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NOMOLOGICAL STATEMENTS STATEMENTS NOMOLOGICAL AND AND
ADMISSIBLE OPERATIONS OPERATIONS ADMISSIBLE
HANS REICHENBACH HANS REICHENBACH Pro/es8or Professor 0/ of Philo8ophy Philosophy in the the University University o/ of CaU/ornia, California, Los Los Angeles Angelm
11954 954
NORTH-HOLLAND N O R T H - H O L L A N D PUBLISHING P U B L I S H I N GCOMPANY COMPANY AMSTERDAM A MSTERDAM
PRINTED IN IN THE TEENETHERLANDS NETHEFXANDS DRUKXZBIJ D R W g E R I J HOLLAND HOLLAND N.y., N.V.. AI4STEBDflI AMSTEmAX
I INTRODUCTION INTRODUCTION
The problem problem of of aa ‘reasonable’ 'reasonable' implication The implication has has frequently frequently occupied occupied
logicians. Whereas in in conversational language this this kind kind of logicians. Whereas conversational language of propro-
positional operationisis regarded regardedas as having having a clear positional operation clear and and wellwelldefined meaning, logicians logicianshave have been been compelled to define define as as defined meaning, compelled to
implication aa term term of much wider wider meaning meaning;;and and it it appears appears extremely extremely implication of much difficult to go go back back from from this this implication in a wider to the difficult to implication in wider sense to the narrower and assumed for for implication implication in in aa narrower and very very specific specific meaning meaning assumed non-formalizedlanguage. language.We We face face here here aa discrepancy non-formalized discrepancy between between usage and and rules: whereas in in actual usage is quite able to usage everyone is to say whether an implication is reasonable, he would be at a loss to say whether an implication is reasonable, he would be at a loss to give rules which give which distinguish distinguish reasonable reasonable implications implications from from ununreasonable ones. ones. The The term term 'reasonable', to reasonable ‘reasonable’, therefore, therefore, is is aa challenge challenge to the logician logician for finding finding rules rules delineating delineating a usage usage that thatfollows follows unconscious rules. unconscious rules. The problem uncovering such such rules rules appears appears even The problem of of uncovering even more more difficult difficult when it is when it is realized realized that that 'reasonable' ‘reasonable’implications implications of of conversational conversational language are are not restricted language restricted to toimplications implications expressing expressing aa logical logical entailment. entailment, but include include what what may be called entailment, but called a physical entailment. For For the the first first kind, kind, we we may may use use as as an anillustration illustration the theimplication, implication, 'if all ‘if all men men are are mortal mortal and and Socrates Socrates is aa man, man, then then Socrates Socrates is is mortal'. may be be ifiustrated illustrated by by the theimplication, implication, mortal’. The The second second kind may 'if metal is is heated, heated, it it expands'. the latter latter kind ‘if aa metal expands’. Since Since the kind of of implication implication expresses what what is is called called aa law law of of nature, nature, whereas the the former may expresses be be said to to express express aa law law of of logic, logic, II have have proposed proposed to to include include both both kinds under the name of nomological implications. under the name of nornological implications. It It isiseasily easily seen seen that thatthe theproblem problemunder under consideration consideration is not specific for for implication, implication, but but concerns specific concerns all all propositional propositional operations operations alike. The The ‘or’, 'or', for instance, can have an an 'unreasonable' as aa alike. for instance, can have ‘unreasonable’as as well well as 'reasonable' meaning.To To say, say, ‘snow 'snow isis white white or or sugar is sour', ‘reasonable’ meaning. sour’, appears as unreasonable as saying saying ‘if 'if snow is not not white, unreasonable as snow is white, sugar sugar is is
22
NTRODUCTfON
sour'; sow’; but but both both statements statements are are true truein inthe thesense senseof of the theoperations operations
in the the truth truth tables reasonable ‘or’ 'or' would defined in tcblesof of symbolic symboliclogic. A reasonable given in in the the Statement statement:: 'there rain in in the winter or be given ‘thereis is sufficient sufficient rain there is aa drought drought in in the thesummer', summer’,an anexclusive exclusivedisjunction disjunction which which for many many aacountry countryexpresses expressesaaconsequential consequential alternative. alternative. Since Since the operation operation is is made made reasonable reasonable by by the the compound compound statement statement whose major major operation operation itit is and which whose which expresses expresses aa law of nature or or of of logic, logic, we we face facehere herethe thegeneral generalproblem problemofofnomological nornological8tatestuteinto the two ments, class of of statements statements which which subdivides subdivides into two subsubments, a class classes of analytic classes of analytic and and synthetic synthetic nomological nomological statements. statements. The The statement confers a certain prerogative statement prerogative upon its its major major operation, operation, which may may be operation. It It will which be called called aa nomological nornological operation. will be be seen, seen, howhowever, that that the to supply ever, the operations operations so so defined defhed are are still still too too general general to supply 'reasonable' propositional propositionaloperations, operations,and and that that such ‘reasonable’ such operations operations must be must be defined defined as as aasubclass subclassofofnomological nornological operations. operations. Based in an earlier Based on these considerations, considerations, II have developed developed in earlier presentation 11 aa theory of statements. Since we are presentation of nomological nomological statements. are here concerned with an an explication of here concerned with of aa term, term, i.e., i.e., with with constructing constructing term proposed proposed to to take take over the functions vague term, term, aa precise precise term over the functions of aa vague we cannot cannot expect expect to arrive we arrive at at results resultswhich which cover cover the the usage usage of of the the vague term term without without exceptions; exceptions;ifif only onlyfor forthe the reason reasonthat that the vague vague term term is differently by different different persons. persons. All Allthat that can can be be vague is used used differently achieved, therefore, constructing aa formal is constructing achieved, therefore, is formal definition definition which which correspondstotothe the usage usageofofthe the vague vagueterm term at at least least in in aa high corresponds high percentage of of cases. cases.For For this this reason, reason, II thank thank those of my percentage my critics critics who have drawn drawn my my attention attention to who have to cases cases where where there there appears appears to to be be aa discrepancy explicans and explicandurn, ifif these are discrepancy between between explicans and explicandum, these terms terms are used to denote used to denote the the precise precise term term and and the the vague vague term, term, respectively. respectively. To their criticisms, criticisms, II added added my my own own and and found found more more such such disdisTo their crepancies. present monograph, monograph, II wish wish to develop an crepancies. In In the the present to develop an improved improved definition nature and andreasonable reasonable operations, operations, hoping hoping definition of of laws laws of of nature that the the percentage percentage of of cases cases of of adequate adequate interpretation interpretation is thus thus increased. the remaining remaining cases, cases, my my definition definition may may be be regarded regarded increased. For For the as should be be glad as aa proposal proposal for for future future usage usage of of the the term, term, and and II should glad if if it it In my book, book,Element3 Elements of Symbolic Logic, In my Log'ic,New NewYork, York, 1947, 1947,ohap. chap. VIII. VIII. This book book will will be be quoted as ESL. This ESL.
INTBODTYCTION INTRODUCTION
33
appears possible to adjust adjust one's to the appears possible to one’sown own usage usage to the proposed proposed definition definition without sacrificing without sacrificing essential essential connotations. connotations. In its its fundamental fundamental idea, idea, the thenew new theory theory corresponds corresponds to the the old oId one; and II will here a short summary one; will therefore therefore give give here summary of of the older older theory insofar insofar as as it is taken theory taken over over into into the thepresent present one. one. The The truth truth tables tablesofofsymbolic symboliclogic logicrepresent representmetalinguistic metalinguistic statements expressing relations between between compound compound statements statements of statements expressing relations these the object language language and their elementary elementary statements. statements. Now Now these tables can be read tables read in in two two directions. directions. Going Going from the compound compound statement to the elementary statements, we we read read the the tables tables as a statement elementary statements, disjunction of disjunction of T-cases, T-cases, for for instance, instance, as asfollows: follows: b' is is true, true, then then ‘a’ 'a' is true and 'b' First F i r s t direction. d i r e c t i o n . If 'a ‘aD 3 b’ ‘b’ is is true, or 'a' ‘a’isisfalse false and and 'b' ‘b’is is true, true, or or 'a' ‘a’isisfalse false and and 'b' ‘b’isisfalse. false. Going from from the the elementary elementary statements statements to the the compound compound stateGoing ment, we ment, we read the the tables tables as asfollows: follows: If 'a' true, then Second ‘a’ is is true and 'b' ‘b’ is true, then 'a ‘a I)b’ Second direction. If is true. b' is is true. true. If 'a' true. If 'a' ‘a’isis false false and 'b' ‘b’ is is true, true, then then 'a ‘aD 3 by ‘u’isis false and 'b' false and ‘b’ is is false, false, then then 'a‘aD 3 b' b’ is is true. true. IIn n the the interpretation interpretation assumed assumed for for mathematical mathematical logic, logic, both both directions of reading are used. I speak here of an adjunctive interdirections of reading are used. I of an adjunctive pretation of correspondingly, of the the truth truthtables tablesand, and, correspondingly,ofofadjunctive adjunctive operations. to omit the possible, however, however, to the second second direction direction operations.1 It is possible, of the truth tables of reading reading the tables and and to to use use only only the thefirst firstdirection. direction. II then interpretation of of the the truth truth tables then speak speak of of aa connective connective interpretation tables and, and, correspondingly, of connective operations. correspondingly, of connective operations. It It is is important important to to realize realize that, that, for for all all'reasonable' ‘reasonabIe’operations operations of conversationallanguage, language,the thetruth truth tables tables are are adequate adequate if we read conversational them only in in the first them first direction, direction, i.e., interpret these these operations operations as as connective. Deviations Deviations from from a reasonable usage occur only when, connective. in addition, addition, the tables tables are are read read in in the the second second direction. direction. In other other words, reasonable reasonable operations operationsare are not not adjunctive, but connective. words, connective. For instance, implication: ‘If 'If a large instance, consider consider the reasonable reasonable implication: large sun spot spot turns turns up on the sun the day day of of the the concert, concert, the the short-wave short-wave radio to such The term 'adjunctive' ‘adjunctive’ corresponds corresponds to such terms terms as as'extensional', ‘extensional’, The 'truth-functional', 'material', which which have have been been used of ‘truth-functional’, ‘material’, used in presentations presentations of 11
logic. But But since since these these terms terms are are often often used used in in various various meanings, meanings,II prefer prefer to to logic. use the precisely precisely defined defined term 'adjunctive'. ‘adjunctive’.
44
nrraOnucTION INTaODUaTION
transmission of the concert will be seriously disturbed'. When we transmission of seriously disturbed’.
regard this statement regard this statement as true, before before the concert concert is given, given, we we
shall be quite willing to admit that willing to that any anyof ofthe thethree threepossible possiblecases cases stated stated for for the thefirst firstdirection directionmay mayoccur. occur.However, However,we wewould would refuse refuse
to regard if, say, no no sun spot turns up regard the the statement statement as as verified verified if, and the of the concert is not disthe short-wave short-wave radio-transmission radio-transmission of concert is dis-
turbed; to regard regard the implication turbed; and andwe we would would not even even be be willing to as verified even ifif aa sun sun spot turns verified even turns up up and andthe theradio radiotransmission transmission is disturbed, unless unless further further evidence evidence for for aa causal causal relation relation between between the the two two phenomena phenomena isisadduced. adduced.11 This This means means that we we use use here a connective implication, implication,but but not not an adjunctive connective adjunctive implication. implication. Similar examples are easily examples are easily given given for for the theother otherpropositional propositionaloperations. operations. It follows cannot be It follows that aadefinition definition of of reasonable reasonable operations operations cannot achieved by by changing the truth truth tables. changing the tables. These These tables tables are are adequate; adequate; however, we we have have to to renounce the the use of the the second second direction direction for for reading the tables. reading the tables. This This program program can can be be carried carried out out as as follows. follows. We We define connective operations operations as as a subclass define connective subclass of of the thecorresponding corresponding adjunctive operations. operations. Then, whenever whenever a connective connective operation operation is is adjunctive operation operation isis also alsotrue, true, and and the true, the the corresponding corresponding adjunctive use of the the first first direction direction of of reading reading the the truth truth tables use of tables is is thus thus assured. assured. However, However, the the second second direction direction is is excluded, excluded, because because aa verification verification
of compound connective connective statements statements requires than a verification of compound requires more more than verification of the the corresponding adjunctive statement. statement. IInn other of corresponding adjunctive other words, words, satissatisfying for an adjunctive fying the requirement requirement for adjunctive operation operation is merely merely aa necessary, not a sufficient for the verification necessary, not sufficient condition condition for verification of of the the corresponding connective connective operation. corresponding operation. Connective operations operations will be defined operations, Connective w i l l be defined as as nomological nomological operations, i.e., as major operations statements. In i.e., operations of of nomological nomological statements. I n a;Inomolognomological statement, all operations are are used, used, first, first, in in the ical all propositional propositional operations adjunctive sense; sense; i.e., i.e., the the statement statement must must be true in an an adjunctive adjunctive interpretation.But But in in addition, the statement has to satisfy interpretation. addition, the statement has satisfy certain another kind. certain requirements requirements of another kind. The The introduction introduction of of suitable suitable 1
The case case that that 'a' The ‘a’is is true true and and'b' ‘b’isistrue trueisissometimes sometimesregarded regarded as aa verifying verifying
a reasonable however, ae as insufficient for a verireasonable implication, implication, sometimes, sometimes, however, insufficient for fication. If If this case is regarded as verifying the implication, implication, II speak speak of a regarded aa verifying the fication. semi-adjunctive ESL, §5 64. 64. serni-adjunctive implication. implication. See See ESL,
INTRODUCTION
55
requirements requirements of this kind constitutes constitutes the the problem problem of of the the present present investigation. investigation. As As far as asanalytic analyticnomological nomological statements statements are are concerned, concerned, the the method outlined outlined here here has has found found an an application application in in Carnap’s Carnap's theory theory method of analytic implication. implication. Carnap Carnap has has pointed pointed out out that ifif an of an implicimplication stands in the place ation place of the major major operation operation of of aa tautology tautology or or analytic statement, itit can analytic statement, can be be regarded regarded as as an an explicans explicans for for the the relation of entailment. This will be be taken relation of logical logical entailment. This conception conception will taken over over into the present present theory. theory. However, However, what is to to be be added added is is aa correscorresponding definition definitionfor for physical physical entailment, entailment,and and with with it, ponding it, quite quite generally, for for synthetic nomological statements. Furthermore, Furthermore, it generally, nomological statements. it will be shown, as mentioned above, above, that that the will theclass classof ofnomological nomological operations is still too too wide wide to to supply supply what what may may be be called called ‘reasonable’ 'reasonable' operations is still operations. This This applies appliesboth bothto to the the synthetic synthetic and and to the analytic operations. analytic case;; in fact, case fact, not not all alltautological tautologicalimplications implications appear appearreasonable. reasonable. For instance, the tautological a3 3 b', For tautological implication, implication, 'a. ‘ a ,ti b’, can can scarcely scarcely be accepted be accepted as as reasonable. reasonable. The general form of of the theory to The general form to be be developed, developed, which which is the the same as as the the form of my previous theory, theory, can now be outlined as same as follows. First, aitclass follows. First, classofoforiginal originalnomological nomolog.ical statements statements is is defined; defined; then statements is is constructed constructed as then the theclass classof ofnomological nomoEogical statements as comprising comprising all those statements that are aredeductively deductively derivable derivable from sets of of statements statements of of the first first class. class. Among Among these, these, aa narrower narrower group group is is defined as nomological inthe thenarrower narrowersense. sense.ItIt isis this this group, group, also defined as nornological in also called the the group whichisis regarded regarded as as called group of of admissible admissible statements, statements, which supplying reasonable propositional propositional operations, operations, while while the the class of supplying reasonable of nomologicalstatements statementssupplies suppliesthe thelaws lawsofofnature nature and and the the laws nomological laws of logic. logic. As As in in the the previous stateof previous theory, theory, the the original original nomological nomological statements ments are included included in in the theadmissible admissiblestatements. statements. Furthermore, Furthermore, analytic, or tautological, tautological, statements statements are are included included in innomological nomological statements, a subclass of them being as in in the older statements, subclass of being admissible, admissible, as older 61 and theory. The 63 in ESL; theory. The new new definitions definitions replace replace §9 61 ESL; the the and §9 63 other sections of chapter chapter VIII VIII in other sections of in ESL ESL remain remain unchanged. unchanged. As As in in ESL, the theory ESL, theory isis developed developed only only for for the the simple simple calculus calcuhs of of functions. An An extension extension to to the the higher be functions. higher calculus calculus can can presumably presumably be constructed, but but would constructed, would require require further further investigation. investigation. Although Although the class class of of 'reasonable' ‘reasonable’ operations operations must must be be defined defined
66
INTRODUCTION
as aa narrower operations, one one must must not narrower subclass subclass of of nomological nornological operations, conclude that the conclude that the latter latteroperations operationsappear appearcompletely completely 'unreason‘unreason-
able'. able’. It seems seems that that there there isisno nounique unique explicans explicans for for the term term 'reasonable';; the the requirements which we we tacitly tacitly include ‘reasonable’ requirements which include in this this
term differ with the context differ with context in in which which the the operation operation is is used. used. The The theory presented accounts for these variations by defining various by defining categories categories and indicating indicating their their specific specific characteristics characteristics and and appliappli-
cations. cations. As an an instrument instrument for As for carrying carrying out out this this construction, construction, aa distinction distinction between three orders of truth truth is between three orders of is introduced. introduced. Analytic Analytic truth truth supplies supplies the highest, truth the highest, or third third order, order, synthetic synthetic nomological nomological truth the second second order, and merely factual factual truth truth the thelowest, lowest, or or first firstorder. order. The The two two higher orders nornological truth truth and and higher orders of truth, truth, which whichconstitute constitutenomological embrace all nomological statements,are are thus thus set above embrace all nomological statements, above merely merely factual truth. This This distinction distinction is is used, used, in in turn, turn, for the factual truth. the definition definition of nomological statementsinin the narrower nomological statements narrower sense, sense, which which are are conconstructed in such a way way that if if their their essential essential parts are true taken separately, they are order than than the the statement separately, they are true true of of aa lower lower order statement itself. itself. By means means of this this method, method, certain certain rather rather strong strong requirements requirements of of reasonableness can be reasonableness can be satisfied. satisfied. An important statements in in the An important application application of nomological nomological statements wider is given given by wider sense sense is by the the definition definitionof of modalities. modalities. These These categories categories are not statements, but are not presupposed presupposed for for nomological nomological statements, are defined defined by their help help and constitute constitute a sort sort of of byproduct byproduct of of the theory theory of of nomologicalstatements. statements. The The modalities modalities are are usually referred, not nomological to a statement, statement, but but to tothe thesituation, situation,or orstate stateof ofaffairs, affairs,denoted denoted by by it; i.e., it; i.e., they they are areused used ininthe theobject objectlanguage. language. We We thus thus define: define: a isis necessary necessary ifif 'a' ‘a’isisnomological. nornological. a is is impossible impossible if 'a' ‘G7isisnomological. nomological. a is (contingent)ififneither neither‘a’ 'a' nor nor 'a' is merely merely possible possible (contingent) ‘6’is is nomononiological. logical. For 'merely ‘merely possible', possible’, the term 'possible' ‘possible’ is is often often used, used, but but somesometimes 'possible' refers to to the the disjunction of 'necessary' and 'merely times ‘possible’ refers disjunction of ‘necessary’ and ‘merely possible'. It It can be seen seen that that the these possible’. can easily easily be the term term 'nomological' ‘nomological’ of of these definitions must must be be interpreted interpreted as in the the wider definitions as nomological nornological in wider sense; sense; if we we attempted attempted to to interpret interpret ititasasnomological nornological in in the thenarrower narrower sense, we we would would be be led led into into serious serious difficulties. difficulties. For For instance, instance, certain sense, certain
INTROJMIO'rION INTRODUCTION
77
analytic analytic statements statements would would then then not notdescribe describenecessary necessarysituations. situations. Whereas the use Whereas the use of of analytic analytic statements statementsfor forthe thedefinition definitionofoflogical logical modalities is obvious, obvious, it it is the modalities is thesignificance significance of of the the given givendefinitions definitions that that they theyalso alsoallow allowfor forthe thedefinition definitionofofphysical physicalmodalities. modalities. These These two result according two kinds kinds of of modalities modalities result according as, in in the the above abovedefinitions, definitions, the term as analytic or term 'nomological' ‘nomological’ is is specified specified as or synthetic synthetic nomonomological, respectively. respectively. Furthermore, Furthermore, a distinction distinction between between absolute absolute logical,
must be be made; made; for for these these points points and and the and relative relative modalities modalities must further theory of modalities to ESL, further modalities II refer to ESL, §Q 65. 65. In In the thefollowing following presentation, we shall refer to to modalities for the the presentation, we shall occasionally occasionally refer modalities for purpose of illustrating illustrating nomological nomological statements. constructed for for the is constructed The The class class of of admissible admissibleimplications implications is purpose of satisfying very strong requirements requirements and thus thus of of expliexplicating reasonable implications implications in in the the narrowest narrowest sense of of the the term. Conversational language has has two two kinds of Conversational language of usage usage for forimplications implications subject to very exacting requirements: they are used subject very exacting requirements: they are used for for prepredictions, or they are contrary to fact. dictions, or are employed employed as conditionals conditionals contrary fact. It unctive implications cannot convey convey important important It isis obvious obvious that that adj adjunctive implications cannot information in in a predictive usage.InInorder orderto to know know that that the predictive usage. the imiminformation plication is true, true, we would would have have to to know know that that aa particular plication is particular T-case, T-case, which verifies verifiesit, it, is is true; true; but which but once once we we know know this this T-case, T-case, we we would wodd lose in in information information ifif we we merely merelystate state the the implication implication and and not not the lose T-case itself. itself. This This applies applieswhether whether we weknow knowthe the truth truth of the T-case T-case of the from past past observations or because we can can predict predict it. it. For because we For instance, instance, from we can can predict predict that that itit will tomorrowand and that that the we will be be Wednesday Wednesday tomorrow this conjunction by the adjunctive sun will will rise; replacing replacing this conjunction by adjunctive imimplication, ‘if 'if it is the sun will rise', we we say plication, is Wednesday Wednesday tomorrow tomorrow the will rise’, less than than we less we know, know, and and therefore therefore such such an an implication implication has no no practical use. practioal use. It It has hasoften oftenbeen beenemphasized emphasizedthat thatfor fora counter/actual a counterfactual usage, usage, likewise, adjunctive implications are completely inadequate. Nolikewise, adjunctive implications NObody would would say, say, 'if not white, sugar would be sour', body ‘if snow were were not would be sour’, although this this implication implication isis true true in the adjunctive although adjunctive sense. sense. But we we heated, it would 'if this metal would say, would say, ‘if metal -were heated, would expand'. expand’. Since Since in the conversational conversational language language is “rather clear clear and unambiguous unambiguous in usage of conditionals contrary to fact, we possess in this usage usage of conditionals contrary to fact, we possess in this usage aa sensitive test test for the sensitive the adequacy adequacy of of the theexplication explication of of reasonable reasonable
8 8
INTRODUCTION INTRODUOTION
implications, and we we shall shall often often make make use use of of it. it. For instance, implications, and instance, it is required for a conditional contrarytoto fact fact that that it required for conditional contrary it be be unique. unique. By By this property II mean mean that, that,ififthe theimplication implication 'a‘aD 3b' 6’ is is used used for aa conditional contrary contrary to fact, fact, the contrary 'a D conditional contrary implication implication ‘a 3 8’ cannot be so cannot so used. used. Obviously, Obviously, adjunctive adjunctive implication implication does not
satisfy the condition when it is used satisfy condition of uniqueness uniqueness when used countercounterfactually, ‘a’is false, both contrary contrary implications implications are factually, because, because, ifif 'a' true in It has out that in the the adjunctive adjunctive sense. sense. It has often often been been pointed pointed out this absence absence of of uniqueness uniqueness makes makes adjunctive adjunctive implications implications ininappropriate for for counterfactual use. In In the theory appropriate count’erfactual use. theory of of admissible admissible implications will therefore therefore be be an an important implications itit will important requirement requirement that two two contrary contrary implications implications cannot cannot be be both both admissible. admissible. The The present present theory theory satisfies satisfies this requirement, requirement , whereas whereas my previous previous theory theory could satisfy itit only to some could satisfy some extent. extent. Since the theory Since the theory to to be be developed developed is is rather rather tecimical technical and andinvolves involves much detail, of which whichisis at at first first not not easily it much detail, the the significance significance of easily seen, seen, it may be to outline the definition may be advisable advisable to outline the the major major ideas ideas on on which which the definition of original nomological statements statements is is based. of original nomological based. These These ideas ideas have have been been developed essentially essentially for for synthetic synthetic nomological nomological statements, statements, because because developed statements of statements of this this kind kind are are in in the the foreground foreground of of this this investigation; investigation; the application to tautologies is then rather easily given. application to tautologies is then rather easily given. The The leading leading idea idea in in the thedefinition definitionof of original originalnomological nomological statestatements ments of of the the synthetic synthetickind kindwifi, will,of of course, course, be be given given by by the the principle principle that that such such statements statementsmust must be be general general statements, statements, or or all-statements, all-statements, and must not be and be restricted restricted to to aasingle single case. case. We We know know from from the writings of David Hume that physical necessity, the necessity writings of David that physical necessity, necessity of the the laws laws of of nature, nature, springs springs from from generality, generality, that thatcausal causalconnection connection differs from mere mere coincidence coincidenceininthat that it it expresses differs from expresses a permanence permanence of of coincidence. Hume is the coincidence. Hume believed believedthat that this this generality generality isis all all that that is required He was was right right when he insisted insisted that that required for for causal causal connection. connection. He when he unverifiable additionsto to this this requirement requirement should should be be ruled ruled out; out; in unverifiable additions in fact, fact, any any belief belief in in hidden hidden ties ties between between cause cause and and effect effect represents represents surplus meaning meaning which which Occam's Occam’s razor razor would shave shave away. away. HowHowa surplus ever, it it turns turns out outthat thatgenerality generalityalone, alone,though thoughnecessary, necessary, is is not not all unreasonable unreasonable forms ruled out. out. sufficient to sufficient to guarantee guarantee that that all forms are are ruled We therefore introduce, introduce,inin addition addition to to generality, generality, aa set set of We shall shall therefore requirements restricting restricting the statement forms forms to be admitted. It goes goes requirements the statement to be admitted. It
ThTRODUCTION INTRODUCTION
99
without saying saying that that these without these additions additions are are formulated formulated as as verifiable verifiable properties of of statements, statements, and and that, that, for aa given properties given statement, statement, we we can can always find find out out whether always whether it satisfies satisfies the the requirements. requirements. In his his early early writings writings on on mathematical mathematical logic, logic, Bertrand Bertrand Russell Russell has pointed ff(x)f (x)3 D g(xfl' has pointed out outthat thataageneral generalimplication implicationofofthe theform, form,'(x) ‘(x)[ g (z )] ’ eliminates the unreasonable eliminates unreasonable properties properties of adjunctive adjunctive implication implication to some extent, extent, but but that to that these these properties properties reappear reappear if if the the implicans implicans '/(x)' is always or the implicate ‘f(x)’ always false false or implicate 'g(x)' ‘g(x)’ is always always true. The The exclusion of of these these two two cases oases will willtherefore thereforebe bean an important important requirerequireexclusion ment within the ment the definition definition of of aareasonable reasonableimplication. implication. However, However, for a general statements, this this requirement for general theory theory of of nomological nomological statements, requirement must be generalized so as as to to be to other other opermust generalized so be applicable applicable likewise Iikewise to ations and totostatements among ations statementspossessing possessing several several operators, operators, among which there there may may be be existential existential operators. operators. It It can that for which can be be shown shown that for the latter latter case case an animplicans implicans which which is is not not always always false false does does not not exclude an an unreasonable unreasonable implication. implication. The construction construction of such aa more requirement is by means more comprehensive comprehensive requirement is achieved achieved by means of of aa formal formal property of of statements, statements,which whichisiscalled calledexhaustiveness exhaustiveness and which which will be defined defined in in group group EE of chapter 2. will be 2. (See (See also the discussion discussion of (4Oae-b) chapter 3.) (4Oa-b) inin chapter 3.) Even ifif an so far mentanimplication implication satisfies satisfies the requirements requirements so mentioned, it can ioned, it can have have forms forms that that are arenot notaccepted acceptedasasreasonable. reasonable. Assume that that during Assume during a certain time it it so so happens happens that that all allpersons persons in a certain in certain room room are are over over 30 30 years years old; old;then thenthe thegeneral generalimplicimplication, 'for all x, x , if x is a person person in in this room at at this this time, time, xx is is over over ation, ‘for all 30 years old’, old', is true in 30 years in the theadjunctive adjunctive sense, sense, and and its itsimplicans implicans is is not not always always false. false. Yet Yet this thisimplication implication does does not not appear appearreasonable, reasonable, as is is seen seen when when it is used used counterfactually: counterfactually: the the statement, statement, 'if ‘if another person had had been been in in this this room at at this time, time, he he would would have been been over over 30 years years old', old‘, would would not be be acceptable acceptable as true. true. This This example shows that that aa reasonable implication has has to to satisfy satisfy further further example shows reasonable implication requirements, which exclude exclude aa restriction requirements, which restriction of of the the implication implioation to to certain times times and and places places and andguarantee guaranteeits itsuniversal universalapplication. application. These requirements w will be explained explained in in group group P, F, chapter These requirements ill be chapter 2. 2. It of this this kind It should should be be noted noted that that requirements requirements of kind are rather strong strong and and are are adhered adhered to, to,ininconversational conversationallanguage, language, only only when when the implicational character of of the the statement is implicational character is explicitly explicitly stated,
10 10
INTRODUCTION INTRODUCTION
for instance instance by 'if-then', 'implies', for by using using terms terms like like ‘if-then’, ‘implies’,etc. etc. No No objection, objection, however, isis raised raised when when the the statement statement is given however, given the wording: wording: 'all ‘all persons in in this room 30 years years old', old’,which which persons room at this this time time were were over over 30 form appears appears quite reasonable. In In the form the disguise disguise of of aa conversational conversational all-statement, therefore, therefore, we we accept accept adjunctive adjunctive implications, fact implications, aa fact which shows showsthat that these these implications implicationsare are not not merely merely aa creation creation of of which the the logician logician but but are arewidely widely used used in inconversational conversational language. language. The The present present investigation investigation into into the the nature natureof ofreasonable reasonable implications implications is is therefore restricted to to an therefore restricted an explicit explicit use use of of this thisoperation. operation. Similar Similar considerations apply apply to other considerations other propositional propositional operations. operations. It turns It turns out out that thatininorder ordertotocarry carrythrough throughthe therequirements requirements mentioned mentioned it is necessary necessary to introduce introduce rules rules which which eliminate eliminate redundant parts procedureof ofreduction, reduction, redundant partsof of statements statementsand anddefine defineaaprocedure by means by means of of which which aa statement statement is is transformed transformed into into simpler simpler forms. forms. This isis necessary, first, because This necessary, first, because a reasonable reasonable statement statement could could easily be be made made unreasonable unreasonableby byadding addingtotoitit redundant redundant parts; parts; for easily instance, ifif aa statement statement contains to a particular instance, contains no no terms terms referring referring to particular space-time region, we we could could add add to to it some space-time region, some tautology tautology containing containing such terms such terms without without changing changing the meaning meaning of of the thestatement. statement.1 Secondly,however, however,ititmay maybe be possible possibletotoinsert insert parts parts that are Secondly, are merely factually factually true true into a nomological statement in in such such a way merely nomological statement that the the requirements the statement statement still still satisfies satisfies the requirements mentioned mentioned previously. with certain previously. IIn n combination combination with certain other requirements, requirements, the reduction rules out out such forms;; and and II have reduction procedure procedure rules such forms have been been able able to to construct a proof that unreasonable parts of construct proof that unreasonable parts of aa certain certain kind kind cannot cannot be contained statements as defined in this contained in in original original nomological nomological statements defined in this presentation presentation (see (see theorem theorem 5). 5). All the the criteria criteria so so far far mentioned mentioned are are of of aa formal formalnature; nature; and and they they All are based on the assumption assumption that we we are able to find find out whether whether these formal relations hold. hold. For For instance, instance, itit is presupposed that formal relations presupposed that we are able able to to find find out out whether whether aa statement statement is to aa given we are is equipollent equipollent to given other statement, statement, whether whether it can be be written written in in syntactical syntactical forms forms of other it can of a certain certain kind, kind, such such as:an all-statement, as much much as as such such all-statement, etc. etc. In In as 1 This This objection objection was correctly aeainst previous theory by was raised raised correctly against my my previous J. C. vol. 55, J. C. C. C .McKinsey, McKinsey,American American Mathematical Mathematical Monthly, Monthly, vol. 5 5 , 1948, 1948, pp. 261-263; and y N. Goodman, Philos. Philos. Review Review 1948, 1948,vol. vol.57, 57,pp. pp.100—102. 100-102. 261—263; andbby N. Goodman,
XN'rRODUCTION INTRODUCTION
11 11
an assumption assumption is made, made, the the present present theory theory presupposes presupposes the the comcom-
pleteness pleteness of of the the lower lower functional functional calculus. calculus. However, However, since since aa general general decision procedure cannot cannot be decision procedure be constructed constructed for for this thiscalculus, calculus,we we cancannot not give give rules rules indicating indicating how how the the test test for forequipollence equipollence is is to to be be made. made. In I n principle, principle, therefore, therefore, there there may may exist exist statements statementsof of complicated complicated
forms for which we are actually forms for which we actually unable unable to decide decide whether whether they satisfy laid down; down; we we then then have satisfy the requirements requirements laid have to put put these these
statements into statements into aa category category under under the the heading, heading, 'at ‘atpresent present unknown unknown whether and hope hope that that some day they whether nomological', nornological’, and some day they will will be taken out out of of this this category, category, because because in in principle principle the thedecision decision can can be be made. made.
In practice, practice, however, however, we we shall shall encounter encounter no nosuch suchdifficulties, difficulties, because scientific laws laws have have rather rather simple syntactical forms and because scientific cannot as to structural cannot compete, compete, as structural form, form, with the the involved involved statestatements which the mathematical likes to to make the the subject which the mathematical logician logician likes subject of his his investigation. investigation. From formal properties properties II will will now now turn to to the thediscussion discussion of of aa
property which property which is independent independent of of form. form. Being Being laws laws of of nature, nature, nomologicalstatements, statements,ofofcourse, course,must mustbe betrue; true; they they must even nomological even be true,which whichisisaastronger strongerrequirement requirement than than truth truth alone. be verifiably verifiably true, alone. Some remarks about about this requirement Some remarks requirement must must now now be beadded. added.1 The requirement of truth isisnot requirement of notsufficient sufficient because because we we wish wish to to exclude from from nomological statements those all-statements exclude nomological statements all-statements which which are merely factually true, true, or or 'true merely factually ‘true by by chance'. chance’. This This kind kind of of statestatement may ment may obtain obtain even even ifif no noreference reference to toindividual individual space-time space-time regions isis made; made; for for instance, the statement, 'all regions instance, the ‘allgold gold cubes cubes are are smaller than than one cubic smaller cubic mile', mile’, may may possibly possibly be true. true. When When we we reject a statement of reject of this this kind kind as as not not expressing expressing a law law of of nature, we mean mean to say we say that that observable observable facts facts do not require require any any such such statement for for their and thus thus do statement their interpretation interpretation and do not not confer confer any any truth, or on it. it. If they or any any degree degree of probability, probability, on they did, did, ifif we we had had good inductive inductive evidence for for the the statement, good statement, we we would would be be willing willing to accept it, it. For For instance, instance, the the statement, statement, 'all ‘allsignals signals are are slower slower than In I n ESL, ESL, p. p. 369, 369, II used used the theterm term'demonstrably ‘demonstrablytrue'. true’.Since Since'demon‘demonstrable' proof, I will will now ngiv use use the the above above strable’ usually usually refers refers only only to deductive deductive proof,
it is now now generally generally term 'verifiable' ‘verifiable’alone alonewould would not not suffice suffice because it term. The term
used as true or used in the the neutral neutral meaning meaning 'verifiable ‘verifiable as or false'. false’.
