barnes . 1968 ISBN 90-277-0030-3 ISBN 90-277-0084-2 N. Rescher , Topics in Philosophical Logic. 1968 R. S. Cohen and M . W. Wartofsky (eds.) , Proceedings of the Boston Colloquium for the , Philosophy of Science, 1966-1968, Part I. [Boston Studies in the Philosophy of Science Vol. IV] 1969 ISBN 90-277-0014-1 R. S. Cohen and M. W . Wartofsky (eds .), Proceedings of the Boston Colloqu ium for the , Philosophy of Science, 1966-1968, Part II. [Boston Studies in the Philosophy of Science Vol. V] 1969 ISBN 90-277-OO15-X J. W. Davis, D. J. Hockney and W. K. Wilson (eds .), Philosophical Logic. 1969 ISBN 90-277-0075-3 D. Davidson and.JHintikka (eds.) , Words and Objections . Essays on the Work ofW. V.Quine. 1969, rev. ed. 1975 ISBN 90-277-0074-5; Pb 90-277-0602-6 P. Suppes, Studies in the Methodology and Foundations ofScience. Selected Papers from 1951 to 1969. 1969 ISBN 90-277-0020-6 J. Hintikka,M odelsfor Modalities. Selected Essays. 1969 ISBN 90-277-0078-8; Pb 90-277-0598-4
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N. Rescheretal. (eds.), Essays in Honor ofCarl G. Hempel. A Tribute on the Occasion of is H 65th Birthday. 1969 ISBN 90-277-0085-0 P. V. Tavanec (ed.) , Problems of the Logic ofScientific Knowledge. Translated from Russian . 1970 ISBN 90-277-0087-7 M. Swain (ed.), Induction , Acceptance, and Rational Belief. 1970 ISBN 90-277-0086-9 R. S. Cohen and R. .JSeeger (eds .), Ernst Mach : Physicist and Philosopher. [Boston Studies in the Philosophy of Science, Vol. VI]. 1970 ISBN 90-277-0016-8 J. Hintikka and. PSuppes, Information and Inference . 1970 ISBN 90-277-0155-5 K. Lambert,Philosophical Problems in Logic. Some Recent Developments. 1970 ISBN 90-277-0079-6 ISBN 90-277-0161-X R. A. Eberle,Nominalistic Systems . 1970 P. Weingartnerand G. Zecha (eds.),Induction, Physics, and Ethics. 1970 ISBN 90-277-0 158-X ISBN 90-277-0173-3 E. W. Beth,A spects ofModern Logic. Translated from Dutch. 1970 . 1971 R. Hilpinen (ed .), Deontic Logic . Introductory and Systematic Readings See also No. 152. ISBN Pb (1981 rev.) 90-277-1302-2 J.-L. Krivine,Introduction to Axiomatic Set Theory. Translated from French . 1971 ISBN 90-277-0169-5; Pb 90-277-0411-2 J. D. Sneed, The Logical Structure ofMathematical Physics. 2nd rev . ed., 1979 ISBN 90-277-1056-2; Pb 90-277-1059-7 C. R. Kordig,The Justification ofScientific Change . 1971 ISBN 90-277-0181-4; Pb 90-277-0475-9 M. Capek, Bergson and Modern Physics. A Reinterpretation and-evaluation Re . [Boston Studies in the Philosophy of Science, Vol. VII] 1971 ISBN 90-277-0186-5 N. R. Hanson,What I Do Not Believe, and Other Essays. Ed. by S. Toulmin and H . Woolf. 1971 ISBN 90-277-0191-1 R. C. Buck and R. S. Cohen (eds.),PSA 1970. Proceedings of the Second Biennial Meeting of the Philosophy of Science Association, Boston, Fall 1970. In Memory of Rudolf Camap. [Boston Studies in the Philosophy of Science , Vol.VIIIj1971 ISBN 90-277-0187-3; Pb 90-277-0309-4 D. Davidson and G . Harman (eds.), Semant ics ofNatural Language. 1972 ISBN 90-277-0304-3; Pb 90-277-0310-8 Y. Bar-Hillel(ed.), Pragmatics ofNatural Languages. 1971 ISBN 90-277-0194-6; Pb 90-277-0599-2 S. Stenlund, Comb inators, "I Terms and ProofTheory. 1972 ISBN 90-277-0305-1 M. Strauss,Modern Physics and Its Philosophy. Selected Paper in the Logic, History, and Philosophy of Science . 1972 ISBN 90-277-0230-6 M. Bunge,Method, Model and Matter. 1973 ISBN 90-277-0252-7 ISBN 90-277-0253-5 M. Bunge,Philosophy ofPhysics. 1973 A. A. Zinov' ev, Foundations ofthe Logical Theory ofScientific Knowledge (Complex Log ic). Revised and enlarged English edition with an appendix . A.by Smirnov G , E. A. Sidorenka, A . M. Fedina and L. A . Bobrova . [Boston Studies in the Philosophy of Science , Vol.IX] 1973 ISBN 90-277-0193-8; Pb 90-277-0324-8 L. Tondl,Scientific Procedures. A Contribution concerning the Methodological Problems of Scientific Concepts and Scientific Explanation. Translated from Czech . Short. by Edited D by R.S. Cohen and M .W. Wartofsky. [Boston Studies in the Philosophy of Science , Vol. X] 1973 ISBN 90-277-0147-4; Pb 9O-277-0323-X N. R. Hanson,Constellations and Conjectures. 1973 ISBN 9O-277-0192-X
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K. J. J. Hintikka, J. M. E . Moravcsik and P. Suppes (eds .), Approaches to Natural Lan guage . 1973 ISBN 90-277-0220-9; Pb 90-277-0233-0 M. Bunge (ed.),E xact Philosophy. Problems, Tools and Goals. 1973 ISBN 90-277-0251-9 R.1. Bogdan andI. Niiniluoto (eds .), Logic, Language and Probability. 1973 ISBN 90-277-0312-4 G. Pearce and P. Maynard (eds .), Conceptual Change . 1973 ISBN 90-277-0287-X; Pb 90-277-0339-6 I. Niiniluoto and R. Tuomela, Theoretical Concepts and Hypothetico-inductive Inference. 1973 ISBN 90-277-0343-4 R. Fraisse,Course ofMathematical Logic - Volume I: Relation and Log ical Formula . Translated from French. 1973 ISBN 90-277-0268-3; Pb 90-277-0403-1 (ForVolume 2 see under No . 69). A. Griinbaum, Philosophical Problems of Space and Time. Edited by R.S. Cohen and M.W . Wartofsky. 2nd enlarged ed. [Boston Studies in the Philosophy of , Vol.XII] Science 1973 ISBN 90-277-0357-4; Pb 90-277-0358-2 P. Suppes (ed.), Space , Time and Geometry. 1973 ISBN 90-277-0386-8; Pb 90-277-0442-2 H. Kelsen,E ssays in Legal and Moral Philosophy. Selected and introduced by O. Weinberger . Translated from German by P. Heath . 1973 ISBN 90-277-0388-4 R. J. Seeger and R . S. Cohen(eds.), Philosophical Foundations of Scien ce. [Boston Studies in the Philosophy of Science , Vol. XI] 1974 ISBN 90-277-0390-6; Pb 90-277-0376-0 R. S. Cohen and M. W . Wartof sky (eds.), Logical and Epistemological Studies in Contemporary Physics. [Boston Stud ies in the Philosophy of Science, Vol. XIII]3197 ISBN 90-277-0391-4; Pb 90-277-0377-9 R. S. Cohen and M. W . Wartofsky (eds.), Methodological and Historical Essays in the Natural and Social Sciences. Proceedings of the Boston Colloquium for the Philosophy of Science,
61.
