Formal Semantics: The Essential Readings (Linguistics: The Essential Readings)

Fot-r

I

The Essential Readings

Edited by

Paul Portner

and

Barbara H. Partee Blackwell Publishing

1

Formal Semantics

Limguistics: Tlme Essential Readings consists of comprcliensivc collectio~~s of classic and contemporary reprinted articlcs in a wide range of fields within linguistics. Thc primary works presented throughout each book in the serics arc complcn~entcdby outstanding- editorial material by key figures in tlze field. Each volume stands as an excellent resource on its own, as n-ell us an ideal companion to an introductory tcxt. 1 Phono/o'yr'c,n/ T h e o q ~ :The Essc~zrilrlRtnr/l'ngs, edited by John :I. Goldsmith 2 fimilctl Stnrarrric-.v: T/IP Essential R~uding.~,cdited by Paul Postncs an3 Hasbara H. Partee

Formal Semantics The Essential Readings Edited by

Paul Portner and Barbara H. Partee

Blackwell Publishers

i' 2002 b! Ulack~vcllPublishers 1-tcl a Bl~lcliwcllPublishing conipanq

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Contents

Acknowledgments Introcluction Paul Portner and Barbara H. Partee

1 The Proper Treatment of

in Ordinary English

Richard Montague

2 A Unified rlnalysis of the Englisll Bare Plural Greg N. Carlson

3

Generalized Quantifiers and Natural Language Jon Barwise and Robin Cooper

4 The Logical Analysis of Plurals and Mass Terms: A Lattice-thcoretical Approach Godehard Link

5 Assertion Robert C. Stalnaker

6 Scorekeeping in a IJanguage Game David Lewis

7 Adverbs of Quantification David Lewis

8 A Theorj- of Truth and Sen~anticRepresentation Hans Kamp

9 File Change Scnlantics and the Familiarity Theory of 1)cfiniteness lrene Heim

10 On the Projection Problem for Presuppositions lrene Heim

1 I Toward a Semantic Analysis of Verb Aspect and the English "Imperfective" Progressive David R. Dowty

viii Contents 12 T h e Notional Catcgory of Modality Angelika Kratzer

1.3 The Algebra of Ei-cnts Emmon Bach

1-1 Gencsalized Conjui~ctioiland Type Ambiguity Barbara H. Partee and Mats Rooth

15 Noun Phrasc Interpretation and Type-shifting Principles Barbara H. Partee

16 Syntax and Seinantics of Questions Lauri Karttunen

17 Type-shifting Rules and the Seii~anticsof Interrogatives Jeroen Groenendijk and Martin Stokhof

18 O n the Notion Afictitle in thc Analysis of Ncgatil-c-polarity Items William A. Ladusaw Index

Acknowledgments

The editors and publishers gratefully ackno\~ledgethe following for permission to reproduce copyright material: l
,!z

Walter dc Gruyter 8i Co. for: "The Logical Analysis of Plurals and h;lass Terms: A 1,iittice-theoretical Approach," by Godehard Link (from Raincr Uiiuerle, Christoph Schmarze and -4rnim yon Stechow (eds), lMcntring, LGe arld 1Iti7 Itzte/pr-t1n~ior7of' Lu~zg~ragc, pp. 303-23, 1983 Walter de Gruyter & Co.); "File Change Semantics and the Familial-itp Theory of Dcfinitencss," by Irene I-Jeim (fi-on1 Rainer Rli~ierle, C:hristoph Schwarze and Arnim von Stecho~v((els), rtleanifrg, L i s p rrnd tlrr Itt~crprc~u~ior~ of'/,llrr~.liugc,pp. 1 6 4 00, 1983 Walter de Gruytcr & Co.); "The Notional Category of hlodalitjl," b!" Angelilia Kratzer (from H.-J. Eikmeyer and M. Kicser (eds), CCitrrls, Il'orldx, rr17J Cnittrsts. Lbi~vr~ :qppronl.hes to R'oril Semnntil-s, pp. 38-74, :(-~:198 1 \&'alter de