12 12
INTRODUCT%ON
or equally equally fast as as light light signals', signals’, is accepted accepted as a law law of of nature nature because observable observablefacts factsconfer conferaahigh highprobability probabilityupon uponit. it. It It is because the inductive not mere mere truth, which inductive verification, verification, not which makes makes an allallstatement statement a law of of nature. nature. In I n fact, fact, ifif we we could could prove prove that gold gold cubes cubes of giant size size would would condense condense under under gravitational gravitational pressure pressure into aasun-like sun-likegas gasball ballwhose whose atoms atoms were were all all disintegrated, disintegrated, we we would be be willing willing also alsoto to accept accept the the statement about would about gold gold cubes cubes among among the laws laws of of nature. nature. The reason is is easily easily explained. The inductive inductive inference inference extends
truth from from 'some' ‘some’ to 'all'; ‘all’;itittherefore thereforeallows allows for for aa predictive predictive as as
well as as counterfactual counterfactual use use of of implications. implications. We We saw saw that that these well these two two kinds kinds of of usage usage are are essential essential for for reasonable reasonable implications; implications ; therefore, therefore, if it qualifies for the the category if an an implication implication is is inductively inductively verified, verified, it qualifies for category of reasonable We discussed discussed the the example of an imreasonable implications. implications. We example of implication which is restricted to persons in a certain room during a plication which is restricted to in
certain certain time; when when we we reject reject such such an an implication implication for for countercounterfactual use, it isis because factual because this this implication implication is is not not verified verified through through inductive extension. extension. The The requirement requirementthat that the all-statement inductive all-statement be be verifiably true, true, therefore, therefore, guarantees guarantees the the kind kind of of truth truth with verifiably with which which we wish we wish to toestablish establishlaws lawsofofnature; nature;it guarantees it guarantees'inductive inductive generality. The word word 'verifiable' ‘verifiable’ includes includes a reference reference to to possibility. possibility. Since Since physical possibility is is aa category to be physical possibility be defined defmed in in terms terms of of nomonomological statements, statements, itit would would be be circular circular to to use, in the definition logical definition of such statements, statements, this category. the term category. For For this reason, reason, II defined defined the 'verifiably true' as verified at at some sometime, time, in in the the past past or or in in ‘verifiably true’ as meaning meaning verified the future. future. It Ithas hasbeen been argued argued against against this this definition definition that there there may be laws laws of of nature nature which which will will never never be be discovered discovered by human human beings. the present present investigation investigationII shall shall show show that that the the latter beings. 1IIn n the statement, indeed, indeed, can can be be given given aa meaning, meaning, and and that thatwe wecan candefine define statement, term veriflably verifiably true in in the the wider wider sense 8eme which covers this this meaning. meaning. aa term which covers But in shallbegin begin with with the the narrower narrower in order order to to define define this term, term, IIshall This objection was by Mr. Mr. Albert Albert This objection was raised raised against against my my theory theory in in a letter by Hofstadter, interesting objections objections answered in Hofstadter, which included some further further interesting the present paper. paper. The The same same objection objection was the present was made made by by G. G. D. D. W. W. Berry, Berry, Journ. of Symbolic Logic,vol. vol. 14, 14, 1949, p. 52. Symbolic Logic, 1949, p. 52. 1
IXTBODUOTION
13 13
term, and later to the and proceed proceed later the introduction introduction of of the the wider wider term term
(chapter (chapter 6). 6). Although Although inductive inductive verifiability verifiability is is presupposed presupposed for fornomological nomological statements, statements, the the definition definition of of such such statements statements can can be be given given without without entering into an an analysis analysis of of the the methods methods of of verification. verification. What What we we statements is is not aa are looking looking for in aa definition definition of of nomological nomological statements
method such statements, statements, but but a set method of of verifying verifying such set of of rules ruleswhich which guarantee is actually guarantee that inductive inductive verification verification is actually used used for for these these statements, in as much as they are synthetic. The requirements statements, as much as they are synthetic. The requirements laid laid down down in in the thedefinition definition of of nomological nomological statements, Statements, in fact, fact, represent represent a set set of of restrictions restrictions which which exclude exclude from from such such statements statements all synthetic forms that can without inductive all synthetic forms that can be be verified verified without inductive extension. extension.
More than that, More than that, the therestrictions restrictions single single out, out,among amonginductively inductively verified statements, a special group of all-statements verified statements, special group of all-statements associated associated with a very of probability; probabifity; and they very high high degree degree of they are are so so constructed constructed
that they they allow allow us us to to assume assume that that these these all-statements all-statements are true without Merelyfactual factualtruth, truth, though without exceptions. exceptions. Merely though in itself itself found found by inductive inductive inference, inference, is is thus thusdistinguished distinguishedfrom fromnomological nomological truth in and the in that that ititdoes doesnot notassert assert an aninductive inductive generality; generality; and
requirements introduced statements are all governed governed requirements introduced for for nomological nomological statements are all by the very very principle principle that factual factual truth truthmust mustnever neverbe besufficient sufficient to t o verify verify deductively deductively aa statement statement of this kind. kind. The is thus The predictive predictive usage usage of of admissible admissible implications implications is thus reduced reduced to the the predictive predictive use use of of inductive inductive inferences inferences equipped equipped with high degrees of ofprobability. probability. Their Their counterfactual counterfactual usage, degrees usage, likewise, likewise, appears appears justified by by this justified this interpretation, interpretation, although although this this usage usage imposes imposes even stronger even stronger requirements requirements upon upon implications implications than aa predictive predictive usage, as as will willbe beshown shownininchapter chapter7.7.ItIt isis its its origin usage, origin in inductive inductive extension, its .inductive generality, that makes an implication extension, its inductive generality, that makes an implication reasonable. reasonable. Since the the function function of of the requirements to be introduced Since requirements to introduced is thus negative negative rather than than positive, positive, inasmuch inasmuch as as these these requirements requirements are are merely restrictive, restrictive, itit is is not not necessary necessary to to give give in in this this presentation presentation a merely detailed discussion discussion of ofinductive inductive verification. verification.That That inductive inductive methods methods detailed exist and and are applied, is a familiar exist applied, is familiar fact; their study study belongs belongs in a theory of of induction induction and and probability, probabifity,and and as as far far as my own theory own conconception of of this this subject subject matter matter is concerned, ception concerned, II refer refer to another another
14 14
INTRODUCTION
publication.'1 However, However,II should shouldlike liketoto add add to to the present publication. present investigation a brief vestigation brief account of of the the methods methods of of inductive inductive verification verification in their relation relation to to general general implication; implication; this account account is given given in in the the appendix. appendix. Those who who have have studied the Those the construction construction of of artificial artificiallanguages languages are often sceptical as as to to the rules that that govern are the possibility possibility of finding rules govern conversational language. language. They They are are disappointed by the conversational the vagueness vagueness of the terms life, and and point point to terms used used in the the language language of everyday everyday life, the apparent inconsistencies in actual usage of language. Yet on apparent inconsistencies in actual usage of language. Yet closer inspection,ititturns turns out out that that aa natural closer inspection, natural language language is by by no no means as inconsistent as is believed. If If it is sometimes sometimes believed. it is is difficult difficult to to find rules, rules, one one must must not conclude find conclude that no no rules rules exist. exist. Physical Physical phenomena, too, do not phenomena, too, not always always openly openly display display the therules rulesfollowed followed by them; able to show them; but but physicists physicists have have been been able show that all all such such phenomena are controlled by very phenomena are controlled by very precise precise rules, rules, though though the the formulation these rules natural formulation of of these rules may may be be extremely extremely complicated. complicated. A natural language is aa complex complex system systemofofpsychological psychologicaland andsociological sociological language is phenomena, and one one cannot cannot expect expect its its laws laws tto phenomena, and o be visible visible tto o the the untrained eye. Those who are not afraid to search for its laws, untrained eye. Those who are not afraid search for its laws, however, have have been been surprised surprised to to discover discover that that rather however, rather precise precise laws laws can be constructed language, and and that, that, once can constructed into actual usage of language, once laws laws have have been been abstracted abstractedfrom fromsingle single examples, examples, they they cover cover large large parts of parts of usage usage practically practically without without exceptions. exceptions. Perhaps it Perhaps it is is possible possible to to explain explain the the hidden hidden precision precision of of language language by the by the fact fact that thatlanguage languagebehavior behavior isiscontinuously continuously tested tested and and corrected by by its its practical practical applications applications;; that, that, in particular, corrected particular, predicpredictions and contrary to tions and conditionals conditionals contrary to fact fact are areof ofgreatest greatestsignificance significance inexact in in the in everyday everyday life, life, and that that aalanguage language which which were were inexact use use of of such such concepts concepts would would soon soon be be led led into intoserious seriousconificts conflicts with with observational experiences. experiences.IfIf itit is required observational required for for aa reasonable reasonable imimplication to be plication to be applicable applicable to to predictions, predictions, the the usage usage of of reasonable reasonable implications isis not not aa matter of implications of taste, taste, or orof of social social convention, convention, but something eminently eminently practical practical;;and and if if we we have have developed developedaanatural natural something To Probability, second second edition, edition, Berkeley To my my book, book, The The Theory Theory of of Probability, Berkeley 1949; 1949; quoted ThP.This Thisbook book includes includes aadiscussion discussion of of induction induction for for predictive predictive quoted as ThP. usage and aa justification justification of of induction, induction, problems problems which which cannot be dealt dealt with with in the the present present monograph. monograph. 1
XN'TRODUCTION
16
feeling for the reasonableness of an implication, we have been feeling for reasonableness of been so so
conditioned by by the exigencies conditioned exigencies of of everyday everyday life. life. Thus Thus practical practical needs zceds have made made language language aa forceful forceful instrument instrument which which owes owes its its efficiency to its precision. The study of natural languages, thereeiEciency to its precision. The study natural languages, fore, offers to the fore, offers to bhe logician logician the the possibility possibility of of making making laws laws explicit explicit which, though unknown which, though unknown to the the language language user, user, implicitly implicitly control control his language language behavior behavior and and make make it it consistent. his consistent. The present study
is intended intended to be be aa contribution contribution to t o this this task. task.
II I1 FUNDAMENTAL TERMS TERMS FUNDAMENTAL
In statements we we shall shall refer refer to I n the thedefinition definition of of nomological nomological statements two kinds of of these these statements. two of properties properties of statements. First, First, we we shall shall speak speak of of properties properties which which remain remain invariant invariant for fortautological, tautological, or orequipolequipollent, lent, transformations, transformations, such such as as truth, truth,or orbeing beingsynthetic. synthetic.These Thesewill will be Terms used used for for the the formulation be called called invariant invariant properties. properties. Terms formulation of of these these properties properties wifi will be be called called I-terms. I-terms. Second, Second, we we shall shall speak speak of of properties whichaa statement statement has has only only in in aa particular properties which particular form form of of writing, do not not remain writing, and and which which do remain invariant invariant for for all all tautological, tautological, or or equipollent, transformations,such such as as being equipollent, transformations, being an an implication, implication, or or containing containing an an all-operator. all-operator. These These will .Rillbe be called calledvariant variantproperties. proprtka. Terms Terms used used for for the the formulation formulation of of such such properties properties will will be be called called V-terms.The The definition definition of of nomological nomologicalstatements statements will be be laid laid down V-terns. down in in certain certain requirements, requirements, which which we we distinguish distinguish correspondingly correspondingly as as I-requirements and and V-requirements. V-requirements. In I n the thebeginning, beginning, we we shall shalldeal dealonly onlywith withoriginal originalnomological nomological statements. statements. For For their their definition definitionboth bothkinds kindsof ofrequirements requirements will will be be used. is thus thus made In used. The The term term 'original ‘original nomological' nomological’ is made a. a V-term. In order to construct the requirements, it is advisable first to define order to construct the requirements, it is advisable first to define certain certain terms terms which which are are to to be beused. used. These These definitions definitions are are ordered ordered by by groups. groups. Notational Sentence name name variables, variables, belonging belonging to to N o t a t i o n a l remark. r e m a r k . Sentence the will beexpressed expressedbybythe theletters letters‘p’, 'p', ’q’, 'q', ‘r’, 'r', etc; the metalanguage, metalanguage, w i l l be etc; combinations such letters letters will will be be interpreted interpreted in in the sense combinations ofof such sense of autonymous such that that 'p. q' is a u t o n p o u s use of of operations operations (Carnap), (Carnap), such ‘p.q’ is the the name name of of the the conjunction conjunction of of pp and andq.q.Sentential Sententialvariables, variables,belonging belonging in in the will be be expressed expressedby bythe theletters letters ‘a’, 'a', 'b', the object object language, Ianguage, will ‘b’, 'c', ‘c’, etc.; etc. ; functional in the the object functional and and argument argumentvariables, variables,likewise likewise belonging belonging in object language, wifi be be expressed expressedby bythe the letters letters ‘f’, 'f', 'g', language, will ‘g’, 'x', ‘d, ‘y’, etc. etc. These These 'y', variables require the the use variables require uae of ofquotation quotation marks maxh within within aacontinuous continuous
FUNDAMENTAL FUNDAMENTAL TERMS TERMS
17 17
on separate lines text; for formulae formulae on lines the quotation quotation marks marks wifi will be be omitted. quotation marks omitted. Likewise, Likewise, quotation marks will will be be omitted omitted after after aacolon colon in in the text. The metalinguistic is made made The metalinguistic variables variables will be be used used when when reference reference is to the inner inner structure structure of of the the sentences sentences denoted denoted by the the individual individual letters. The object language variables variables will will be be used, used, first, letters. fist, when no referenceisis made madeto to the the inner reference inner structure of the the sentence, sentence, or the the function, abbreviated abbreviated by by one one letter, letter, and all the structure function, structure referred referred to is is expressed expressed by by combination combination of of letters. letters.Secondly, Secondly,however, however, object language variables will will be be used used in in a mixed object language variables mixed conteftt, where where the structure structure of of the thesentences sentences isispartially partiallyexpressed, expressed, partially partially described in in words. described words. The between these these cases cases may may be ifiustrated The distinction distinction between illustrated by by examples. II shall shall write write:: 'a' b';; the examples. ‘a’ is is derivable derivable from 'a. ‘a.b’ the variable variable 'x' ‘z’ in '(x)t(x)' ‘ ( x ) f ( ~ )is ’ bound; etc. In I n these these cases, cases, no no reference reference is made made to the inner structure structure of of the the expressions expressions abbreviated abbreviated by one letter, letter, and the truth truthofofthe themetalinguistic metalinguistic sentence sentence is is visible visible from the structure expressed by by the the symbols. IInn contrast, to inner inner structure contrast, reference reference to structure of the referred to to is made the individual individual sentences sentences referred made in in such such statements as, statements as, 'p ‘ pisisderivable derivablefrom fromq', q’,for for which which II use use metalinguistic metalinguistic variables. The The truth truth of of such not visible visibIe from the variables. such a statement statement is not structure indicated by the statements can the symbols; symbols; therefore such statements only occur form, such such as: as: if p isis derivable only occur in conditional conditional form, derivable from from q, q, then ...; assume Slssume that pp isisnomological; nornological; etc. A A mixed mixed context context is is then ...; D g(x)' and ‘f’ '/' given by by aa statement of the form: given form: if if 'f(x) ‘ f ( x )3 g(z)’ is is analytic and with ‘g’, 'g', then then 'f' is identioal with ‘f’ or or 'g' ‘g’ is is composed composed of elementary elementary is not identical functions. IfIf such were formulated formulatedby by the the help functions. such statements statements were help of of metalinguistic have to to be be extended extended metalinguistic variables, variables, autonymous autonymous use use would would have to the to the parentheses; parentheses; although although this thiscould couldof of course course be beconsistently consistently done, done, II prefer prefer to to use useobject objectlanguage language variables variables and and quotes. quotes. The The decision forone oneoror the the other other method methodisis aa matter decision for matter of of style style and and personal taste) not not of who do do not not like like the the rather rather personal taste, of correctness. correctness. Those Those who wide use of of quotes quotes may may regard regard the the expression, wide use expression, “"/' f ’ is is composed composed of of elementary elementary functions', functions’, as as an an abbreviation abbreviationfor forthe thelonger longerexpression, expression, “f’ is is interpreted by by aafunction functionwhich whichisiscomposed composed of of elementary elementary "/' functions'. Likewise, the expression, "a' is an implication', functions’. Likewise, the expression, ‘‘a’ an implication’, can can be regarded "a' isis regarded as an abbreviation abbreviation for the longer longer expression, expression, “a’
18 18
rUNDAMENTAL 1E1%MS FUNDAMENTAL TERMS
interpreted by an implication'. interpreted implication’. In In this this way, way, the thewider wider use use‘of quotes can be regarded can regarded as an an abbreviated abbreviated mode mode of of speech speech translatable translatable into a narrower of quotes. Note that the into narrower use of quotes. Note the wider wider use use of of quotes quotes can occur only can only in in conditional conditional sentences. sentences. In aasynthetic synthetic statement, statement,sentential sentential and andfunctional functional variables variables express uninterpreted constants, constants, i.e., i.e., such such statements statements are are true express uninterpreted true only for specific specificvalues valuesof ofthese thesevariables. variables.IIn an analytic analytic statement, statement, only for n an sentential and functional in the functional variables represent represent free variables variables in sense that any value sense that value may be be given given to them them while while the statement statement remains true. true. A remains A notational notational distinction distinction between between these these two two cases casm will not not be made, the same letter may represent will made, because because the same letter represent a free free variable for the whole formula and and an uninterpreted variable for whole formula uninterpreted constant constant for for a part part of of it. it.Bound Boundfunctional functional variables variables will will not be be used used since since the the presentation presentation remains remains entirely entirely within within the thelower lowerfunctional functionalcalculus. calculus. A sentence is called propositionalterm term ifif it it has called an elementary elementary propositional has no no inner inner structure structure expressible expressible by by the the use use of of propositional propositional symbols; symbols ; otherwise A function together with with its otherwise it it is called called compound. compound. A function together its variables, such such as 'f(x, variables, ‘f(x, y)’, is called called aafunctional. functional.11 A function is is A function y)', is called elementary ifif itit does not stand for called elementary does not for aa combination combination of of other other functions; notational distinction functions ;otherwise otherwise it it is is called called compound. compound. A notational distinction between elementary and and compound compoundterms terms will will not not be be made; in between elementary in fact, owing such a owing to the thevagueness vagueness of of conversational conversational language, language, such distinction can scarcely be carried it is distinction can scarcely be carried out uniquely. uniquely. However, However, it usually to assume assume that, that, in usually sufficient sufficient to in aa certain certain context, context,some somerule rule has has been laying down down this this distinction; the rule itself been introduced introduced laying distinction; the itself is is irrelevant. irrelevant. Furthermore, Furthermore,ifif by by regarding regarding certain certain terms terms as as elementary, elementary, a statement statement can can be be shown shown to to be be tautological, tautological, or or to to be bederivable derivable from some some other other statement, statement, these relations will not be changed if if from will not the assumed assumed elementary elementary terms are are further further subdivided. subdivided. GROUP A. TRUTH AND GROUP A. TRUTH ANDTRANSFORMATIONS TRANSFORMATIONS Definition D e f i n i t i o n 1. 1 . A statement as statement isisverifiably verifiably true if if itit isisverified verifiedas practically true at some practically true some time time during during the past, past, present, present, or or future future history history of mankind. mankind. (I-term). (I-term). If time, but but regarded Ifaa statement statementisisregarded regarded as as verified verified at at some some time, regarded as at aa later later time, as falsified falsified at time, then then the thelater laterdecision decisiontakes takesprecedence, precedence, 1
1
ESL, p. p. 81. 81. ESL,
FUNDAMENTAL TEnMS TERMS
19 19
being based based on on aa more more comprehensive comprehensivebody body of of evidence. evidence. The The earlier earlier being
decision is regarded decision is regarded as as erroneous. erroneous. say that Definition When we we say that aastatement statementppcan canbc bewritten written D e f i n i t i o n 2. 2 . When asp' top', meant that that pp and as p’or orthat thatppisisequisignificant equisignificant to p’, it it is is meant and p' p’ contain contain certain elementary terms terms and and that, that, in certain elementary in these these elementary elementary terms, terms, p' p’ is is tautologicaily equivalenttotopp,, or or is equipollent to p (see tautologically equivalent equipollent to (see ESL, ESL, p. 108). (I-term). (I-term). p. GROUPB. REDUCTION GROUP B. REDUCTION
The procedure procedure of of reduction reduction serves serves to to eliminate eliminate redundant redundant parts parts The from formavoiding avoidingunnecessary unnecessarycompcompfrom aa statement statement and andto togive giveititaaform lications. It is lications. It is obvious obvious that the thedefinition definition of of such such aa procedure procedure is to some extent a matter that the some extent matter of of taste. taste. However, However, it it will will be seen that the definition given leads leads to to statement statement forms definition given forms which which appear appear appropriate appropriate both from standards of of taste and from general general standards and from from the the viewpoint viewpoint of
constructing propositional propositional operations operationsthat that appear reasonable, constructing reasonable, in particular, that can particular, implications implications that can be be interpreted interpreted asasconditionals conditionals contrary to fact. contrary fact. However, However, the thelatter latterconsequence consequencewill willbecome become visible only only in in later chapters visible chapters of of this this presentation. presentation. In order order to tocarry carrythrough through the thereduction reduction procedure procedure we we first first
define redundant redundant parts, parts, and define and then then define define aa procedure procedure of of contraction contraction which serves serves to to diminish diminish the the number number of of binary in a which binary operations operations in statement. The statement. The contraction contraction procedure procedure is is subdivided subdivided into into two two forms, forms, according as as the expressions referred referred to to are synthetic or analytic. according analytic. The term 'analytic' ‘analytic’will will always always be be used used synonymously synonymously with the the term ‘tautological’. term 'tautological'. Definition contained inina statement D e f i n i t i o n3.3A . unit A unit contained a statementp pisisany anycomcombination of of signs signs in in pp such that, that, ifif this in thiscombination combination is is enclosed enclosed in bination parentheses within the statement, the resulting total expression parentheses within the statement, resulting total expression is equisignificant to p . (V-term). (V-term). is equisignificant to p. Definition is closed includes, A unit is closedif ifit it includes,for forevery every arguarguD e f i n i t i o n4.4A . unit ment variable in it, it, aa corresponding operator. An operator operator ment variable occurring occurring in corresponding operator. is redundant ifif its its variable variable does does not not occur occur in in any any functional functional within within is its scope. its scope. For instance, in '(x)/(x)' For instance, in ‘(x)f(x)’the unit unit 'f(x)' ‘f(x)’isis not notclosed, closed, whereas whereas the total total expression is aa closed closed unit. unit. In In '(x)/(y)' the operator '(x)' the expression is ‘(x)f(y)’the ‘(x)’ is redundant. is
20 20
FWNDAMEN!PAL TERMS
Double negation negation lines lines are are redundant, redundant, except Definition D e f i n i t i o n 5. 5 . Double except if if such their scope is aa unit which is is binary-connected binary-connected to to aa unit unit u, such u,which their scope is unit u1 that u1 that u, isis equisignificant equisignificant to to u2. u,. The term 'binary-connected' ‘binary-connected’ refers to connection connection by means means of of of the exception a binary binary operation. operation. The The significance significance of exception made made in definition will be be explained explained presently. presently. Note Note that that the definition 55 will the term term 'scope ‘scope of line.'1 of a negation' negation’ is meant meant to to include include the negation negation line. Definition 6 . If uu is is aasynthetic synthetic unit, unit,then thenan anelementary elementary D e f i n i t i o n 6. propositional term, or or an elementary in u is propositional term, elementary function, function, occurring occurring in is redundant can be be written written without without this this term, or function, redundant ifif uu can function, and without replacing replacingitit by by some term or function without some term function not already already used used in in u. u. (V-term). (V-term). If than once If the the elementary elementary term, term, or orfunction, function, occurs occurs more more than onoe in in u, u, the phrase 'without the ‘without this term' term’ is is to tomean mean that thatall alloccurrences occurrences of of the term addition about about replacing the term term by by the term are a m eliminated. eliminated. The The addition replacing the another one another one is is necessary necessary because because variables variables can can of of course course be be given given c)' the the term term 'c' redundant, different names. names. For For instance, instance, in different in 'a. ‘ a .(a (avv c)’ ‘c’ is is redundant, whereas ‘a’ 'a' is not, although although 'a' ‘a’could could be be eliminated eliminated by by replacing replacing whereas it by 'b', which latter term, however, does not occur in the it by ‘b’, which latter term, however, does not occur in theoriginal original statement. statement. Definition 7 . A synthetic synthetic unit unit u,u,isiscontractible contractible ifif canceling canceling D e f i n i t i o n 7. binary-connectedunits unitswithin withinu1 u, together together with with the the sign sign of of their their binary-connected connecting operations operations leads leads to to a unit u2which which isisequisignificant equisignificant connecting unit u2 with If and and only with u1. %. If only if if adding adding negation negation lines lines on on units units inside inside %makes it possible to cancel other units, units, it it is possible to cancel other is admitted admitted and and required required for for the process areredundant. redundant. process of of contraction. contraction. The The canceled canceled units units are ((V-terms). V-terms). This definition of contraction, contraction, which which applies appliesonly only to to binary This dehition of binary operations, may be illustrated in application to the statement operations, may be illustrated application the statement (1)
Here u, u1 isis the the whole wholeformula. formula.IfIfthe the term term ‘a.F, 'a. which which is Here is binarybinaryconnected by the preceding implication sign, is canceled together connected by the preceding implication sign, is canceled together with sign and and aa negation negation line line is is added added on on the the term term with the the implication implication sign 1
ESL, ESL, p. 25. 25.
TEEMS FUNDAMENTBL TERMS
21 21
the resulting 'c.d', ‘c.d’, the resulting form form (2) (2)
(a 3 b ) 3 c . d
represents the represents the the unit unitu2 u,which which is is equivalent equivalent to to u1. %. The The form form (2) (2) is is the reduced form of of (1). Another Another example example is is given given by by the the contraction reduced form contraction into 'aD of of '(aD ‘(a3b)b—_ ) = b' b’ into ‘63 b'; b ’ ; or or in in the thecontraction contraction of of 'a.(b ‘a.(bv 6)’ 'a'. Note that is not not a unit before into ‘a’. that in inthe thefirst firsttwo twoexamples examples u2 u,is before into the canceling, whereasitit is is so in in the the last lastexample. example. canceling, whereas The term inapplicable to to The term 'binary-connected' ‘binary-connected’makes makes definition definition 77 inapplicable expressions like‘a’, 'a', within within which whichthe the unit unit ‘a’ 'a' isis not not binary binary conconexpressions like nected. The by canceling nected. The reduction reduction of of such such expressions expressions is is achieved achieved by canceling
the double negation negation lines, lines, which whichare arenot not units, units, but but are are redundant redundant the according to to definition according definition 5. 5. It It isispossible possible to to set setup upeven evenstronger strongerrequirements requirements for for concontraction; traction ;for for instance, instance, the theintroduction introductionof ofparentheses parentheses may may enable enable us to to cancel cancel aa unit, unit, m as in in the the transition transition from from ‘a. 'a.bb v a.Ô' us a. C’ ttoo 'a.(b ‘a.( b vye)'. c)’. But definition But definition 77 appears appears sufficient sufficient for for our our purposes. purposes. Whereas definitions6-7 6—7refer referonly onlytoto synthetic synthetic units, units, the Whereas definitions following definition gives rules of contraction for analytic units. following definition gives rules of contraction for A separate separate treatment treatmentofofthese thesetwo twocases casesisisunavoidable, unavoidable,because because all all analytic units units are are equivalent to one another analytic equivalent to another and therefore therefore the condition of of equisignificance equisignificance does doesnot not supply supply aa sufficient sufficient restriction restriction condition for the reduction fact, if the reduction process. process. IIn n fact, the word word 'reduction' ‘reduction’ is is not not carefully modified,every everyanalytic analyticstatement statementcan canbebe ‘reduced’ 'reduced' to carefully modified, some such such simple form as as 'a some simple form ‘av a'. 6’.Although the operation operation of of concontraction, introduced 7, can can be be taken taken over for analytic analytic in definition definition 7, over for traction, introduced in units, itit will be modified soas as to to apply apply merely merely to to the will therefore therefore be modified so the major unit,asasfollows: follows: major operation operation of aa unit, Definition An analytic 8. An analytic unit unit u1 u, whose whose major major operation operation is is D e f i n i t i o n 8. binary is cancelingone one major major term, term, possibly after after contractible, ifif canceling is contractible, canceling or adding adding aa negation negation line line on on the the other other major major term, term, leads leads canceling or The canceled to an analytic canceled term term is is redundant. redundant. (V-terms). (V-terms). to analytic unit ug.The The following definitionapplies appliesboth bothto to synthetic synthetic and and analytic The following definition analytic statements. statements. Definition D e f i n i t i o n 9. 9. AAstatement, statement, or or aa unit unit in in aastatement, statement, is is reducedififititcontains containsno nocontractible contractible units units and and no no redundant redundant elemenreduced elementary terms, functions, or negation negation lines. lines. ((V-term). V-term). tary functions, operators, operators, or
22 22
FUNDAMENTAL TERMS
Examples for for synthetic statements: Examples statements : non-reduced form non-reduced form
reduced reduced form form
(3)
(aDb).(aDb)
a
(4)
{aD(cJb)].[aD(öDb)]
aDc
(5)
(6)
(7)
.
g(x) v
.
D a].
g(y) D
a] (x)[f(x) D g(x)]
D g(x). [h(y) D h(z)]}
The will here here be be understandable. understandable. On On the the The application application of of definition definition 8 will left-hand side is the the whole statement, and left-hand side of of (3), (3)) u1 is whole statement, and u2 u2results results by by adding everything adding aa negation negation line line on on 'a'. ‘a’.On On the theleft-hand left-handside sideof of (4), (4), everything is canceled followingthe thefirst first occurrence occurrenceofofthe theletter letter ‘c); 'c'; then then the canceled following the negation line on on top top of negation line of this this letter letter isiscanceled. canceled. This This statement statement can can also be be reduced by the help also reduced by help of of definition definition 6, 6, because because it contains contains term ‘b’. 'b'. In the redundant redundant elementary elementary term I n (5) (5)the the redundance redundance of an elementary function functionisis visible visibleonly onlyafter after the the statement statement is transelementary formed into aa one-scope one-scope form (also (also called prenex form). This and and the examples (6)—(7) show that the proof of the equivalence of the the examples (6)-(7) show that the proof of the equivalence of the reduced form form to to the original reduced original form form may involve tautological transtransformations concerning concerning operators. operators. In In (6), for instance, instance, the the operators formations (6)) for operators are moved to their functionals, and the statement moved close close to functionals, and statement then then assumes the form assumes the form (3). (3). Double negation lines lines are in Double negation in general general redundant, redundant, according according to to definition 5, and and thus aa reduced definition 5, reduced expression expression carries carries in in general general no no double negation lines. lines. An An exception exception isis given given by by the tautology double negation tautology
‘z .
23 23
FUNDAMENTAL TERMS
conjunction ofof aa synthetic synthetic statement statement and and aa tautology is A conjunction is not reduced, because the the tautology reduced, because tautology can can be be canceled canceled according according to to definition 7. A A conjunction of tautologies is not reduced, definition 7. conjunction of tautologies is reduced, either, either,
because definition 8 applies. because definition applies. This This definition definition also also rules rules out out forms forms like 'a.ã 3 b' and 'a 3 by which represent unreasonable like ‘a.6 3 b’ ‘a 3 b v 6’, which represent unreasonable imimplications. Note that that these forms ruled out if they occur plications. Note forms are also also ruled occur as units units within within aamore morecomprehensive comprehensiveexpression expression which which is anaanalytic, since the term 'major lytic, since the ‘majoroperation' operation’ in in definition definition 88 refers refers only only to the to the unit unitconsidered. considered. However, However, the the contraction contraction defined defined in defidefi-
nition 8 differs from the the contraction contraction defined inasmuch nition differs from defined in in definition definition 7 inasmuch as definition requiresthat that the the unit u2 be identical with aa major as definition 88 requires u2 be identical with term, or Furthermore, because or the the negation negation of aa major major term, term, of of u1. ul.Furthermore, because term, can be be applied aa tautology tautology includes includessynthetic syntheticunits, units,definitions definitions6—7 6-7 can applied to its its inner inner structure, structure, and and ititcan cancontain containcontractible contractible synthetic synthetic units or, within within aa synthetic synthetic unit, unit, redundant redundant elementary elementary terms or or functions. Finally, Finally, aa tautology tautology can can have have redundant redundant operators. functions. operators. The The reduction process process for for tautologies tautologiesrequires requiresthat that all all these these redundant redundant reduction parts be be eliminated. eliminated. Examples for Examples for tautologies: tautologies :
non-reduced form (8)
(10)
a.b3a
a.b.(avc)Da v f(x)]
(9) V
reduced form (x)[f(x) v f(x)}
3 a]. [1(y) 3 a]
Note that that the Note the form form on on the the right right in in (8) (8)is is not contractible, contractible, although 'b' can ‘b’ can be becanceled canceled while while the thestatement statementremains remainstautologically tautologically equivalent. only to synthetic equivalent. But this kind kind of of contraction contraction applies only units, according according to to definition definition 7, 7, while while definition definition 8 cannot be be used used here. For For the treatment here. treatment of of (10) (10) we we refer refer to (6). (6). Since in in the the contraction Since contraction process, process, according according to to definitions definitions 7 and and 8, 8, the number number of of binary binary operations operations becomes becomes smaller, smaller, a repeated repeated application of the process cannot lead lead back back to the application of process cannot the original original unit unit and and must must come come to to an anend. end.Likewise, Likewise,canceling canceling of of double double negation negation lines cannot cannot lead lines lead back back to to the theoriginal originalformula. formula. These These results results guarantee that for guarantee for every every statement statement there there exists exists aareduced reduced form. form. Furthermore, itit is easily seen if aa synthetic synthetic statement statement conoonFurthermore, seen that, if
24
FUNDllMENTAL TERMS
tains an analytic, or a contradictory, unit, this unit unit can can always always be eliminated by by the use eliminated use of of the the operations operations described described in in definition definition 7. 7. Since double double negation negation lines, lines, as as was was mentioned, Since mentioned, are exempt from from the cancelation processonly onlyififthey theystand stand over over aa major major term term of of aa cancelation process tautological unit, it follows that aa reduced tautological unit, follows that reduced synthetic synthetic statement statement cannot contain double negation negation lines; lines; such such lines lines can can remain remain only cannot within tautologies. Moreover, no no reduced within tautologies. Moreover, reduced statement, statement, whether whether or synthetic, can analytic or can contain contain aa contractible contractible unit unit (definition (definition 77 or 8), S), because because the the reduction reduction process process would would eliminate eliminate such units. We thus thus arrive We arrive at the the theorem: theorem: Theorem For every there exists T h e o r e m 1. 1 . For every statement, statement, there exists aatt least least one one reduced form. form. In In a reduced every unit unit is reduced reduced synthetic synthetic statement, statement, every is reduced; applies likewise likewise to reduced reduced analytic analytic statements statements reduced; this applies except except for possible possible double negation negation lines on certain certain units. units. An of aa reduced An example example of reduced analytic analytic statement statement containing containing aa nonnonreduced synthetic unit is given by the statement a', where the statement ‘aD 3 a’, where the synthetic unit 'a', taken is not not reduced. synthetic unit taken separately, separately, is reduced. In synthetic synthetic statements, statements, such nonreduced nonreduced units cannot cannot occur. occur.