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1969-1972 . [Boston Studies in the Philosophy of Science , Vol. XIV] 1974 ISBN 90-277-0392-2; Pb 90-277-0378-7 R. S. Cohen,J. J. Stachel and M. W. Wartofsky (eds.), For Dirk Struik. Scientific, Historical and Political Essays. [Boston Studies in the Philosophy of Science, Vol.XV] 1974 ISBN 90-277-0393-0; Pb 90-277-0379-5 . 1974 K. Ajdukiewicz , Pragmatic Logic. Translated from Polish by O. Wojtasiewicz ISBN 90-277-0326-4 S. Stenlund (ed.) , Logical Theory and Semantic Analysis. Essays dedicated to Stig Kanger on His 50th Birthday . 1974 ISBN 90-277-0438-4 K. F. Schaffner and. R S. Cohen (eds.), PSA 1972. Proceedings ofthe Third Biennial Meeting of the Philosophy ofScience Association. [Boston Studies in the Philosophy of Science , Vol. XX] 1974 ISBN 90-277-0408-2; Pb 90-277-0409-0 H. E. Kyburg, Jr ., The Logical Foundations of Statistical Inference. 1974 ISBN 90-277-0330-2; Pb 90-277-0430-9 M. Grene,The Understanding ofNature. Essays in the Philosophy of Biology. [Boston Studies in the Philosophy of Science, Vol. XXIII] 1974 ISBN 90-277-0462 -7; Pb 90-277-0463-5 J. M. Broekman,Structuralism: Moscow, Prague, Paris. Translated fromerman.1974 G ISBN 90-277-0478-3 N. Geschwind, Selected Papers on Language and the Brain. [Boston Studies in the Philosophy of Science, Vol. XVI] 1974 ISBN 90-277-0262-4; Pb 90-277-0263-2 R. Fraisse, Cours e ofMathematical Logic - Volume2: Model Theory. Translated from French. 1974 ISBN 90-277-0269-1; Pb 90-277-0510-0 (ForVolume I see under No. 54)
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A. Grzegorczyk, An Outline ofMathematical Logic. Fundamental Results and Notions explained with all Details . Translated from Polish. 1974 ISBN 90-277-0359-0; Pb 90-277 -0447-3 F. von Kutschera, Philosophy ofLanguage. 1975 ISBN 90-277-0591-7 J. Manninen and R. Tuomela (eds .), Essays on Explanation and Understanding . Studies in the Foundations of Humanities and Social Sciences. 1976 ISBN 90-277-0592-5 J. Hintikka(ed.), Rudolf Carnap , Logical Empiricist. Materials and Perspectives. 1975 ISBN 90-277-0583-6 M. Capek (ed.), The Concepts ofSpace and TIme. Their Structure and Their Development. [Boston Studies in the Philosophy of Science , Vol. XXII] 1976 ISBN 90-277-0355-8; Pb 90-277-0375-2 J. Hintikka and .URemes, The Method of Analysis. Its Geometrical Origin and Its General Significance.[Boston Studies in the Philosophyof Science, Vol. XXV] 1974 ISBN 90-277-0532 -1; Pb 90-277-0543-7 J. E. Murdoch and .ED. Sylla (eds .), The Cultural Context of Medieval Learning. [Boston Studies in the PhilosophyScience,Vol. of XXVI] 1975 ISBN 90-277-0560-7; Pb 90-277-0587-9 S. Amsterdamski,Between Experience and Metaphy sics. Philosophical Problems of the Evolution of Science. [Boston Studies in the Philosophy of Science, Vol. XXXV] 1975 ISBN 90-277-0568-2; Pb 90-277-0580-1 P. Suppes (ed.), Logic and Probability in Quantum Mechanics. 1976 ISBN 90-277-0570-4; Pb 90-277-1200-X H. von Helmholtz : Epistemological Writings. The Paul Hertz I Moritz Schlick Centenary Edition of 1921 with Notes and Commentary by the Editors. Newly translated from German and Bibliography , by R. S. Cohen and Y. Elkana. by M. F. Lowe. Edited, with anIntroduction [Boston Studies in the Philosophy of Science , Vol. XXXVII] 1975 ISBN 90-277-029O-X; Pb 90-277-0582-8 J. Agassi, Science in Flux. [Boston Studies in the Philosophy of Science, Vol. XXVIII] 1975 ISBN 90-277-0584-4; Pb 90-277-0612-2 S. G. Harding (ed .), Can Theories Be Refuted? Essays on the Duhem-Quine Thesis. 1976 ISBN 90-277-0629-8 ; Pb 90-277-0630-1 S. Nowak,Methodology ofSociological Research . General Problems. 1977 ISBN 90-277-0486-4 J. Piaget, J.-B. Grize, A. Szeminsskaand V. Bang, Epistemology and Psychology ofFunctions. Translated from French . 1977 ISBN 90-277-0804-5 M. Grene and .EMendelsohn (eds .), Topics in the Philosophy ofBiology. [Boston Studies in the Philosophy of Science, Vol. XXVIII 1976 ISBN 90-277-0595-X ; Pb 90-277-0596-8 E. Fischbein,The Intuitive Sources ofProbabilistic Thinking in Children. 1975 ISBN 90-277-0626-3 ; Pb 90-277-1190-9 E. W. Adams, The Logic of Conditionals. An Application of Probability to Deductive .Logic 1975 ISBN 90-277-0631-X M. Przelecki and R. Wojcicki (eds.), Twenty-Five Years of Logical Methodology in Poland. Translated from Polish. 1976 ISBN 90-277-0601-8 J. Topolski, The Methodology ofHistory. Translated from Polish by O. Wojtasiewicz . 1976 ISBN 90-277-0550-X A. Kasher (ed.),Language in Focus: Foundations, Methods and Systems. Essays dedicated to Yehoshua Bar-Hillel. [Boston Studies in the Philosophy of, Science Vol. XLIII] 1976 ISBN 90-277-0644-1 ; Pb 90-277-0645- X