(c,

x Acknowledgments Grupter 81 Co.); and "Generalized Conjunction and TJpc Ambiguity," by Barbara H. Partee and Mats Root11 (from Rainer Ha~~erle, Christopli Scli\varze and Arnim von Stechow (eds), ~Mraning,CJ.w nnd ~ h Irlterpr~tatiorr r ~ ~ ' L N ~ S Ipp. I I I301-03, ~P, !(: 1983 Walter de Gruyter & Co.); Acadc~nicPress for "Assertion," by Robert C:. Stalnakel- (from Pctcr Cole (ed.), Prllginnlit.~,pp. 3 15-32, I:(_;) 1978 Academic Press Inc.); rcprintcd by permission of the publisher; Cambridge University Press for "Adverbs of @lantification," by David 1,ewis (from E. L. Keenan (cd.), Forrrznl Sew7awtics of'N~~turil/ L,(lngrr(lp, pp. 3 1 5, LC'! 1975 Cambridge University Press); reprinted with the permission of Cambridge University Press; Foris Publications for "A Theory of Truth and Semantic Reprcscntation," by I-lans Kamp (from Jeroen Groenendijk, Theo Janssen and Martin Stokhof (eds), T'ril~h, In/erpretcrtion, I~zjbrma~ion (GRASS 2 ) , pp. 1-41, >Jy 1984 Foris Publications, originally publislled in Jcroen Groenendijk, Theo Janssen and hlartin Stokhof (eds), For~~lu/ M~'t/ro~I's ill thr Stu4)t r!/'I,a11guage, Mathematische Centrum, Uni~ersityof Amsterdam, 135); and "Noun Phrase Interpretation and Type-shifting Principles," by Barbara M. Partee (from J. Groenendjik, D. de Joi~ghand Rfl. Stokhof (eds), Stlrilits ilr Di.s~~rntrl:ve R~~pres~wtutio~z Theo?)~arzd the Th~'o/:yqf Genercl/i:ed @iant[/i~.rs, pp. 11.5 -43, 1086 1'~- Foris Publications); ((''1

Irene Heim for "On the Projection Probleni for Presuppositions" (from hZ. Barlow, D. Flickingcr and kl. Wescoat (eds), ITCCFL 2: Second rlnnttal W ~ s tCULLS/ Co~!fl~rcnce on Formal LinLquisti~.s, pp. 114-25, :'(::I 1983 Irene EIeim); William Ladusabv for "On the Notion 'Affective' in the Analysis of Negative Polarity L P1-1 ' . ~0, , (!: 1080 William L,adusatv). Items" (Jountal q/ Lirzguzstic R L ~ S ~ C I, '

1

T h e publislicrs apologize for any errors or oii~issionsin the above list and would be grateful to be notified of any corrections that should be incorporated in the next edition or reprint of this book.

Introduction

Our purpose here has been to put together a collection of "classics" that have shaped the field of formal semantics in linguistics which can serve both as a reader for graduate-level semantics courses and as a reference collection for researchcrs in semantics and related fields. These works are widely recommcndcd to virtually evcry student of formal semantics, and we believe they are of value for anyone interested in the semantics of natural language.