‘z’,
NORMAL FORMS C. NORMAL GROUPC. FORMS
Assume aa synthetic written in aa reduced reduced and and Assume synthetic statement statement pp is is written closed one-scope one-scopeform formwith with aa minimum minimum of of argument argument variables. variables. The closed The argument all assumed assumed to to be be bound; bound;i.e., i.e.,special special argument variables variables are are thus thus all constant values of these variables do not occur. Each elementary constant values of these variables do not occur. Each elementary functional regarded as propositionalvariable variable'e1', ‘el’, 'e2', ‘e2’, ..., functional may may be be regarded as aapropositional in such such aa way way that thatdifferent differentpropositional propositional variables variables are are used used even even for different functionals of of the the same function ifif they they differ in the different functionals same function differ in the argument variables, ‘f(x)’ and and '/(y)'. ‘f(y)’. We now now cancel cancel the variables, such as 'f(x)' operators; is called operators; the resulting resulting formula formula is called the matrix rnutrix of of p. If aa statement statementpp ininthe thelower lowerfunctional functional calculus calculus is is given given in in any any If form, to write form, it it is is always always possible possible to write ititin in aaone-scope one-scope form, form, according according to aafamiliar familiar theorem. theorem. Although Although the the rule rule of of using using aaminimum minimum number of argument argument variables variables restricts restricts the the possible possible forms forms of of number matrices thus resulting for statement, there there may may exist exist several several matrices thus resulting for a statement, matrices for itit which matrices for which are arenot notpropositionally propositionally equivalent. equivalent. For For instance, instance, when when p is is given given as as aa conjunction conjunction of of two two statements statements each each of has two two major these two two statements statements can of which which has major all-operators, all-operators, these can be be
FUNDAMENTAL FUNDAM3ZSl”T TERMS TERMS
25 25
into aa one-scope form by by identifying identifying the the variables merged into merged one-scope form variables governed governed by all-operators; but this identification identification can be done done in two two ways, ways, and thus different and different matrices matrices may may result. result. By making By making derivations derivations in in the the calculus calculus of of propositions propositions from from the the matrix matrix of p, p, ifif this this statement statement isisgiven given in inaacertain certainreduced reducedoneonescope form, we arrive propropositionally po8itionally scope form, arrive at atstatements statementsq qwhich whichareare derivablefrom fromp.p. For For instance, instance, the the statement derivable (11) (11)
(x)(y)[/(x) (x)(y)[f(dD 3 g(x, g(x, y)] Y)I
is propositionafly is propositionally derivable derivable from from h(x)) J f(y).g(x, (x)(y)[f(x) v vW (x)(y)[f(x) 3 f(y).g(x,y)] Y)1 whereas whereas the formula formula (12) (12)
(13) (13)
((x)/(x) x ) f(4
though derivable from (12), (12), is not not propositionally propositionally derivable derivable from from it. We from (12). (12). IfIf q is will say say that that(13) (13)isisoperator-derivable operator-derivable from is We will propositionally derivablefrom from p, p, the the implication implication from from the the matrix propositionally derivable of to the of p to the matrix matrixof of qq is is aa tautology tautologyin inthe thecalculus calculusof ofpropositions. propositions. Since (13) is derivable from (12), we can add it conjunctively to Since (13) is derivable from (12), we can add it conjunctively to (12); and and we we can can also also add add it it in in aa form form in in which which the the variable variable 'x' ‘x’isis replaced by by 'y'. the two two scopes, we arrive arrive at at the replaced ‘y’.Merging Merging the scopes, we the form form (14)
(x)(y){[f(x) v h(x) D f(y).g(x, y)].f(x).f(y)}
This form form is is tautologically tautologically equivalent equivalent with with (12). (12). But But the the operand This operand of (14) is is not tautologically with the operand of tautologically equivalent equivalent with operand of of (12). (12). Furthermore, whereas is reduced, is not not reduced, Furthermore, whereas (12) (12) is reduced, (14) is reduced, because because we can cancel the the factors factors '/(x)' the state‘f(x)’ and and '/(y)' ‘f(y)’while while keeping keeping the statement tautologically ment tautologically equivalent. equivalent. It It isis possible possible to make make further further derivations derivations in terms terms of of operator operator rules and thus to to construct construct further furtheroperator-derivable operator-derivable statements. We can can interchange interchangethe the variables variables‘x’ 'x' and and ‘y’ 'y' in (12) (12) and add the result the variables 'x' and result to to (14); (14); and and we we can can make make the variables ‘2’ and 'y' ‘y’ identical. identical. Thus (14) the form: (14) assumes msumes the form: (x)(y){{f(x) v h(x) D /(y) .g(x, y)J, [f(y) v h(y) 3 f(x).g(y, x)] {/(x) v h(x) 3 /(x).g(x, x)]. {/(y) v h(y) 3 /(y).g(y, y)]}
26 26
FUNDAMENTAL FUNDAMENT& TERMS TERMS
Here we have omitted omitted the the factors factors'1(x)' ‘f(x)’and and '/(y)' ‘f(y)’ofof(14), (14),because because they are from the operand are now now propositionally propositionally derivable from operand of of (15). (15). The contains several several elementary elementary terms terms not not contained The form form (15) (15) contains contained in in 'h(y)', etc. etc. Note (12), such such as (12), as 'g(y, ‘g(y, x)', x)’,‘h(y)’, Note that (15) (15) is is not not reduced, reduced, because it is equivalent because it equivalent to the the shorter shorter form form (12). (12). When we we restrict restrict derivations derivations by by the the condition that that the the operator operator set of (12) (12) must not be be changed changed although although some some of of the operators operators may become redundant, that that the may become redundant, the number number of of argument argument variables variables
must not be and that no must be increased, increased, and no elementary elementary terms terms must must be be introduced other than resulting from the functions contained in introduced other resulting from the functions contained in (12) by inserting no (12) by inserting the the argument argument variables variablesinindifferent differentways ways1,l, no further addition addition to to (15) (15) can can be be made, made, except except for for terms terms which which are are propositionally derivable from from the operand propositionally derivable operand of of (15). (15).Regarding Regarding the functionals ... ‘en’, functionals of of (15) (15) as as separate separate elementary elementary terms terms 'e1' ‘el’ ... we shall call the operand of (15) the complete matrix of we shall the operand of (15) the complete matrix of (12). (12).
From From itit all all the theexpressions expressions subject subject to to the the restrictions restrictions mentioned mentioned are are propositionally derivable derivable;;and and with the propositionally with respect respect to to these these expressions, expressions, the complete matrix matrix is is the the full full propositional propositional equivalent equivalent of of the the original complete original statement. statement. For the the definition definition of the the complete complete matrix matrix we shall shall add add the the requirement that that ititbe requirement beconstructed constructedfrom fromaaone-scope one-scopeform form which which has aa minimum minimum number of of variables. variables. This This requirement requirement is is satisfied satisfied has by (12). by (12). It It was was mentioned mentioned above above that that the theminimum minimum requirement requirement does not not uniquely uniquely determine determine the the matrix. matrix. Now it it is easily seen seen that that does for minimum forms which whichhave havethe the same same operator operator set, set, or or can be for minimum forms made to have made have the the same same set set by bysuitable suitable naming naming of of variables variables and and arranging the the order commutative operators, operators, the the complete matrix arranging order of of commutative complete matrix is the the same. is that that these is same. The The reason reason is these forms forms lead lead to to identical identical classes classes of derivable statements possessing possessingthe thesame sameoperator operatorset set as as the of derivable statements original form, form, or or aa subset subset of of it, it. Therefore, Therefore, ifif the the operator operator set set of the original of the minimum one-scope one-scopeform formisis given, given, the the complete matrix is decomplete matrix deminimum termined. termined. The question forms of of the question arises: arises: if if two two minimum minimum one-scope one-scope forms same are given, given, must must they they possess possess the the same same same synthetic synthetic statement statement are
'* Note Note that that this thiscondition condition also also excludes excludes the substitution substitution of of special special the variables. variables. constant values for the
TERMS FUNDAMENTAL TERNS
27
operator set? The The Iatter latter term term is to mean mean that that the operator set? is to the operator operator sets sets can can be made be made identical identical by suitable suitable naming naming of variables variables and interinterchangingofof commutative commutativeoperators. operators.I Ido donot not know know of of a proof changing proof answering this question question in in the the affirmative, affirmative,and and II do not not know any answering this know any instance to the the contrary. instance to contrary. II think think this this is is aaproblem problem that that should should attract the the attention attention of of the thelogicians. logicians. Until it it isis solved, solved, IIshall shall proceed on the assumption proceed on assumption that there there are aredifferent differentminimum minimum operator operator sets, sets, and and thus thusalso alsodifferent differentcomplete complete matrices, matrices, for for aa given given synthetic synthetic statement. statement. If If ititshould shouldbe be possible possible to to prove prove the the contrary, contrary, of the present but the the application application of present theory theory will will be be simplified, simplified, but theory will not be invalidated. will be invalidated. Since there is is only only aa finite number of possible operator operator sets sets for for aa Since there finite number of possible minimum one-scope one-scope form, form, the the class minimum class of of complete complete matrices matrices is is finite finite and constitutes constitutes a characteristic characteristic for for aa given given synthetic synthetic statement. statement. It may matrices; and It may sometimes sometimes be difficult difficult to find the complete complete matrices; we have have no no general general method method to to prove prove that that aa certain we certain form form represents represents such aa matrix. such matrix. This This question question isisclosely closely connected connected with with the the decision decision problem, for for which which even even in the problem, the simple simple calculus calculus of of functions functions no no general solution solutionexists. exists.But But for for statements statements of of a none general none too too comcomplicated form the class of complete complete matrices matrices can can be be found; found; and and it it is is plicated form the class of often possible possibleto to prove prove that that there is only matrix, as as often only one complete matrix, for We only only have have to to go through the for the the example example (12). (12). We go through the various various other other possible forms formsofofoperator operator sets sets in in two two variables variables and and to to show that that possible such operator such operator sets sets do do not not furnish furnishequivalent equivalentone-scope one-scope forms. forms. The The present theory is restricted present theory restricted to formulae formulae for which which the class class of complete matrices matrices can can be be determined; complete determined; this this is is aapermissible permissible restricrestriction, of nature nature and tion, because because laws laws of and 'reasonable' ‘reasonable’operations operations will will always always be represented represented in in rather rathersimple simpleformulae. formulae. Using matrix of of aa statement, we Using aa supplemented supplemented matrix we can expand it it disjunctively in in elementary T-cases (ESL, (ESL, p. p. 52, p. 361), disjunctively elementary T-cases 361), or or conconjunotively junctively in negated negated F-eases. F-cases. We We now now define: define: Definition By a D-/orm D e f i n i t i o n 10. 1 0 . By D-form of of aa synthetic synthetic statement statement we we understand the elementary terms of understand the the formula formula resulting resulting when when the elementary terms of aa completematrix matrix ofof the the statement complete statement are written written as as aadisjunctive disjunctive expansion T-cases and and the the original original operator operator set set is is put put expansion in in elementary elementary T-cases before the expansion. ofaasynthetic synthetic statement statement we before the expansion. By By aa, C-form of we underunderstand the formula when the the complete matrix of of the the stand formula resulting resulting when complete matrix
28 28
FUNDAMENTAL TERMS
statement is in negated is conjunctively conjunctively expanded expanded in negated elementary elementary FPcases. cases. (V-terms). (V-terms).
Although 'D-form' and and 'C-form' are V-terms, to aa Although ‘D-form’ ‘C-form’ are V-terms, referring referring to mode of writing writing aa statement, statement, itit is an mode of an I-property I-property of of aa synthetic synthetic statement statement to have have aacertain certain class class of of D-forms, D-forms, or or C-forms. C-forms. If the synthetic statement synthetic statement contains contains no no variables, variables, or or only only propositional propositional variables, the the forms variables, forms are also defined; defined; they then contain contain no no operoper-
ators. ators. If the the elementary elementary terms of the complete complete matrix are abbreviated combinations ofof these these terms terms can be all 2" by ‘el’ ... ... 'en', ‘e,,’, all 2” possible possible combinations be by 'e1' written as asfollows: follows: (16) (17) (17)
(( ))
... (( )[(el. ...
(( )) ... ... (( )[(elv
...v(ë1 ejv(e1 ... .en) v (el. ... . ë,,)v En) v ... v (El. ... . E n ) ] ...ve,,) (ë1v...vë,,)] ... vYe,,). en) . (e,v ... v En) . ... . (El v ... v en)] e,,)]
...
The The D-form D-formresults results from from(16) (16) by byomitting omittingcertain certain terms termsof of the the disdisjunction. junction. of of
omitting certain The results from The C-form G-form results from (17) (17) by by omitting certain factors factors the conjunction. The variables variables ‘x’, 'x', ‘y’, 'y', etc., are contained conjunction. The contained in the
The D-form results from terms ... 'e,,'. ‘el’ ... ‘en’. The D-form results from the terms abbreviated abbreviated by by 'e1' C-form by by 'multiplying out', and vice C-form ‘multiplying out’, vice versa. versa. Furthermore, Furthermore, the D-form can be be constructed we take take the the D-form can constructed from from the the C-form C-form as as follows: follows: we conjunction of the the omitted omitted factors factors of of (17) and negate negate it; it; breaking conjunction of (17) and breaking this long line until until we we arrive arrive aatt the this long negation negation line the shortest shortest negation negation lines, we we find find the the D-form. This follows follows because becausethe the omitted omitted factors factors lines, D-form. This of of (17) (17) are the negated negated T-cases T-cases of the statement. statement. If we forms to to aa we restrict restrict disjunctive disjunctive and and conjunctive conjunctive normal normal forms given set of of elementary elementary terms terms and and add the condition given set condition that that no no mere mere duplicationsofofterms terms or or factors factors are are admitted, admitted, aa D-form is the duplications D-form is longest version version of of aa disjunctive normal form, form, and longest disjunctive normal and aa C-form C-form is the the longest version version of of a conjunctive longest conjunctive normal normal form. form. Shorter Shorter versions versions result by merging terms, or or factors; factors; as for instance result merging terms, instance by using using the equivalences equivalences
= e1.e2
(18)
e,.e2.e3v el.e,.E3
(19) (19)
((e1ve2ve3).(e1ve2vë3) e,ve,v%)*(e,ve,vQ?
e1ve2 e,ve2
Definition D e f i n i t i o n 11. 1 1 . If aatautology tautology isisgiven givenininone-scope one-scope form, form,
we we construct construct its D-form D-form by expanding expanding its matrix matrix in in elementary elementary
rUNDAM:ENrAL TERMS FUNDAMENTAL
29 29
T-cases, using using those those elementary elementary terms terms which which occur occur in in the given T-cases, given statement. (V-term). (V-term). The D-form D-form of of aa tautology The tautology is is the the complete complete expansion expansion (16). (16). Note Note that aa tautology tautology has has no no C-form, C-form, because because it has no no F-cases. F-cases. The The complete expansion (17) (17) is is a contradiction. contradiction. complete expansion we add certain terms to Definition D e f i n i t i o n 1 22.. When When we certain elementary elementary terms those contained in the the complete matrix matrix and and then then construct expanthose contained in sions sions in elementary elementary terms equivalent equivalent to to D-form D-form or or C-form, C-form, we we obtain an obtain anelongated elongated D-/orm, D-form,or orelongated elongated C-form. C-form. (V-term). (V-term). For instance, 'a = = b’ b' is given, instance, ifif ‘a given, its D-form D-form is aa.bvã.ii .bvG.&
(20) (20)
An elongated D-form An D-form is is (21) (21)
GROUP D. GROUP D.
a.b.cva.b.FvG.6.cvii.b.C ALL-STATEMENTS ALL-STATEMENTS
Definition D e f i n i t i o n 11 33.. A synthetic synthetic statement statement pp isisan anall-statement all-statement if and only only if if the the operator operator set setof ofits itsD-form D-form(or (orC-form) C-form) contains contains at least least one one all-operator. all-operator. (I-term). (I-term). Definition is written asasanan all-statement D e f i n i t i o n14. 1 4A . statement A statement is written all-statement if if and only if itit isis written written with with aanon-redundant non-redundantall-operator all-operator whose whose scope is is the whole scope whole statement. statement. (V-term). (7-term). Note that that definition definition 14 14 can can be beapplied appliedtototautologies, tautologies,wherewhereas definition definition 13 1 3 cannot be be so so applied. applied. GROUP E. RESIDUALS GROUP E. RESIDUALS
Residuals are are definable definablefor for all all forms forms of of statements. statements. They They are Residuals used to characterize certain properties properties of of the statement, in particparticular, which term term applies ular, exhaustiveness, exhaustiveness, which applies when when all all the thepossibilities possibilities opened up by by aa statement statement are the physopened up me exhausted exhausted by by the the objects objects of of the physical world, world. A A non-exhaustive non-exhaustive statement statement includes includes empty empty parts. parts. This property for all-statements, but is property is of of special special importance importance for all-statements, but is also also defined for other statements. Furthermore, residuals will be used defined for Furthermore, residuals used to define define a kind kind of of generality generality that that refers refers to to all-statements all-statements only. only. Definition residual in in elementary A disjunctive residual elementaryterms terms of of aa D e f i n i t i o n15.1 5A. disjunctive statement statement pp is is any any statement statementresulting resulting when when pp is is written written in in D-form D-form
30 30
PUNDAMJLN%"T TERMS
and one one or or several several terms of the operand operand are are canceled. canceled. (I-term (I-term if if pp synthetic, V-term V-term if pp isisanalytic.) analytic.) is synthetic,
Definition statement D e f i n i t i o n 16. 1 6 .A A statementp pwhich whichisisverifiably verifiably true true is is
exhaustive in elementary etementarl,terms termsififnone noneof of its its disjunctive exhaustive in disjunctive residuals residuals
in elementary terms, for for any of elementary terms, of its its D-forms, D-forms, is is verifiably verifiably true. synthetic, V-term V-term if if pp isisanalytic.) analytic.) (I-term if pp isis synthetic, For For instance, instance, the statement statement (22a) (22a)
((x){f(x).g(x) d C f ( x ) . 9 ( 4= 3 h(x)J W4l
has the following following D-form: D-form : has the
---
----
.g(x) (x) .g(x) (22b) (22b) (x)[/(x).y(x) ( M ( 4* g ( 4.h(x)v/(x).g(x) W . ) v f ( 4 g. ( 4.h(x)vt(x) *W)vf(4 . 9 ( .h(x)v/ 4.W)vf(x). g ( 4 -.h(x)] 441
The statement (22a) (22a) is not exhaustive exhaustive in in elementary elementary terms terms if if cancanThe
celing any one, one, or or any any two, two, or or any any three, three, of of the the four terms in in the celing any four terms the brackets of (22b) leads to a statement which is verifiably true. brackets of leads to a statement which is verifiably If aa tautology tautology contains contains oniy only variables, variables, it it is is always always exhaustive exhaustive in elementary terms. elementary terms. A disjunctive disjunctive residual residualin in major major terms terms of of pp is Definition 17. Definition 17. A any statement statement resulting resulting when when pp is is expanded expanded in aa disjunction disjunction of of major T-cases (ESL, p. 52, p. 362) and one or several terms of the major T-cases (ESL, p. p. 362) and one or several terms of the operand are operand are canceled. canceled. (V-term). (V-term). A statement 8. A D e f i n i t i o n 118. statement p which which is is verifiably verifiably true is is Definition
exhaustiveiin major terms termsififnone noneofofits itsdisjunctive disjunctive residuals residuals in in major major exhaustive n major terms is veriflably verifiably true. true. (V-term). (V-term). terms is Note that aaconjunction conjunction as as well well as as aastatement statement whose whose major major Note operation is a negation, operation is negation, if they they are areveriflably verifiably true, true, are are always always exhaustive in in major exhaustive major terms, terms, because because they they have have no no disjunctive disjunctive residuals in in major terms. residuals terms. D e f i n i t i o n 19. 19.A A statementPifiwhich whichisisveriflably verifiably true Definition statement true is exhaustive except /or p (in major or elementary terms), if all its exhaustiwe except for p (in major or elementary terms), if all veriflably true true disjunctive disjunctive residuals residuals in in major major or or elementary terms, verifiably elementary terms, if are any, any, are are derivable derivable from from p. (I-term (I-termfor for elementary elementary terms, terms, if there there are V-term major terms.) terms.) V-term for for major The use of this this definition be seen seen in in the the discussion discussion of of definition definition The use of definition will will be 35. Note Note that that if 36. if Pi pl is is also also exhaustive is exhaustive, exhaustive, it it is exhaustive except except for for pp,, any p. p. It Pi is for any It can can happen happen that that aamajor majordisjunctive disjunctive residual residual of of fi derivable pa while while fi is derivable from Pi is exhaustive exhaustive in in elementary elementary terms; terms; then then
R'UNDAmNTA% TERMS TERMS
31 31
p1 is is exhaustive exhaustive in in major major terms termsexcept except for for itself itself (see (see the the discussion discussion (43)). of (43)). of Definition residual ofofaasynthetic A conjunctive residuaE syntheticstatement statement D e f i n i t i o 20. n 2 0A. conjunctive p is is any any statement statementresulting resulting when when pp isiswritten writtenininC-form C-formand and one one
or several several factors factors of of the the operand operand are arecanceled. canceled. (I-term). (I-term). Definition A disjunctive, or or conjunctive, A disjunctive, conjunctive,extension extension of of aa D e f i n i t i o n21. 2 1. statement statement results results from from its its D-form, D-form, or or its its C-form, C-form, by by adding adding terms, terms, or factors, factors, from the the total totalexpansion expansion (16) (16) or or (17). (17). (V-term). (V-term). Obviously, canceling canceling terms terms in in the D-form is the same the same as as adding adding factors factors to the theC-form, C-form, and andvice-versa. vice-versa. Therefore, Therefore, aa disjunctive disjunctive residual equivalent to aa conjunctive residual is tautologically tautologically equivalent conjunctive extension, extension, and aa conjunctive conjunctive residual residual is is tautologically tautologically equivalent equivalent to aadisdisjunctive extension. Note Note that that an analytic junctive extension. analytic statement has no no conconjunctive residuals residuals because it has has no noC-form. C-form. Using conjunctive extensions, we can can find out whether extensions, we whether aa given given expansion is exhaustive exhaustive in in elementary terms. For instance, expansion is elementary terms. instance, if we we expand can prove prove that that expand the the matrix matrixof of (12) (12)in in elementary elementary T-cases, 5"-cases, we we can it it is is not not exhaustive, exhaustive,because because the theconjunctive conjunctiveextension extension(14) (14)is is true. true.
This shows that it is shows that is important important to to construct construct the theD-form D-form from from the the
complete matrix. matrix. Otherwise many statements statements would not be be exhaustexhaustcomplete Otherwise many would not ive or aa disive in in elementary elementary terms, terms, because because aa conjunctive conjunctive extension, extension, or dis-
junctive junctive residual, residual, would would be derivable derivable from the statement statement itself. itself. The definition of exhaustiveness exhaustiveness in in terms terms of of the the complete matrix The definition of complete matrix excludes this this possibility. excludes possibility. From familiar familiar properties properties of of conjunctive conjunctive normal forms forms we derive the following theorem: theorem: the following Theorem T h e o r e m 2. 2 . If aa synthetic synthetic statement statement qg is is derivable derivable from a synthetic statement a subset of that that synthetic statement p, p ,while while the the operator operator set set of of qq is a subset of of pp,, both referred and the elementary of referred to C-forms, C-forms, and elementary terms of of qq are are contained among among those those of of pp., then then q can can be be written written as contained as a conjunctive conjunctive residual of of pp,, and and pp can be written as as aa disjunctive disjunctive residual residual of of q, q, residual or of or of an elongated elongated D-form D-form of of q. If the the C-form of pp is is tautologically tautologically equivalent equivalent C-form of Definition D e f i n i t i o n 22. 2 2. If to aa conjunction of two of its residuals, each of these residuals conjunction of two of its residuals, each of these residuals is is called self-contained self-containedwithin within p. p. (I-term). called (I-term). With the help With help of of theorem theorem 11 we we derive: derive: If p contains contains aa closed closed unit which which is is derivable derivable Theorem T h e o r e m 3. 3 . If
32 32
rTINDAMENTAL FUNDAMENTAL TERMS TERMS
then this from pp,, then from this unit unit is is tautologically tautologically equivalent equivalentto to aa self-contained self-contained conjunctive residual residual of of p. conjunctive p. Definition p isp general in in self-contained D e f i n i t i o n 23. 2 3A . statement A statement is general self-contained /actor8if, if, for for any of its its C-forms, C-forms, each each of of its itsself-contained self-contained conconfactors junctive residuals, after being reduced, possesses at least one nonresiduals, after being reduced, possesses least one nonredundant all-operator. all-operator. (I-term). (I-term). For instance, instance, the statement statement D g(x)].h(y)}
(23)
is not general in self-contained factors, factors, because because itit can be be written as the conjunction conjunction of of its its two tworesiduals residuals (24) (24)
(x)[f(x) (.)[f(z) D
f7(4l*(3Y)h(Y)
the no all-operator. all-operator. However, the 8econd second of of which which possesses possesses no However, (23) (23) would would be general in self-contained factors if if the second self-contained factors second operator operator were were an an all-operator, or if 'h(y)' all-operator, or ‘h(y)’ were were replaced replaced by by 'h(x, %(x,y)'. y)’.
Combining theorem theorem 22 and definition Combining definition 23, we we derive: derive:
Theorem synthetic statement statement p. p , which which is is general general in T h e o r e m 4. 4 . If aa synthetic
self-contained factors, can can be be written self-contained factors, written in in aa reduced reduced form form containing containing a closed unit unit q that that isis derivable derivable from p, p , then then qq is is an an all-statement. all-statement.
This theorem theorem shows shows that definition definition 23 23 formulates formulates aa property property which ensures generality generality for for closed closed units units within within aa statement. which ensures statement. UNIVERSAL STATEMENTS GRouP GROUPF. F. UNIVERSAL STATEMENTS
Definition individual-term is isa aterm D e f i n i t i o n24.2 4An . An individual-term termwhich whichisisdefined defined with reference to to a certain certain space-time space-time region, or or which can be so so defined without change defined without chmge of meaning. meaning. (I-term). (I-term). The term can The can be be aa proper proper name name or oraadefinite definitedescription. description. The The addition in which terms like like 'terrestrial' addition refers refers to languages languages in which terms ‘terrestrial’ are axe used as primitive terms. If If the thelanguage languageisisrich rich enough enough to topossess possess space-time coordinates,itit can can be be shown shownthat that the meaning space-time coordinates, meaning of of the term can can be be equivalently equivalently defined defined by by reference reference to the the space-time space-time region 'Earth'. Languages region ‘Earth’. Languageswhich which do do not not possess possess the the means means to to formformulate are disregarded in the the theory ulate space-time space-time coordinates coordinates are disregarded in theory of of nomnomological operations, which which are are intended intended to to account ological operations, account for for the the language language of The phrase be defined' of science. science. The phrase coan ‘can be defined’ refers refers to to logical logical possibffity, possibility, and therefore only the definition therefore presupposes presupposes only definition of of tautologies, tautologies, or or
FTJNDAMENTAL FUNDAMENTAL TERMS TERMS
33 33
logical formulae, but but does logical formulae, does not not presuppose presupposesynthetic syntheticnomological nomological statements. statements. Definition statement is is un,iver8at A synthetic statement universal if it it cancanD e f i n i t i o n25. 25.A synthetic
not not be be written written in in aa reduced reduced form form which which contains contains an an individual-term. individual-term. (I-term). (I-term). Definition 5a. AAstatement statement is is written as aa universal universal statestateD e f i n i t i o n 225s. if itit contains ment contains no no individual-term. individual-term. (V-term). (V-term). ment if Note Note that that definition definition 25a 25a can can be be applied applied to to tautologies, tautologies, whereas whereas definition 25 cannot so applied. applied. definition 25 cannot be so Since the the units Since units of of measure measure are areusually usuallydefined definedwith withreference reference to to an an individual individual standard, standard, such such as as the the meter meter standard standardpreserved preserved in in Paris, Paris, the thenumerical numerical values values of of physical physical constants constants are areexcluded excluded by from the the content by definition definition 25 25 from content of of physical physical laws. laws. But But this this conseconsequence appears appears reasonable these constants quence reasonable because because these constants merely merely express express relationships to the the standard. 3.1010 relationships to standard. That That the thespeed speedofoflight lightisis3.1010 centimetersper per second secondisis aa relation light, the meter centimeters relation between between light, meter standard in states in Paris, Paris, and and the the revolving revolving earth. earth. The The physical physical law law states standard merely that the speed merely that speed of light is a constant. constant. ItItisisdifferent different when when standards standards are axe defined defined through through class class terms, terms, for for instance, instance, when when the the unit of by the of the the cadmium cadmium line. line. unit of length length is is defined defined by the wave wave length length of Measured in such standards, Measured in standards, the the numerical numerical values values of of physical physical constants belong belongto to the the content constants content of of the law. law. The The omission omission of of universal universal values values defined defined with reference reference to to individuals not represent represent any individuals does does not any disadvantage disadvantage for for our our conception conception of laws laws of of nature. nature. For as well well as as practical practical applications, of For all all scientific scientific as applications, the numerical value of of aa quantity quantity has has only only an an intermediate intermediate use: use: itit the numerical value serves for for computing relationshipsbetween betweenthis this quantity quantity and serves computing relationships others. For instance, others. instance, the the numerical numerical values values in the the decimal decimal system system may serve to compute the ratio the speed of light light and and the the may serve to compute the ratio between between the speed of speed of sound. sound. We speed of We can can imagine imagine the the system system of of knowledge knowledge as given given by a in the form form of of the totality totality of of laws laws of of nature, nature, supplemented supplemented by table of numerical constants. for practical practical table of numerical constants. This This table table is is indispensable indispensable for applications, applications, but but does does not not belong belong in in the thenomological nomological part part of of knowknowledge. ledge. Furthermore, definition definition 25 25 excludes excludes the the use of Furthermore, of definitions definitions of of class terms by reference to samples. For instance, copper might class reference samples. For instance, copper might be defined as anything that is defined as anything that is like like aa certain certain sample sample in in certain certain
34 34
TERMS FUNDAMENTAL TERMS
respects, respects, say, say, that that has hasthe thesame sameatomic atomicweight weight as asthe thesample. sample. Such Such
definitions are are not not used definitions used in in the theformulation formulation of ofscientific scientific laws. laws. This This is illustrated illustrated by by the thediscovery discovery of of isotopes; isotopes; ifif the thegiven givendefinition definition of aa substance used, there there could not be substance were were used, could not be any any forms forms of of the the subsubstance having a different atomic weight. If interpreted in the sense stance having a different atomic weight. If interpreted in the sense of the usual usual laws of this this and and the the preceding preceding qualification, qualification, the laws of of nature nature cannot cannot be be written written in in aareduced reduced form form which which contains contains an an individualindividual-
term. 1 term. certain ambiguity arises because because aa natural natural language is often A certain ambiguity arises language is often A capable of of different different rational rational reconstructions. reconstruotions. The The term term 'polar capable ‘polar bear', bear’, for can be be interpreted as meaning bear living living in in the the for instance, instance, can interpreted as meaning aa bear polar of the the Earth, Earth, in polar regions regions of in which which interpretation interpretation it it would would be be an an individual-term. individual-term. It It could couldalso also be bedefined defined as as aabiological biological species species of certain general general characteristics, characteristics, for for instance, instance, as as aa bear bear with with aa white certain white skin, etc. In such cases, we have two rational reconstructions which skin, etc. I n such cases, we have two rational reconstructions which are equivalent, though though perhaps are not not logically logically equivalent, perhaps practically practically equivalent. equivalent, As aa consequence, statement which rational reconstruction consequence, aa statement which in in one one rational reconstruction is nomological, may not not be be so in is nomological, may in another another reconstruction reconstruction of of conconversational language. versational language.This Thisambiguity, ambiguity,however, however,offers offersno no difficulties, difficulties, since the the class statements isis defined defined only only for for a since class of of nomological nomological statements certain reconstruction language. If If aa statement statement of certain reconstruction of of language. of conversational conversational language is is given, given,itit would would be be meaningless meaninglesstotoask: ask:isisthis this statement statement language really There is is no no such such thing thing as really nomological? nomological? There as an an absolute absolute meaning meaning of of the the terms terms of of aanatural naturallanguage. language.AAclassification classification of of statements statements such such as as expressed expressed in in categories categories like like 'analytic' ‘analytic’oror'nomological' ‘nornological’ refers to to aa given Whether this this refers given rational rational reconstruction reconstruction of of language. language. Whether reconstruction is adequate, reconstruction is adequate, is to to be beinvestigated investigated separately. separately. when we we say say that that aa statement IInn other other words, words, when statement like like 'copper ‘copper has has an is aa law an atomic atomic weight weight of of 63.5' 63.6’ is law of of nature, nature, this this should should be be underunderstood meaning, ‘there 'there exists stood as as meaning, exists an an adequate adequate rational rational reconstruction reconstruction of language language in in which whichthis thisstatement statement isis aa law law of ofnature’. nature'. That That there there of may also exist aa rational may also exist rational reconstruction reconstruction in in which which copper copper is is defined defined by reference reference to an individual individual sample sample and and for for which which the the statement statement by to an 1 I think this this answers answers N. Goodman's Goodman’s remark my definition definition of of remark that, ifif my universal ‘virtually all all apparently apparentlyuniversal universal sentences universal statement statement is is used, 'virtually will be be ruled out'. out’. (l.c.p. (1.c.p.101). 101). will
FUNuAMENTAL FUNDAMENT& TERMS TERMS
35
is not aa law law of of nature, nature, isisirrelevant. irrelevant. In Infact, fact,the thesecond secondreconrecon-
struction may may appear appear less less adequate adequate for for the the very very reason reason that that itit does struction does not account for the statement not for the the usage usage of of calling calling the statement considered considered aa law law of of nature. nature. If aalanguage language is is to to be berationally rationally reconstructed, reconstructed, it is is advisable not not merely to look for definitions that coincide advisable merely to look for definitions that coincide extensionextensionally with usage terms, but but also to the ally with usage of terms, also to to adjust adjust definitions dehitions to the usage usage of such such categories categoriesas as ‘Law 'Lawof ofnature’, nature', ‘analytic’, 'analytic', etc. the total total of etc. Only Only the reconstruction, as can be be judged judged as as adequate. adequate. For reconstruction, as aa whole, whole, can For instance, instance, defining 'human being' defining ‘human being’ as as 'featherless ‘featherless biped' biped’ would would correspond correspond extensionallyto to usage; usage; but but it extensionally it would would contradict contradict such such statements statements as 'featherless ‘featherless bipeds are not not necessarily necessarily human beings', beings’, which which we we also find among also among linguistic linguistic usage. usage. It It may may even evenbe bepossible possible to toconstruct construct aadefinition definitionof oforiginal original nomological statements which for the nomological statements which allows allows for the occurrence occurrence of certain certain individual-terms; for such such a reconstruction individual-terms; for reconstruction it it would would be bepossible possible also also to admit, admit, in in such such statements, statements, terms terms defined defined by by reference reference to samples, etc. Interpretations Interpretations of of this this kind kind may be as samples, etc. as adequate adequate as as the one presented here. All the the present investigation is intended to to achieve is one one adequate adequate reconstruction reconstruction of of the the term term 'law of nature’, nature', achieve is ‘law of without claiming that this is the without claiming that the only only one. one. The The addition addition concerning concerning aa reduced reduced form, form, given given in indefinition definition25, 25, is necessary because otherwise otherwisewe wecould couldinsert insertinto into any any statement statement necessary because redundant for instance, instance, by by adding adding a tautology redundant individual-terms, individual-terms, for tautology containing an individual term. The containing an individual term. The phrase phrase 'cannot ‘cannot be be written' written’ excludes hidden individual-terms. For instance, we excludes hidden individual-terms. For we could could define define an individual (ESL, p. p. 257) 257) and and individual by the the use use of of class class terms terms alone alone (ESL, eliminate the iota-operator; the resulting eliminate the iota-operator ; the resulting statement statement would would not directly contain an individual-term. the individual-term. This This is is excluded excluded because because the resulting statement statement could be be transformed transformed so so as as to to contain an iotaiotaoperator. An for an excluded An example for excluded statement is given given by 'for ‘for all x, if x1: is is aa man man that thathas hasseen seenaaliving living human human retina, retina, and and no no other person such a retina before person has seen seen such before x, x, then xx contributed contributed to to the the establishment establishment of of the the principle principle of of the the conservation conservation of energy'. energy’. This is true, true, contains This sentence, sentence, which which is contains in in aa hidden hidden form form aa description description of H. H. v. v. Helmholtz, Helmholtz, and and isistherefore therefore not not universal. universal. A further further condition condition must must now now be be added. added.IfIfan anindividual-term individual-termis is introduced by means introduced by means of aa definite definite description, description, it isispossible possible to to
36
kTJNDAXF,NTAL TERUd6
eliminate eliminate the iota-operator and then to to derive derive aa statement statementwhich which is no no longer longer tautologically tautologically equivalent equivalent to the the original original statement. statement. Since this statement into aa Since this statement is is not not tautologically tautologically transformable transformable into statement containing an iota-operator, it is universal in the sense of statement containing it definition 25;; but itit still definition 25 stillincludes includes aa reference reference to an individual in aa hidden form and does does not not possess possess the the kind kind of of generality generality which which isis to be be required required for for aa law law of of nature. nature.1 Consider the statement, Consider statement, 'all ‘all stars stars which which H. H. v.v.Helmholtz Helmholtz saw saw were aatt least 1-th magnitude were least of of the the111-th magnitude 2'• 2’. This This statement cannot cannot be be regarded as a law of nature, nature, because because it it merely merely expresses the techregarded
nical limitations limitations of telescopes available available at at the the time nical of telescopes time when when Helmholtz Helmholtz lived. IIn lived. n the form form given, given, it is is ruled ruled out out as asnot notbeing beinguniversal, universal, according to definition 25. However, However,replacing replacingthe theterm term ‘H. 'H. v. v. according to definition 25. Helmholtz' by the definite description mentioned, we can now Helmholtz’ the definite description mentioned, can now derive derive the statement, 'all ‘allstars starsseen seen by by any any man man who who saw saw aa living living human retina before were at at least of the before any other man saw one, were 11-th magnitude’. magnitude'. The The latter latter statement does not express 11-th express the fact that there back that there was was such such a, man, man, and and it it thus thus cannot cannot be be transformed transformed back into the form containing an iota-operator. This statement, thereform containing iota-operator. statement, therefore, is is universal fore, universal in in the the sense Bense of of definition definition 25. 25. add a In order order to rule rule out out statements statements of of this kind, we we shall shall add further condition further condition to to the the requirement requirement of of exhaustiveness, exhaustiveness, introduced introduced in definitions 16 and and 18. 18. The The statement statement contains in definitions 16 contains a function function ‘4 saw saw aa living human retina before any other man saw one', which may before other man saw one’, which may This function function is is satisfied satisfied by by one one and and only be abbreviated abbreviated by be by ‘st@)’. This only one argument argument x, x,which existed only during a restricted restricted space-time space-time therefore have have the relation, region rr;; we we therefore region relation, which which is verifiably verifiably true, (24a) (244
(x){sr(x) 3 r(x)] (”x) r(4l
Let us us now now write write the the statement statement under Let under consideration considerationin in the thefollowing following
*' I am indebted to to C. C. Hempel am indebted Hempel and and D. D. Kalish Kalish for for having having drawn drawn my my attentattention to this ion this problem. problem. The term term '11 -th magnitude’ magnitude' contains, The ‘11-th contains, in its its usual usual definition, definition, aa reference couldbe be to the the earth earthand andthus thuswould wouldrepresent representan anindividual-term. individual-term.But Butititcould replaced by by a term expressing light intensity intensity at the expressing light the point point of of observation, observation, replaced using a memure mea8ure not not depending on on the the earth as aa reference point, for instance instance photons per per unit of photons of time. time.