SYNTHESE LIBRARY J. Hintikka,The Intentions ofIntentionality and Other New Modelsfor Modalitie s. 1975 ISBN 90-277-0633-6; Pb 90-277-0634-4 , Collected Papers on Epistemology, Philosophy of Science and History of 91. W. Stegmuller Philosophy. 2 Volumes. 1977 Set ISBN 90-277-0767-7 92. D. M. Gabbay, Investigations in Modal and Tense Logics with Applications to Problems in ISBN 90-277-0656-5 Philosophy and Lingui stics . 1976 ISBN 90-277-0649-2 93. R. J. Bogdan, Local Induction . 1976 94. S. Nowak, Understanding and Prediction . Essays in the Methodology of Soeial and Behavioral Theories. 1976 ISBN 90-277-0558-5; Pb 90-277-1199-2 95. P. Mittelstaedt, Philosophical Problems ofModern Physics . [Boston Studies in the Philosophy of Science, Vol. XVIIIll976 ISBN 90-277-0285-3; Pb 90-277-0506-2 96. G. Holton and W. .ABlanpied (eds .), Science and Its Public : The Changing Relationship . [Boston Studies in the Philosophy of Science, Vol. XXXIII] 1976 ISBN 90-277-0657-3; Pb 90-277-0658-1 ISBN 90-277-0671-9 97. M. Brand and D . Walton (eds .), Action Theory. 1976 98. P. Goehet,Outline ofa Nominalist Theory ofPropositions . An Essay in the Theory of Meaning and in the Philosophy of Logic. 1980 ISBN 90-277-1031-7 99. R. S. Cohen, P.K. Feyerabend, and .M W. Wanofsky (eds .), Essays in Memory oflmre Lakatos. [Boston Studies in the Philosophy of Science, Vol. XXXIX] 1976 ISBN 90-277-0654-9; Pb 90-277-0655-7 , Selected Papers of Leon Rosenfield. [Boston Studies in 100. R. S. Cohen andJ. J. Stachel (eds.) the Philosophy of Science, Vol. XXI] 1979 ISBN 90-277-0651-4 ; Pb 90-277-0652-2 101. R. S. Cohen, C. A. Hooker, A. C. Michalos and J. W. van Evra (eds .), PSA 1974. Proceedings ofthe 1974 Biennial Meeting ofthe Philosophy ofScience Association. [Boston Studies in the Philosophy of Science , Vol. XXXIIl 1976 ISBN 90-277-0647-6; Pb 90-277-0648-4 102. Y. Fried andJ. Agassi, Paranoia. A Study in Diagnosis . [Boston Studies in the Philosophy of Science, Vol. L] 1976 ISBN 90-277-0704-9; Pb 90-277-0705-7 103. M. Przelecki , K. Szaniawski and R. Wojcicki (eds .), Formal Methods in the Methodology of Empirical Sciences . 1976 ISBN 90-277-0698-0 104. J. M. Vickers,B eliefand Probability. 1976 ISBN 90-277-0744-8 105. K. H. Wolff , Surrender and Catch. Experience and Inquiry Today. [Boston Studie s in the Philosophy of Science , Vol.LIl1976 ISBN 90-277-0758-8; Pb 90-277-0765-0 106. K. Kosik, Dialectics ofthe Concrete. A Study on Problems of Man and World . [Boston Studies in the Philosophy of Science, Vol. LII] 1976 ISBN -277-0761-8; 90 Pb 90-277-0764-2 107. N. Goodman, The Structure of Appearance. 3rd ed. with an Introduction by G. Hellman. [Boston Studies in the Philosophy of Science, Vol. LIII] 1977 ISBN 90-277-0773-1; Pb 90-277-0774-X 108. K. Ajdukiewicz,The Scientific World-Perspective and Other Essays , 1931-1963. Translated from Polish. Edited and with an Introduction by J. Giedymin . 1978 ISBN 90-277-0527-5 ISBN 90-277-0779-0 109. R. L. Causey, Unity ofScience. 1977 110. R. E. Grandy , Advanced Logicfor Applications. 1977 ISBN 90-277-0781-2 Ill. R. P. McArthur , Tense Logic . 1976 ISBN 90-277-0697-2 . Translated from Swedish by P. 112. L. Lindahl,Position and Change. A Study in Law and Logic Needham. 1977 ISBN 90-277-0787-1 113. R. Tuomela, Dispositions. 1978 ISBN 90-277-081O-X 114. H. A. Simon, Models ofDiscovery and Other Topics in the Methods ofScience . [Boston Studies in the Philosophy of Science, Vol. LIVl1977 ISBN 90-277-0812-6; Pb 90-277-0858-4
90.