What Is Formal Semantics? The roots of formal semantics lie in logic and the philosophv of language. Its first appearance as part of a theory extending to natural language semantics was in the form of "hlontague grammar," originally developed by the logician Richard M o n t a g ~ ~ e (1930-71) and subsequen~lymodified and extended hy linguists, philosophers, and logicians. It quick]! became influential in linguistics, and linguists have played il large role in its evolution into contemporary formal semantics. T h e most constant 6eatur.e~of the theory ovcr time have been the focus on truth-conditional aspects of meaning, a model-theoretic conception of semantics, and the methodological centrality of the Principle oJ'Cornpositiunality: "The meaning of a whole is a function of thc mcanings of its parts and their mode of syntactic con~bination." Formal semantics contrasts on a number of dimensions with other aypl.oaches to meaning within linguistics, psycl~olog.y,and philosophy. 1;ormal semantics originates within the non-psychologistic tradition of "objective" (though abstract) meanings (1;regc 1982; Tarski 1944; Carnap 1956; Montague 1970b), which contrasts wit11 thc psj-chologistic view of meanings "in the head" (Fodor 197.5; Jackcndoff 1983, 190h; Higginbotham 1985; Lakoff 1987, and all psychologists). Do expressions rckr to objects or to concepts? Is semantics a branch of mathematics, or is it (as on thc Chomshj-an view of all of linguistics) a branch of psychology? Classical formal scmanticists, n-11o took 1-he first disjunct in these choices, distinguished scmantics from knowledge of semantics (I,ew+is 1975b; Cresswell 1978), making semantic competence

2 Introduction interestinglj- different from syntactic competence. Many today seek an integration of these two perspectives b y studying inind-internal intuitions of inind-external relations such as reference and truth conditions (see Chierchia and McConnell-Ginet 1900). Formal semantics diflers from most previous linguistic thcorics of semantics on another dinlension as well: it is model-theoretic rather than representational. Many linguists have thought of semantics in terms of a "level of reprcscntation" of' cxprcssions analogous to a syntactic or phonological le\rel, and this mas the way semantics was approached in the theories of Katz and Fodor (1963), Katz and Postal (1964), in generative sen~anticsand interpretive semantics (see Newn~cyer1980; Harris 199.1)) and in later work positing a level of "Logical Form" within gcncrati\:c grammar (Higginbotham 1983b; May 1985; Larson and J3udlow 1993). Psychologists, who generally think of semantics as relating expressions to concepts, often regard concepts as something like elements of a "languagc of' thought." A representational kiew of semantics is quite congenial to the popular coml.>utationaltheory of mind (Jachcndoft' 1983). A pure model-theoretic view sees semantic interpretation relating expressions to elements of models (possibly mental models) defined in terms of constituents such as possible situations, entities, properties, truth-values, etc. Intensional objects may be modeled, for instance, as functions from possible worlds or situatiot~sto extensions. T h e question of the mental representation of such model-theoretic constructs is o$cn (see Johnson-Laird 1983); the inclusion of h4arrian "2 $ - I 1 sltetches" in Conceptual Structure in Jackendoff (1987) suggests the possibility of mixed approaches. And Iieim (1982) and many other formal semanticists have found it fruitful to ivork within a Chomskyan syntactic framework that does include a syntactic level of "Logical Form" or "LF" w-hich is then taken as thc input to the kind of modcl-theo~aeticcomposilional senlantics characteristic of formal semantics. Many current rcscarchers seek an integration of model-theoretic and representational approaches, and not all contenlporary formal semanticists emphasize a inodel-theorctic pcrspcctivc in tlreir worh.

Historical Perspective Formal semantics as a part of linguistic theory was born of two parents: philosophical logic and generative grammar. Like any child, this child inherits some features from each parent, learns others from the parents during life, follows its peers in certain ways, and eventuallj- develops an independent personality and makes its own uniqiic contributions. This volume provicles a highlight-album style view o i classics that mark the , early development of the field, fi-om the end of the 1000s to the late 1980s, emphasizing the linguistically important developments, while recognizing the crucial interconnectedness of linguistic, philosophical, and logical perspectives. As such, it is explicitly about a variety of linguistic topics, drawn from among thosc for \vhich this approacll to the study of meaning has proven particularly insightf~~l, and implicitly about the foundations and history of the field itself As mentioned above, the approach to the analysis of meaning nhich is pivsued in formal semantics has its origin in the development of formal logic. The ~nilestoncssct 1 by Frege, Tarski, Carnap, Dat-idson, Kriplce, Kanger, Hintikka, Montaguc, Kaphn, and others developed the ideas of a truth-conditional, modcl-thcorctic, and intcnsioilal 1