37
rUNDAMENTAL TBRMS FUNDAMENTAL TERMS
form, using ‘st’ 'st' for for ‘star’, 'star', ‘s’ 's' for for (see’, 'see', and and ‘my 'm' for for ‘at 'at least least of of 11-th ll-th form, using magnitude': magnitude’ : (24b) (24b)
(x)(y)[81(y).$(x, y).sr(x) y).Br(x) 3 D m(y)I m(y)] (x)(Y)[w).s(z,
We will will abbreviate abbreviate the the implicans iniplicans by by ‘g(x,y)’ 'g(x,y)' and and expand expand the the stateWe
ment in major however,that that the the major T-cases, T-cases, with the the modification, modification, however, term 'r(x)' is term ‘~(x)’ is added: added: (24c) (240)
-
”- -
g(x, Y)*m(Y) y).m(y) V g(x, (x)(y)[r(x) v v g(x, (x)(y)[r(x) g(x, y).rn(y) Y).,W/) vv 9(x, g(z9 Y)-m(?l)l
this term The term (g(z, y) y)..m(?,)' m(y)’ can can here here be be canceled, canceled, since since this term implies implies to (24a). (b D a)' is ‘'r(x)' ~ ( x )according according ’ (24a). This This follows follows because because '(a ‘(av bb)) ..(b 3 u)’ is equivalent to to 'a'. ‘u’.Note Note that thatthis thisderivation derivationwould wouldalso alsobe bepossible possible equivalent if the operators if only if operators in in (24c) (24c) were were existential existential operators, operators, if only the operator of operator in in (24a) (24a) is an an all-operator. all-operator.Calling Calling (240) (24c) an an r-expansion r-expansion of (24b), we wethus thus find find that that aa disjunctive (24b), disjunctive residual residual of of the r-expansion r-expansion is verifiably true; i.e., i.e., that that the exhaustive. We is verifiably true; the r-expansion r-expansion is not exhaustive. understand here here by a residual, when a understand residual, as above, above, a form form resulting when term stemming term stemming from the the original original statement statementisiscanceled; canceled ;canceling canceling of ‘~(x)’is is not of not regarded regarded as as constituting constituting aaresidual, residual,since since such such cancanceling celing is is always always possible. possible. The fact fact that that statement The statement(24b) (24b)contains contains an an unreasonable unreasonable function, function, namely whose extension extension is is restricted restricted to to aa certain namely ‘m(&)’,whose certain space-time space-time region r,r, thus thus finds its expression expressionininthe the result result that that the the statement region finds its is not not exhaustive is exhaustive in in its its r-expansion. r-expansion. This This criterion, criterion, which which can can likelikewise be be applied applied to to an expression in in D-form, D-form, enables enables us us to to rule rule out wise statements which are are non-equivalently non-equivalently derived from from statements statements containing individual-terms.Note Notethat thatwe wethus thus also also rule containing individual-terms. rule out out functions whose extension contains more than one individual, functions whose extension contains more than individual, if only the extension restricted to to aa certain only extension is veriulably verifiably restricted certain space-time space-time region. We region. We now now define: define: Definition A statement p ispunre8trictedly A statement is unrestrictedlyexhaustive, exhawlive, in D e f i n i t i o n26.26. major major or elementary elementary terms, terms, if there there exists exists no norestricted restricted spacespacetime region region r, r, for any of of its its variables, variables, such that that the ther-expansion r-expansion time of p isis not of not exhaustive. exhaustive. we understand understand an undivided By a restricted restricted space-time space-time region we undivided is not not identical with the part of of the theuniverse, universe, which which however however is identical with
38 38
FUNDAMENTAL TERMS FUNDAMENTAL TERMS
universe. Such Suchaa region regionmight mightbe begiven givenby byaa part part of of the earth's universe. earth’s surface during during aa certain certain time, time, or being surface or by by a galaxy. galaxy. The The condition condition of of being rules out which are unrestrictedly unrestrictedly exhaustive exhaustive rules out all-statements all-statements which are verified by by examining examining all all individuals individuals of of aa certain kind within such verified region. It It amounts to to requiring that the aa region, requiring that the statement statement be be exhaustive exhaustive is excluded from the range of its all-operator. even ifif any even any region region r is from all-operator. Note that aastatement statementwhich which isisunrestrictedly unrestrictedly exhaustive exhaustive is is also also Note exhaustive. exhaustive. The The question question may may be be posed posed whether whether the the condition condition of of being being unrestrictedly exhaust,ive exhaustive can can be be used used to to replace replacethe the condition conditionthat, that unrestrictedly the statement should contain no individual-terms. The answer statement should oontain no individual-terms. The answer is is negative;; both conditions negative conditions are are needed. needed. The The example example concerning concerning stars up to the the 11th 11th magnitude magnitude has has shown shown that that the the condition condition of being unrestrictedly exhaustive cannot cannot be dispensed being unrestrictedly exhaustive dispensed with. The The followingexample example shows showsthat that the the other following other condition condition is is also also indispensindispensable. Assume Assumethat that Peter weighs as much much as as Paul. The statement, able. weighs as statement, 'for all all x, x, if if and and only if x weighs weighs as as much much as as Peter, Peter, x weighs weighs as as much much ‘for only if as Paul', Paul’, is then then true; true; and and ititisisunrestrictedly unrestrictedly exhaustive. exhaustive. But it it cannot be regarded as as aa law of of nature; nature; itit represents represents merely merely a comcomplicated way of of saying plicated way saying that Peter Peter weighs weighs as much much as as Paul. Paul. 1 In In order order to rule rule out outthis thisstatement statementfrom fromnomological nomological statements, statements, we we need need the condition condition that the the use use ofofindividual-terms individual-terms is not not admitted for statements, a condition for original original nomological nomological statements, condition which which this statement statement violates. violates. With With the the exclusion exclusion of of individual-terms, individual-terms, the theory theory developed developed is greatly from aa formal point of is greatly simplified simplified from formal point of view. view, Within Within the the chapter on nomological statements, statements, the the argument chapter on original original nomological argument variables variables 'x', ‘x’,'y', ‘y’, etc., etc.,can canalways always be be assumed assumed to t o be be either either bound bound variables, variables, or with respect respect to to the statement, i.e., or free free variables variables with the whole whole statement, i.e., variables variables for which which any any value value may be substituted. for substituted. Since Since such such variables variables can always always be bound, bound, the the theory theoryofoforiginal originalnomological nornological statements statements can, in principle, be restricted to bound can, bound argument argument variables. variables. The The Note, however, however, that that the transitivity, transitivity,symmetry, symmetry, and and reflexivity reflexivity of of the tlicrelation 'weighs as much much as’ as' must must be regarded, not as analytic, relation ‘weighs as regarded, not analytic, but as as verified by experience, when ‘weighing 'weighingas as much much as’ as' is defined by aa scale verified by experience, when defined by scale being for deriving the being in balance. balance. Since Since these properties are presupposed presupposed for deriving thr, equivalence of the the two statements, equivalence of statements, this this equivalence equivalence is not not analytic. analytic.
FUNDAMENT&
TERMS TERMS
39
use of free variables, use variables, however, however, is is sometimes sometimes advisable advisable for for reasons reasons of of convenience. convenience. Consequently, we we need need not not consider consider in in this this chapter Consequently, chapter derivations derivations consisting in in the the substitution of values for for argument variconsisting of special special values varito’ 'f(x1)', where ables, as as for for instance, the transition transition form form '(x)/(x)' ‘ ( z ) f ( ~ )to ‘/(q)’, where 'xi' refers to to some specified specifiedindividual. individual.ItIt is isfor forthis this reason reasonthat that the the ‘5’ is sufficient concept of of aa complete matrix, matrix, introduced in group concept group C, C, is sufficient to cover cover all all possible possible derivations derivations considered. considered. Even for for derivative derivative nomological statements, statements, where where individual-terms individual-terms will will be be admitted, admitted, nomological it follows that such it follows that such terms termsalways alwaysoccupy occupy places places which which could could also also be occupied occupied by free free variables. variables.
III I11 ORIGINAL ORIGINAL NOMOLOGICAL NOMOLOGICAL STATEMENTS STATEMENTS
We now now possess possess the the notational notational means to lay lay down down the the requirerequirements defining original nomological statements. Under I-requiredefining nornological statements. I-require-
ment we we classify classify those those requirements requirements which, which, for forsynthetic synthetic statements, statements, ment requirements, formulate invariant properties, though most of these requirements, with the only with only exception exception of of 1.1, 1.1, formulate formulate variant variant properties properties for for tautologies.ItIt is is therefore understoodthat, that, ifif pp is analytic, therefore understood analytic, the tautologies. requirementsrefer referonly onlytoto the the form form of of pp as it is requirements is written. written. For For ininstance, requirement requirement 1.3, 1.3,then then means meansthat that pp must must be be written written as an all-statement. Requirement Requirement 1.5 drops out for analytic analytic statements. statements. I-REQUIREMENTS FOR NOMOLOGICAL I-REQUIREMENTS FORORIGINAL ORIGINAL NOMOLOGICAL STATEMENTS STATEMENTS
Requirement R e q u i r e m e n t 1.1 .1. 1 . The statement pp must must be be verifiably verifiably true.
(Definition (Definition 1). 1). The statement Requirement R e q u i r e m e n t 1.2. 1 . 2 . The statement p p must must be beuniversal. universal. (Definitions 25, 25, 25a, (Definitions 25a, 26). 26). The statement statement p must must be be an an all-statement. all-statement. Requirement R e q u i r e m e n t 1.3. 1 . 3 . The (Definitions 13-44). 13-14). (Definitions The statement p must Requirement R e q u i r e m e n t 1.4. 1 . 4 . The must be be unrestrictedly unrestrictedly exhaustive exhaustive in elementary elementary terms. terms. (Definition (Definition16). 16). Requirement R e q u i r e m e n t 1.5. 1 . 5 . If If the the statement statement pp is is synthetic, synthetic, it it must must be be general 23). general in in self-contained self-contained factors. factors. (Definition (Definition 23).
Of these these requirements, only 1.5 1.5was was not not used used in in my Of requirements, only my earlier earlier presentation. I wifi of this presentation. will therefore therefore explain explain the the significance significance of this requirement. In In combination with the the other requirement. combination with other requirements, requirements, it it serves serves to ensure, nomological statements statements of of the the synthetic synthetic kind, to ensure, for for original original nomological kind, generality may be generality in in aareasonable reasonablesense; sense;ininfact, fact,1.1—1.5 1.1-1.5 may be regarded regarded proper all-statement. all-statement. For stateas requirements requirements defining defining aa proper as For analytic analytic statements, generality factors is is not not defined, but no generality in self-contained self-contained factors defined, but no
ORIGINAL NOMOLOGICAL NOMOLOQICAL STATEMENTS STATEMENTS
41 41
requirement of this kind requirement kind appears appearshere here necessary necessary because because analytic analytic statements are are always always general. general. Not Not always always is is lack lack of of generality generality an an indication indication of of non-nomological non-nomological character. There statements, i.e., i.e., statements statements character. There are are purely purely existential existential statements, which in in D-form D-form or possess only only existential existential operators, which or C-form C-form possess operators, which which we would wouldnot not hesitate hesitate to to recognize recognizeas aslaws lawsofofnature. nature.IIn the theory theory n the we here presented, presented, such such statements statements are are constructed by a derivation here constructed by derivation from original nomological from original nomologicalstatements statementsand andare are therefore, therefore,inin a terminology to to be Consider, terminology be explained explained in in definition definition 28, 28, nomological. nomological. Consider, for instance, instance, the the statement, statement, 'all ‘allthings things contain contain elementary elementary particles'. particles’. This is true even even for for elementary elementary particles, particles, if 'contain' ‘contain’isisregarded regarded as aa reflexive such that aaparticle reflexive relation, relation, such particle contains contains itself. itself. The The statement can be written, with 'ep' ‘ep’for for 'elementary ‘elementary particle' particle’ and 'c' ‘c’ for 'contains', ‘contains’, in the the form form ((25) 25)
(x)(Jry)ep(!I).C(X’ y) Y)
This is an statement, since it satisfies an original original nomological nornological statement, satisfies all the the requirements 1.1—1.9. containsthe thepurely purely existential existential conjunctive conjunctive requirements 1.1-1.9. ItItcontains residual residual (26) (26)
(3Y)eP(Y)
But its variable But this this is is not not aaself-contained self-containedresidual residual of of (25), (25), because because its variable 'y' also in the other ‘y’ also occurs occurs in other functional functional of of (25); (25); thus (25) ( 2 5 ) is not not
tautologically equivalent equivalent to to the conjunction tautologically conjunction (27) (27)
(3YkP(Y)* (s)(3Y)c(x’y) Y)
Therefore requirement requirement 1.5 1.5isis not not violated, violated, and (25) Therefore (25) is general general in self-contained factors. self-contained factors. Now since since we we can can derive derive from from (25) (25)the thestatement statement (26)’ (26), this this purely purely existential statement, reading, 'there are elementary particles', existential statement, reading, ‘there are elementary particles’, nomological.But Butitit appears appears reasonable reasonableto to regard regard this this statement statement is nomological. as aa law nature, because to the as law of of nature, because it it is is indispensable indispensable to the formulation formulation of of the law that all elementary particles. particles. This This illustration illustration the law that all things things contain contain elementary shows why why we we restricted, restricted, in requirement shows requirement 1.5, 1.5, the thenon-redundance non-redundance of an an all-operator residuals. If a variable of all-operator to to self-contained self-contained residuals. variable governed governed by an by an existential existential operator operator occurs occurs in in two two functionals functionals connected connected by an 'and', an ‘and’, this this conjunction conjunction states states more more than aa conjunction conjunction of of
4422
OIUGINAL NOMOLOGICAL NOMOLOGIOAL STATE-NTS STArEMENTS ORIGINAL
existential statements statements with with different such as given existential different variables, variables, such given in in (27). existential residual residual may may therefore (27). A non-self-contained non-self-contained existential therefore be tolerated in statements and becomes, in turn, in original original nomological nomological statements becomes, in turn, nomological. nomological. In contrast contrast to to the thestatement statement'there ‘thereare areelementary elementaryparticles', particles’, a statement statement like like 'there ‘there isis copper' copper’isismuch much more more specific; specific; and we we would consider this this statement statement as as referring referring merely merelyto to aa matter of would consider of fact, fact, thus thus refusing refusing to to regard regard itit as asnomological. nomological. It It appears appears therefore therefore satisfactory not nomo— satisfactory that thatin inthe thepresent presenttheory theorythis thisstatement statementis is not nomological because it cannot be derived from original nomological logical because cannot be derived from original nomological statements. statements. The following consideration may may make make these these relations The following consideration relations clear. clear. We We might might try to to make make the thestatement statement about aboutthe theexistence existence of of copper copper nomological by inserting inserting it it in nomological by in an anoriginal originalnomological nomological statement. statement. For instance, we might use the form (23), with 'h' for instance, we might use form (23)) ‘h’ for 'copper', ‘copper’, while the the first part while part of of (23) (23) might might be bethe thereasonable reasonableimplication, implication, expands (g)'. 'if is heated heated (f), (f), it expands (9)’.IIn n this interpretation, (23) (23) ‘if aa metal metal is satisfies all the requirements Butitit does does not satisfy satisfies all requirements 1.1—1.4. 1.1-1.4. But satisfy requirement 1.5, the self-contained requirement 1.5, because because it it possesses possesses the self-contained conjunctive conjunctive Therefore we cannot in this way residual the residual ‘ ( ~ y)h(y)’.Therefore we cannot in this way take the statement statement about about the theexistence existenceofofcopper coppernomological. nomological. This result, result, however, however, isis not not yet yet satisfactory. satisfactory. It It only that, This only shows that, if p isis original-nomological, cannot contain contain a purely original-nomological, itit cannot purely existential existential closed unit which closed unit which is is derivable derivable from from p (theorem (theorem 4). 4). However, However, there there are other forms of statements p, which contain a closed existential forms of statements p , which contain a closed existential unit unit in in such such aa way way that that this thisunit unitisisnot notderivable derivablefrom fromp, p ,whereas whereaspp appears unreasonable Consider the the statement unreasonable nonetheless. nonetheless. Consider (28) (28)
(x){f(x)D (W(4 3 b(4 (flY)h(Y)lI *
From From it we we can can derive derive the the all-statement all-statement (29) (29)
(x)[/(x)3J ( m ) h ( ~ , ) l (s)[f(=)
Though purely existential existential closed closed unit Though here here the the purely unit ‘(3y)h(y)’ is not derivable, derivable statement statement (29) (29) appears appears unreasonable, unreasonable, as as is is derivable, the the derivable easily seen seen ifif the the above interpretations exeasily interpretations f = heated heated metal, metal, gg = = expands, h = = copper, pands, copper, are are used. used.
43 43
ORIQIN’AL NOMOLOGICAL NOMOLOGICAL STATEMENTS STATEMENTS
now be be shown shownthat that this this kind kind of of statement statement is will now is ruled ruled out out by by It wifi requirement requirement 1.4. 1.4. This is is achieved achieved by by the thefollowing following theorem: theorem : T h e o r e m 5. 5 . If aa statement statement pp can canbe bewritten written in inaaform formwhich which Theorem
a,
is reduced and then either closed unit unit q, then either q, q, or or is is derivable derivable is reduced and contains containsaa closed from from p, p, or or pp isisnot notexhaustive exhaustiveininelementary elementaryterms. terms. The ofthis this theorem theorem for for synthetic synthetic statements statements p and and The significance significance of q is If q is is as as follows. follows. If is derivable derivable from from p and and pp satisfies satisfies requirement requirement 1.5, we weknow knowfrom fromtheorem theorem 44that that q is is an an all-statement; all-statement; therefore 1.5, therefore qq does does not present present an an existential existential statement statement of of the themerely merely factual factual kind, whichcannot cannotbe be regarded regarded as as a kind, like like 'there ‘there exists exists copper', copper’, which law nature. IfIfç7 isis derivable, derivable, we weknow knowthat that qq is is false, false, and and thus thus q law of of nature. is no no longer longer dangerous. dangerous. In this this case, case, requirement requirement 1.5 1.5 guarantees guarantees the conditions conditions appropriate appropriatefor for aa law law of of nature; nature; its satisfies the that cj satisfies negation negation qq is not subject subject to to such such conditions. conditions. The The negation negation of of aa nomological statement isis not not bound nomological statement bound to to structural structuralrequirements; requirements ; for instance, the negation of a proper for instance, the negation of proper all-statement all-statement is aa merely merely factual existential existential statement. statement. If nor is factual If neither q nor p, is derivable derivable from from p, theorem 55 says says that that the the statement theorem statement pp is is not not exhaustive exhaustive in in elementary elementary terms and An example example for for the terms and therefore therefore not not original-nomological. original-nomological. An latter case latter case is is given given by by (28) (28)where where qq isis'(Ey)h(y)'. ‘(Ey)h(y)’. If is analytic, analytic, it it is derivable from from every every pp,, and and therefore If q is is derivable therefore theorem theorem 5 is is satisfied. If p is analytic 5 satisfied. If analytic and and qq is is synthetic, synthetic, qq is is not not derivable derivable from pp;; but from but ininthis thiscase casethe thefollowing followingproof proof of ofnonexhaustiveness nonexhaustiveness remains valid. valid. For For these theorem 5S is is not restricted remains these reasons, reasons, theorem restricted to to synthetic statements, but to analytic synthetic but applies applies likewise likewise to analytic statements. statements. It It;is is true true that, that,ififqqisisanalytic, analytic,we wecannot cannotuse usetheorem theorem4.4.However, However, according to to aa remark the formulation requireaccording remark above above (following (following the formulation of of requirement 1.5), analytic statements statements are not need ment 1.5), analytic are always always general general and and do do not need a specific concerning generality. generality. specific requirement requirement concerning The proof The proof of theorem theorem 55 isissomewhat somewhat complicated, complicated, because because exexhaustiveness in in elementary elementary terms terms refers refers to to a one-scope form of of pp.. haustiveness one-scope form If q includes includes bound bound variables variables (and this is the very If (and this is the very case case which which interinterests us), us), its its operators operators will will thus thus be be separated separated from from its its operand, operand, and and ests the D-form the D-form of of p will will include include indefinite indefinite expressions expressions stemming stemming from from q, like like 'h(y)', are capable truth values. q, ‘h(y)’,which which are capable of different different truth values. To prove prove theorem theorem 5, we we first first write write pp in To in aa reduced reduced form form which which contains the the definite definite unit unit q. q. Now Now qq is is treated as contains as an an elementary elementary
44 44
NOMOLOGICAL STATEMENTS ORIGINAL NOMOLOGICAL STATEMENTS
term, and the statement statement pp is is written written like like aa D-form D-form with the difdifference that that the definite ference definite unit qq is is not not decomposed, decomposed, but figures figures as one among the elementary terms. terms. We We call call this this form form the theq-expan8ion q-expansion of pp.. It It is of is written written with with aaminimum minimum number number of of argument argument variables, variables, among which, which, however, however,the the variables variables of of qq are are not contained. among contained. This form form can be be written written ( )...( )...(
(30) (30)
(
)[(SIV
... v s,).qv ...
(tlV
... v tm).4]
tk are are conjunctions conjunctions of of the the elementary elementary terms where terms si and and tk terms where the the terms of p, p , but do do not not contain contain qq or orthose thoseelementary elementary terms terms which which occur occur exclusively exclusively inside inside q. q. Since qq isis closed, closed,itit isis either either true true or or false, false, and and thus thus one Since one of of the two terms inside the brackets two brackets of of (30) (30) can be be canceled. canceled. In other other words, the the q-expansion of pp is not exhaustive, q-expansion of exhaustive, one one of of its itsresiduals residuals being true. Exception to be be made for the case being true. Exception isis to made for case that either either no no q-terms, or no no @terms, i-terms, occur occur in in (30). But But then y-terms, or then p', or or q, q, is is derivable derivable from p, and and thus thus theorem theorem 55isissatisfied. satisfied.We We therefore therefore proceed proceed on on the assumption assumption that this this case case does does not not occur. occur. We now now shall prove We prove that that the thenon-exhaustiveness non-exhaustiveness of the the q-exq-expansion (30) is is transferred transferred to the pansion the D-form D-form of of p. p . This Thisproof proof isissomesomewhat complicated Some of of the the si what complicated by the the following following consideration. consideration. Some identical; let say that that rr such and the t, may and the may be be identical; let us us say such terms terms are are identical, identical, terms, and let us and us write write these these rr terms terms as asthe thelast lastn—r, n-r, or or m r , terms, respectively. Then (30) assumes the form respectively. Then assumes form
v(fiv ...vt, _,). qv(s,_,+lv...vs,).(qvq) but apart The tautologyqqvv!7 in in the third term term can can be be canceled; canceled; but The tautology from special cases to to be will then no from special cases be discussed discussed presently, presently, (31) (31) will no longer be a q-expansion. Furthermore, one, or two, of the three longer be q-expansion. Furthermore, one, or two, of the three terms of may vanish, vanish, i.e., i.e., need need not not occur occur in in the the expansion. expansion. Let Let terms of (31) may us begin by studying us studying these these simpler simpler cases; cases; after after this this discussion discussion we shall examine shall examine the general general case. case. (31) (31)
((
)...( )...( )[(sIv...vs, _,).q
If two terms and one If terms vanish vanish and one of of them them is the last last term, term, there there
remains either the the q-term, or or the the +term. i-term. Then q, or 4, is remains either derivable is derivable from pp,, which from which case case was was already already excluded excluded as aa satisfying satisfying theorem theorem 5. 5. If the first in the the remaining third first two two terms terms vanish, vanish, we we can cancel in remaining third term the term the tautology tautology qq v q. This shows that can be be written This shows that pp can written without without
45 45
ORIGINAL NOMOLOGICAL ORIGINAL NOMOLOGICAL STATEMENTS
the the unit unit q. q. Then Then qq is is redundant redundant and andpp isisnot notreduced. reduced.This Thiscase casewas was
excluded in theorem theorem 5. 5. It It follows that we have to consider excluded in follows that consider only only cases where where one term vanishes. cases vanishes. Let us first assume that that the the third third term termvanishes. vanishes. We We then have have r = and (31) assumes the the form (30). In In order to = 0, 0 , and (31)assumes to transform transform (30) (30)
into we write write qq in inD-form D-formand and then then move move the the operators operators into aa D-form D-form of of pp,, we of of q to to the thefront frontofofthe thewhole wholeexpression. expression. Those Those variables variables of of qq which are bound which are bound by by all-operators all-operators must must thereby thereby be be given given different different names in in q and q, because of the the disjunctive form of of the the brackets because of in those variables variables of of qq which whichare arebound boundby by existential existential operators operators in (30); (30) ;those are given But all all these these variables are given the the same same names names in in qq and and ç7. q. But variables have have names different different from from those those of of the the variables in the the si and and names variables occurring occurring in because the latter tk, because latter variables variables are are already already bound bound by by operators. operators. tk, When we denote denote the the operands operands of of qq and and4by by'q1' ‘ql’and and'q2', ‘qa),respectively, respectively, When we we thus arrive we thus arrive at the Ohe form form ( )...( ). ..( )[(s1v )[(sl v
(32) (32)
(
.*. v sJ.41
v (tl v
... v t,)
.q2]
This form is not exhaustive exhaustive for for the the same same reasons reasons as as explained explained for for (30); one one term in the the brackets brackets of of (32) (32) can can be be canceled, canceled, since since the
remaining statement statement is tautologically equivalent with with the corremaining tautologically equivalent correspondingremaining remainingstatement statementofof(30). (30).But But this this result result is is not responding not sufficient for our our purpose, purpose. It It is possible that the possible that the number number of of arguargusufficient for ment ment variables variables stemming stemming from from qq in in (32) (32) can can be be reduced reduced to to aa smaller smaller number; then (32) is not not aa D-form. therefore must must prove prove that that number; (32) is D-form. We therefore the statement remains non-exhaustive for a reduction of the number the statement remains non-exhaustive for a reduction of the number of the of the variables. variables. But But this this proof proof isis easily easily given. given. The The number number of of variables variables can can only only be reduced by identifying be identifying some some of of the the variables variables of ofq1 ql with with some some variables in the the s, or variables occurring occurring in or by by identifying identifying some some of of the the variables variables of q2 with those occurring in the tk• This must be a tautological of q2 with those occurring in t,. This be a tautological transformation; thus thus the residual transformation; residual of (32) (32) which which is true true remains remains true after the that for the transformation. transformation. We thus find that for the the case ca8e r = =0 0 the D-form D-form of of p is is not notexhaustive. exhaustive. Second, let us assume Second, let assume that that the thesecond second term termofof(31) (31)vanishes. vanishes. Then Then (38) can be written: written: (33) (33)
( )...( ).”( ){(s1v )C(%V
(
*..v s , - , ) . q v
(%-r+lv ’ . *
V8,).(PV!al
46 46
O1UGINAI. ORIUINAL NOMOLOGIOAL NOMOLOOICAL STATEMENTS STATEMENTS
we infer infer immediately immediatelythat that the the first If q is false, we f%st term can can be be cancan-
celed and (33) is not not exhaustive. exhaustive. So So let let us us assume that q is true. celed and (33) is assume that true. Before Before further further discussing discussing the the form form(33) (33) let let us us consider consider the the possibility possibility that qq contains contains only only elementary elementary functions functions already already contained contained in the the si; then then it might might happen happen that that moving moving the operators operators of q to to the front q, the the resulting resulting front and and reducing reducing the the number number of of variables variables in in q, elementary terms in the elementary terms the operand operand q1 ql are all all identical identical with with those those be absorbed absorbed in in the the si. In contained in in the the si, so contained so that tbat q1 q1 would would be I n this this case, however, however,there there would would remain remain in in (33) (33) only only terms terms not not concase, conso that qq taining v can taining q, 8ince since the tautology tautology qqvrj can be canceled, canceled, so would be shown to be aa redundant would be shown to redundant unit. unit. This Thiscase caseisisexcluded excluded because it was assumed assumed that that p is because it is reduced reduced in in the the form form in in which which itit contains considerations hold hold ifif q includes contains q. Corresponding Corresponding considerations includes factors factors which can can be absorbed in the which absorbed in the si; then these these factors factors would would be be redundant redundant in p. p. So we may may assume assume that that the So we the elementary elementary terms terms contained contained in in qq do do not the operators are moved moved to to the the front front and and not vanish vanish even even when when the operators of of qq are the number to aa minimum. the number of of variables variables is is reduced reduced to minimum. For For this this reason, reason, the tautology tautology qq v Sin the the last last term term cannot cannot be be canceled; canceled; the the resulting resulting ç7 in the expression would would not not be be aa D-form D-form of since some some of ofthe the elementary elementary expression of pp,, since terms of of the first terms first term, term, namely namely those those contained contained in q, q, would would not not, occur in in the last term. occur term. Now we we can can write in the form Now write (33) in form (34) (34)
... v s , ) . q l ... v;).qv (( )...( ) [ ( S I V ... v s , ) . q v (sn-r+lV ... )...( )[(s1v
Using we can write this Using the same same methods as described described for (32), we (35) (35)
V -.* ... V (( )..4 v )...( )"% v
%)41V
(sm-*+lv.*....vs,j.q2] V%)421
and can then then conclude that, ifif q is true, the and can conclude that, the second second term caii can be canceled, the residual residual canceled, so so that that the (36) (36)
(( I...( )...(
)WIV
... vsm).Qll
is true. Reducing the the number number of Reducing of variables variables cannot cannot change change this result, result, except for for the the possibility of the the following manipulation.Instead Instead of of except possibility of following manipulation. going f i s t replace replace the the tautology tautologyqqVv going from from (33) (33) to to (34), we we might might first q*, which by some some other other tautology tautology q v q*, which possesses possessesthe the same same elementary elementary by
ORIGINAL NOMOLOGICAL ORIGINAL NOMOLOGICAL STATEMENTS STATEMENTS
47 47
terms, terms,
but not the but where where q* is not the negation negation of q. (The (The term term q cannot cannot be be changed because becauseitit is is also also contained contained in in the the first term of changed of (33) (33) and must thus thus have the same elementary elementary terms and and the the same operators operators must as q in the latter as latter term; term;otherwise otherwisethe thenumber number of ofvariables variables would would the tautology be increased.) increased.) For For instance, be instance, if if qq is is given given as as ‘(gy)h(y)’, v q, namely, namely, qq V ( 3 Y ) m f ) vv ( m M Y )
(37) (37)
might might be replaced replaced by the the tautology tautology (3Y)hL(Y)v ( 3 Y ) W
(38) (38)
This procedure procedure would wouldin in fact fact reduce reduce the number This number of variables variables in (33), when we we proceed proceed tto the D-form. D-form. Instead Instead of then have (33), when o the of (35) (35) we we then have the form form (39) (39)
)...( )[(s1v (( I.-.( )[(%V
... *.*
v
%I).%V (%-,+1V
...