SYNTHESE LIBRARY 115. R. D. Rosenkrantz, Inference. Method and Decision. Towards a Bayesian Philosophy of Science. 1977 ISBN 90-277-0817-7; Pb 90-277-0818-5 116. R. Tuomela,Human Action and Its Explanation. A Study on the Philosophical Foundations of Psychology . 1977 ISBN 90-277-0824-X The Language of Philosophy. Freud and Wittgenstein . [Boston Studies in the 117. M. Lazerowitz, Philosophy of Science, Vol. LVj1977 ISBN 90-277-0826-6; Pb 90-277-0862-2 118. Not published 119 . J. Pelc (ed.), Semiotics in Poland. 1894-1969. Translated from Polish. 1979 ISBN 90-277-0811-8 . 1977 ISBN 90-277-0846-0 120. I. Porn,Action Theory and Social Science. Some Formal Models 121. J. Margolis,Persons and Mind. The Prospects of Nonreductive Materialism. [Boston Studies in the Philosophy of Science , Vol.LVIIj1977 ISBN 90-277-0854-1; Pb 90-277-0863-0 Saarinen (eds .), Essays on Mathematical and Philosophical 122. J. Hintikka,I. Niiniluoto, and. E Logic. 1979 ISBN 90-277-0879-7 123. T. A. F. Kuipers,Studies in Inductive Probability and Rational Expectation. 1978 ISBN 90-277-0882-7 I. Niiniluoto and .M P. Hintikka (eds .), Essays in Honour ofJaakko 124. E. Saarinen, R. Hilpinen, ISBN 90-277-0916-5 Hintikka on the Occasion ofHis 50th Birthday. 1979 125. G. Radnitzky and G. Andersson (eds .), Progress and Rationality in Science. [Boston Studies in the Philosophy of Science, Vol. LVIIIj1978 ISBN 90-277-0921-1; Pb 90-277-0922-X Quantum Logic . 1978 ISBN 90-277-0925-4 126. P. Mittelstaedt, 127. K. A. Bowen, Model Theory for Modal Logic. Kripke Models for Modal Predicate Calculi. 1979 ISBN 90-277-0929-7 128. H. A. Bursen,Dismantling the Memory Machine . A Philosophical Investigation of Machine Theories of Memory . 1978 ISBN 90-277-0933-5 Models . Representation and the Scientific Understanding. [Boston Studies 129. M. W. Wartofsky, in the Philosophy of Science, Vol. XLVIIIj1979 ISBN 90-277-0736-7; Pb 90-277-0947-5 . [Boston Studies in the Philosophy 130. D. Ihde, Technics and Praxis . A Philosophy of Technology of Science, Vol. XXIV] 1979 ISBN 9O-277-0953-X; Pb 90-277-0954-8 in 131. J. J. Wiatr (ed .), Polish Essays in the Methodology of the Social Sciences . [Boston Studies the Philosophy of Science, Vol. XXIX] 1979 ISBN-277-0723-5; 90 Pb 90-277-0956-4 ISBN 90-277-0958-0 132. W. C. Salmon(ed.), Hans Reichenbach: Logical Empiricist. 1979 133. P. Bieri, R.-P. Horstmann and L. Kruger (eds .), Transcendental Arguments in Science . Essays in Epistemology. 1979 ISBN 90-277-0963-7; Pb 90-277-0964-5 134. M. Markovic and G. Petrovic (eds .), Praxis. Yugoslav Essays in the Philosophy and Methodology of the Social Sciences. [Boston Studies in the Philosophy of, Vol. Science XXXVI] 1979 ISBN 90-277-0727-8; Pb 90-277-0968-8 . 135. R. Wojcicki,Topics in the Formal Methodology ofEmpirical Sciences. Translated from Polish 1979 ISBN 90-277-1004-X 136. G. Radnitzky and .GAndersson (eds .), The Structure and Development of Science. [Boston Studies in the Philosophy of Science , Vol. LIX] 1979 ISBN 90-277-0994-7; Pb 90-277-0995-5 137. J. C. Webb, Mechanism , Mentalism and Metamathematics. An Essay on Finitism. 1980 ISBN 90-277-1046-5 .), Body. Mind and Method. Essays in Honor of Virgil 138. D. F. Gustafson and. BL. Tapscott (eds C. Aldrich. 1979 ISBN 90-277-1013-9 139. L. Nowak,The Structure of Idealization. Towards a Systematic Interpretation of the Marxian Idea of Science. 1980 ISBN 90-277-1014-7
SYNTHESE LIBRARY 140. C. Perelman,The New Rhetoric and the Human ities. Essays on Rhetoric and Its Applications. Translated from French and German. With an Introduction . Zyskind. by H 1979 ISBN 90-277-1018-X; Pb 90-277-10 19-8 141. W. Rabinowicz,Universalizability. A Study in Morals and Metaphysics . 1979 ISBN 90-277-1020-2 142. C. Perelman,Justice . Law and Argument. Essays on Moral and Legal Reasoning. Translated from French and German. With an Introduction. by Berman H.J . 1980 ISBN 90-277-1089-9; Pb 90-277-1090-2 143. S. Kanger and S.Ohman (eds.), Philosophy and Grammar. Papers on the Occasion of the Quincentennial ofUppsalaUniversity . 1981 ISBN 90-277-1091-0 144. T. Pawlowski,Concept Formation in the Humanities and the Social Sciences. 1980 ISBN 90-277-1096-1 145. J. Hintikka, D. Gruender and. Agazzi E (eds.),Theory Change. Ancient Axiomatics and Galileo's Methodology. Proceedings of the 1978Pisa Conference on the History and Philosophy of Science, Volume I. 1981 ISBN 90-277-1126-7 146. J. Hintikka, D . Gruender and . EAgazzi (eds.), Probabilistic Thinking . Thermodynamics . and the Interaction of the History and Philosophy of Science . Proceedings of the 1978 Pisa Conference on the History and Philosophy of Science , Volume11.1981 ISBN 90-277-1127-5 147. U. Monnich (ed.),Aspects ofPhilosophical Logic. Some Logical Forays into Central Notions of Linguistics and Philosophy. 