~

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Introduction 3 seniantics for formal languages. T h e classics of this literature may be found in tlie many excellent readers on philosophy of language, and so we have not reproduced them here. The idea that the techniques developed for artificial formal languages could he applied to natural language was first pursued systematically in the late 1960s and early 1970s by Richard Montague (Montaguc 1970a, b, 1973), David Lewis (Lewis 1970), Max Cresswell (C:rcsswell 1973), and Terence Parsons (Parsons 1972). O n l ) the most influential of these works, hlontague's paper on "The proper trcatment of qui~ntificationin ordinary English" (Montague 1973), is reproduced here, but it shoulcl be stressed that the others also made significant contributions to the basic form which formal semantic theories take. Montague's paper is hcrc because it is a historically important starting point for Montague grammar and hence for formal semantics, and is not available except in tlie out-of-print collection (Montague 1974). It is not easy to read, however. (Introductions to its content can be found in Dowty et al. (1981), Gamut (1991), Link (1979), and Partee (1973, 197%)) A more readable and also important early work is Lewis (1970), which is more widely available and hence not included licrc. At thc most basic level, a formal semantic analysis postulates a compositional, hnctional pairing between syntactically analyzed sentences of il language and their truth-conditional meaning. From the time of these earliest works, including Montague's, it has becn usual (though by no means universal) for the expression of truth a~nditionsto be mediated by an intensional logic (such as Montaguc's own typed intensional language 11,) with an underlying model structure utilizing thc notion of possible worlds. In its reliance on having an explicit syntactic analysis for each sentence under anal!-sis, a semantic theoly requires some sort of syntactic theorj- to build on. Wontaguc's o\vn ivork followed the logicians' tradition of stating the sFntnx in the form of a reciusivc definition of the set of wcll-formed expressions of each syntactic category. 'Thc "analysis trees" corresponding to the steps of the derivation of particular sentences provided the syntactic structilrcs that were the input to the compositional semantic interpretation. While some linguists follo\ved up on the development of some aspects of Montague's syntax, such as his use of a modified categorial grammar (see Bach ct al. 1987), othcr linguists preferred to scek ways to integrate Montague's semantics svith the kinds of approaches to syntax that had bccn developed within linguistics in the 1960s and 1970s (Partee 1973, 197%; Cooper 1975; Bach and Cooper 1978; Bach 1979). Explicit comparisons between Montaguc grammar and generative semantics, as well as the earliest systematic attention to thc lexicon in formal semantics, can be found in Dovvty (1979). Montague was doing his work on natural language a t the height of the "linguistic wars" bctwcen generative and interpretive semantics (see Fodor 1980; Newrneycr 1980; Harris 1993), though Montague and the semanticists in linguistics had no am-arencss of one another. (A4ontague was aware of Chomsk~'swork and respected its aim for rigor but was skeptical about the fruitfulness of stuclying syntax in isolation from semantics.) -4s a]-gued by Partee (1973, 1975b), one of thc potential attractions of Montague's work for linguistics was that it offcred an interestingly different view of thc relation between syntax and semantics that might be able to accommodate the best aspccts of both of the warring approaches. T h e instantiation of Wontaguc's algebraic theory in Montaguc (1973) illustrates what Bach (1976) christcncd the "rule-by-rule"