..*
V%)*q,*l
q* to the where qz q2 is is the the operand contributed where contributed by q* the D-form. D-form. Now Now
we cannot conclude that the it (39)is is false, false, because because it we cannot conclude that the second second term term in in (39)
1 does not not contain does contain the the operand operand q2, q2, or or @. This procedure, procedure, however, does not not make This however, does make our our previous previousinference inference invalid, which which showed showed that that (36) is true. true. 2 Now Now this this is is aa residual residual of invalid, (36) is of (39), too, and and so (39) is not not exhaustive. This shows showsthat that pp is not (39), too, (39) is exhaustive. This exhaustive for for the the case (33). A similar similar result result is is exhaustive case formulated formulated in (33). derivable ifif the first derivable first term term of of(31) (31)vanishes. vanishes. Finally, we have to to prove prove that that in no term term Finally, we have in the the general general case, case, when when no of of (31) (31) vanishes, vanishes, pp is not not exhaustive. exhaustive. This This proof proof isiseasily easilygiven, given,
A D.form D-form of the form (39) ( 3 9 ) may result naturally naturally when when pp is is transformed transformed into D-form directly, without going going through the theq-expansion. q-expansion. This This is is always always the case its propositional contains no no equivalence equivalence operation among its propositional the case when p contains operations. for this this theorem, theorem, but but will not give it proof for will not operations. I have constructed a proof here, since since it is is unnecessary unnecessary for the the proof proof of of theorem theorem 5. 5. 2 2 This can also also be be shown shown as as follows. follows. Using the tautology tautology This can a.q1v Day a . q l v b.q2* b.q2* 3 a v bb are canceled, the remaining we can can derive derive that, that, ifif in we in (39) (39)the thefactors factorsq1 qI.and and pa* are canceled, the 1
formula is true. true. But But the content content of of the brackets brackets then is is equivalent equivalent to t o the the formula is term in the if qq is is true, (36) the first first parentheses. parentheses. Therefore, Therefore, if (36) is true. The The case case
that treated with transition from is false was already treated with reference to (33). The transition from that qp is
can be be proved (35) to (36) (35) to (36) can proved in the the same same way. way.
48 48
OEIGINAL NOMOLOGICAL ORIGINAL NOMOLOUICAL STATEMEIcTS STATEMENTS
because either either the the first or the the second second term term of of (31) (31) is is false and thus thus aa residual true, aa result residual is true, result transferred transferredto tothe theD-form. D-form.This Thisconcludes concludes the proof proof of of theorem theorem 5. 5. the significance of the the I-requirements With the the theorems theorems 4—5, 4-5, the significance of now turn to is made clear. We now to the thesecond secondkind kind of of requirements. requirements. V-REQUIREMENTS FOR V-REQUIREMENTS FOR ORIGINAL ORIGINAL NOMOLOGICAL NOMOLOGICAL STATEMENTS STATEMENTS
The statement pp must 1.6. The must be bereduced. reduced.(Definition (Definition 9). 9). 1.6. is aadisjunction, disjunction, or or if if pp isisaaconjunction conjunction having having aa 1.7. 1.7. If pp is
major factor which is is aa disjunction, major factor p1 p , which disjunction, then then p, p ,or orp1 pl respecrespectively, must tively, must satisfy satisfy the thefollowing following cancelation cancelation condition: condition : If one or both of the major major terms of p, p , or of of Pi, plyare are transtransformed in any conjunction which, which, taken taken formed in any way way into into a aconjunction separately, is is reduced, separately, reduced, then then every every statement statement resulting resulting from factor in this conjunction in this conjunction must must be be reduced. reduced. from canceling canceling aa factor
animplication, implication, or if if pp isisaaconjunction conjunction having having a 1.8. 1.8. If pp isisan
major factor factor fi Pi which is an implication, then pp,, or Pi' major which is implication, then pl, respectively, after being transformed into into aa disjunction respectively, after being transformed disjunction by by negating the the implicans, negating implicans, must satisfy satisfy the the cancelation cancelation concondition of requirement dition requirement 1.7. 1.7.
1.9. 1.9.
If p contains at any place If contains an an equivalence equivalence at place and and this this equivequivalence is is replaced alence replaced by double double implications, implications, the resulting resulting statement must must satisfy statement satisfy 1.7 1.7 and 1.8, 1.8, with with the the exception exception that that one one of of the theimplications implications may may be be redundant. redundant.
We now introduce the We the definition: definition: Definition statement original nomological 2 7 .A A statementp is p an is an original nornological D e f i n i t i o n 27. statement ifif and only if the requirements requirements1.1—1.9. 1.1-1.9. satisfies the statement and only if itit satisfies (V-term). (V-term). The significance of requirement requirement 1.6, 1.6, on on the one one hand, hand, follows follows The significance of from the the above of reduced reduced statements. statements. T This requirement from above discussion discussion of his requirement eliminates unnecessarycomplications complicationsofofthe theform form in in which which the eliminates unnecessary statement statement is is written. written. On On the the other otherhand, hand,this thisrequirement requirementassumes assumes an important important function function through through its itscombination combination with with requirement requirement 1.4: it it can that exhaustiveness in major can be be shown shown that exhaustiveness in major terms, terms, as as far far as as it isis needed, needed, can can be bederived derivedfrom fromthis thiscombination. combination. AAspecific specific
O1UGINAL NOMOLOOICAL ORIGINAL NOMOLOGICAL STATEItEENTS
49 49
concerning this this kind kind of requirement of exhaustiveness, exhaustiveness, as as was was used used requirement concerning in 368, is is therefore in 1ESL, ESL, p. therefore dispensable. dispensable. p. 368, Exhaustiveness in major terms terms formulates formulates an an important importantrequirerequirement ment for for what what we we call call aa reasonable reasonable use use of of propositional propositional operations. operations. If a statement If statement is is not not exhaustive exhaustive in in major major terms, terms, its its major major operation operation is used the possibilities openedby by itit are is used inappropriately, inappropriately, because because the possibilities opened are not exhausted. For instance, instance, consider consider the two two statements: statements : exhausted. For (40a) (40a) (40b) (40b)
(x)[f(x) (4 tf(4 D 3 Bg(xfl (4l
y) D g(x, Y)l y)] Y> 2 9@, Their expansions in major T-cases have the forms, Their expansions in forms, respectively, respectively, (41a) (41a) (41b) (41b)
(X)(fi!Iy)[f(G
--
(x)[f(x) v f(x) .g(x) (4 [f (4..g(x) g(4 vf(x) .g(4 v v f(x).g(x)] f ( z )g(4l . -/(x, y).g(x, f(x, Y).9(X, y).g(x, y) y) ( s ) ( z y ) [ f (y).g(x, y).g(x, z, 9)v f(% 9)vv f(z, Y)*9(X,y)] Y)1
Now assume assume that that the of (40a) is always always false, false,i.e., i.e., that that Now the implicans implicans of (40a) is the relation the relation holds: holds : (42a) (424
-
((x)/(x) X)f(X)
It was above, that that It was pointed pointed out out by by Bertrand Bertrand Russell, Russell, as as mentioned mentioned above, on this (40a) can can lead lead to to unthis condition condition the general general implication implication (40a) unreasonable consequences. consequences.For Forinstance, instance,we wecan canthus thusassert assertthat that if reasonable z is aa centaur z centaur xx isis aabanker. banker. When When we we require require that that (40a) (40a) be be exexhaustive in in major haustive major terms, terms, however, however, the case case (42a) (42a) isis ruled ruled out, out, becauseifif (42a) (42a)were weretrue, true,we wecould couldcancel cancelthe thefirst first term term in in the because the brackets the resulting true. brackets of (41a), (41a), while while the resulting residual residual is true. It that it It isis the the advantage advantage of of the the requirement requirement of of exhaustiveness exhaustiveness that it is applicable to other other statement it would not help is applicable to statement forms forms where where it would not help us us Forinstance, instance, to require require that that the theimplicans implicans be be not notalways alwaysfalse. false. For the form the form (40b) (40b) can can lead lead to to unreasonable unreasonable implications implications of of a similar similar kind as resulting if the implicans of (40b) iis kind resulting for (40a) (40a) even even if implicans of s not always false. false. Suppose Supposethat that ‘f(z, '/(x, y)’ y)' means, 'x is the the father father of always means, ‘x of y', g’,and and that 'g(x, y)' means, 'x is taller ‘g(z, y)’ means, ‘z taller than than y'. y’.In I nthis thisinterpretation, interpretation, 'f(x, y)' is is not ‘f(z,y)’ not always always false. false. But But since since here here the the relation relation holds holds
-
(42b) y) 142b) (d(z?df(z, Y) the first the first term term in in (41b) (41b) can can be be canceled, canceled, and and (40b) (40b) is is therefore therefore not not exhaustive in in major major terms. terms. Thus which in in the the given given exhaustive Thus the the form form (40b), (40b), which
50 50
ORIGINAL NOMOLOOICAL NOMOLOGICAL STATEMENTS STATEMENTS ORIGINAL
interpretation is is unrewnable unreasonable for for the the very very reaaon reason that that only interpretation only (42b) (42b) makes it it true, makes true, is is ruled ruled out out by bythe therequirement requirementofofexhaustiveness. exhaustiveness. This consideration shows showsthat that exhaustiveness exhaustiveness represents represents aa suitable suitable T his consideration generalization of of Russell’s Russell's rule rule according according to to which which an an always false generalization always false implicans should should be be excluded. excluded. Similar Similar considerations considerationsapply applyto to the the implicans case of of an always case always true true implicate. implicate. Likewise, Likewise, the requirement requirement of of exhaustivenesscan can be be applied exhaustiveness applied to to other other propositional propositional operations operations than implications. than implications. It will that exIt w i l l now now be be shown shown for for the the individual individual operations operations that exhaustiveness in major haustiveness in major terms terms can can be be derived derived from from requirements requirements 1.4 1.4 and 1.6 1.6 to toaasufficient sufficientdegree. degree. Assume, for instance, instance, that that in Assume, for in the thestatement statement(40a) (40a)implicans implicans and implicate are composed of elementary elementaryterms, terms,and andthat that the and implicate are composed of whole statement is further whole statement is exhaustive exhaustive in in elementary elementary terms. terms. Assume Assume further that that the theimplicans implicansisis always always false, false, i.e., i.e., that that (42a) (42a)is is true. true. IfIf(42a) (42a)isis not derivable derivable from (40a), (40a), it represents represents aa certain certain restrictive reNtrictiveconcondition for for the the elementary terms of of (40a) which isis added added to to this dition elementary terms (40a) which this statement. admitted statement. But But this this means means that thatnot notall allelementary elementary T-cases T-cases admitted by and thus by (40a) (40a) are are satisfiable, satisfiable, and thus (40a) (40a) is is not not exhaustive exhaustive in in eleelementary mentary terms. terms. Therefore, Therefore, (42a) (42a) must be be derivable derivable from from (40a). (40a). Since, vice versa, versa, (40a) is derivable Since, vice (40a) is derivable from from (42a), (42a), both both statements statements are are tautologically tautologically equivalent. equivalent. Consequently, Consequently, (40a) (40a) is is not not reduced; reduced ; namely, we add add aa negation namely, we we can cancancel cancel the the implicate implicate when when we negation line line on the the implicans. implicans. (Definition (Definition 77 for for synthetic, synthetic, definition definition 8 for for analytic analytic statements.) statements.) In I n aa similar similar way way we can prove that (40a) (40a) is not reduced reduced if the implicate implicate is is always always true. true. Canceling Canceling the the implicans implicans in in (40a), (40a),we we then then arrive at at aatautologically tautologicallyequivalent equivalentform. form. With With respect respect to to exhaustiveness exhaustiveness in in major major terms, terms, these these two two cases cases are as follows, when we we refer refer to to the the expansion are classified classified as follows, when expansion in in major major T-cases given by by (41a). If (40a) T-cases given (4la). If (40a) has has an an always always false false implicans, implicans, the the first first term term in in the the brackets brackets of of (41a) (4la) can can be be canceled. canceled. If If (40a) (40a) has has an always true implicate, the last term always true implicate, the term in in the the brackets brackets of of (41a) (41a)can can be These are are the the two be canceled. canceled. These two cases cases already already discussed. discussed. It It remains remains to to discuss discuss the the case case that thatthe themiddle middleterm terminin(41a) (41a)can canbebecanceled. canceled. However, it is However, it is unnecessary unnecessary to discuss discuss the the possibility possibility that that two two terms because this this leads leads back back to to the the first terms in in (41a) (41a)can can be be canceled, canceled, because first
ORIGINAL NOMOLOGICAL NOMOLOGICAL STATEMENTS STATEMENTS
51 51
two For instance, instance, ifif the the last two cases. cases. For last two two terms terms in in (41a) (41a)can can be be cancanceled, we we can can reduce celed, reduce (40a) by canceling canceling the implicate. implicate. Since Since we can D3 g(x)', can go go from from 'g(x)' ‘g(x)’to to 'f(x) ‘f(x) g(x)’,we we can can rederive rederive (40a), (40a),and andsince since this statement statement was was assumed assumed to to imply imply the theconsidered considered residual residual of of (41a), we we thus thus derive derive ‘(x)f(s).g(z)’. '(x)f(x) .g(x)'.This Thisshows showsthat thathere herethe the latter latter (41a), statement is is replaceable replaceable by the the reduced reduced form form of of(40a). (40a). Now ifif the the middle Now middle term in in (41a) (4la) is is dispensable, dispensable, the the implication implication can be replaced replaced by by an anequivalence. equivalence. Here Here the theimplication implication (40a) (40a)isis reduced, because because itit cannot cannot be by reduced, be transformed transformed into into an an equivalence equivalence by canceling terms. It It cannot canceling terms. cannot even even be be written writtenasasdouble doubleimplication, implication, because the converse implication is is then then redundant, redundant, being because the converse implication being derivable derivable from the first from the first implication; implication;this thisderivability derivabilityfollows followssimilarly similarly as as was was explained explained for for (42a). (42a). With respect to With to this thisconverse converseimplication implication we we have have to to distinguish two according as as this this implication implication is is synthetic synthetic or two cases, cases, according or tautological. tautological. An example of the the first kind is given by the statement example of statement (43) (43)
((x)(y)[/(x, x) ( y) [ f ( x,y) Y)D 3 ff(y, ( Y , 4x)]1
from which which we can can operator-derive operator-derive the converse converse implication implication by If we interchanging 'x' and and 'y'. interchanging ‘x’ ‘y’. If we add addthis thisimplication, implication, and and write write (43a) (434
(x)(y){[f(x, D f(Y, f(y, 4x)]. ( z ) ( y ) { [ f ( x Y) ,y) 3 1 * [f(y, x) 4D 3 f(x, f(x, y)]} Y)l)
this this statement statement isis not not reduced, reduced, because because the the second second brackets brackets can can be be canceled. So So itit appears appears reasonable reasonable to to accept is reduced. reduced. canceled. accept (43), (43),which which is
This because This statement statementisisexhaustive exhaustiveininmajor majorterms termsexcept exceptfor foritself itself,, because the the statements statementsp1 p1and andpp of of definition definition19 19here herecoincide. coincide.The The exclusion exclusion of a major becauseitit is is said of major T-case T-case is here here acceptable acceptable because said by the the statement itself. is given given by by the the statement: statement itself. An illustration illustration of of (43) (43)is statement: 'if is a sibling of y, y, then then yy is of x’. x'. We We do do not not hesitate ‘if x is sibling of is aa sibling sibling of hesitate to to accept this this implication as reasonable although although it can accept can be be replaced replaced by an by an 'if ‘if and and only only if', if’,i.e., i.e., by by aa double doubleimplication implicationor or an anequivalence. equivalence. Note that (43), Note (43), if the the terms terms 'f(x, ‘f(z,y)' y)’ and and 'f(y, ‘f(y,x)' x)’are areelementary elementary terms, is exhaustive terms, exhaustive in in elementary elementary terms, terms, because because definition definition 16 16 refers to to the complete refers complete matrix matrix of of the the statement, statement, given given here here by by (43a). In In contrast, contrast, exhaustiveness in major major terms terms refers refers to to the the form (43a). exhaustiveness in form (43), in in which the statement is given. which statement is given.
52
ORIGINAL NOMOLOGICAL NObiOLOCrCAL STATEMENTS STATEBfENTS
An example example where the the converse converse implication implication is is tautological tautological is is given by by the statement given statement (44) (44)
”
(x)f/(x) v g(x) Df f(x).g(x)] (z)Cf(z) g ( 43 (s)*g(z)l
Here, too, we Here, we cannot cannot add add the theconverse converse implication implication because because the statement then would statement would not be be reduced. reduced. But But since sincethis thisconverse converse implication is tautological, tautological, itit can can be be said to be implication is be derivable derivable from the statement itself; itself; thus thus (44) (44) is is exhaustive exhaustive in major major terms terms except except for for itself. An example example isis given given by by the the statement: statement: 'if itself. An ‘if one one of of two two concondenser plates is electrically then both denser plates electrically charged, charged, then both are areelectrically electrically charged'. charged’. There these forms There is an an essential essential difference difference between between these forms and the the forms previously, in in which the falsehood the implicans, forms discussed discussed previously, which the falsehood of of the implicans, or the truth the statement or truth of of the theimplicate, implicate, can can be be derived derived from the itself. The implication itself. implication which which is replaceable replaceable by an anequivalence, equivalence, exemplified by (43) (43) and and (44), (44), isis both both reduced exemplified by reduced and exhaustive exhaustive in major the implication with an major terms except except for itself. itself. However, However, the implication with always faIse false implicans, implicans,or or an an always true implicate, always always true implicate, though exexhaustive in major terms except haustive except for itself, itself, is not reduced. reduced. We We see see that the combination of the two conditions, being reduced and the combination of the two conditions, being reduced and being exhaustive exhaustive in in major major terms, terms, except for itself, guarantees being except for guarantees aa form of of implication which we we are to accept form implication which are willing willing to accept as as reasonable. reasonable. But the But the latter lattercondition conditionisisderivable derivablefrom fromexhaustiveness exhaustivenessin inelemenelementary terms; tary terms; therefore, therefore, the thecombination combinationof of requirements requirements 1.4 1.4 and and 1.6 1.6 furnishes precisely precisely the the kind furnishes kind of of implication implication which which we we are are willing willing to to accept. accept. For these these reasons, reasons, the the requirement requirement of of exhaustiveness exhaustiveness in major major terms, used used in in ESL, p. 369, can be dropped. dropped. Its function terms, 369, can function is taken over by by the over the requirement requirement that the the statement statement must must be bereduced, reduced, though in in aa more and the other though more tolerant tolerant way. way. For Forimplications implications — - and other operations will be operations be discussed discussed presently presently — - this this means: when when 'if ‘if and only if‘ if' is the grammatical that makes only grammatical conjunction conjunction that makes the statement statement exhaustive in in major major terms, terms, the the form exhaustive form 'if' ‘if’is is still still admitted. admitted. This This is is one of of the the changes changes of of the the present present theory theory as as compared compared with with that that of one of ESL. Only the cases and an always ESL. cmes of an an always always false false implicans implicans and always both for for synthetic true implicate implicate are still excluded, excluded, both synthetic and analytic statements. We may that, when when '/(x)' ‘f(z)’and and'g(x)' ‘g(z)’ are are elementary elementary statements. We may add add that,
ORIGINAL NOMOLOGICAL NOMOLOGICAL STATEMENTS STATEMENTS
53
terms in (40a), (40a), (this can only only occur in synthetic statements), the requirement 1.4 requirement 1.4 of of exhaustiveness exhaustiveness in elementary elementary terms directly directly rules out forms is always false or or the the imrules forms in which which the implicans implicans is always false plicate is is always always true true;; here the not plicate the requirement requirement of of being being reduced reduced is is not applicable and not not necessary. necessary.And Andititthen thencannot cannothappen happenthat that the applicable and converse implication implication of of (40a) (40a) is is derivable derivable from from (40a) (40a)or or is is tautotautoconverse logical, becausethis this would would require requireaa substructure substructure of of the terms. logical, because terms. For For this this reason, reason, aa synthetic syntheticimplication implication like like (40a) (40a)which which could could be be replaced by an equivalence is not accepted accepted as as an anoriginal originalnomonomoreplaced equivalence is logical statement statement if if the the terms '1(x)' and 'g(x)' are elementary elementary terms. terms. logical ‘f(s)’and ‘g(z)’ are Such implications, implications, however, however, will willbe beadmitted admitted later later into into aa somewhat Such somewhat wider category wider category (see (see the the discussion discussion following following theorem theorem 13). 13). It It isis different different when when the the implication implication is is analytic. analytic. The The implication implication (45) (45)
(x)[f(x) D f(x)J
satisfies even ifif '/(x)' satisfies all requirements requirements 1.1—1.9, 1.1-1.9, even ‘&)’ is is an an elementary elementary
term. In I n particular, particular, itit isisexhaustive exhaustive in in elementary elementary terms, since it has only one elementary elementary term; and and ititisisreduced. reduced. Its Itsdispensable dispensable major T-case (which is not an elementary T-case) 't(x).f(x)' major T-case (which is an elementary T-case) ‘f(z).f(z)’ is is contradictory; contradictory ; thus thus the thecorresponding corresponding residual residual is is tautological tautological and and can be regarded as derivable from the the statement itself. Statement derivable from (45) isistherefore therefore exhaustive exhaustive in in major major terms terms except (45) except for for itself. itself. Because Because
of .9, statement statement (45) of the the satisfaction satisfaction of ofthe therequirements requirements1.1—i 1.1-1.9, (45) is an original nomological statement. original nornological statement. It It will will now now be be shown shown that thatsimilar similarconsiderations considerations can can be be carried carried through for for the the other operations, if they they stand stand in the through operations, if the major major place. place.
We begin begin with with the inclusive 'or'. Assume that in the We inclusive ‘or’. Assume that the statement statement (46)
(x)[/(x) v g(x)]
the major major terms terms stand stand for for combinations combinations of elementary terms, and assume further that that the statement the statementisisexhaustive exhaustivein inthese these elementary elementary terms. If the the statement statementremains remainstrue truewhen whenone oneof of the theterms termsisiscancanceled, the the resulting statement statement must mustbe bederivable derivablefrom from(46), (46),because because
otherwise otherwise it it would would represent represent an an additional additionalcondition conditionfor forthe theelemenelementary tary terms terms and andmake make(46) (46)nonexhaustive nonexhaustive in elementary elementary terms. But But if one term can be canceled, (46) is not reduced, whether it is synone can be canceled, (46) is synthetic or the possibility possibifitythat that the the 'or' or analytic. analytic. There There remains remains the ‘or’in in
54 54
NOMOLOGICAL STATEMENTS ORIGINAL NOMOLOGICAL STATEMENTS
(46) can can be be replaced replacedby bythe theexclusive exclusive 'or',i.e. ,that we can can add to (46) -‘or’, -i.e. ,that the operand the operand of of (46) (46) the relation relation '/(x) ‘f(x)v g(x)'. g(x)’. If (46) (46) is exhaustive exhaustive in elementary elementary terms, terms, this relation relation must must be be derivable derivable from from (46); (46); so we add it, it, the resulting so if we resulting statement would would not be reduced. But in this case, is reduced. reduced. Although Although itit might might then then appear appear in this case, at at least, least, (46) (46) is preferable to write write (46) with the sign preferable to (46) with sign of the the exclusive exclusive 'or', ‘or’, we we would not object would not object to to using using an an inclusive inclusive 'or' ‘or’when when we we know know from from the the statement itself itself that itit can can be bereplaced replaced by by an anexclusive exclusive 'or'. ‘or’, For For instance, we are are willing to accept instance, we willing to accept such such statements statements as 'a ‘acomet comet moves on on an open or aa closed section', even even when when the the 'or' closed conic section’, ‘or’isis not to be In this not expressly expressly marked marked to be exclusive. exclusive. In this example, example, the the relation relation _ . that can ‘f(z)v v g(x)', g(s)’, would would be tautological; tautologicrtl; in in can be be added, added, namely, namely, '/(x) other examples examples it will will be be synthetic, synthetic, though though derivable derivable from from (46). (46). For For the the equivalence equivalence as major operation, the situation situation is is slightly slightly different. In order different. order to toexclude excludeunreasonable unreasonable forms, forms, requirement requirement 1.8 has has been added. Consider the statement 1.8 Consider the (47) (47)
(x)[/(x)= = g(4l (W(4
that the and assume and assume that the two two terms terms stand stand for forcertain certaincombinations combinations of of elementary terms. terms. If II (47) elementary (47) is not exhaustive exhaustive in major terms, but but,is is -exhaustive in . exhaustive in elementary elementary terms, terms, then then either either'/(x). ‘ f ( x ).g(x)' g(z)’or or 'f(x) ‘ f ( x ).g(x)’ is dispensable. Assumethe the latter latter is the case; is dispensable. Assume case; then then we we can canreplace replace we now the brackets .g(x)'. When When we brackets in in (47) (47) by bythe thecombination combination'/(x) ‘f(z).g(x)’. now write (47) write (47) in the form form (48) (48)
(x)([f(x) ( x ) ( [ f ( x )D 3 g(x)]. !Wl*{g(x) [9(4D 3f(43
we see that that neither their we see neither of of these these implications implications is reduced, reduced, because because their
implicans can be canceled. For the other case implicans can canceled. For case an an analogous analogous result can can be be proved. proved. Thus Thus requirement requirement 1.8 1.8 makes makes itit impossible impossible to to use use an an
equivalence as major major operation operationifif itit is not equivalence as not exhaustive exhaustive in in major major terms. terms.
If one one of of the theimplications implications in in (48) (48) is is tautological, tautological, or or derivable derivable from the the other, this other not reduced. from other, while while this other one one is is synthetic, synthetic, (48) (48) is is not reduced. For this For this case, case, requirement requirement 1.8 1.8 admits admits the form form (47), (47), which which is reduced reduced and and exhaustive exhaustive except except for for itself. itself. If If the theimplicational implicational form form is used, the the redundant redundant implication to be be canceled. canceled. Examples Examples is used, implicationin in(48) (48) is is to
ORIGINAL ORIUINAL NOMOLOGICiAL NOYOLOQICAL STATEMENTS STATEMENTS
55 55
are given given by (43) (43) and (45). (45). IIn n both both these these instances instances it would would be be permissible to replace replace the the implication permissible to implication by an anequivalence. equivalence.
Finally, the 'and' exhaustive Finally, if the ‘and’is is the the major major operation, operation, itit is is always always exhaustive in do not not consider in major major terms, terms, because because it it has has only only one one T-case. T-case. We We do consider here the exclusive 'or', because a special sign for this operation here the exclusive ‘or’, because a special sign for this operation is is seldom used. If If it it were to be requirement by seldom used. were to be used, used, aa special special requirement by analogy analogy with 1.8 would wouldhave haveto tobe beintroduced, introduced,demanding demandingthat that the the statement statement with be reduced reduced if if the the exclusive 'or' is by aa conjunction be exclusive ‘or’ is replaced replaced by conjunction of of two two inclusive disjunctions, disjunctions,inin the the form inclusive form '(a ‘(av b). b ) .(a (6 v v 8)’.
results may be These results These be summarized summarized in in the thefollowing following theorem, theorem, which holds holds both both for synthetic and for analytic which analytic statements: statements: is Theorem T h e o r e m 6. 6. If aa statement statement is is original-nomological, original-nomological, it it is exhaustive in in major major terms except exhaustive except for itself. Thus the only only cases cases where the the statement statement can in major major terms terms are are the be non-exhaustive non-exhaustive in the where can be following ones:the the statement statement may following ones: may be be an an implication implication which which can can be be replaced by by an replaced an equivalence; equivalence; or or it it may may be bean aninclusive inclusivedisjunction disjunction which can can be which be replaced replaced by by an anexclusive exclusive one. one. These showthat that the the major These considerations considerations show major operations operations of of original original nomological statements statements are are used used in in aa way nomological way we we regard regard as aa reasonable. reasonable. They apply when the the operation occurs as as the major They apply likewise likewise when operation occurs major operation of of a major operation major factor factor of of aaconjunction conjunction which which isisoriginaloriginalnomological, because because the the given given proofs proofs can can be be constructed nomological, constructed for for each each such factor separately. However, an an important important distinction is to to be such factor separately. However, distinction is be made. If the made. the conjunction conjunction is is exhaustive exhaustive in in elementary elementary terms, terms, we we cannot cannot infer infer that that each each factor factor is is so, so, although although each each factor factor is is asserted asserted as true and and the theconcept conceptofofexhaustiveness exhaustiveness is is at atleast leastapplicable. applicable. But that each But we we can can conclude conclude that each factor factor isis exhaustive exhaustive in in elementary elementary terms through the total total statement statement(definition (definition19). 19). Going Going through terms except except for the the previous with this this qualification, we find find that that the the previous proofs proofs with qualification, we the cases cases where the the statement statement is not where not reduced reduced are are the the same same as as before; before; but but for for the the two two cases cases where where the the converse converse implication, implication, or or the theexclusiveness exclusiveness of of the disjunction, disjunction, is derivable, derivable, we we find that that these these relations relations are are derivable, not from from the the factor factor alone, alone, but but only from from the the total total statestatederivable, not ment. ment. This result result is is transferred transferred to tomajor majorterms. terms.Consider, Consider, for for instance, the statement instance, the statement (49)
(x){(/(x) :D g(x)]. [g(x)
J
56
ORIGINAL NOMOLOGICAL NOMOLOGICAL STATEMENTS STATEMENTS
in which the given given terms may be be elementary elementary or stand stand for for combicombinations of elementary terms. When we take take the elementary terms. When we the first first implication implication alone, it is not exhaustive in major for itself, itself, but but only alone, it is not exhaustive in major terms terms except except for only except for for the the total total statement. except statement.We We can can still still accept accept such such implications implications as reasonable reasonable ifif they they are are used used in in the the context context of of the the total total statement. as statement. We We shall shall later study study this this kind kind of of statement statement in inmore more detail detail (see (see definition definition 34). 34). An An illustration illustration where where the the equivalence equivalence operation operation occurs occurs in in major major
factors is given by the the formulation formulation of of an an exclusive exclusive disjunction disjunction of three terms ESL, p. p. 45): 45): terms (see (see ESL, --(50) D ffj(x)./3(xfl} (50) ( W I ( 4 D3 /2(x).f3(x)]. f z ( 4 .f,C41. [/2(x) 3 i ( 4 . f3(41> In contrast contrast to to (49), (49), this this statement statement cannot cannot be be replaced replaced by by aasingle single equivalence. Neither factor factor of is exhaustive exhaustive in in the the given given terms, terms, equivalence. Neither of (50) is but either except for for the the total statement. either one one is is exhaustive exhaustive except statement. The The statement and itit remains if the statement (50) (50) is is original-nomological; original-nomological; and remains so if equivalences are replaced replaced by by double implications. We We arrive arrive at at the equivalences are double implications. the
following following theorem: theorem : Theorem 7. Theorem 7. If aa conjunction conjunction is is original.-nomological, original-nomological, each each factor is exhaustive in major and elementary factor elementary terms except for the the
total statement. total stat-ement.Thus Thus the the only only cases cases where where a factor factor can can be be nonnonexhaustive in in major terms are the ones: the the factor may exhaustive the following following ones: be an implication which can can be be replaced by an equivalence; be implication which replaced by equivalence; or it it may be an which can can be be replaced by an may an inclusive inclusive disjunction disjunction which replaced by exclusive one. exclusive one. Summarizing, we may may say say that that propositional Summarizing, we propositional operations operations can can be be called reasonable reasonable when when they they stand stand as called as major major operations operations of of original original nomological statements, or as as major nomological statements, major operations operations of of major major factors factors of of such statements. statements. To such To go go beyond this result result and and define define reasonable reasonable operations for for secondary secondary places places of of aa statement, statement, appears operations appears extremely extremely difficult. Though Though the the requirement requirement of difficult. of being being reduced reduced controls controls every every unit within if it it is unit within the statement statement (theorem (theorem 1), l), it is is not not sufficient sufficient if is not combined combined with with the the requirement requirement of of exhaustiveness. exhaustiveness. But the the units in which which secondary units in secondary operations operations occur occur are are not not asserted asserted as as true true;; therefore criterion of of exhaustiveness exhaustiveness is is not not applicable, applicable, being being therefore the the criterion defined for statements only only (definitions (definitions 16 16 and and 18). 18). defined for true true statements have made made many many attempts attempts to togeneralize generalize this this requirement requirement in II have
ORIGINAL NOMOLOGICAL NOMOLOGICAL STATE)IENTS STATEXENTS
57 57
such a way such way that that every every operation operation within within a statement statement is is made made reasonable;; but but this reasonable this aim aim seems seems unattainable. unattainable. Consider, Consider, for for instance, instance, an implication in in a secondary place, place, i.e., i.e., standing standing as as the the implicate an of a major major implication. implication.ItIt is is not not asserted asserted to to be be true, true, since sinceits its truth truth of depends on on the major depends major implicans. implicans. But how how could could we we characterize characterize as reasonable without assuming assumingthat that itit is true? an implication implication as reasonable without true? The methods methods used used for for the thedefinition definitionofofnomological nomological statements statements could not not be be used, used, because because they they presuppose presupposetruth truth of of the certainly could statement in an statement an essential essential manner. manner. A A false false implication implication cannot cannot be be reasonable. Or assume assumethat that the statement in a negated reasonable. Or statement consists consists in negated implication. Then Then the the negation negation is is the the major major operation and and thus thus is a connective operation;but but the the implication implicationisisnot. not. It It is merely connective operation; merely a negated adjunctive The negation negated adjunctive implication. implication. The negation of of aa connective connective implication would be be something very different 380). different (see (see ESL, ESL, p. p. 380). It will It will be shown shown in chapter chapter 8 that for for certain certain special special cases, cases, at operations can be made reasonable in a somewhat least, secondary operations wider sense;; they are wider sense are then thencalled calledrelative relativenomological nomologicaloperations. operations. For For instance, instance, aa secondary secondary implication implication can can be be made madereasonable reasonable for for the case that itit is that the the case that is true, true, and and thus thus for for the the case case that the major major implicans implicans is true. However, However, within the the theory theory of of absolute, absolute,or orunconditional, unconditional, nomologicalstatements, statements,with withwhich whichwe we are are concerned in the nomological concerned in the present and the two chapters, the only way to present the following following two to improve improve the status is to to strengthen the methods the status of of secondary secondary operations operations is methods of reducing. Whether Whether aa unit unit is reduced reduced depends depends often often on on the the statement statement reducing. rules for stronger methods in which it is is imbedded. imbedded. Therefore, Therefore, rules methods of reducing reducing have have been been formulated formulatedininrequirements requirements1.7—1.9, 1.7-1.9, which which must now be be studied. studied. Let Let us use an implication for for an an illustration: illustration: must
(51)
(x)[f1(x) v 12(x) D g1(x).g2(x)]
Here the four Here four terms terms written written down down may be elementary elementary or or stand stand for for combinations of of elementary elementary terms. terms. Negating combinations Negating the the implicans implicans we we have have
(51a) (51a)
--
q1(x).g2(x)J (x){f1(x)./2(x) ( X ) [ f l ( 4 * f 2 ( 4V v g1(z) * g2(@1
to requirement When we When we now cancel cancel factors factors according according to requirement 1.7 and transform the resulting transform resulting disjunction disjunction into into implications implications we obtain obtain four implications of the the form four implications of form (Sib) (51b)
( W z ( 4
D 3 gk(Z)l
i,i, kk
=
1, 22 1,
58 5s
ORiGINAL 0 ItlGIXAL NOMOLOGICAL NOMOLOCICAL STAI'EMENTS STATEMENTS
guaranteethat that these four Requirements 1.7—1.8 Requirements 1.7-1.8 guarantee four implications, implications, whose conjunction conjunctionisis equivalent equivalentto to (51), (51), are are reduced; reduced; they they thus whose thus satisfy theorem theorem 7, and can satisfy can be be accepted accepted as as reasonable. reasonable. This result, conversely,makes makesthe the ‘or’ 'or' in conversely, in the theimplicans implicansof of(51) (51)reasonable. reasonable. 'f1(x) this implicans implicans is is written writtenininthe theform form ‘fi(Z)D3f2(x)', f 2 ( x ) ’ , this this aAnd n d if this secondary implication implication appears appears thus thus reasonable reasonable to to a certain secondary certain degree. degree. Another illustration isis given given by by the statement Another illustration statement of of an animplied implied equivalence, or double implication: equivalence, double implication : (52) (52)
(x){f(x) 3 D [d4 [g(x) 3 D W)l* h(x)]. [[h(x) (.){f(.) h ( 4 2D Yg(x)]} (4l)
If this this implication implication satisfies satisfies requirement requirement 1.8, 1.8, each each implication implication resulting from from canceling cancelingone one of of the the brackets brackets is reasonable in the resulting reasonable in sense explained sense explained previously. previously. An which violates violates 1.8 1.8 is is given given by by the statement An example example which statement (53) (53)
(x)[/(x) g(x) (z)[f(s)v vg (43 3 /(x).h(x)j f(s)-W9l
This statement is This is exhaustive in major and elementary elementary terms, and and it is reduced. But if we cancel the the term term 'h(x)', it reduced. But ‘h(x)’,the resulting resulting statement is not reduced, to the statement ment reduced, because because itit is equivalent equivalent to statement '(x)f(x)'. ‘(z)f(z)’.Thus Thus requirement requirement 1.8 1.8 rules out out unreasonable unreasonable forms forms like like (53), which which the the other other requirements requirements do do not not eliminate. eliminate. Note Note that that (53) (53) also ruled ruled out out if if the the implicans is is written written in in the the form '/(x) is also ‘ f ( x )3 3 g(s)’, because referto to possible possible transformations transformations of because requirements requirements 1.7—1.8 1.7-1.8 refer of the major major terms; terms; thus thus this thissecondary secondary implication implication is regarded regarded as unreasonable. In unreasonable. I n this thismanner, manner,requirements requirements1.7—1.8, 1.7-1.8, in achieving achieving a certain certain mutual mutual reducing reducing of of implicans implicans and implicate, implicate, contribute contribute to to the the reasonableness reasonableness of of secondary secondary operations. operations. An illustration illustration of is given, given, when when we we use use for for ‘g(z)’ 'g(x)' a, a negative negative An of (53) (53)is form, by the the statement form, by statement 'if ‘ifmasses masses do do not not have have attractive attractiveforces forces or or galaxies do not recede, galaxies do recede, then masses masses have attractive attractive forces forces and and galaxies have have an an enormous enormous gravitational gravitational potential’. potential'. This statement statement satisfies all the but violates violates 1.8. 1.8. Note Note that that satisfies all the requirements requirements 1.1—1.6, 1.1-1.6, but the statement statement is is ruled ruled out out by by 1.7 1.7 ifif the theimplication implication is is converted converted into aa disjunction: disjunction : 'masses ‘masses have have attractive attractive forces forces and andgalaxies galaxies recede, or masses have attractive forces recede, or masses have forces and galaxies galaxies have have an an enormous gravitational gravitational potential’. potential'. The duplication of the first part first is here is here regarded regarded as as an anunnecessary unnecessarycomplication. complication.