1981 ISBN 90-277-1201-8 148. D. M. Gabbay, Semantical Investigations in Heyting 's Intuitionistic Logi c. 1981 ISBN 90-277-1202-6 149. E. Agazzi(ed.) ,Modern Logic -A Survey. Historical, Philosophical, and Mathematical Aspects of Modern Logic and Its Applications . 1981 ISBN 90-277-1137-2 150. A. F. Parker-Rhodes, The Theory of Indistinguishables. A Search for Explanatory Principles below the Level of Physics. 1981 ISBN 90-277-1214-X 151. J. C. Pitt,Pictures. Images . and Conceptual Change . An Analysis of Wilfrid Sellars' Philosophy of Science. 1981 ISBN 9O-277-1276-X; Pb 90-277-1277-8 152. R. Hilpinen (ed.),New Studies in Deontic Logic. Norms, Actions, and the Foundations of Ethics. 1981 ISBN 90-277-1278-6; Pb 90-277-1346-4 153. C. Dilworth,Scientific Progress. A Study Concerning the Nature of the Relation between Successive Scient ificTheories. 3rd rev. ed . • 1994 ISBN 0-7923-2487-0; Pb 0-7923-2488-9 154. D. Woodruff Sm ithand R. Mcintyre , Husserl and Intentionality. A Study of Mind. Meaning. and Language. 1982 ISBN 90-277-1392-8; Pb 90-277-1730-3 155. R. J. Nelson.The Logic ofMind. 2nd. ed., 1989 ISBN 90-277-2819-4; Pb 90-277-2822-4 156. J. F. A. K. van Benthem, The Logic ofTime. A Model-Theoretic Investigation into the Varieties of Temporal Ontology , andTemporal Discourse. 1983; 2nd ed.• 1991 ISBN 0-7923-1081-0 157. R. Swinburne (ed.) , Space . Time and Causality. 1983 ISBN 90-277-1437-1 158. E. T. Jaynes,Papers on Probability. Statistics and Statistical Physics. Ed. by R. D. Rozenkrantz . 1983 ISBN 90-277-1448-7; Pb (1989) 0-7923-0213-3 159. T. Chapman,Time: A Philosophical Analysis. 1982 ISBN 90-277-1465-7 160. E. N. Zalta,Abstract Objects. An Introduction to Axiomatic Metaphysics. 1983 ISBN 90-277-1474-6 161. S. Harding and M . B. Hintikka (eds.) , Discovering Reality. Feminist Perspectives on Epistemology, Metaphys ics. Methodology, and Philosophy of Science . 1983 ISBN 90-277-1496-7; Pb 90-277-1538-6 162. M. A. Stewart (ed.), Law. Morality and Rights. 1983 ISBN 90-277-1519-X
SYNTHESELIBRARY 163. D. Mayr and G. Siissmann (eds.) , Space , Time, and Mechanics. Basic Structures of a Physical Theory. 1983 ISBN 90-277-1525-4 , Handbook of Philosophical Logic. Vol. I: Elements of 164. D. Gabbay andF. Guenthner (eds.) Classical Logic. 1983 ISBN 90-277-1542-4 ( eds.), Handbook of Philosophical Log ic. Vol. II: Extensions of 165. D. Gabbay andF. Guenthner Classical Logic. 1984 ISBN 90-277-1604-8 , Handbook ofPhilosophical Logic. Vol. III: Alternat ive to 166. D. Gabbay andF. Guenthner (eds.) Classical Logic. 1986 ISBN 90-277-1605-6 , Handbook of Philosophical Logic. Vol. IV: Topics in the 167. D. Gabbay andF. Guenthner (eds.) Philosophy of Language . 1989 ISBN 90-277-1606-4 168. A. J. I. Jones, Communication and Meaning. An Essay in Applied Modal Logic. 1983 ISBN 90-277-1543-2 ISBN 90-277-1573-4 169. M. Fitting,ProofMethods for Modal and Intuitionistic Logics. 1983 . 1984 170. J. Margolis , Culture and Cultural Entities. Toward a New Unity of Science ISBN 90-277-1574-2 ISBN 90-277-1703-6 171. R. Tuomela,A Theory of Social Action. 1984 172. J. J. E. Gracia.E. Rabossi, E. Villanueva and. Dascal M (eds.) , Philosophical Analysis in Latin ISBN 90-277-1749-4 America. 1984 . 1984 173. P. Ziff,Epistemic Analysis . A Coherence Theory of Knowledge ISBN 90-277-1751-7 174. P. Ziff,Antiaesthetics. An Appreciation of the Cow with the Subtile Nose. 1984 ISBN 90-277-1773-7 , Examples, 175. W. Balzer , D. A. Pearce, and H.-J. Schmidt(eds.), Reduction in Science. Structure Philosophical Problems. 1984 ISBN 90-277-1811-3 176. A. Peczenik,L. Lindahl and .Bvan Roermund (eds .), Theory ofLegal Scien ce. Proceedingsof the Conference on Legal Theory and Philosophy of Science , Sweden (Lund, December 1983). 1984 ISBN 90-277-1834-2 Is Science Progressive ? 1984 ISBN 90-277-1835-0 177. I. Niiniluoto, Analytical Philosophy in Comparative Perspective. Explor178. B. K. Matilal and J. L. Shaw (eds.), atory Essays in Current Theories and Classical Indian Theories of Meaning and. Reference 1985 ISBN 90-277-1870-9 ISBN 90-277-1894-6 179. P. Kroes, Time: Its Structure and Role in Physical Theories. 1985 180. J. H. Fetzer.Sociobiology and Epistemology. 1985 ISBN 90-277-2005-3; Pb 90-277-2006-1 181. L. Haaparanta and. Hintikka J (eds .), Frege Synthesized. Essays on the Philosophical and Foundational Work of Gottlob Frege. 1986 ISBN 90-277-2126-2 182. M. Detlefsen , Hilbert's Program. An Essay on Mathematical Instrumentalism. 1986 ISBN 90-277-2151-3 183. J. L. Golden and. JJ. Pilotta (eds .), Practical Reasoning in Human Affairs. Studies in Honor ofChaimPerelman . 1986 ISBN 90-277-2255-2 184. H. Zandvoort , Models ofScientific Development and the Case ofNuclear Magnetic Resonance. 1986 ISBN 90-277-2351-6 Truthlikeness. 1987 ISBN 90-277-2354-0 185. I. Niiniluoto, ist 186. W. Balzer, C . U. Moulines and. JD. Sneed, An Architectonic for Science. The Structural Program. 1987 ISBN 90-277-2403-2 ISBN 90-277-2414-8 187. D. Pearce,Roads to Comm ensurability. 1987 l Structure s in Cognitive Neuroscience. 188. L. M. Vaina (ed.), Matters of Intelligence. Conceptua 1987 ISBN 90-277-2460-1