4 Introduction approach to syntax-semantics correspondence: syntactic ri~lcsput exprcssions (or : bracketed expressions; see Partee 197%) together to form more complex expressions, i and corresponding semantic rules interpret the whole as ;l function of the interpretations of the corresponding parts. ?'his is quite different from both gencrativc and 1 interpretive semantics, which were framed in terms of the prevailing conception of 1 syntactic derivations from some kind of phrase-structure-gcneratcd ul~derlyingstructures via transformations to surface structures, with the debate centered on which level(s) of syntactic representations provided the basis for s e l ~ ~ i ~intcrpretation. ~~tic Cooper and Parsons (1976) providcd "conversioi~algorithn~s"to sl~owthe descriptivc equivalence and theoretical differences among (particular versions of) A/fontague grarnmar, generative semantics, and interpretive semantics wit11 respect to the fvagment of English generated in Montague (1973). Once it was shown hour productively the basic ideas of a formal semantics for natural languagc could be applied, a flurry of work started in the 1970s as many scholars tried to see what insights were to be gained by looking at a widc variety of linguistic phenomena. At first, the theoretical forinat used was typically closcly modeled on Montague's, and the theory as a whole was k n o w as "Montague grainniar." Some important figures from this era arc Michael Bennett, Barbara Pnrtee, Greg Carlson, Robin Cooper, David Dowry, Lauri Karttuncn, Richmond 'I'liomason, and Lmmon Bach; the earliest collections were Kodman (1972) and Partec (1976). 13ou.t~-ct al.'s (1981) classic textbook Introduction to Mowtrigue Scrnnntir.~includes iln extensive bibli, papers by C~arlson, ography of work in this framcworli. Within the prescnt ~ o l u m e thc Dowty, and K,~rttunenrepresent this pcriod. Related independent work was done by Edward keenan, hl. J. Cresswcll, Terence Parsons, C:. I,. Hamblin, Kenatc Bartsch, and others. A4ontague's compositionality requirement mas in principle compatible wit11 many different forms of syiitactic thcory, and linguists soon began to explore a range of options that were opened up by the h c t that with a morc powerful semantics, a less pow-erfirl syntax might be adequate. For instance, therc have been intercsting dcbates about whether control of infinitival complements is to be analyzed by having an cinpty category PRO as subject of' an embedded clause (the classic syntactic view, successor to "Equi-NP deletion"), or with a bare infinitival complenient plus a theory of control as a semantic entailment associated uith the embedding verb (Cl~ierchia1984; 13owty 1985). Similar issues arise with "Raising" and other "NP-nioveinent" rilles (Sag 1982; Jacobson 1990); Dowty (1978) in fact argued that all "gotcrncd transti)rinationsl' should be recast as lexical rules, with lexico-syntactic effects on the argument structure of the governing item and a corresponding con~positionalchange in its semantics. A nunlbcr of new syntactic theorics developed as a part of this trend. Gcneralizcd Phrase Structure Grammar (GPSG: Gazdar 1982; Gazdar et ill. 1985) was the first of a number of proposals for "monostratal" (lion-transforrnation;ll) gr;imniars conibined with coi~lpositionalformal semantics. Other proposals for monostratal syntax to be combined w-ith compositional formal senlantics inclucled extensions of C:atcgorii~l Grammar (CG: Rach 1984; Chicrchia 1984; Rach ct al. 1987), Heacl-Uriken P11rasc Structure Grammar (IIPSG: Pollard and S;lg 1994), and t11c morc recent TrccAdjunction Grammars (TAGs: Joshi 1985). T A G s wcre originally motivated by purely syntactic consideratioils b ~ l twere later cstended to includc con~l>ositio~ial scrnantics,