ORIGINAL NOMOLOGICAL ORIGINAL NOMOLOQICAL STATEMENTS STA!PEMEN!CS
59
The reducing process can of course The process required required by by1.7—1.9 1.7-1.9 can course always always be carried be carried out. The The terms terms violating violating these these requirements requirements are not not redundant; but redundant; but they they can canbe becanceled canceled if they are are added, added, in suitable suitable form, form, outside outside the implication. implication. For For instance, instance, (53) (53) then then assumes assumes the form form (53a) (53a)
(x){[g(x) D 4341. h(x)].f(x)} (4{[B(4 3 f(41
This form form is is equivalent equivalentto to(53) (53)and andsatisfies satisfiesrequirements requirements1.6—1.8. 1.6-1.8. The sentence used for for the the illustration thus read: 'masses The sentence used illustration would would thus ‘masses have attractive forces, and galaxies recede or have an enormous have an enormous have attractive forces, and galaxies recede gravitational potential'. potential’.
Iv IV DERIVATIVE DERIVATIVENOMOLOGICAL NOMOLOGICAL STATEMENTS STATEMENTS
We now now turn We turn to toderivative derivativenomological nomological statements, statements, beginning beginning with the the definition: definition :
also called is nornological, nomological, also Definition D e f i n i t i o n 28. 28. AA statement statement is called it is nomological iinn the nornological the wider wider 8ense, sense, if it is deductively deductively derivable derivable from a set of statements. The The major major operation of aa of original original nomological nomological statements. nomological statement statement is called connective operation. operation. (I-terms). (I-terms). nomological called aaconnective Definition 28 isis the the same Definition 28 same as given given in ESL, ESL, p. p. 371. 371. But But since since definition 27 lays down definition 27 down narrower narrower rules rules for fororiginal originalnomological nomological statements than does given in ESL, does the thecorresponding corresponding definition definition given ESL, 371, requirements 1.5—1.9 being added, both the classes p. 371, requirements 1.5-1.9 being added, both the classes of p. original statements and of original nomological nomological statements of nomological nomological statements statements are down. However, However, as as in in the the previous presentation, the the are narrowed narrowed down. previous presentation, class statements includes includes all all tautologies. tautologies. In In parclass of of nomological nornological statements ticular, written in the ticular, tautologies tautologies written the calculus calculus of of propositions propositions are are introduced with definition 28 into the class of derivative nomointroduced with definition 28 the class of derivative nomological statements. They are logical statements. are not not original originalnomological nomological statements because they possess possess no no all-operators all-operators;; but but they they result from tautobecause they tautologies containing containing all-operators all-operators and and satisfying satisfying the the requirements for logies requirements for original nornological nomologicalstatements statements by by aa derivation derivation process process in which which functionals are replaced functionals replaced by by constants. constants. 1 statement may A nomological A nomological statement mayviolate violatethe therequirements requirements1.2—1.9, 1.2-1.9, but itit cannot cannot violate violaterequirement requirement 1.1, 1.1, since since itit isisverifiably verifiably true. true. Dropping of major major all-operators all-operators and and transition transition to free variables is Dropping of regarded as abandoning requirement 1.3; statements thus regarded abandoning requirement 1.3 ; statements thus resulting resulting are nomological, but not original-nomological. nomological, but not original-nomological. Some remarks remarks about about conjunctions conjunctions must must now now be be added. added. If two Some two original nomological nomologicalstatements statements pp and and q, q, which which are are synthetic, are are combinedinto into the the conjunction combined conjunction p.q, p.q, this thisconjunction, conjunction, though though nomological,need need not not be A set set of nomological, be original-nomological. original-nomological. A of original original 1
See ESL, p. See p. 139, 139, rule (a). (a).
DERNATIVE NOMOLOGICAL NOMOLOQICAL STATEMENTS
61 61
nomologicalstatements, statements, therefore, therefore, is is not not always nomological always equivalent equivalent to to one one nomological statement, though though it is nornological statement, is equivalent equivalent to toaanomological nomological statement, namely, to the conjunction of the the statements of the the set. statement, conjunction of However, it it will will now now be be shown shownthat that this this conjunction However, conjunction deviates deviates from from an original original nomological nomologicalstatement statementonly onlyininminor minorpoints, points,and andthat that itit an can be statement (theorem can be replaced replaced by by an an original original nomological nornological statement (theorem9). 9). is easily easily seen seen that thatrequirements requirements1.1—1.3 1.1-1.3 and and1.7—1.9 1.7-1.9 are are It is satisfied by by the conjunction. that requirement conjunction. We shall now show that requirement 1.5 is is satisfied by the 1.5 satisfied by the conjunction. conjunction. Let two nomologicalstatements statements be be given; given; we we write write them them Let two original original nomological in in C-form C-form (definition (definition 10): 10) : (54) (54) (55) (55)
(e1 v V ... ... vV e,),] ... . (el )...( (efvv ... v e&] )...( )[(ev )[(e; v ... ... v eL)l. ... . (e;
)...( . .( (( ). ((
)[(el v ... ... v
In the the parentheses, parentheses, some some of the the elementary elementary terms terms occur occur in in negated negated form;; instead form instead of of writing writing corresponding corresponding negation negation lines, we we have have added to the the parentheses, that each added subscripts subscripts to parentheses, so so that each subscript subscript indicates indicates aa particular particular combination combination of of positive positive and and negative negative elementary elementary terms. terms. Some of the the elementary elementary terms Some of terms of of (55) (55) may may be be identical identical with with elemenelementary terms terms of of (54). (54). We We now now write write the the conjunction conjunction of of the the two two statements statements in in one-scope one-scope form, form, as asfollows. follows. Argument Argument variables variables governed by by all-operators all-operators are are given given the the same same names names in in both both statestategoverned ments, ments, as far far as as this thisisispossible; possible;however, however, variables variables governed governed by existential operators in (55) existential (55) are given given names names different different from those those of the existential variables in in (54). (54). We Wethus thus arrive arrive at at the form: of existential variables form: (56) )...( )[(e1V...v )[(e,v ...ve,),. (56) (( )...( Cm)j
.... (elv...ve,),.(e;v ...ve;),. ... . (e;v ...ve&]
This is is not not aa C-form, but aa shorter This C-form, but shorter conjunctive conjunctive normal normal form. form. Yet Yet if (54) and and (55) if (55) satisfy satisfy requirement requirement 1.5, 1.5, (56) (56) cannot cannot possess possess aa selfselfcontained residual which which is is purely contained conjunctive conjunctive residual purely existential. existential. Such Such aa residual residual can can only only result result when, when, by by the the use use of of relations relations (19), (19), certain certain factors merge merge in in such factors such aa way way that thatterms termscontaining containingvariables variables governed by by all-operators drop out. out. But since governed all-operators drop since either either (54) (54) or or (55) (55) contains (56) could contains at least least one one existential existential operator operator — - otherwise otherwise (56) could the elementary not have elementary terms not have an an existential existentialresidual residual— - the terms stemming stemming from same as as those from one one formula formula cannot cannot be be throughout throughout the the same those stemming stemming
62
DERIVATIVE NOMOLOGICAL NOMOLOGICAL STATEMENTS STATEb6ENTS
from from the other other formula, formula, because because existential variables variables in the two two formulae have different formulae different names. names. However, itit may be However, be possible possible to to overcome overcome this this difference difference by merging of of variables, variables, using using such tautologies as merging (57) (57) (58) (58) (59) (59)
y) = (3Y)(x)f(z, x) (%yy)(@f(x, Y) 4
y) Y)
(qx)f(x) pP ( 3 5 W Y ) f ( 4./(y). f(Y) * pP = (34f(4* v 9(Y) = ( 3 4 9 ( 4*.pP g(y).p] ( 3 4 ( m ) [ av g(x).pJ. g ( 4 . P I * [a v *PI= .
By By the use use of of such such formulae, formulae, it may may be be possible possible to transform transform (56) (56) equivalently into into aa form in which at at least equivalently least one one statement statement contains contains smaller number number of of elementary elementary terms, terms, so so that that we we arrive arrive at at the the form aa smaller form (60) (60)
... vvem_8)i 5 ( )...( )[(elvv ... e,,&. )...( )[(e1 (
(e1 v v ... ... . (el ... v em-Jp-Q
( .(ev .(e;v ... ... v e i - ~ ... ~ .. (ev (e;v ... ... vei-t)lY-o
?
This This statement, statement,in inwhich which 8s > both,may maypossess possessaaselfself> 00 or t >>00ororboth, contained conjunctive conjunctive residual residual which which is is purely existential. contained existential. To have To have an an illustration illustration for for elimination elimination of of elementary elementary terms, terms, consider the two statements consider the statements (61) (61)
(62)
(4(3Y)Cf(% Y) v d Y ) 1 (x) (4(y) (Y)[/(x, [f@7 Y) vv g(y)] dY)l
Their conjunction conjunction is in in one-scope one-scope form form (63) (63)
(x)(ffy)(z){/(x, (4(3Y)(z)Cf(2, y) Y) vv g(y)]. g(Y)l- [f(x, C f k z) 4 vv y(z)] g(4l
Using the tautology tautology (57) (57) and and putting putting there there ‘2’ 'z' for 'x', Using the ‘x’,we we see that (63) is equivalent (63) is equivalent to (64) (64)
-
( 4 ( 3 Y ) ( 4 C f ( G y) v S W 1 - [f(G y) Y)vv g(y)]. 9(Y)l*[f(x, [f(%z)2 )Vv g(z)] s(z)l
Here the first first brackets brackets represent represent aa factor factor not notcontained contained in in (63), (63), and this factor with the the second factor in in such such aa way way that that factor merges merges with second factor we we arrive arrive at at the thefollowing following statement, statement, which which is is equivalent equivalent to to (63) (63) (65) (65)
(4(3Y)(z)(g(Y)
*
{/(x, Ef(% z) 4 vv g(z)]} s(z)l)
Whereas in the Whereas (63), in the form form written, written, does does not notpossess possess any anyself-conself-contained conjunctive conjunctive residual residual which which is purely existential, existential, the the equivequivtained is purely form (65) (65) does does possess possess such residual, namely, namely, the part alent form such aa residual, the part
DERIVATIVE NOMOLOGICAL NOMOLOUICAL STATEMENTS STATEMENTS
63 63
(63) is is not not general ‘($y)g(y)’. Therefore, Therefore, (63) general in in self-contained self-contained factors. factors.
23 in in such such aa way way that that itit The latter term term was was defined d e h e d in definition definition 23 refers to to the complete refers complete matrix of of aa statement; statement;this thisfollows follows because because the term 'conjunctive t!ie ‘conjunctive residual' residual’ used used in in definition definition 23 refers refers to the C-form of of aa statement. Since C-form Since (65) (65) is not not propositionally propositionally derivable derivable from (63), but but merely from merely operator-derivable operator-derivable (see (seethe thediscussion discussionof of (12) (12) and (14)), (63) does doesnot not contain contain the the complete matrix matrix of the the stateand (14)), (63) statedoes not not represent the ment. Although even (65) does the complete complete matrix matrix of (63), it shows of (63), it shows already already the the existence existence of of aaconjunctive conjunctive residual residual contrary to 23. contrary to definition definition 23. Now Now it is is easily easily seen seen that that (61) (61) is is not not exhaustive, exhaustive, because because its disjunctive residual disjunctive residual ‘(Ry)q(y)’is true. true. Therefore, Therefore, (65) (65) does does not represent a conjunction conjunction of of two twooriginal originalnomological nomological statements. statements. represent We shall shall now now show show that that whenever, the the conjunction can be given a We form like like (65), form (65), aatt least one one of of the the individual individual statements statements is not exhaustive. exhaustive. This the transformation This is is shown shown as as follows. follows. IIn n the transformationleading leadingfrom from(56) (56) to we follow followthe therule rule always always to to keep keep the the statement statement in to (60) (60) we in one one scope scope form; furthermore, if operators redundant, we we do do not form; furthermore, if operators become become redundant, cancel them. them. Then Then the the operator set remains the same on each step; step; and since we use only equivalent transformations, each and since we use only equivalent transformations, each step step represents an an equivalent equivalent version version of of the the original original conjunction. conjunction. Now Now of there must be be one one step step(Or (or several) several) on which through merging merging of variables aa factor variables factor is is produced produced which which contains contains only only such such elementary elementary terms as terms as occur occur in in the the first first (or (orsecond) second) formula, as for instance on the step (64), where the first brackets step (64), first brackets represent represent such such an an emergent emergent term. term. The The statement statement (56) (56) thus thus isisequivalently equivalently transformed transformed into into (66) (66)
(e1v ... . (el v .,. ... vern)i,.(cjv v e m ) p (el . v ... ... v em),i+l ... v ( )...( )[(el v ... (evv .. . v e;)’] .(el’ v ... v e& . ... (e;
.
we cancel here the factors When we When cancel here factors containing containing terms terms with with prime prime marks, marks, we we derive derive a statement statement which which after aftercanceling canceling redundant redundant operators has has the the operator operator set set of of (54). But this this statement is a operators (54). But statement is conjunctive conjunctive extension extensionof of (54), (54), and and according according to to the the remarks remarksfollowing following definition 21, is is thus thus equivalent definition 21, equivalent to to aadisjunctive disjunctiveresidual residualof of(54). (54). We thus thus have shown that that aa disjunctive is true, true, or disjunctive residual residual of (54) (54) is in other words, words, that that (54) (54) is is not not exhaustive exhaustive in in elementary elementary terms. terms.
64 64
NOMOLOcICAL STATEMENTS DERIVATIVE NOMOLOCICAL
The proof proof is is easily easily extended extended to to aa conjunction of three three or more The conjunction of more statements, because there must always certain statements, because there always be a step step on which aa certain factor is is added added to to one one of ofthe the statements statements in in such such aa way waythat that aa certain certain factor elementary term drops drops out; through elementary term through this addition, addition, aa conjunctive conjunctive extension of that statement we have have the extension of statement is is derived. derived. Therefore, Therefore, we following following theorem: theorem : Theorem T h e o r e m 8. 8. If aa conjunction conjunction of two or more synthetic statestatements ments is is not not general general in in self-contained self-contained factors, factors, whereas whereas each each statestatement is so, then at least so, then least one one of the statements statements is is not exhaustive exhaustive in elementary terms. terms. This theorem is of great importance theorem is importance within the theory theory of of synsynthetic nomological statements.ItIt shows showsthat that a conjunction thetic nomological statements. conjunction of original nomologicalstatements statements cannot cannot violate requirement original nomological requirement 1.5. 1.5. If the the theorem theorem were were not not true, true, we we could could transcribe transcribe any any purely purely existential statement into existential into aaconjunction conjunctionofoforiginal originalnomological nomological statements, and statements, and thus thus make make the theexistential existentialstatement statementnomological. nomological. in (65) the the part ‘(gp)g(y)’ might For instance, instance, in might mean mean 'there ‘there exists exists copper', whereasthe the other other part part may copper’, whereas may be be given given any any suitable suitable interinterpretation by a true pretation true statement. statement. Then Then the the statements statements (61) (61)and and (62) (62) would be true, being derivable form (65), and they would thus would be true, being derivable form (65), and they would thus be be original nomological nomologicalstatements. statements.I In this way, way, the the statement statement 'there original n this ‘there exists copper’ copper' would would be be made made aa law law of of nature. nature. Such is exists Such aa procedure procedure is excluded excluded by by theorem theorem 6. 6. In In our our example, example, the the procedure procedure breaks breaks down down because (61) (61) is is not not exhaustive because exhaustive and and thus thusnot notoriginal-nomological. original-nomological. The only requirements which a conjunction requirements a conjunction of of original original nomonomological statements can can violate violate are are requirements and 1.6. logical statements requirements 1.4 and 1.6. The latter possibility possibility is obvious; obvious; for instance, the the two two statements statements may may have a common would be be redundant redundant in their common factor, which which would their conconjunction. junction. But it it isisof ofcourse coursealways always possible possible to reduce reduce the conconjunction, junction, for for instance, instance, by by canceling canceling the the common common factor factor once. once. That aaconjunction conjunction ofoftwo twooriginal originalnomological nomological statements, statements, though exhaustive exhaustive in major terms, terms, need need not not be be exhaustive exhaustive in in eleelethough in major mentary terms, terms, is is shown shown by by the thefollowing following example. example. Consider Consider the the mentary two statements statements (67a)
(z)[/(x) D g(x)]
(67b)
(x)[h(x) J lc(x)]
DERIVATIVE NOMOLOGICAL NOMOLOGICAL STATEMENTS STATEMENTS
65 65
Assumethat that furthermore They They may may be be original-nomological. original-nomological. Assume furthermore the the relation relation holds holds (67c) (67c)
(x)[ /(x) 3 D M41 h(x)] (W(4
Then is not not exhaustive Then the the conjunction conjunction of of the thestatements statements(67a-.-b) (67a-b) is exhaustive in in elementary elementary terms. terms. However, this deficiency can be be remedied. remedied. We We have have merely However, this deficiency can merely to add, to to the theconjunction conjunction of of the the statements, statements, the thecondition condition which which
establishes the elementary terms; then the establishes aa relation relation between between the elementary terms; the resulting statement isis exhaustive in elementary terms. In In the resulting statement exhaustive in elementary terms. the example given we we thus thus construct of the three example given construct the conjunction conjunction of three
statements whichisis original-nomological. original-nomological.ItIt is is not not always statements(67a—c), (67&c), which always possibleto to make make the the addition addition in in the the form form of of aa separate separate statement; statement; possible if the thetwo twooriginal originalnomological nomological statements statementspossess possess existential existential operators, it it may to make make the the addition addition directly to the operators, may be be necessary necessary to directly to the the one-scope form of of their their conjunction. conjunction. We We thus thus arrive operand of the one-scope form arrive at the the theorem: theorem : Theorem T h e o r e m 9.9.A conjunction A conjunctionofoftwo twosynthetic syntheticoriginal originalnomonomological statements either logical statements either isisoriginal-nomological, original-nomological, or or can can be be made made so so by adding some part and reducing. and reducing. Thus Thus far far we we have have studied studied the thetransition transitionfrom fromtwo twooriginal originalnomonomological statements to logical statements to their their conjunction. conjunction.Oonsidering Considering the the converse converse transition, that ifif aasynthetic transition, we we see see easily easily that syntheticconjunction Conjunctionisisoriginaloriginalnomological, itsfactors factorsneed neednot not be be so. so. But But from theorem nomological, its theorem 7 we we know that that its factors are, at least, reasonable to some degree, being its factors axe, at least, reasonable to some degree, being exhaustive the total total conjunction. this property property exhaustive except except for for the conjunction. A A study study of of this may be may be postponed postponed to to the thediscussion discussion of of definition definition 35. 35.
V V
ADMISSIBLE ADMISSIBLE STATEMENTS STATEMENTS
We now introduce introduce aa new new notation, notation, in in which which an an order order of of truth truth is defined, and which will be useful defined, and which will useful for our our further further investigations. investigations. tautology is true of third order; order; aa synthetic synthetic Definition D e f i n i t i o n 29. 2 9 . A tautology of third
nomological statement is order; aa statement statement which nomological statement is true true of second second order; which is is verifiably true true but of first first order. order.IfIf pp is is true true verifiably but not notnomological, nomological, is is trtie true of is false of k-th order, order, pj5 is false of of k-th k-th order. order. (I-terms). (I-terms). of
Note that that the thesecond-order second-order character is not restricted restricted to to original original nomologicalstatements, statements, but but also applies nomological applies to all all derivative derivative nomonomological statements. Likewise, the class of third-order statements logical statements. Likewise, the class of third-order includes both both those and includes those tautologies tautologies which which are are original-nomological original-nornological and those which those which are not. Definition 30.3 0A. set has order Definition A of set statements of statements hasthe t,he orderofofthe thelowestlowestorder in it. (I-term). order statements statements contained contained in (I-term). Theorem statement has the order Theorem 10. 1 0A . true A true statement has the orderofofthe thehighesthighestorder order sets from from which which it it isisderivable. derivable. This theorem theorem follows followsimmediately immediatelyfrom fromthe the nature nature of of analytic This and definitions 29—30. and synthetic synthetic statements, statements,and andfrom from definitions 29-30. that nomological 65, that It It was was shown shown in in ESL, ESL, §3 65, nomological statements can be used to define modalities. This chapter used define modalities. This chapter of of my my previous previous theory theory remains remains unchanged, unchanged, when when the thenew newdefinition definitionof of original originalnomological nomological statements is for it it and statements is assumed assumed for and it it is is thus thus referred referred to to the thenew new class class of nomological statements. However, However,itit was was also also pointed pointed out out that nomological statements. for the for the definition definition of of reasonable reasonable operations, operations,the theclass classof ofnomological nomological statements is too statements too wide, wide, while while the theclass classofoforiginal originalnomological nomological statements is too narrow. For this reason, of reason, an intermediate class of nomological statements statements in in the nomological the narrower narrower sense sense was was defined defined in ESL, ESL, 1. This This definition definition will will now now be be changed, changed, for for the the reasons reasons explained explained p. 37 1. p. above. above. The new new intermediate intermediate class, class, which which isisless lesscomprehensive comprehensive than the the older olderone, one,will willlikewise likewisebe becalled callednomological nornological iin n the the narrower sen8e; in order to have a brief name, we shall speak of narrower sense; in order to have a of
ADMISSiBLE STATEMENTS ADBIISSIBLE STATEMENTS
67 67
admissible statements. statements.This Thisclass classisissubdivided subdivided into into the the two admissible two classes classes of fully admissible and semi-admissible statements, which of admissible and semi-admissible statements, which will will be be defined in in order. order. The The term 'admissible defined ‘admissible operation', operation’, which which refers to the the major major operations operationsof of both both subclasses, subclasses, is is proposed proposed as as the the explicans explicans of the vague vague term term'reasonable ‘reasonableoperation'. operation’. The definition will again again be be given The definition will given by means means of of I-requirements I-requirements and V-requirements. First, some and V-requirements. First, some new new terms must must be bedefined. defined. Definition A statement p which D e f i n i t i o n31. 3 1. A statement p whichisistrue trueofofthe theorder order kk is quasi-exhaustive (inmajor majorororinin elementary elementary terms) terms) ifif none none of of its quasi-exhaustive (in disjunctive residuals residuals (in (in major major or or in in elementary elementary terms) terms) is is true true of of an an disjunctive order (I-term for for elementary elementary terms, terms, V-term V-term for for major major terms.) terms.) order 2 k. (I-term It It isis not not necessary necessary here here to to write write 'verifiably ‘verifiably true', true’, because because the qualifier ‘verifiably’ 'verifiably' is is already already included included in in the the definition definition of of ‘true 'true of qualifier of the order order k'. k’.Note Note that that aastatement statementwhich which isisexhaustive exhaustive is is also also quasi-exhaustive. quasi-exhaus tive . Definition p, pwhich D e f i n i t i o n32. 3 2A . statement A statement , whichisistrue trueofofthe theorder order kk is is quasi-exhaustive exceptfor forpp(in (inmajor major or or elementary elementary terms) quasi-exhaustive except terms) ifif none none of its its disjunctive residuals residuals is is derivable derivable from from pp,, but but all those of its of disjunctive residuals residuals which which are are true true of of an an order are derivable derivable disjunctive order 2 ic k are from p. p. (I-term from (I-term for for elementary elementary terms, terms, V-term V-term for for major major terms.) terms.) Note that that aastatement statementwhich whichisisquasi-exhaustive quasi-exhaustive is is also also quasiquasiexhaustive except except for for pp,, for for any p. exhaustive p. We to the We now now proceed proceed to the definition definition of of fully fully admissible admissible statements, statements, introducing the introducing the following following I-requirements. I-requirements. I-REQUIREMENTS FOR I-REQUIREMENTS FORFULLY FULLYADMISSIBLE ADMISSIBLESTATEMENTS STATEMENTS
2.1. 2.1. The statement pp must must be be deductively deductively derivable derivable from a set
of of original original nomological nomological statements. statements.
2.2. 2.2. The statement pp must must be bequasi-exhaustive quasi-exhaustive in in elementary elementary terms. (Definition 31). terms. (Definition 31).
It is not not necessary necessary to tointroduce introducespecific specific V-requirements; V-requirements; we we :it simply take over simply over the theV-requirements P-requirementsofoforiginal originalnomological nomological statements, We thus thus define: statements, i.e., i.e., requirements requirements 1.6—1.9. 1.6-1.9. We dehe: Definition p ispfully admissible A statement is fully admissibleififititsatisfies satisfies D e f i n i t i o n33. 3 3A. statement requirements 2.1—2.2 and 1.6-1.9. 1.6—1.9.The Themajor majoroperation operationof of aa fully requirements 2.1-2.2 and fully admissible statement is called a fully admissible operation. called a fully admissible operation. (V(Vadmissible statement terms). terms).
68 88
ADMISSIBLE STATEMBNTS ADMISSIBLE
It the remarks that every It is is easily easily seen seen from from the remarks added addedto todefinition definition31 31 that every
original nomologicalstatement statement is fully original nornological fully admissible. admissible. The subclass subclass of fully admissible admissible statements statements which which are arenot notoriginal-nomological original-nomological will be be called called fully fully admissible admissible by by derivation. derivation.
The use use of of the the concept concept ‘quasi-exhaustive’ 'quasi-exhaustive' is seen in in its its application application The is seen to implications. If 'a‘aD implications. If 3 b' b’ is true true of of second second order, while 'a' ‘a’isis false false of first first order, order, the the implication, of implication, though though not not exhaustive, exhaustive,is is reasonable, reasonable, being quasi-exhaustive. Ofthis this kind kind are the being quasi-exhaustive. Of the usual usualconditionals conditionals contrary contrary to fact. 'If ‘If there there had had been been aacurrent current in inthe thehigh highvoltage voltage wire, the the man who wire, who touched the the wire wire would would have been been killed.' killed.’ The The
implicansrefers refershere heremerely merelytotoaa matter matter of of fact which implicans which did not occur, whereas whereas the implication implication expresses expresses a physical physical law. law. In I n contrast, contrast, an anunreasonable unreasonable implication implication can can be be constructed constructed as as follows. 'a' and 'b' follows. IfIf ‘a’ ‘b’ are are true true of of the the 2nd 2nd order, order, 'a‘aD 3 b' b’ is also also of 2nd order, order, being being derivable derivablefrom from‘a’'a'and and ‘by. 'b'. But But 'a‘aD is not 2nd 3 b’b' is quasi-exhaustive, becauseits itsresidual residual‘b’, 'b', which which isis the same quasi-exhaustive, because same as as true of the same order as as 'aD ã.b', ‘'a.b a .b v 6. b’, is true ‘a 3b'. b’. ItItisisdifferent differentwhen when 'a 3 D b’b' isis true true of of 3rd 3rd order, order, while while ‘a’ 'a' and and 'b' then ‘a ‘b’ are are of of 2nd 2nd order; then is quasi-exhaustive and, ifif also ‘'aa D 3 bb'’ is quasi-exhaustive and, also the further further requirements requirements are satisfied, in the restricted restricted sense. sense. A A tautological tautological satisfied, nomological nomological in implication is thus implication between between synthetic synthetic nomological nomological statements statements is considered as reasonable. reasonable. For For instance, instance, the the implication, implication, 'if Newton's considered as ‘if Newton’s law law of gravitation gravitation holds, Kepler's Kepler’s laws laws of of planetary planetary motion motion are true', appears true’, appears reasonable. reasonable. An An unreasonable unreasonable implication implication is given given by by the the statement, statement,'if‘ifNewton's Newton’slaw lawof ofgravitation gravitationholds, holds,Ohm's Ohm’slaw law of of electric electric circuits circuits is is true'. true’.This Thisimplication, implication,though thoughnomological nornological because derivable from from the the truth truth of taken separately, is because derivable of these laws taken not it is not quasi-exhaustive, quasi-exhaustive, since since it is of of the the same sameorder orderas asits itsimplicate, implicate, and thus isis not not admitted admittedas asreasonable. reasonable. It presently that that for It will will be be shown shown presently for tautologies tautologies the the class class of of semisemiadmissible statements is is empty; therefore, admissible statements therefore, the the terms termsadmissible admissible and fully for tautologies. However, not not all fully admissible admissible coincide coincide for tautologies. However, all tautologies are admissible. Now requirements requirements 2.1 2.1 and and 2.2 tautologies are admissible. Now 2.2 are are always satisfied satisfied for for tautologies, the latter always tautologies, the latter requirement requirement because because canceling a term in the D-form leads to a synthetic and thus canceling a term in the D-form leads to a synthetic and thus to a lower-order statement. tautology is is lower-order statement.This This holds, holds, too, too, when when the tautology specialized. Therefore, Therefore, only specialized. only the theV-requirements 7-requirements1.6—1.9 1.6-1.9 select, select,
ADMISSIBLE STATEMENTS ADMISSIBLE STATEMENTS
69 69
among the the tautologies, ones. Note Note that that conjunctions among tautologies, the the admissible admissible ones. conjunctions of tautologies are never admissible, because they they are not reduced admissible, because reduced
(definition 8); the the same same holds holds for for a conjunction of a synthetic (definition 8); conjunction of synthetic statement with with aa tautology tautology (definition (definition 7). 7). Examples Examples of of inadniisinadmissible and admissible tautologies are are given given on on the sible and admissible tautologies the left-hand left-hand sides sides
and and right-hand right-handsides sidesofof(8)—( (8)-(10), lo), respectively. respectively. In the was shown shown that that for In the discussion discussion of of the theforms forms(50)—(60) (50)-(60) itit was for reduced statements statements the exhaustiveness in in elementary elementary reduced the requirement requirement of of exhaustiveness
terms leads that these terms leads to the the consequence consequence that these statements statements are are exexhaustive in in major major terms terms to to aa sufficient extent. It It is seen that that haustive sufficient extent. is easily easily seen similar conclusions conclusionscan can be be drawn similar drawn with respect respect to toquasi-exhauquasi-exhaustiveness. If there there exists exists aa relation of the same order as, or of aa stiveness. If relation of order as, higher order than, than, the the statement, and if higher order if this this relation relation restricts the major terms, terms, it must major must be bederivable derivable from from the thestatement, statement,because because otherwise itit would would also alsorestrict restrict the the elementary elementary terms, terms, and and thus thus the the statement statement would would not be be quasi-exhaustive quasi-exhaustive in in elementary elementary terms. terms. For of the For these these reasons, reasons, we we can can derive derive the thefollowing following analogues analogues of theorems whichagain againhold holdboth bothfor forsynthetic synthetic and and for for analytic theorems 6—7, 6-7, which analytic statements: statements : Theorem If aa statement it is 1 1 . If statement isis fully fully admissible, admissible, it is quasiquasiT h e o r e m 11. exhaustive exhaustive in major terms except except for for itself. itself. Thus Thus the theonly onlycases cases where the statement where the statement can can be benon-quasi-exhaustive non-quasi-exhaustive in in major major terms terms are ones: the the statement statement may are the the following following ones: may be be an animplication implicationwhich which can or it may can be replaced replaced by an an equivalence; equivalence; or may be bean aninclusive inclusive disjunction can be replaced disjunction which which can replaced by by an anexclusive exclusive one. one. If aa conjunction conjunction is is fully fully admissible, admissible, each each factor factor Theorem 12. 1 2 . If is quasi-exhaustive in major major and and elementary terms except except for the the quasi-exhaustive in elementary terms total statement. factor can can be be nonnonstatement. Thus Thus the the only only cases cases where where a factor quasi-exhaustive in major major terms terms are the ones: the the factor factor quasi-exhaustive in the following following ones: may be an an implication implication which can be replaced replaced by an an equivalence; equivalence; or itit may may be be an aninclusive inclusive disjunction disjunction which which can can be be replaced replaced by by an an exclusive one. exclusive one. The synthetic acceptedby by theorem theorem 11 11 as as fully The synthetic implications implications accepted fully admissible resembleas asto totheir theirstructure structure those those admitted admitted by by theorem theorem admissible resemble nomologicalstatements. statements.IIn particular, aa synthetic 66 into original original nomological n particular, implication which can can be be replaced by an implication which replaced by an equivalence equivalence can can be be fully fully admissible onlyifif its its major if they are admissible only major terms terms are compound; compound; if are
70
ADMISSIBLE STATEMENTS STATEMENTS
elementary terms, terms, such elementary such an implication implication is is still still ruled ruled out. out. These These implications will will be be admitted later implications later (see (seethe thediscussion discussionfollowing following theorem 13). 13). IIn theorem n contrast, contrast, it it was was explained explained for formula formula (45) that analytic implications implicationsofofthis this kind kind are are accepted. analytic accepted. IIn n addition addition to statement (45), statement (45)) which which isisoriginal-nomological, original-nomological, we we accept accept as fully fully admissible the form form 'a a', which which because because of of its its missing missing allalladmissible the 'a. D 3 a', operator operator is is not notoriginal-nomological. original-nomological. We now We now turn to the the definition definition of of semi-admissible semi-admissible statements, statements, for which for which purpose purpose we we introduce introduce the thefollowing following definitions definitions and and distinguished from from fully are distinguished requirements. These requirements. These statements statements are fully admissible ones in in that that they they are admissible ones are not not quasi-exhaustive quasi-exhaustive in in elementary elementary terms; i.e., Of this this kind kind are certain terms; i.e., requirement requirement 2.2 is abandoned. abandoned. Of statements referred to in the last referred to last theorem. theorem. In I n contrast contrast to tooriginal original nomological statements, the the V-requirements V-requirements 1.6—1.9 nomological statements, 1.&1.9 are are not not sufficient for semi-admissiljle semi-admissible statements. statements. We We need need certain sufficient for certain requirerequirements concerning in major major terms, which concerning quasi-exhaustiveness quasi-exhaustiveness in which is is no longer longer derivable derivable when when 2.2 2.2 is is given given up. up. Definition statement pp is not quasiD e f i n i t i o n 34. 34. If aa nomological nomological statement quasiexhaustive in in major exhaustive major terms except except for for itself, itself, it isis called called supplesupplementableififititcan canbe bemade, made,without without any any change, change, aa factor factor in in aa reduced reduced mentable conjunction such such that that p is conjunction is quasi-exhaustive quasi-exhaustive in major terms except for for the the conjunction. conjunction. (V-term). (V-term). Of the following has no Of following requirements, requirements, 1.10* 1.10" has no analogue analogue among among previous requirements. requirements. In In contrast, previous contrast, 1.7*_i 1.7"-1.9" are merely stronger stronger forms of forms of the thecorresponding correspondingrequirements requirements1.7—1.9. 1.7-1.9. V-REQUIREMENTS FOR FORSEMI-ADMISSIBLE SEMI-ADMISSIBLE STATEMENTS STATEMENTS
The statements The statements described described in in the thecancellation cancellation condition condition of requirement requirement 1.7 1.7 must must not not only only be be reduced, but must of reduced, but also be quasi-exhaustive in major major terms, terms, or be be so so except except also be quasi-exhaustive in for themselves, for themselves, or be be supplementable. supplementable. 1.8*. 1.8*. The The statements in requirement requirement 1.8 1.8 must must not statements described described in only satisfy but also only satisfy requirements requirements 1.6—1.7, 1.6-1.7, but also 1.7*. 1.7". 1.9*. 1.9". The The statements statements described described in requirement requirement 1.9 1.9 must must not not l.7*_1.8*. only satisfy but also only satisfy requirements requirements 1.6—1.8, 1.6-1.8, but also 1.7"-1.8". 1.7*. 1.7".