SYNTHESE LIBRARY 189. H. Siegel,Relativism Refuted. A Critique of Contemporary Epistemological Relativism . 1987 ISBN 90-277-2469-5 190. W. Callebaut and R.inxten, P Evolutionary Epistemolog y. A Mult iparadigmProgram, with a Complete Evolutionaryistemology Ep Bibliograph . 1987 ISBN 90-277-2582-9 191. J. Kmita,Problems in Historical Epistemology. 1988 ISBN 90-277-2199-8 192. J. H. Fetzer (ed .), Probability and Causality . Essays in Honor of Wesley. Salmon, C with an Annotated Bibliography. 1988 ISBN 90-277-2607-8; Pb 1-5560-8052-2 193. A. Donovan , L. Laudan andR. Laudan (eds.),Scrutinizing Science. Empirical Studies of Scientific Change. 1988 ISBN 90-277-2608-6 194. H.R. Otto and.A. J Tuedio (eds.), Perspectives on Mind. 1988 ISBN 9O-277-264O-X 195. D. Batensand J.P. van Bendegem (eds.),Theory and Experiment. Recent Insights and New Perspectiveson Their Relation. 1988 ISBN 90-277-2645-0 196. J. Osterberg, Self and Others. A Study of Ethical Egoism. 1988 ISBN 90-277-2648-5 ificialIntelligence, Cognitive 197. D.H. Helman (ed.) , Analogical Reasoning. Perspectives of Art Science, and Philosophy. 1988 ISBN 90-277-2711 -2 198. J. Wolenski , Logic and Philosophy in the Lvov-Warsaw School. 1989 ISBN 90-277-2749-X 199. R. Wojcicki, Theory ofLogical Calculi . Basic Theory of Consequence Operations. 1988 ISBN 90-277-2785-6 200. J. Hintikka and M .B. Hintikka,The Logic of Epistemology and the Epistemology of Logic. Selected Essays. 1989 ISBN 0-7923-0040-8; Pb 0-7923-0041-6 201. E. Agazzi (ed.),Probability in the Sciences. 1988 ISBN 90-277-2808-9 202. M. Meyer (ed.), From Metaphysics to Rhetoric. 1989 ISBN 90-277-2814-3 203. R.L. Tieszen, Mathematical Intuition . Phenomenology and Mathematical Knowledge . 1989 ISBN 0-7923-0131 -5 204. A. Melnick,Space, TIme, and Thought in Kant. 1989 ISBN 0-7923-0135-8 205. D.W.Smith,The Circle ofAcquaintance. Perception , Consciousness , and Empathy. 1989 ISBN 0-7923-0252-4 206. M.H. Salmon (ed.),The Philosophy of Logical Mechanism. Essays in Honor of Arthur W. Burks. With his Responses, and with a Bibliography of Burk's Work. 1990 ISBN 0-7923-0325-3 207. M. Kusch,Language as Calculus vs. Language as Universal Medium . A Study in Husserl , Heidegger , and Gadarner, 1989 ISBN 0-7923-0333-4 208. T.C. Meyering , Histori cal Roots of Cognitive Science. The Rise of a Cognitive Theory of Perception from Antiquity to the Nineteenth Century. 1989 ISBN 0-7923-0349-0 ISBN 0-7923-0389-X 209. P. Kosso, Observability and Observation in Physical Science. 1989 210. J. Kmita, Essays on the Theory ofScientific Cognition. 1990 ISBN 0-7923-0441 -1 211. W. Sieg (ed.),Acting and Reflecting. The InterdisciplinaryTum in Philosophy . 1990 ISBN 0-7923-0512-4 212. J. Karpinski,Causality in Sociological Research. 1990 ISBN 0-7923-0546-9 213. H.A. Lewis (ed.), Peter Geach: Philosophical Encounters . 1991 ISBN 0-7923-0823-9 214. M. Ter Hark,Beyond the Inner and the Outer. Wittgenstein 's Philosophy of Psychology . 1990 ISBN 0-7923-0850-6 215. M. Gosselin,Nominalism and Contemporary Nominalism. Ontological andistemological Ep Implicationsof the Work ofW.V.O. QuineofN. and Goodman. 1990 ISBN 0-7923-0904-9 216. J.H. Fetzer, D . Shatz and G. Schlesinger (eds.) , Definitions and Definability. Philosophical Perspectives . 1991 ISBN 0-7923-1046-2 217. E. Agazzi and A . Cordero(eds.), Philosophy and the Origin and Evolution of the Universe. 1991 ISBN 0-7923-1322-4
SYNTHESE LIBRARY 218. M. Kusch,Foucault's Strata and Fields. An Investigation into Archaeological and Genealogical Science Studies . 1991 ISBN 0-7923-1462-X 219. C.J. Posy, Kant's Philosophy ofMathematics. Modem Essays. 1992 ISBN 0-7923-1495-6 220. G. Van de Vijver , New Perspectives on Cybernetics. Self-Organization, Autonomy and Connectionism.1992 ISBN 0-7923-1519-7 221. J.C. Nyiri, Tradition and Individuality. Essays. 1992 ISBN 0-7923-1566-9 222. R. Howell,K ant's Transcendental Deduction. An Analysis of Main Themes in His Critical Philosophy. 1992 ISBN 0-7923-1571-5 The Logical Foundations of the Marxian Theory of Value. 1992 223. A. Garcia de la Sienra, ISBN 0-7923-1778-5 224. D.S. Shwayder,Statement and Referent. An Inquiry into the Foundations of Our Conceptual Order. 1992 ISBN 0-7923-1803-X 225. M. Rosen, Problems ofthe Hegelian Dialectic. Dialectic Reconstructed as a Logic of Human Reality. 1993 ISBN 0-7923-2047-6 226. P. Suppes, Models and Methods in the Philosophy ofScience: Selected Essays . 1993 ISBN 0-7923-2211-8 227. R. M. Dancy (ed.), Kant and Critique: New Essays in Honor ofW. H. Werkmeister. 1993 ISBN 0-7923-2244-4 228. J. Wolenski (ed.),Philosophical Logic in Poland. 1993 ISBN 0-7923-2293-2 229. M. De Rijke (ed.), Diamonds and Defaults. Studies in Pure and Applied Intensional .Logic 1993 ISBN 0-7923-2342-4 230. B.K. Matilal and A. Chakrabarti (eds.) , Knowingfrom Words. Western and Indian Philosophical Analysis of Understanding and Testimony . 1994 ISBN 0-7923-2345-9 231. S.A. Kleiner , The Logic ofDiscovery. A Theory of the Rationality of Scientific Research. 1993 ISBN 0-7923-2371-8 232. R. Festa,Optimum Inductive Methods. A Study in Inductive Probability , BayesianStatistics , and Verisimilitude . 1993 ISBN 0-7923-2460-9 233. P.Humphreys (ed.), Patrick Suppes: Scientific Philosopher. Vol.1: Probability and Probabilistic Causality. 1994 ISBN 0-7923-2552-4 , 234. P. Humphreys (ed.), Patrick Suppes: Scientific Philosopher. Vol. 2: Philosophy of Physics Theory Structure , and Measurement Theory. 1994 ISBN 0-7923-2553-2 235. P. Humphreys (ed.), Patrick Suppes: Scientific Philosopher. Vol. 3: Language, Logic, and Psychology . 1994 ISBN 0-7923-2862-0 Set ISBN (Vols 233-235) -07923-2554-0 236. D. Prawitzand D. WesterstAhl (eds.), Logic and Philosophy of Science in Uppsala. Papers from the 9th International Congress of Logic, Methodology, and Philosophy of Science. 1994 ISBN 0-7923-2702-0 237. L. Haaparanta (ed .), Mind, Meaning and Mathematics. Essays on the Philosophical Views of Husserl and Frege . 1994 ISBN 0-7923-2703-9 238. J. Hintikka(ed.), Aspects ofMetaphor. 1994 ISBN 0-7923-2786-1 239. B. McGuinness and G . Oliveri (eds .), The Philosophy ofMichael Dummett. With Replies from Michael Dummett. 1994 ISBN 0-7923-2804-3 is, In Honor 240. D.. Jamieson (ed.), Language, Mind , and Art. Essays in Appreciation and Analys of Paul Ziff . 1994 ISBN 0-7923-2810-8 241. G. Preyer,F. Siebelt and .AUlfig (eds.), Language, Mind and Epistemology. On Donald Davidson's Philosophy. 1994 ISBN 0-7923-2811-6 242. P. Ehrlich(ed.), Real Numbers, Generalizations ofthe Reals, and Theories of Continua. 1994 ISBN 0-7923-2689-X
SYNTHESE LIBRARY 243. G. Debrock and M . Hulswit (eds .), Living Doubt. Essays concerning the epistemology of Charles Sanders Peirce. 1994 ISBN 0-7923-2898-1 244. J. Srzednicki,To Know or Not to Know. Beyond Realism and Anti-Realism . 1994 ISBN 0-7923-2909-0 ISBN 0-7923-3171-0 245. R. Egidi (00.),Wittgenstein : Mind and Language . 1995 246. A. Hyslop,Other Minds. 1995 ISBN 0-7923-3245-8 247. L. Palos and M . Masuch (eds.),A pplied Logic: How , What and Why . Logical Approaches to Natural Language . 1995 ISBN 0-7923-3432-9 248. M. Krynicki, M. Mostowski and L.M . Szczerba (eds .), Quantifiers : Logics , Models and Com. 1995 ISBN 0-7923-3448-5 putation. VolumeOne: Surveys 249. M. Krynicki, M. Mostowski and L.M . Szczerba (eds .), Quantifiers : Logics, Models and Com. 1995 ISBN 0-7923-3449-3 putation. VolumeTwo: Contributions Set ISBN(Vols248 + 249) 0-7923-3450-7 250. R.A. Watson,Representationol Ideas from Plato to Patricia Churchland. 1995 ISBN 0-7923-3453-1 251. J. Hintikka(ed.), From Dedekind to Giidel. Essays on the Developmentof the Foundations of Mathematics. 1995 ISBN 0-7923-3484-1 252. A. Wisniewski,The Posing ofQuestions. Logical Foundations of Erotetic Inferences . 1995 ISBN 0-7923-3637-2 . 1995 253. J. Peregrin,Doing Worlds with Words. Formal Semantics without Formal Metaphysics ISBN 0-7923-3742-5 254. LA. Kieseppa, Truthlikeness for Multidimensional, Quantitative Cognitive Problems. 1996 ISBN 0-7923-4005-1 255. P. Hugly and .CSayward : lntensionality and Truth. An Essay on the Philosophy of .N.APrior. 1996 ISBN 0-7923-4119-8 256. L. Hankinson Nelson and J. Nelson (eds.) : Feminism , Science, and the Philosophy ofScience. 1997 ISBN 0-7923-4162-7 257. P.L Bystrov and.N. V Sadovsky (eds .): Philosophical Logic and Logical Philosophy. Essays in Honour of Vladimir. A Smirnov. 1996 ISBN 0-7923-4270-4 258. A.E. Andersson and N-E . Sahlin (eds .): The Complexity of Creativity. 1996 ISBN 0-7923-4346-8 259. M.L. Dalla Chiara , K. Doets, D. Mundici andJ. van Benthem (eds.) : Logic and Scientific Meth Methodology Logic, and Philosophy ods. VolumeOne of the Tenth International Congress of of Science, Florence , August 1995 . 1997 ISBN 0-7923-4383-2 Structures and Norms 260. M.L. Dalla Chiara, K. Doets, D. Mundici and . van J Benthem (eds.): in Science. Volume Two of the Tenth International Congress of Logic, Methodology and Philosophy of Science, Florence , August 1995. 1997 ISBN 0-7923-4384-0 Set ISBN (Vols259+ 260) 0-7923-4385-9 261. A. Chakrabarti: Denying Existence. The Logic, Epistemology and Pragmatics of Negative Ex.istentials and Fictional Discourse. 1997 ISBN 0-7923-4388-3 262. A. Biletzki:Talking Wolves. Thomas Hobbes on the Language of Politics and the Politics of Language. 1997 ISBN 0-7923-4425-1 ISBN 0-7923-4630-0 263. D. Nute (ed.) : Defeasible Deontic Logic. 1997 264. U. Meix.ner:Axiomatic Formal Ontology. 1997 ISBN 0-7923-4747-X 265. I. Brinck: The Indexical 'J'. The First Person in Thought and Language. 1997 ISBN 0-7923-4741-2 Contemporary Action Theory. Volume I: 266. G. Holmstrom-Hintikka and R. Tuomela .): (eds Individual Action. 1997 ISBN 0-7923-4753-6; Set: 0-7923-4754-4