1.10*. 1.10'.
The statement statement p must in major major The must either either be be quasi-exhaustive quasi-exhaustive in
ADMISSIBLE STATEMENTS STATEMENTS
71 71
terms, or be into terms, or or be be so except except for for itself, itself, or be supplementable supplementable into aa conjunction conjunctionwhich whichsatisfies satisfies1.6, 1.6,1.7*_1.1O*, 1.7"-1.10*, 2.1. 2.1.
In refers to In the thefollowing following definition, definition, the theterm termnon-con4unetive non-conjunctive refers to statements, or to factors which do do not not have have a statements, or factors in aa conjunction, conjunction, which conjunction as their major conjunction as major operation. operation.
Definition statement pp is D e f i n i t i o n 35. 3 5 . AAnon-conjunctive non-conjunctive statement is semiadmissibleifif it it satisfies 2.1 but but is admissible satisfies requirements requirements 1.6, 1.6, 1.7*-1.10*, 2.1 is not fully if it it satisfies not fully admissible. admissible. A conjunction conjunction pp isissemi-admissible semi-admissible if satisfies requirement each of of its itsnon-conjunctive non-conjunctive factors factors isissemisemirequirement 1.6 and each admissible but not fully admissible. The major operation of a semiadmissible but not fully admissible. The major operation a semiadmissible statement statement isis called operation. ((V-term). admissible called aasemi-admissible semi-admissible operation. V-term). It It isis easily easily seen seen that thatrequirements requirements1.7*_1.1O* 1.7"-1.10" are are always always satisfied satisfied by fully 11—12. by fully admissible admissible statements; statements;this thisfollows followsfrom fromtheorems theorems 11-12. However, according according to to definition 35, fully admissible However, admissible statements statements are not the latter latter class so as to not semi-admissible, semi-admissible, the class being being defined defined so to exclude the former. exclude the former. The use use of of the thenew newrequirements requirements for forsemisemiadmissible statements will will be illustrated admissible statements illustrated presently. presently. The concept applies only only to synthetic concept of of semi-admissibility semi-admissibility applies synthetic statements, because because tautologies tautologies always always satisfy satisfy requirement requirement 2.2. 2.2. For synthetic statements, statements, the the introduction introduction of of the the wider wider category category of semi-admissible statements statements appears advisable of advisable for for the thefollowing following reasons. Exhaustiveness Exhaustiveness in in elementary elementary terms terms is used for reasons. for original original nomologicalstatements statements essentially essentially to eliminate nomological eliminate forms forms containing containing closed existential units units (theorem 5). Once closed existential (theorem 5). Once the class class of of original original nomological statements statements has nomological has been been defined, defined, the the class classof of nomological nomological statements does not need need such such aa protection, protection, because because these these statestatements are sufficiently protected by by the requirement ments sufficiently protected requirement of of being being derivable from original ones. For admissible derivable original nomological nomological ones. admissible statestatements, quasi-exhaustiveness in in elementary elementary terms serves serves mainly to to ensure the corresponding property for major terms to a sufficient corresponding property to a sufficient extent. If, however, the latter property extent. however, the property is is guaranteed guaranteed by other other means, the means, the possibility possibility arises arises to to renounce renouncequasi-exhaustiveness quasi-exhaustiveness in in elementary terms completely. elementary completely. Consider, forinstance, instance,the thestatement, statement,'if'if aa metal metal is heated, it Consider, for expands'. We symbolized expands'. symbolized it in inpropositional propositional variables, variables, because because operators are are not operators not required required for for derivative derivative nomological nomological statements, statements,
72 72
STATEMENTS ADMISSIBLE STATEMENrS
and It then and can can be be dispensed dispensed with for for the the following following considerations. It then
the form form has the
aDb a3b
(68) (68)
From it we we can derive the statement, 'if ‘if a metal metal is is heated heated and is is red, itit expands', to be symbolized expands’, to be symbolized
a.cDb a.c3b
(69) (69)
This implication, which like like (68) (68) isis true true of second implication, which second order, order, is not not
quasi-exhaustive in quasi-exhaustive in elementary elementary terms, terms,because because relation relation(68) (68)excludes excludes
false of of second second order. order. When When we we are are still the T-case T-case 'a. ‘a.b.6’ as as being being false still the willing to accept (69) as as reasonable, reasonable, this this may may be accounted for for by willing to the fact fact that that(69)is (69) isquasi-exhaustive quasi-exhaustivein in major major terms, terms,because because the the -major T-case 'a. c. b' is not faLse of second ‘a.c.b’ false of second order; the the implicans implicans major can be be made made false by making can false by making 'a' ‘a’false. false. For For this thisreason, reason,(69) (69)satisfies satisfies requirement 1.10* The slightly requirement 1.10* and and is is semi-admissible. semi-admissible. The slightly disparaging disparaging connotation the latter connotation of of the latter term termmay mayreflect reflectthe thefeeling feelingof of uneasiness uneasiness we we have have about aboutthis thisimplication, implication,which whichcontains containsthe theunnecessary unnecessary referenceto to the the color colorofofthe themetal. metal. A A technical technical reason reason for for setting reference setting off this this category categoryfrom fromthat that of offully fully admissible admissiblestatements statements will wifi turn turn off up presently. up presently. .
A somewhat more general general case case than than (69) somewhat more (69) is given given when when the the statement is merely in major major terms except merely quasi-exhaustive quasi-exhaustive in except for for itself. An itself. An illustration may be be constructed constructed from from (43) (43)by by assuming assuming that the the interpretations interpretations of of the the terms terms '/(x, ‘ f ( x ,y)' y)’ and and 'f(y, ‘f(y,x)' x)’ of of this this statement are of elementary elementaryterms termsin in such such aa way way that that statement are composed composed of the statement the statement is is not not exhaustive exhaustive in in them. them. In order order to to construct construct an an example example for for the the most most general general case, case, considerthe the statement: statement: 'a‘aliving consider livingorganism organism receives receives energy energy from
digesting remnants remnants of of other other organisms, or it it is digesting organisms, or is aa plant plant and andreceives receives energy from from direct direct assimilation of light’. light'. The energy assimilation of The 'or' ‘or’isishere hereinclusive, inclusive, because there there are insect insect catching catching plants plants which which do do both. both. The statebecause statement is ment is symbolized symbolized in propositional propositional variables variables (70) (70)
aaDbvc.d 3 b v c.d
Each propositional variable stands stands here for aa certain Each propositional variable certaincompound compound structure of elementary terms. But we structure elementary terms. we need need not not introduce introduce these these
ADMISSIBLE srATEMEN'rs ADlKISSIBBLE STATEBfENTS
73 73
elementary terms in order to see elementary terms see that that (70) (70)isis not notquasi-exhaustive quasi-exhaustive in elementary elementary terms, because because the relation relation holds holds ((70a) 704
cc=—d =a
which restricts the elementary terms, but isis not which restricts elementary terms, not derivable derivable from from (70). Furthermore, Furthermore, in in the given interpretation, (70). interpretation, (70) (70) is not quasiquasiexhaustive exhaustive in in major major terms, terms, because because itit can can be be replaced replaced by by an anequivaequivalence. Yet we lence. Yet we can can add add the theconverse converseimplication, implication, which which is not not
derivable from (70), derivable from (70), and thus thus construct constructa asemi-admissible semi-admissibleconconjunction in major terms except junction such that (70) (70) is is quasi-exhaustive quasi-exhaustive in is supplementable and thus for this conjunction. conjunction. Therefore Therefore (70) (70) is supplementable and
semi-admissible. Wesee seethat that the the statement statement about semi-admissible. We aboutliving livingorganisms, organisms, which we we would which would regard as as aareasonable reasonableimplication, implication, possesses possesses a formal structure formal structure which which allows allows for for this thisclassification. classification. We We could could also also add (70a) (70a) to (70) (70) and and thus thus make make ititquasi-exquaai-exhaustive terms. But this haustive in elementary elementary terms. this isisnot notalways alwayspossible. possible, Consider, for instance, instance, (69) (69)and and assume assumethat that not only only (68), (68), but but Consider, for also the relation relation (71) (71)
bbDa.c 3 a.c
holds. Then (69) is is no no longer longer quasi-exhaustive quasi-exhaustive in in major major terms. terms. But holds. Then But it it is is still still supplementable supplementable and andthus thussemi-admissible, semi-admissible,because because we we can can add and thus thus arrive conjunction such such that that add (71) (71) and arrive at at aa semi-admissible semi-admissible conjunction (69) is is quasi-exhaustive quasi-exhaustive in in major major terms terms except except for (69) for this this conjunction. conjunction. However, we cannot cannot make make it it quasi-exhaustive However, we quasi-exhaustive in in elementary elementary terms. terms. When also add add (68), in in order order to to include include the the restrictive restrictive condition condition When we we also
for terms, we we see see that in (69) is is for elementary elementary terms, in this thisconjunction conjunction (69) redundant, because it is (68). For this this reason, reason, we we because it is derivable derivable from from (68).
do not require 34 that that it be require in definition definition 34 be possible possible to make make the
statement pp aa factor that pp is statement factor in in aa conjunction conjunction such such that is quasi-exhaustive quasi-exhaustive
in elementary elementary terms except except for for this this conjunction; conjunction;we we only only require require
that this this can can he be done done for formajor major terms. terms.
Finally, assume that that we we have have an an implication implication of of the the second second order order
(72) (72)
a3b
in which order. Here Here the terms which the implicans implicans is false false of second second order. terms abbreviated by by ‘a’ 'a' and abbreviated and 'b' ‘b’may, may,or ormay maynot, not,be becomposed composed of of eleele-
74 74
ADMiSSIBLE STATEMENTS STATEMENTS ADNISSIBLE
mentary terms. not quasi-exhaustive in major mentary terms. Now Now (72) (72) is is not quasi-exhaustive in major terms, terms, though it still is reduced. we add the condition 'a', or though reduced. But when when we condition ‘Z, or some equivalent equivalent of of it, it, we can cancel, in the the resulting some we can cancel, in resulting conjunction, conjunction, the implicans other words, the implicans of of (72); (72); in in other words, within within this this conjunction, conjunction, (72) (72) is not not reduced. cannot be be made, is reduced. Therefore Therefore (72) cannot made, without without aa change, change, aa factor factor in in aa reduced reduced conjunction conjunction as as required required in in definition definition 34, 34, and and is thus not not supplementable supplementable and andnot notsemi-admissible semi-admissible (requirement (requirement 1.10*). The same same consideration consideration applies applies ifif the the implicate of (72) 1.10*). The (72) is is true of of second second order. order. We We see see that that requirement requirement 1.10* 1.10* rules out out implications of second second order order ifif their their implicans is false false of of an an order implications of implicans is order 2, and and likewise ifif their their implicate implicate is is true true of 2 2, of an an order order 2 2. HowHowever, it admits ever, it admits implications implicationswhich which can can be be replaced replacedby byequivalences. equivalences. Finally, second-order implications implications of of the the form 'a‘aD c', in Finally, 3 b .c’, in which which 'c' ‘c’ is is true true of of second second order, are ruled ruled out out by by requirement requirement 1.8*. 1.8*. The The same implicationsofofthe the form form 'a same holds holds for second-order second-order implications ‘av bb 3 J c'c’ in which which 'c' ‘c’ is false false of of second second order. order. Similar considerations can can easily easily be be carried carried through through for for the the other other Similar considerations operations, and and we we have operations, have the theorem theorem analogous analogous to theorem theorem 11: 11 : Theorem 13. 13. If If aa statement statementisissemi-admissible, semi-admissible,the the only only cases cases where the statement can where the can be benon-quasi-exhaustive non-quasi-exhaustive in major major terms terms are ones: the the statement are the the following following ones: statement may may be be an an implication implication which which can or it may can be be replaced replaced by an an equivalence, equivalence, or may be bean aninclusive inclusive disjunction disjunction which which can be replaced replaced by by an anexclusive exclusiveone. one. For statements can For these these reasons, reasons, semi-admissible semi-admissible statements can be be accepted accepted as reasonable. They either either include the condition reasonable. They include the condition restricting restricting their major T-cascs, or this condition T-cases, or condition can be added added without without changing changing the statement. statement. In I n the thelatter lattercase, case,the thesemi-admissible semi-admissibIe statement statement appears appears reasonable reasonable within aa certain certain context; context ; the the restrictive restrictive condition condition for may be for the the major major T-cases T-cases may be kept kept in in the the background, background, so so to to speak, speak, ready to be know that that this be added, added, while while we know this addition addition can can be be done done without without cutting cutting down down the the statement. statement.Reasonableness Reasonableness appears appears here here as property;aastatement statement taken taken alone is in in this this case as aa wholeness wholeness property; alone is case not not reasonable, reasonable, but but is is so so if if it it can can be be made made aa part partof ofaareasonable reasonablewhole. whole. This is explicated This kind kind of of reasonableness reasonableness is explicated by by the the term termsemi-admissemi-admissible. There are are many many illustrations illustrations for for this this kind kind of sible. There of statement. statement. For For instance, instance, we we say: say: 'if ‘if aanumber number is is divisible divisible by by 10, 10, it it is is divisible divisible by by 2 and and 5', 5 ’ )although although the thegrammatical grammatical conjunction conjunction 'if ‘if and only only if' if’ .
ADMISSIBLE STATEMENTS STATEMENTS
7.5 75
would be more appropriate. Or we we say: say: 'if would be more appropriate. ‘if aa body body gets gets warmer, warmer, its its molecules are We would would regard regard this this implication implication as as molecules are speeded speeded up’. up'. We reasonable,because becausewewethink thinkofofthe thetacit tacit addition: addition: ‘and 'and ifif the reasonable, molecules ofthe the body body are are speeded up, it it gets molecules of speeded up, gets warmer'. warmer’. Again, Again, an an 'if and only if' would be more appropriate; it would raise the ‘if and only if’ would be more appropriate; it would raise the statement from to fully status. statement from semi-admissible semi-admissible to fully admissible admissible status. With these we have have finally finally admitted admitted the the synthetic With these implications, implications, we synthetic forms ruled ruled out previously, forms previously, where where implicans implicans and implicate implicate are are elementary terms terms and elementary and the the converse converse implication implication is is synthetic synthetic nomonomological. For we may logical. For instance, instance, we may very very well well consider considerthe the terms terms ‘magnetic 'magnetic north pole' north pole’ and and 'magnetic ‘magneticsouth southpole' pole’asaselementary elementaryterms; terms;nonenonetheless, the statement theless, t,he statement 'if ‘ifthe theearth earthhas hasaamagnetic magnetic north north pole, pole, it it also has aamagnetic magneticsouthpole' southpole’isisaccepted acceptedasassemi-admissible, semi-admissible, also although nomological and and is is not not although its its converse converse implication implication is is likewise likewise nornological derivable from from it. it. It we add add the the corresponding corresponding derivable It is is derivable derivable only only when when we law of magnetism.With With the the addition law of nature concerning concerning magnetism. addition of the the category of semi-admissible statements,we wethus thus have have admitted category of semi-admissible statements, that can aall LI forms of implications implications that can be be replaced replaced by byequivalences. equivalences. Since fully fully admissible admissible statements statements include include both both original originalnomonomoSince logical statements and those statements which are fully admissible logical statements and those statements which are fully admissible by derivation, whereas whereas semi-admissible semi-admissible statements include by derivation, statements do do not not include fully admissible statements, we statements as admissible statements, we define define admissible admissible statements as their joint class: their class : D e f i n i t i o n36. 3 6A . statement Definition is admissible A statement is admissibleififititisiseither either fully fully admissible or or semi-admissible. semi-admissible. The The major major operation operation of admissible of an an admissible admissible statement is called statement called an an admissible admissible operation. operation. (V-term.) (?‘-term.) The classification of nomological statements introduced The classification of nomological statements introduced by by definitions 27, 28, 29, 33, 35, 35, 36, is isrepresented represented in in the the following following table. table. definitions 28,29,33, The categories categoriesnamed namedinside insidethe therectangles rectanglesdrawn drawnin in the the diagram diagram do do The not overlap overlap with with other other categories; categories ;the theoverlapping overlappingcategories categories are are not named are defined named outside. outside. The The nonoverlapping nonoverlapping categories categories are defined by positive requirements the addition addition of of negative negative requirements requirements positive requirements and and the stating stating that that a aprevious previouscategory categoryisisexcluded. excluded.However, However, every every category satisfies satisfies the the positive category positive requirements requirements of of every every nonovernonoverlapping categoryto to the the right right of it. lapping category it. We shall shall now now derive derive some some theorems for admissible admissible statements in theorems for statements in general. see from from definition definition 35 35 that that a factor factor of of aa semi-admissible semi-admissible general. We We see
76
ADMISSIBLE STATEMENTS
TABLE OF NOMOLOGICAL STATEMENTS
nomological and true of second order achnissible
fully admissible C)
0
.4
U
c, C)
C) 0
.4
U
original original nomological nomological
fully admissible fully admissible semi-admissible semi-admissible by derivation by
................ ....................
I ..........................
h
7S3
sS
I
. ....... .....................................
no statements statements in this this category category
admissible admissible
merely merelyp] nomological nomological
1' l
.............................
i
I
11
I
......_.__
nomological and true true of of third order nomological and order
conjunction is is at Combining this this result result with conjunction at least least semi-admissible. semi-admissible. Combining with theorem theorem 12, 12, we derive the theorem: theorem: Theorem T h e o r e m 114. 4 . If If aa conjunction conjunction is is admissible, admissible, its factors factors are are
admissible. admissible. Conversely, when when we we combine combinetwo two admissible admissiblestatements statements into into aa Conversely, conjunction, may happen happen that the conjunction, itit may the conjunction conjunction is is not not reduced. reduced. But we we can can prove prove the the theorem: theorem: Theorem unction ofofadmissible T h e o r e m15. 15.A conj A conjunction admissiblestatements, statements,possibly possibly after after reducing, reducing, isisadmissible. admissible. This seen to to be This theorem theorem is is easily easily seen be true true apart apartfrom fromthe thefollowing following possibility. IIn process one one of of the the statements may be possibility. n the reducing process so changed changed that that its its major major T-cases T-cases are are not not the the same same as as before; before; and and we have to show we have show that the the statement statement remains, remains, at atleast, least,supplesupplementable. This This is is shown shown as asfollows. follows. Assume that the statement Assume that statement p, p , which which is to be be included included in the the conjunction, is of of second secondorder order and and that that in the it conjunction, is the reducing reducing process it has been into p'. p'. Let the major been changed changed into major operation operation of of pp be aa disdisjunction. junction. It Itisisimpossible impossible that thatone oneof of the the major major terms termsof of pp is is canceled canceled in the because then then we we could could derive derive the the truth the reducing reducing process, process, because of the other of the other major major term term from from the the conjunction, conjunction, and and this this term term would would thus be true p' is thus true of of the the same same order order as as p. p. Therefore, Therefore, p' is aa disjunction disjunction
ADMISSIBLE STATEMENTS ADMISSIBLE
77 77
whose major major terms terms result result from canceling units within within the the major whose canceling units terms of terms of p. p . If If the thecanceled canceled units units were were connected connected to to the the remaining remaining p' can part of the term by an an 'or', ‘or’,none none of of the the major major terms of p‘ can be be true of second order or higher, because then it would have been true of second order or higher, because then it would have been so so before the the canceling. If the to the before canceling. If the canceled canceled units units were were connected connected to the remaining part of of the the term term by an 'and', remaining part ‘and’,none none of of the the major major terms terms of p’ p' can of can be be true trueofofsecond secondorder, order,either, either,because because of of requirement requirement 1.7 *•The Thesame sameresult result follows followsfrom from 1.8* 1.8*ififthe the equivalence equivalence was was the the 1.7”. connecting relation. Therefore, p’ is still supplementable. A similar connecting relation. Therefore, p' is still supplementable. similar proof is given if the the major operation proof is given if operation of p is is an an implication implication or or an an equivalence. Thus theorem equivalence. Thus theorem 15 15 is is proved. proved. Turning back back ttoo the by analogy Turning the study studyof of implications, implications, we we derive derive by analogy with the the discussion discussion of (51) (51) the theorem: theorem: Theorem Theorem 16. 16. If the the implication implication (73a) a1va2Db1.b2 a, v a23 b,.b2 (734 is admissible, is admissible, each of the the implications implications
(73b) (73b)
a i 3 b, aDbk
i,k= i, k =1,2 1,2
and vice is admissible; admissible; and vice versa, versa, if the the four four implications implications (73b) (73b) are admissible of the the same after admissible of same order, order, the the implication implication (73a), (73a), possibly possibly after reducing, is reducing, is admissible. admissible. The The first first part part of of the thetheorem theoremfollows follows directly directly from from requirement requirement 1.8*. The second secondpart part is is easily easily seen seento to be be true. true. If 1.8*. The If the the implications implications (73b) are are of of second secondorder, order,itit cannot cannot happen happen that that the implicans (73b) implicans of of (73a) isis false false of of second secondorder, order,or orthat that the the implicate (73a) implicate of of (73a) (73a) is is true true of second second order, order, because because otherwise otherwise the the same same would would happen happen to to the of the relations (73b). The same applies ifif the the implications relations (73b). The same inference inference applies implications are are true of of third third order. order. Let us us now now study study contraposition contraposition in the two two forms: forms: (74) (74) (75) (75)
from 'a‘aD 3 b' b’ from from‘a.c 'a.cib' from 3 b’
to ‘5 3 D li’ a' to to to ‘a.b 3 15’
It It isis easily easily seen seen that, that,for forsynthetic syntheticstatements, statements,these thesetransitions transitions have no influence influence upon upon the thesatisfaction satisfactionofofrequirements requirements2.1—2.2 2.1-2.2 have no and 1.6-1.9. Thus theorem: and 1.6—1.9. Thuswe wehave have the the theorem:
Theorem T h e o r e m 17. 1 7 . If If the is applied applied to to aa the contraposition contraposition (74) (74) or (75) (75) is
78 78
ADMISSIBLE STATEMENTS ADMISSIBLE
synthetic fully fully admissible admissible implication, implication, the resulting resulting implication, implication, possibly possibly after after canceling canceling double double negation negation lines, lines, is is fully fullyadmissible. admissible. However, we we can However, can make make this thisinference inferencefor forsemi-admissible semi-admissible impliimplications only with respect to the transition (74), because here the cations only with respect the transition (74)) because the major T-cases are are the the same same before beforeand andafter afterthe the transition. transition. For For the the major T-cases transition this is not transition (75), this not the thecase. case.Consider, Consider, for for instance, instance, the the example (69). (69). If If we apply to example to ititcontraposition contraposition according according to to (75), (75), the resulting order, and and the the resulting implicans implicans is is false false of of second second order, the implication implication is thus is thus not notsupplementable supplementable and and not notsemi-admissible. semi-admissible. Using Using the illustration given given for for (69), we we would have here the illustration would have the implication: implication: 'if aa metal is heated and does ‘if does not expand, it is is not not red', red’,which which is is certainly certainly unreasonable. unreasonable. We see see that thatthe theconcept conceptsemi-admissible semi-admissible takes account account of of this fact fact and and characterizes characterizes aa category category of of impliimplications require precaution precaution as as to contraposition. cations which which require contraposition. We have have the theorem: the theorem: Theorem T h e o r e m 18. 1 8 . If the the contraposition contraposition (74) (74) is is applied applied to to aasemisemiadmissible implication, the the resulting admissible implication, resulting implication, implication, possibly possibly after canceling But ifif the canceling double negation negation lines, lines, is semi-admissible. semi-admissible. But contraposition (75) is applied to a semi-admissible implication, the contraposition (75) is applied to a semi-admissible implication, the resulting implication resulting implication may not not be besemi-admissible. semi-admissible. For For admissible admissible tautologies, tautologies, contraposition contraposition is is likewise likewise restricted restricted to the tautoto the form form (74); (74);the the form form (75) (75)may may lead lead to to non-admissible non-admissible tautologies, as is is shown shown by by the the tautology bb J3a', logies, as tautology 'a. ‘a. a’,which whichisisadmissible, admissible, whereas its contrapositive ‘a.ED 3 b' 6’ is nonreduced nonreduced and nonad. nonadwhereas contrapositive 'a.ã
niissible.. missible Some Some further theorems theorems concerning concerning implications implications wifi will now be be studied. With studied. With respect respect to to transitivity transitivitywe wehave havethe thefollowing following theorem: theorem : Theorem T h e o r e m 19. 19. If If the theimplications implications (76a) aDb (76b) bDc a3b b3c (76b) (76a)
are admissible of the the same order, then then the theimplication implication are admissible of (76c) (76c)
aDc u 3c
possibly possibly after reducing, reducing, is is admissible. admissible. That reducing reducing may may be be necessary necessary is is seen seen when when we we put c’ for for 'c'. ‘c’. That put 'a‘a.. c' The proof proof of of the the theorem theorem isisgiven givenasasfollows. follows. Let us are both both true true of us first first assume assume that that(76a—b) (76&b) are of second second order; then (76o), being derivable derivablefrom fromthe thetwo twostatements, statements,isistrue true of of at at (76c), being
19
AbMISSIBLB SPATEMENTS
least Furthermore, least second second order. order. Thus Thus requirement requirement 2.1 2.1 is is satisfied. satisfied. Furthermore, requirement 1.10* is satisfied satisfied when when neither neither ‘u’ 'a' isis false requirement 1.10” is false of of second second order nor nor 'c' true of second order order;; but but this order ‘c’ true of second this must must be be the the case, case, because because otherwise 1.10* relations otherwise 1.10” would would be beviolated violatedby byone oneofofthe t’he relations(76a—b). (76a-b). We must study 1.8”. If If ititwere wereviolated violatedby by(76c), (76c), We now now must study requirement requirement 1.8*.
we could could write write the the latter latter relation we relation in the form form a1va2Dc1.c2 a, v u23 c1.c2
(77a) (77a)
such that that each each side, side,taken takenseparately, separately,isisreduced, reduced,whereas whereas (77b) (77b)
a1 3 c1 a, 3 is not reduced In the latter reduced or not not supplementable. supplementable. In latter case, case, either either 'a1' is false of of second or higher higher order, order, or 'c1' ‘ul’ second or ‘c,’ is true of of second second or or higher order. But But then one higher order. one of of the the relations relations ( 7 6 e b ) violates 1.8*. 1.8”. So there there remains only the possibility So remains only possibility that (77b) (77b) is not not reduced. reduced. we have the If If reducing reducing this this relation relation leads leadsto toeither either'a1' ‘6,’or or'c1', ‘cl’, we have the same same case as as before before and and thus thus conclude that this case conclude that this cannot cannotoccur. occur. Therefore, Therefore, reducing can only only concern concern the the inner inner structure structure of the terms reducing can of one one of of the terms of of (77b). But But if if some some unit unit within within these these terms terms can the same (77b). can be be canceled, canceled, the same can and therefore, also within within can be be done done within within (77a), (77a), and therefore, as as is is easily easily seen, seen, also (76c). Now the latter form was assumed to be reduced. This proves (76c). Now the latter form was assumed to be reduced. This proves theorem are true of theorem 19 19 for the the case case that that(76a—b) (76a-b) are of second second order. order. proof isis given given ifif these these two two statements statements are are true true of third third A similar similar proof order. order. If relations are true true of different If relations (76a—b) (76a-b) are different orders, orders, exceptions exceptions to to theorem 19 can arise. For instance, (76a) may be true of second theorem 19 instance, (76a) true of second true of order, while (76b) (76b)isis true true of of third third order order, while order and 'c' ‘c’ is is true of second second order. is not admissible, becauseitit is true of order. Then (77c) (77c) is admissible, because of second second order and its implicate of the the same same order. order. Here 'b' ‘b’ cannot cannot implicate is true of be true of of higher higher than than first first order, order,because because (76a) (76a) is true true of of second second order. But 'b' can also be false of first or second order. Such ‘b’ can also be false of first or second order. Suchcases cases cannot be excluded by the theory. cannot excluded by theory, It isis not notpossible possible to to derive derive theorem theorem 19 19 with with respect respect to fully fully admissible implications implications alone. alone. If If (76a—b) are fully fully admissible, admissible, it it admissible (76a-b) are may happen may happen that that(76c) (76c)isismerely merelysemi-admissible. semi-admissible. Consider, Consider, for instance, the instance, the implications implications
(78a) (78a)
aDb a 3 b
b3c1vc2 b3qvc,
(78b) (78b)
80 80
ADMISSIBLE STATEDNTS ADMISSIBLE
which will assume assume fully fully admissible admissible of of second secondorder. order. The The derivable derivable which we we will implication 1ication imp
aDc1vc2 a3)vc, is true true of order, and and need not not be If 'a‘aD need be fully fully admissible. admissible. If 3 <’ is of second second order, (78c) isis true true of (78c) of second second order, order, (78c) (78c) is not not quasi-exhaustive quasi-exhaustive in (78c) (78c)
elementary terms. sentences:: elementary terms. An An illustration illustrationisisgiven given by by the thefollowing following sentences
(79) (79)
When the the telephone rings, rings, II go to the telephone. When go to telephone. When I go to the telephone, I turn the dial When I go telephone, I turn dial or II answer answer the the call. call. When the the telephone telephonerings, rings,IIturn turn the the dial dial or or II When answer the call. call.
is merely and appears appears not not quite The conclusion conclusion is merely semi-admissible semi-admissible and reasonable because, because,when whenthe thetelephone telephonerings, rings,I Ido donot not turn turn the reasonable dial. It It isisnot notquasi-exhaustive quasi-exhaustive in inelementary elementary terms. terms. We We go go on on with withthe thestudy studyofofimplications. implications.When Whenan animplication implication (80) (80)
aaJb 3b
of the the order k, is admissible admissible of k, the the implication implication (81) (81)
aa.ejb .c3b
is certainly being derivable derivable from from (80). Now Now (81) may may is certainly nomological, nornological, being be admissible, as is is seen from the the illustration be admissible, as seen from illustration given given for for (69), (69), where where this form be admissible, admissible, because because this form is is semi-admissible. semi-admissible. But But(81) ( 8 1 )need need not be the conjunction 'a. c' may may be false of of an order 2 k. For the conjunction ‘a.c’ For instance, instance, let (80) 'when an an ice cube let (80) be be the thesecond-order second-order implication, implication, ‘when cube is is heated, it contracts'. heated, it contracts’. For For (81) (81) we we put put the the implication, implication, 'when ‘when an an ice cube cube of of 86’ 86° isisheated, heated, it it contracts’. contracts'. Since ice Sinceaa physical physicallaw law excludes excludes of an ice cube at the existence existence of at 86°, 86O,the the implicans implicans is is here here false false of of the same the same order as the the implication implication is true, and and this thisimplication implication is is therefore not therefore not admissible. admissible, The rule rule leading leading from (80) (80) to to (81) (81) may may be be called called the theinvariance invariance of implication. implication. We We see see that that this principle, pprinciple h c i p l e of principle, though indispensable for all forms does not not hold pensable for forms of of logical logical implication, implication, does hold for for admissible implications. The Thereason reasonisisthat that the the latter latter are admissible implications. are dependent, dependent, not only on truth truthconditions, conditions, but but also also on on conditions conditions for the form of writing. writing.
ADMISSIBLE STATEMENTS STA!CEMENTS
81 81
must be excluded, That the implication That implication (81) must excluded, in the the example example
considered, from from admissible admissible statements statements is shown shown by by the thefollowing following consideration. We We know know the the physical physical law law that that water consideration. water of of 86° 86" when when
heated, expands, expands, which which can can be be written heated, written as (82) (82)
C 3 Z
and we we thus thus derive derivethe thesecond-order second-orderimplication implication (83) (83) a.c 3 6 words this this means, means, 'if 'if an ice cube of 86° In words 86" is heated, it expands'. expands'. Since (83) is as good as (81), these implications cannot be Since (83) is as good as (81), these implications cannot be used used for for the expression of aa conditional conditional contrary contrary to to fact, fact, because they state, the expression of because they state, for the case for case that thatthe theimplicans implicanswere weretrue, true,contradictory contradictoryconseconsequences. quences. Implications of the form Implications of form (84a) (84a)
u 3 6 (84b) (84b) aDib may be implication8.IfIfthey they are are true true of of the the order may be called called contrary contrary implications. order k, k, we derive that that their their implicans implicans 'a' 'a' isisfalse false of of the the order order k; k ;therefore therefore neither of them neither them isisadmissible. admissible. a3b aDb
are true true of the Theorem T h e o r e m 20. 2 0 . If two two contrary contrary implications implications are the
same order, neither neither one one isisadmissible. admissible. This result result does not not apply, however, however, to contrary contrary implications implications of different orders. orders. The The implicans implicans is is then then false false of of the the lower order and and different lower order implication can can be be admissible. Referringto to the the higher-order higher-order implication admissible. Referring above example example concerning above concerning aa man touching touching aa high-voltage high-voltage wire, wire, we find find that that the we the contrary contrary implication, implication, according according to to which which he he would would not have been killed not killed ifif there had been current in the the wire, wire, is is ininadmissiblebecause becauseitit isis true true merely admissible merely of of first fist order, order, while while the imimplicans is is false of of the the same plicms same order. order. Similar Similar results results hold hold for for higherhigher-
order implications. implications.For Forinstance, instance,the the statement statement 'an 'an ice order ice cube cube of of 86° is is not not spherically shaped' appears 86" spherically shaped' appears reasonable. reasonable. This This implication implication is of of third third order, order, while while its its implicans implicans is is false false of of second second order, and and thus itit isisquasi-exhaustive thus quasi-exhaustive and andadmissible. admissible. Obviously, the the lower-order lower-order implication implicationcan canthen then not not be admisObviously, admissible, because its its implicans implicans is is false false of of the the same same order. order. For For instance, instance, sible, because the implication implication 'an 'an ice ice cube cube of of 86° 86" is is spherically sphaped", though though true of of second second order, order, is is not not admissible. admissible. We have have the the theorem: theorem:
82
ADMISSIBLE Sl’ATEMENTS
cannot be be both Theorem 21. Theorem 2 1. Two Two contrary contrary implications implications cannot both admissible. admissible. considerthe the statement, statement, ‘if 'if a signal To have another example, example, consider signal travels faster than light, it arrives at a distant point earlier travels faster than light, srrives d.istant point earlier than a light with it’. it'. This light ray ray departing departing simultaneously simultaneously with This statement, statement, too, too, is admissible, becauseitit is is true true of is admissible, because of third third order, order, while while its its implicans implicans is false false of of second secondorder. order.The Thecontrary contrary implication, implication,stating statingthat that the the is signal would would arrive arrive later later than than the signal the light light ray, ray, is is true trueof of second second order order and thus inadmissible. This example exampleshows showsthat, that, in in the theory inadmissible. This theory here presented, presented, we we can can state in here in an anadmissible admissible form form tautological tautological implications of assumptions assumptionsthat that are are ruled out implications of out by by physical physical laws. laws. But we But we cannot cannot admissibly admissibly state synthetic synthetic implications implications of of such such assumptions. IInn fact, the first first kind kind are are regarded regarded assumptions. fact, only only implications implications of of the as reasonable reasonable in in the the practice practice of of the the scientist. In as I n order order to to compare compare an accepted accepted theory with with aanon-accepted non-accepted one, one, the the physicist physicist often often computes of the the non-accepted computes mathematical mathematical consequences consequences of non-accepted theory, theory, and conditionals contrary contrary to to fact. and regards regards his his results results as as reasonable reasonable conditionals fact. With this With this solution solution of of the the problem problem of of contrary contrary implications, implications, my my present theory appears superior to my previous theory, which present theory appears superior my previous theory, which could exclude contrary contrary implications implicationsonly onlyfor forthe the case case that that the could exclude the implicans was was false false of of first first order. order. II used there implicans there 1 a 'qualification ‘qualification of derivation' in in order derivation’ order to to define define aa reasonable reasonable implication; implication; an imimplication was was accepted accepted as as reasonable reasonable ifif it it was derivable from from aa set set s plication was derivable statements from the falsehood of original of original nomological nomological statements from which which the falsehood of of its its implicans was was not not derivable. implicans derivable. Although Although this this rule rule appears appears plausible plausible at first first sight, sight, it can can be be shown shown that ititcannot cannot rule rule out out contrary contrary implicationswhose whoseimplicans implicansisisfalse falseofofthe the same same order order as as the implications implicationisis true. true. Assume implication Assume that the the above above implication implication (80) (80) is is derivable from a set of original nomological statements; then derivable from s, of original nomological statements; then (81) is is derivable derivable from from the the same Assume, furthermore, furthermore, that that same set set sl. Assume, the implication the implication (82) (82) is is derivable derivable from from aa different differentset set82 sz of of original original nomological statements, which, which, however, however, isis true true of of the the same nornological statements, same order order as 9; then Each of of the the implications as then (83) (83) is is also also derivable derivable from from 82. s., Each implications 1 ESL, Whenwe wereplace replaceininthe the example example (20) given given there ESL, p. 370—371. 370-371. When the last by‘.'....implies implies 2 sinks down', down’, the resulting resulting contrary contrary imimthe last words words by x sinks plication would also satisfy satisfy the qualification of derivation, derivation, and would plication would also qualification of would thus be accepted by the be the older older theory theoryas asnornological nomological in the the narrower narrower sense. sense.
.
ADMISSIBLE STATEMENTS ADMISSIBm STATEMENTS
83 83
(81) and and (83) (81) (83)is is thus thus derivable derivable from from aaset setofoforiginal originalnomological nomological
statements from which which the the falsehood of the the implicans is not statements from falsehood of implicans is not
derivable, and and hence derivable, hence each each of these these two two contrary contrary implications implications is is accepted. Their Their implicans, implicans, of of course, course, isis false false of of the the same order as accepted. same order the implications implications are true. true. This consideration shows that that the mentioned This consideration shows mentioned qualification qualification of derivation contrary to derivation is is inappropriate inappropriate to to supply supply conditionals conditionals contrary to fact fact which are safe from including contrary implications. implications. If If it it appears which are including contrary appears desirable desirable to introduce introduce synthetic synthetic counterfactuals counterfactuals whose whose implicans implicans contradicts a physical law, one would have to proceed otherwise. physical one would to proceed otherwise. It may It may be befeasible feasible to to define define aasubdivision subdivisionof of synthetic syntheticnomological nomological statements in in such aa way way that that aa further further stratification stratification of orders of truth for which which II see at present truth is is constructed. constructed. Such Such a procedure, procedure, for no specific need, would would once once more more represent represent aa theory no specific need, theory of of reasonable reasonable implications in terms of orders of truth. For these reasons implications in terms of orders of truth. For these reasons it appears that an to solve the appears that an order-theory order-theory of of truth truth isisindispensable indispensable to solve the problem of of conditionals contrary to to fact. problem conditionals contrary fact. The contrary to to fact isis The inability inability to understand understand conditionals conditionals contrary often for aa low often regarded regarded as evidence evidence for low capacity capacity of of abstraction abstraction in children and primitive people. There Thereisis aa story story told told about a South primitive people. American Indian Indian who, in a class American who, in class on on arithmetic arithmetic given given by by aa missionmissionary, was unable to answer answer the question: question: if aa white white man man shoots shoots 66 bears in one day, how many bears would he shoot shoot in in 5 days? On bears wouId he further inquiry, received the the answer: further inquiry, the missionary missionary finally finally received answer: a white man man cannot shoot 66 bears bears in in one day. It white cannot shoot one day. It isisan aninteresting interesting fact that, fact that, on oncertain certainconditions, conditions, the the theory theory of of conditionals conditionals contrary contrary to fact back to to the the attitude to fact leads leads back attitude of of the the primitive primitive man man who who rejects rejects a nomological nomological implication implication because because the implicans implicans isisimpossible; impossible ; this is the case false of of the same order as as the this case if the the implicans implicans is false same order implication is true. The correct reply to the Indian who had his implication is true. The correct reply to who his doubts about aa white white man's man’s ability ability of of hunting hunting would would have have been: been: according to your the implicans is false false of of second secondorder order;;but but according to your views, views, the implicans is the question statement of question concerns concerns aa statement of third order order and and is is therefore therefore meaningful. meaningful. II doubt, however, however, whether the the missionary's missionary’s capacity capacity have been high enough to understand this for abstraction would would have enough to this answer. answer.
VI VI
EXTENSION OF OF VERIFIABILITY EXTENSION VERIFIABILITY
After the definition statements in in the the wider and definition of nomological nomological statements wider and in the narrower narrower sense, sense, a certain certain extension extension of of the the definitions definitions must now be be considered. considered. The use use of of admissible admissible statements is is not not always alwayssufficient sufficient for for the practice the practice of of science. science. In I n order order totodiscuss discussscientific scientific theories, theories, we we sometimes use use procedures procedures in in which which empirical empirical truth truth is sometimes is deliberately deliberately disregarded. For For instance, we disregarded. we may dispense dispense with some some established laws the system of laws and consider consider the of the the remaining remaining laws; laws; or or we we may may even replace established laws by by contradictory established laws contradictory ones ones and and consider consider the resulting 8, though this system resulting system system S, system is known known to be be false. false. The theory of statements isis then then applicable with the of nomological nomological statements applicable with
followingchanges. changes.Instead Insteadofofthe the term term 'verifiably true' we use following ‘verifiably true’ use the term 'accepted for S’; 8'; and the ‘accepted for and the the term term 'admissible' ‘admissible’is is replaced replaced by by statements the term in 8’. 8'. Within the term 'admissible ‘admissible in Within the the system system 8, S, certain certain statements will then then appear will appear as aa reasonable reasonable which which are are considered considered unreasonable unreasonable within the the system which is is based based on onthe theconcept concept ‘verifiably 'verifiablytrue’. true'. within system So, which For instance, instance, we can construct a theory theory of of time time without without using using Einstein's law law that that light is the Einstein’s the fastest fastest signal, signal, but keeping keeping to the the synthetic law law that that ifif a signal synthetic signal makes makes aa round round trip, its arrival arrival is is later than than its itsdeparture departure(which (whichexcludes excludesinfinite infinite velocities velocities for for signals).IIn such aa system, system, the the statement statement 'if signals). n such ‘if aa signal signal travels travels faster faster than light, trip, its arrival is later later than than its light, and makes makes aa round round trip, arrival is it is indeparture', is admissible, although in in the departure’, admissible, although the system system So it inadmissible, being being true true of admissible, of second second order, order, while while the the implicans implicans is is false false of second second order. order. Or Or we wecan canmake makethe the fictitious fictitiousassumption assumption that that there there of were negative negative as as well well as as positive positive masses, masses, whose whosemutual mutual attraction attraction were follows rules analogous analogoustoto those those holding holdingfor for the the attraction follows rules attraction of of electric charges; then then we we could could derive derivethe the statement, statement, ‘of 'of any any three three electric charges; masses, at at least masses, least two two repel repel each each other'. other’. This This would would be be an an admissible admissible statement system, although although it false in 8,. statement in in this system, it is false This that the statements can can be This shows shows that the theory theory of of nomological nomological statements
EXTENSION OF VERIFIABILITY VERIFIABILITY
85 85
carried carried through for any any system system SSwhich whichsupplies supplies aasufficient sufficient subsubstitute for true'. The theory of for the the concept concept 'verifiably ‘verifiably true’. of the system system among these these theories; theories;itit isis the the one that applies So is included included among applies to actual science. actual science. II will will now now show show that thataaconcept conceptverifiably verifiably true true in inthe the wider wider sense sense can be defined, which which makes makesitit permissible permissibletotosay say that that there there may can be be laws laws of of nature which which will never never by found found by by human human beings. beings. Beginning with the we define define modalities modalities as as explained Beginning with the system system So,we explained The term physically possiblethen then refers refers to to any in ESL, §5 65. 65. The physically possible in ESL, occurrence not excluded by aa nomological statement, i.e., nomological statement, i.e., not not occurrence not excluded by excluded by a statement excluded by statement of of S0. IS’,,. An An occurrence occurrence denoted denoted by q is is physically denotedby bypp ifif neither neither qq iuhysically possible relative relative to the occurrence occurrence denoted nor derivablefrom frompp by by the use of the nor is deductively deductively derivable the class class IS’,, (ESL, p. 396). 396). Furthermore, the system leads to definitions of kinds Furthermore, system So leads definitions of kinds of of physical objects and and quantities, physical objects quantities, such such as as temperature, temperature, voltage, voltage, wave length, etc. etc.;; but wave length, but also also including including such such things things as as trees, trees, planets, planets, human cats, etc. etc. For the the observation observation of of such such objects objects we we human beings, beings, cats, have developed procedures.For For instance, instance, looking looking at have developed observational observational procedures. at a cat cat isis an anobservational observational procedure procedure to ascertain ascertain the color color of its its skin, but but unsuitable unsuitable to to ascertain the content its stomach; stomach; directing skin, ascertain the content of of its directing aa telescope to the telescope to the night night sky sky isis an anobservational observational procedure procedure suitable suitable of stars, but unsuitable for the observation observation of unsuitable for the observation observation of radio waves. The result is an radio waves. The result of of an an observational observational procedure procedure is an observationaldatum. datum.There Thereare are actual actual and possible observational possible observational observational data; for for instance, instance, the temperature temperature of aa certain certain room room during during aa certain night certain night is is an an actual actual or or aa possible possible observational observational datum, datum, dependepending on whether by means of aa thermometer ding whether observations observations by thermometer were were made. made. Speaking Speaking of of possible possible data has has the thefollowing followingmeaning. meaning. Using Using actual nature, we can derive derive the the implication: implication: actual data dataand andknown knownlaws laws of of nature, we can 'if at the a acertain ‘if the time timet tand andthe theplace placex ~t: certainknown knownobservational observational procedure had been been used, used, aa datum of aa certain procedure had certain kind kind would would have have been is nomological relativeto to the been observed'. observed’. This This implication implication is nomological relative actual data on actual on which which it is is based, based, and and thus thusderivable derivable from from these these Usually, the the implication implication does doesnot not tell tell us us the data in terms terms of S,,. Usually, specificvalue valueofofthe thedatum, datum,but but only only the the kind kind tto which the the datum specific o which
86 86
EXTENSION OF VERIFIABILITY VERIFIABILITY
belongs. For instance, instance, we we can can say say that using belongs. For using aa thermometer thermometer aa certain would have have been been observed, observed,but but we we do do not certain temperature temperature would
know we can can even even derive value; for know which. which. Sometimes Sometimes we derive the the specific specific value; for instance, we can say, say, 'if ‘ifancient ancientastronomers astronomershad hadused usedtelescopes, telescopes,
they would would have seen seen the moons moons of of Jupiter'. Jupiter’. With the With the discovery discovery of of new new laws, laws,new new observational observational procedures procedures are constructed. constructed. For For instance, instance, the thediscovery discoveryof of the thephoto-electric photo-electric
effect led to to using as observational observational instruments. instruments. effect led using photographic photographic films films as Futhermore, by extending Futhermore, new new laws laws are are sometimes sometimes discovered discovered by extending the the range of known observational procedures. Before 1896, it had not of procedures. Before 1896, it had
been known known that that wrapping been wrapping a photographic photographic plate in opaque paper and putting aa piece piece of ore on top of of it, it, isis aamethod method producing producing a blackening of of the the emulsion (after development); development); in in the the year menblackening emulsion (after mentioned, Becquerel thus discovered the radiation of tioned, Becquerel thus discovered the of uranium uranium ore. ore. This discovery discovery was wasdue dueto to chance chance;; no no one onehad had anticipated anticipated that that the This the range of the the observational possibilities contained in in aa photographic photographic range possibilities contained emulsioncould couldthus thus be be extended. Extending the the range emulsion extended. Extending range of of an an using the speaking, using observational observational procedure procedure means, means, logically logically speaking, procedure even even when when we wehave haveno noimplication implicationstating statingthat that a datum procedure of a certain new kind will be observed, but merely know of certain new kind will be observed, merely know that at at least aa datum though perhaps perhaps aa trivial trivial least datum of of aa known known kind kind will will occur, occur, though one. Becquerel knew, knew, of ofcourse, course,that that putting putting the one. Becquerel the ore ore on on the the wrapped wrapped plate must lead to some he expected it would lead to to plate some observation; he expected it would lead a state state of no blackening. blackening. a of the the plate platewhich, which,on ondeveloping, developing,would would show show no When we we speak speak of of possible data, we we shall shall include includethe the use use of of When possible data, observational procedures procedures in in an extended observational extended range. range. The if-statements am making making here here are areconditionals conditionals contrary contrary The if-statements II am to fact; fact ;but butthey theypresuppose presupposemerely merelynomological nomological statements conconis clear clear because because the the implications implications do do not not state state the tained tained in in So.This is the specificvalue valueofofthe the observation; observation; they they merely refer to to 'the specific merely refer ‘the result result of the observation', and that there of observation’, and there will will be be some some definite definite result This shows, however,that that a follows from laws laws contained follows from contained in So. This shows, however, merely possible possible datum datum is not determined S0,but but depends depends merely is not determined by by R0, R,, P0, Po,So, on the the nature on nature of of the thephysical physicalworld. world.The Thecombination combinationB0, R,,P0, Po, So, determines definite definitedescriptions, descriptions,and andaadatum datum isis given givenby by aa statedetermines ment assigning somefurther further property property to to the description, ment assigning some description, such as, 'the result of is a black spot'. ‘the of the the observational observational procedure procedure is spot’. This This
87 87
EXTENSION EXTENSION OF OF VERIFIABILITY VERIFIABILITY
statement statement is synthetic synthetic and and usually usually cannot cannot be be asserted asserted without without actual observation. In other other words, words, we we may may say say that the actual observation. In the comcomdetermines the the observational bination bination B0, R,,P0, Po, So, determines observational questions questions we we can can ask; by physical physical reality. reality. For For this this reason, reason, II shall ask; the the answer answer is is given given by shall regard regard the theclass classB1 R,asasaawell-defined well-definedclass, class,although althoughwe wedo do not not know, know, and and never never shall shall know, know, all all the theobservational observational data databelonging belonging to toR1. R,. Given the class R1,II shall shall regard regard it it as to speak of Given the class R,, as permissible permissible to speak of the statements verifiable on the the class class S, of of nomological nomological statements verifiable on the basis basis B1. R,. The latter latter term term means means that that ititisislogically logicallypossible possible to verify verify these statements inductively Rp The class class S, inductively on on the the observational observational basis basis B1. using requirements 1.1—1.10 is thus defined just like defined just like the the class class So, using requirements 1.1-1.10 and definitions 27-28, 27—28,with withthe thequalification, qualification,however, however,that that the the and definitions term 'verifiably true' is replaced by the term on the replaced by term 'verifiable ‘verifiable on term ‘verifiably true’ may be basis basis R1'. R,’.The The class class S, may be regarded regarded as as supplying supplying the the explicans explicans of the the term term 'all of nature’ nature' in contradistinction to the of ‘all existing existing laws of contradistinction to term 'all term ‘all laws laws of of nature nature known known at at some some time', time’, which which is is explicated explicated by means means of of So. The The class class S, might include, for instance, laws of a by certain certain kind kind of of radiation radiation which which forever forever has has escaped, escaped, and and will will escape, the the observations observations of of our our physicists. physicists. And And the the term term 'verifiable escape, ‘verifiable on the basis basis R1' R,’ may be be considered considered as the explicans explicans of the term term 'verifiably true in the ‘verifiably true thewider wider sense'. sense’. appears not not comprehensive enough, an an iteration If the the class class S, appears comprehensive enough, of the definition We can can speak of the class of definition may be employed. We class P1 P, of observational which itit is logically of observational procedures procedures which logically possible possible to conconstruct for for s, and and R1, B,, and and then then define define in interms termsofofB1, R,,P1, PI, a class class observationaldata data and and a class B2 of possible R, of possible observational class S, of of nomological nomological statements, the form statements, repeating repeating the form of of definition definition used used before. before. This This iteration may be to speak speak of of the joint be continued. continued. Are we allowed allowed to This is presumably of all the class class S , (union) of the classes classes S,, S,, ...? This presumably permissible. Howeverthis thismay may be, be, the the statement permissible. However statement '8 ‘8 is is a law law of of . . there nature' nature’ may may be be interpreted interpretedas as meaning, meaning, 'in ‘in the the series series S,, S,, ... there includes8’.s'. And And the the term term 'verifiably true in in the is an an Xi which which includes ‘verifiably true wider sense' may may then then be interpreted as meaning, 'there wider sense’ ‘there is a basis basis it is possibletotoverify verifythe thestatement’. statement'. For For all Ri on which it is logically logically possible is a sufficient approximation approximation to to the practical practical purposes, purposes, the the class class S, is concept 'all existing laws of nature'; and the concept 'verifiable ‘all laws of nature’; the concept ‘verifiable on the basis B1', to the the concept 'verifiablytrue true in in the wider the bash B i , to concept ‘verifiably wider sense', sense’.
s,,
VII VII COUNTERFACTUALS COUNTERFACTUALS OF NONINTERFERENCE NONINTERFERENCE
The conditionals contrary to to fact conditionals contrary fact so so far farconsidered, considered,also also called called contrary totofact, fact,are areused usedwhen when‘a’ 'a' is is fahe false and and 'b' ’regular conditionals conditionals contrary ‘b’ is false, and and we we assert assert that that if if 'a' true. There is false, ‘a’were were true, true, 'b' ‘b’ would would be be true. There is a second contrary to to fact. It second kind of conditional conditional contrary It isis used used when when 'a' is falseand and‘b’ 'b'isistrue; true; we we then then assert assert that that if 'a' ‘a’isfalse ‘a’had had been been true, true, 'b' would still be true. Statements ‘b’ would still Statementsof of this thiskind, kind,which which we we will will call call counter factualsofofnoninterference, noninterference,must mustnow now be be investigated. investigated. counterfactuals It for this this kind kind of of counterfactual counterfactual assertion Itisis aa sufficient sufficient condition condition for assertion that 'a‘aD’I)b'b’ be this presupposes that 'a‘Z D that be admissible. admissible. Obviously, Obviously, this presupposes that ’I) b' b’ is not not true be is true of of the thesame sameorder, order, because because otherwise otherwise 'b' ‘b’ would would also be true of true of this this order, order, and and neither neither one one of of the thetwo twoimplications implications would would be admissible. be admissible. Calling Calling such such statements statements implications implications with with contrary contrary antecedents,we we have have the the following antecedents, following theorem: theorem : Theorem 22. 2 2 . If If an an implication implication isis admissible, admissible, the the implication implication with contrary antecedent with antecedent is is not notadmissible. admissible. But statements of of this kind are rather trivial trivial if if they they depend depend on on the admissibility of ‘a 'a 3 D bb'. For instance, if an the admissibility of y . For instance, if an airplane airplane arrives arrives safely and and one of has died of heart failure safely of the passengers passengers has failure on on the trip, we may say, say, 'if ‘if the the plane plane had had crashed crashed into into the themountains, mountains, the man be dead’. dead'. We are usually usually not not interested the man would would also also be We are interested in in this this trivial implication; we wish wish to to assert trivial implication; we assert such counterfactual counterfactual imimplications as, as, ‘if 'if the the plane had had aa different plications different pilot, the the passenger passenger would also also be be dead’. dead'. IIn of this this form, would n an an implication implication of form, we we do not assert implication,ororthat that ‘a’ 'a' assert that 'a' ‘a’ implies implies 'b' ‘b’ in in aa connective connective implication, entails ‘b’, 'b', but merely entails merely that 'a' ‘a’does does not not interfere interfere with with 'b'. ‘b’. It It can can be beshown shownthat thatthese thesecounterfactuals counterfactualsofofnoninterference noninterference allow for for a rather allow rather simple simple treatment, treatment, which which does does not not presuppose presuppose the theory of admissible implications but can be given completely the theory of admissible implications but can be given completely within the frame within frame of of the the theory theory of of probability. probability. For have to to introduce introduce probability probability expressions expressions For this this investigation investigation we we have Since probabilities refer to concerning 'a' and concerning statements statements ‘a’ and 'b'. ‘b’. Since probabilities refer
comn'ERFACTUALS Or COUNTERFACTUALS ORNONrNTEBrERENOE NONINTERFERENCE
89
classes, whereas whereasthe the present present theory classes, theory concerns concerns statements, statements, there exists aa certain ambiguity in finding for these these statements suitable exists finding for classes which which afford afford the possibility classes possibility of of constructing constructing meaningful meaningful probability values valuesfor forthe the statements. statements. It It will be assumed assumedthat that this this probability will be construction can can be In particular, suitablereference reference construction be carried carried out. out. 1 In particular, aa suitable class has to to be class has be constructed constructed which which refers refers to to the the general general situation situation G G in which the statement which the statement 'b' ‘b’has hasbeen beenverified verified and andwhich which confers confers aa degree of of probability probability on 'b'. ‘b’.This Thisgeneral generalreference reference class classG C will will be be omitted in the formulae; formulae; we thus use use the the absolute absolute notation notation 2 and write probability expressionsinin the the form ‘P(b)’,'P(a, ‘P(a,b)'. b)’.Since Since write probability expressions form 'P(b)', these probabilities, probabilities, in in a transfer of these of meaning, meaning, refer refer to to single single eases, cases, they can be We will will always always assume assumethat that these these as weights. weights. We they be regarded regarded as probabilities are genuine definition VI). probabilities genuine (see the appendix, definition VI). Using for the Using these these considerations considerations concerning concerning reference reference classes classes for the assignment of of weights weights to to individual assignment individual statements, we we now now proceed proceed to define defhe noninterference. noninterference. D e f i n i t i o n 37. 3 7 .A A statement'a'‘a,’ doesnot notinterfere interfere with with aa Definition statement does statement statement 'b' ‘b’ if and and only only if if (87)
P(a, b)
P(b)
If the the equality equality sign sign holds, 'a' ‘a’ is is irrelevant irrelevant to 'b'; ‘b’;otherwise otherwise it it is is positively relevant. Noninterference is thus thus defined positively relevant. Noninterference is defined as as the the absence absence of negative relevance. For instance, if relevance. For if we we have reasons reasons to to believe believe that aacertain certainperson person isisintelligent intelligent (statement (statement 'b'), ‘b’), it will will be be irrelevant of of we we get get the further information (statement ‘a’) 'a') that irrelevant information (statement he is 20 20 years years old. old.Furthermore, Furthermore, the the information information that that he he is is aa trained trained he is mathematician be positively positively relevant relevant and and thus mathematician will will be thus does does not not interinterfere with statement the information that he fere with statement 'b', ‘b’,whereas whereas the information that he repeatedly repeatedly failed be negatively failed in in academic academic examinations examinations would would be negatively relevant relevant and and would thus interfere would thus interfere with with 'b'. ‘by.To To give give another another illustration, illustration, imagine imagine that aatennis tennisplayer player who who isisnot notiningood goodform form wins wins aa match. match. We We then would would say: say: ifif he he had had been been in in better betterform, form,he hewould wouldalso also have have won relevance). But But we say, likewise: if there there had won (positive (positive relevance). we would would say, likewise: if had 1
I refer refer to to the thediscussion discussionin in TIiP. ThP. For Forthe theproblem problemof ofthe thereference referenceclass class
see p. 47 47 and p. p. 374. 374. The The term term 'weight' ‘weight’is is there therediscussed discussed on p. p. 378 378 and and
p. 408. The The notation notationof of this thisbook bookisisused usedfor forthe thefollowing followingpresentation. presentation. p. 408. 2 ThP, ThP, p. p. 106. 106.
90 90
OF NONINTERFERENCE COUNTERFACTUALS OF NONINTERFERENCE
been more been more spectators, spectators, he he would would also also have havewon won(irrelevance). (irrelevance). Using the the theorem the calculus calculus of of probability: probability: Using theorem of the P(a,b) — P(b) ( 88 P(b,a) P(a) we derive from we from (87) (87) (89) (89)
P(a) PP(b, ( b , a) a) 2 w -4
Noninterferenceisisthus thus aa symmetrical symmetrical relationship: relationship:ifif ‘a’ 'a' does not not Noninterference interferewith with ‘b’, 'b', then then 'b' not interfere with ‘a’. 'a'. interfere ‘b’ does does not interfere with We now introduce We introduce the the following following definition: definition: Definition ‘6’ is is true true and and 'a' ‘u’is is false, false, then then the the countercounterD e f i n i t i o n 38. 3 8 . If 'b' factual of 'if ‘u’ 'a' had been of noninterference, noninterference, ‘if been true, 'b' ‘b’would would also also be be true', true’, isispermis8ible permissible if relation relation (87) (87) isissatisfied. satisfied. We use We use here the term term 'permissible' ‘permissible’because because the counterfactual counterfactual considered neednot not be be an admissible statement; in fact, itit need considered need admissible statement; need not even and can not even be be aanomological nomological statement statement and can be be aafirst-order first-order statement. The The class class of of reasonable reasonable conditionals contrary to fact fact isis thus somewhat somewhat wider wider than the theclass classofofadmissible admissibleimplications; implications ; this result this result will will be be confirmed confirmed by by the the investigations investigations of of the the following following chapter and will will be be formulated formulated in in definition definition 42. 42. We We must must now now explain explain why why the the counterfactual counterfactualof of noninterference noninterference is permissible permissible oii on these these rather rather simple simple conditions, conditions, whereas whereas the regular contrary to to fact, or regular conditional conditional contrary or counterfactual counterfactual of of interinterference, requires the involved ference, requires involved theory theory of ofadmissible admissible statements. statements. Assume weknow knowthat that the probability P(a, 6) b) is very high, and Assume we probability P(a, that that ititsatisfies satisfiesthe theconditions conditionsof ofaagenuine genuineprobability. probability. We Wewould would then be has if 'a' ‘a’ hasbeen beenobserved, observed, be willing willing to use it it for for predictions; predictions; if we shall predict predict 'b'. use it it for aa regular we shall ‘b’.Would Would we we also also use regular conditional conditional contrary to fact? that neither neither 'a' fact Z Assuming Assuming that ‘u’nor nor 'b' ‘b’are are true, true, would would we be willing willingto to say, say, ‘if 'if ‘a’ 'a' had we be had been been true, true, 'b' ‘b’would would have have been been true'? true’? II think thinkwe we would would hesitate hesitate to to do do so; so ;since since aa high high probability probability admits admits of exceptions, we might might argue argue that this exceptions, we this particular particuIar instance instance might might have have been been aa case case of of exception. exception. Why Why do do we we thus thus distinguish distinguish between between a predictive predictive and and aacounterfactual counterfactualusage? usage? The predictive by the predictive use use of of high highprobabilities probabilities isis justified justified by the
'1 ThP, ThP,p. p.
112, 112, formula formula (32). (32).
COVNTERFACTUALSOF OF NONINTERFERENCE NONu4IEItFERENOE COUNTERFACTUALS
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interpretation when frequency frequency interpretation when we we regard regard predictions predictions as as posits; posits ;our our posits will will then then be be true true in in aa high high percentage percentage of cases. cases. IInn the the same same way way we could could argue argue that that regular regular counterfactuals counterfactuals of of high high proprobabilities could be be asserted asserted as as posits posits and and then then would would be be true true in in a babilities could high the latter high percentage percentage of cases. cases. Although Although the latterconclusion conclusion appears appears correct, correct, there there is is an an important importantdifference difference between between these these two two kinds kinds of of posits. posits. For aa predictive predictive use, use, we we can can verify verify the the individual individual posit posit later, by waiting waiting until until observation observation tells us whether whether the posit posit was was true. For For counterfactual counterfactual use, use, no no such such individual individual verification verification is possible;inin fact, fact, although we know know that that the majority possible; although we majority of of our our counterfactual posits would be true, we would never know which posits true, we would never know which of them them are this majority. of are true, true, i.e., i.e., which which of of them them are are members members of of this majority. For this reason, For reason, we we hesitate hesitate to apply apply the the counterfactual counterfactual in an an individual prefer to individual case. case. We We would would prefer to use use counterfactuals counterfactuals only only when when there that they It is there is i,s evidence evidence that they are are not not subject subject to to exceptions. exceptions. It is this this very condition statements, or condition which which is is satisfied satisfied for for nomological nomological statements, laws of of nature nature;;and and for for this thisreason, reason, we we restrict restrict regular regular conditionals conditionals contrary to fact to in the precontrary to admissible admissible statements statements as defined defined in ceding ceding chapters. chapters. Now itit can be Now be shown shown that that for for counterfactuals counterfactuals of of the the noninternoninterference kind this difficulty can be circumvented in a certain ference kind this difficulty can be circumvented in a certain way. way. Although, of course, such conditionals conditionals contrary contrary to to fact cannot Although, of course, such cannot be be verified individuallyeither, either,ititturns turns out out that there is verified individually is no no objection objection to assuming that they to assuming that they are are always always true, true, without without exceptions. exceptions. This is seen as seen as follows. follows. Assumethat that we we have have aa certain of event, event, for for which which ‘b’ 'b' is Assume certain series series of sometimes true, and and a parallel sometimes true, parallel series series containing containing 'a': ‘a’: (90)
. .aa.a .a a . ~ . . . . . . bbbbbbbbbb. bb66b6bb ...
For the cases For cases where where 'a' ‘u’is is false false we have put put aa dot. dot.We Wewill will assume assume that P(a, P(u,b) b) < <1,1, although although itit may may be be aa high high value; value; then then there there will will will also also assume be cases where an an ‘u’ 'a' is by aa %’. 'i'. We be cases where is accompanied accompanied by We will assume that the the probabilities probabilities P(a) P(u)and and P(b) P ( b )exist. exist. The The question question arises: arises: there were more cases cases ‘u’, 'a', i.e., P(a) higher, can we arrange arrange if there were more i.e., if P ( a ) were were higher, can we the cases 'a' in such a way that that every by an an ‘u’ 'a' the cases ‘u’ every 'b' ‘b’ is is accompanied accompanied by without changing the values values P(b) P(b) and and P(a, without changing the P(u,bb)? ) ? IIn n other other words: words:
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COITNTERFACTUALS NOxINTERFERENCE COUNTERFACTUUS OF OF NONINTERFERENCE
can we replace replaceevery everydot dotabove aboveaa ‘b’ 'if by by an an ‘u’, 'a', and and then can we then by by replacing replacing some, or or all, all, of of the the remaining remaining dots dots likewise likewiseby bycases cases‘u’ 'a' arrive arrive at at aa some, distribution such such that the distribution the percentage percentage of of cases cases 'a' ‘u’accompanied accompanied by 'b' 'a' is by ‘b’ among among all oases cases ‘a’ is the the same same as as the theprevious previous percentage? percentage? terms this means: In probability probability terms means: can we we add add the thecondition condition 'b 3Da’, a', or or its its statistical substitute P(b, = 1, 1, to to given ‘b P(b,aa)) = given values values of of P(b) P(b) and and P(a, P(a,b)? b ) ?The Theanswer answerfollows follows from from relation relation (88). (88). If we put If we put 1, we there P(b, we infer, there P(b,a) a)= = 1, infer, because because P(a) P(u) 5 1, the resulting resulting 1, that that the left-hand value P(b) P(b) on on left-hand side side P(u, P(a, b) b) isis smaller smallerthan than or or equal equal to to the the value the right-hand and thus thus that that the must hold. In the right-hand side side and the inequality inequality (87) (87) must hold. In other words, formulates the (87) formulates condition which which the the given given other words, relation relation (87) the condition probabilities P(b)and andP(a, P(a,bb) mustsatisfy satisfyinin order orderthat that it be probabilities P(b) ) must be possible to put put an an 'a' It was that possible to ‘u’above above every every 'b' ‘b’ in in (90). (90). It was explained that we we then shall also have to put put some some cases cases 'a' ‘a’above above eases cases ‘5’in in order to keep the given probablities unchanged; but we can thus order to keep the given probablities unchanged; but we can thus at least a'. at least satisfy satisfy the the conditon conditon 'b‘bD 3 a’. This means: if condition condition (87) (87) is satisfied, satisfied, it isis permissible permissible to to This means: assume that, that, ifif there there were were more more cases cases 'a', ‘a’,every every case case 'b' ‘b’would would be be accompanied by an an ‘u’. 'a'. This not contradict the the accompanied by This assumption assumption would would not observed statistical values of P(b) observed statistical P(b)and P(a, P(a,b). b ) . Since Since our our countercounterfactual merely merely states states that the factual the occurrence occurrence of of 'b' ‘b’ would would not be be ininvalidated ‘a’ had had come come true, true, we we conclude conclude that we are are allowed allowed to validated if if 'a' that we to say: if all say: if only only there there had had been been more more cases cases 'a', ‘u’, allour ourcounterfactuals counterfactuals would have have been true. would true. This This assumption assumption isispermissible permissible although although
P(a, P(u,b) b) < <1.1. Note proof that thatall allthe thecounterfactuals counterfactuals Note that we we do do not not have have aa proof would have have been been true. true. Such What would Such aa proof proof is is of of course course impossible. impossible. What we can prove prove is is merely merely that that we proof to to the the contrary, we can we have have no no proof contrary, i.e., i.e., that that the thestatement: statement :'all ‘allour ourcounterfactuals counterfactualswould would have have been been true true if there there had had been been more more cases cases 