SYNTHESE LIBRARY 267. G. Holmstrorn-Hintikka and . Tuomela R (eds.): Contemporary Action Theory. Volume 2: Social Action. 1997 ISBN 0-7923-4752-8; Set:0-7923-4754-4 268. B.-C. Park:Phenomenological Aspects ofWittgenstein's Philosophy. 1998 ISBN 0-7923-4813-3 ionof Classical Logic . 1998 269. J. Pasniczek:The Logic ofIntentional Objects. A Meinongian Vers Hb ISBN 0-7923-4880-X; Pb ISBN 0-7923-5578-4 270. P.W. Humphreys and J.H. Fetzer (eds.): The New Theory of Reference. Kripke, Marcus, and Its Origins. 1998 ISBN 0-7923-4898-2 271. K. Szaniawski, A. Chmielewski and . Wolenski J (eds .): On Science, Inference , Information and Decision Making. Selected Essays in the Philosophy of Science . 1998 ISBN 0-7923-4922-9 272. G.H. von Wright: In the Shadow ofDescartes. Essaysin the Philosophy of Mind . 1998 ISBN 0-7923-4992-X 273. K. Kijania-Placekand J. Wolenski (eds .): The Lvov-Warsaw School and Contemporary Philosophy. 1998 ISBN 0-7923-5105-3 . 1998 274. D. Dedrick: Naming the Rainbow. Colour Language, Colour Science, and Culture ISBN 0-7923-5239-4 - Onto 275. L. Albertazzi (ed.) : Shapes of Forms. From Gestalt Psychology and Phenomenology to logy and Mathematics. 1999 ISBN 0-7923-5246-7 276. P. Fletcher : Truth, Proofand Infinity. A Theory of Constructions and Construct iveReasoning . 1998 ISBN 0-7923-5262-9 277. M. Fitting and.L. R Mendelsohn (eds.): First-Order Modal Logic. 1998 Hb ISBN 0-7923-5334-X; Pb ISBN 0-7923-5335-8 278. J.N. Mohanty : Logic, Truth and the Modalities from a Phenomenological Perspective. 1999 ISBN 0-7923-5550-4 279. T.Placek:M athematical Intiutionism and Intersubjectivity. A Critical Exposition of Arguments for Intu itionism.1999 ISBN 0-7923-5630-6 280. A. Cantini, E . Casari and .PMinari (eds .): Logic and Foundations ofMathematics. 1999 ISBN 0-7923-5659-4 set ISBN 0-7923-5867-8 281. M.L. Dalla Chiara , R. Giuntini and F. Laudisa (eds.):L anguage, Quantum, Music. 1999 ISBN 0-7923-5727-2; set ISBN 0-7923-5867-8 282. R. Egidi (ed.): In Search ofa New Humanism. The Philosophy of Georg Hendrik von Wright. 1999 ISBN 0-7923-5810-4 283. F. Vollmer:A gent Causality. 1999 ISBN 0-7923-5848-1 284. J. Peregrin (ed.) : Truth and Its Nature (ifAny). 1999 ISBN 0-7923-5865-1 s on Donald Davidson's Philo285. M. De Caro (ed.):Interpretations and Causes. New Perspective sophy. 1999 ISBN 0-7923-5869-4 286. R. Murawski:Recursive Functions and Metamathematics . Problems of Completeness and Decidability, GOdeI'sTheorems. 1999 ISBN 0-7923-5904-6 287. T.A.F. Kuipers: From Instrumentalism to Constructive Realism. On Some Relations between Confirmation , Empirical Progress, and Truth Approximation . 2000 ISBN 0-7923-6086-9 288. G. Holmstrom-H intikka(ed.): Medieval Philosophy and Modern Times . 2000 ISBN 0-7923-6102-4 289. E. Grosholz and. HBreger (eds.):The Growth ofMathematical Knowledge. 2000 ISBN 0-7923-6151-2
SYNTHESE LIBRARY 290. 291. 292. 293. 294. 295. 296. 297. 298. 299. 300. 301. 302. 303.
304.
305. 306.
G. Sommaruga: History and Philosophy ofConstructive Type Theory. 2000 ISBN 0-7923-6180-6 1. Gasser(ed.): A Hoole Anthology. Recent and Classical Studiesin the Logic of George Boole. 2000 ISBN 0-7923-6380-9 V.F. Hendricks, S.A.Pedersenand K.F. Jergensen(eds.): Proof Theory . History and PhilosophicalSignificance.2000 ISBN 0-7923-6544-5 w.i, Craig: The Tensed Theory of1ime. A CriticalExamination . 2000 ISBN 0-7923-6634-4 W.L. Craig: The Tenseless Theory of Time. A CriticalExamination . 2000 ISBN 0-7923 -6635-2 L. Albertazzi (ed.): The Dawn of Cognitive Science. EarlyEuropeanContributors . 2001 ISBN 0-7923-6799-5 G. Forrai: Reference. Truth and Conceptual Schemes. A Defense of Internal Realism. 2001 ISBN 0-7923-6885-1 V.F. Hendricks, S.A. Pedersenand K.F. Jergensen(eds.): Probability Theory. Philosophy, ISBN 0-7923-6952-1 Recent History and Relationsto Science.2001 M. Esfeld:Holism in Philosophy ofMind and Philosophy of Physics . 2001 ISBN 0-7923-7003-1 E.C. Steinhart : The Logic ofMetaphor. Analogous Parts PossibleWorlds of . 2001 ISBN 0-7923-7004-X To be published. T.A.F. Kuipers: Structures in Science Heuristic Patterns Based on Cognitive Structures. An 2001 ISBN 0-7923-7117-8 Advanced Textbook in Neo-Classical Philosophy of Science. G. Hon and S.S. Rakover (eds .): Explanation. Theoretical ApproachesandApplications.2001 ISBN 1-4020-0017-0 G. Holrnstrom-H intikka,S. Lindstromand R. Sliwinski (eds.): Collected Papers ofStig Kanger with Essays on his Life and Work. Vol.I. 200 I ISBN 1-4020-0021 -9; Pb ISBN 1-4020-0022-7 G. Holmstrom-Hintikka, Srl.indstrom andR. Sliwinski (eds.):Collected Papers ofStig Kanger with Essays on his Life and Work. Vol. II.2001 ISBN 1-4020-0111 -8; Pb ISBN 1-4020-0112-6 C.A. Anderson and M. Zeleny (eds.) : Logic , Meaning and Computation . Essays in Memory of Alonzo Church. 2001 ISBN 1-4020-0141-X P. Schuster , U. Bergerand H. Osswald(eds.): Reuniting the Antipodes - Constructive and ISBN 1-4020 -0152-5 Nonstandard Views ofthe Continuum . 2001
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