I

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Introduction 5 both clirectly (Subrahmanyam 1989; Joshi and Vi jay-S hanker 1999) ancl via the simultaneous derivation of' an LF-like representation which can serve as input to modeltheoretic interpretation (Schabes and Shieber 1994). An independent but in some ways related line of development is found in Bresnan and Kaplan's Lexical Functional Grammar (Brcsnan 1982; Kaplan and Bresnan 1995). While debilt~sconcerning the notion of LLautonomoussyntax" (including debates about what the tcrm should bc taken to mean) have continued since the issues were raised by Chomsky in thc 1970s (Chornsky 1975; Partee 1975a), and similarly fi)r the status of the compositionalitv constraint (Partee 1984; Janssen 1997), most lii~guistsby the end of the 1980s werc agreed that proposals for syntax must be judged at least in part by their compatibility ivith a coherent semantics and vice ircrsa. In the meantime, other scmanticists found new ways to integrate formal scmantics with mainstream Chomshyan approaches to syntax. Hein1 (1982) combined innovativc formal semantic proposals with innovations in a GB-style lcvel of l~,ogical Form, making it clear for the first time how these approaches could be fit together. Kooth (1985) proposed that the syntactic operation of Quantifier Raising ((LR) was triggered by tvpe mismatch, i.e. by the impossibility of interpreting quantiticational NPs in ordinary argument positions. In general, semantic ~vorkof this kind raises issues of tradeoffs t~etweenthe use of a semantic module enrichcd with devices such as typeshifting (Partee and Rooth 1983; Partee 1986) versus the crucial use of ;-I level of Logical Form distinct fiom surfacc syntax (cf. Heim and ICratzer 1998). A second major movement that paralleled the development of formal scmantics is the dcvclopinent of formal pragmatics; the two movements often influenced one anothcr and a number of scholars contributed to both. Within philosophy, the works by Stalnaker (1978) and Lewis (1979) reprinted in this volume, as well as others by, for csample, Montague, Michael Bennett, Kaplan, and Soames (see Davis 1991), were to have a trerncndous in~pacton semantic theory. Within linguistics, pragmatics grew greatly as a separate subfield closelj- tied to both semantics and syntax, a development which was to some extent launched by generative semanticists like James McCawlcy, Georgia Green, Jerry Morgan, and Larry Horn, and also ineludcd scholars with a more direct intcrest in pragmatics like killen .Prince and Gerald Gazdar. Some of thc work was closely ticd to Montague grammar, like that of Lauri Karttunen and Stanley Peters on lxesupposition (Karttuncn and Peters 1979); Keenan's earlies work on presupposition reflects a similar interest in thc semantics-pragmatics interf~ce(Keenan 1971). Reseilrch at the hordes bctwccn linguistics and philosophj on such topics as prcsupposition, implicature, indexicality, and speech acts continues to have an important iinpact on the ficld of formal scmantics. One effect of the subsequent tlevelopn~entof d y n i ~ m i ~ scmantics, discussed bclom-, \\.-asto bring formal semantics and formal pragmatics closer together, as can bc seen in such works as Heiln (1983b), included here. The late 1070s and early 1980s saw a grcat expansion and diversification in thc rangc of theoretical frameworks which fall into the formal semantics tradition. Montague grammar evolved into a less monolithic thcorctical framework, with formal semanticists modifying or replacing various aspects of the syntax, the scmantics, or the architecture of the interface, while continuing to use many of the tools from Montaguc's "toolbox." 'I'his time saw the de\~elopmentof dynamic approaches to meaning, that is an;~l!ses n-hich vie\$-meaning as the contribution which an expression can make to the increase

6 Introduction of information in a context. This trend was inspired by Stalnakcr's (1978) work on pragmatics and Lewis's (19753) on adverbial quantification, both included here, ils well as by linguistic and philosophical work on anaphora and the sen~anticsof indefinite and dcfinite noun phrases (Evans 1977, 1980; Fodor and Sag 1982), and especially the problems of discourse anaphora (Karttuncn 1976; Wcbber 1979). Stalnakcr's contrihution was crucial because he developed the basic ideas of the evolving conversational context which allows tlie dynamic view to be formalized; JJewis's work led to a better understanding of the seniantics of indefinite noun phrases and anaphora, tlie crucial laboraton for most of the progress within the dynamic franictvork. l,etvis (1979), illso included Ilere, opened broader perspectives on the range of phenomena for which a dynamic perspective could be iniportant. Dynamic semantics also had iniportailt predecessors in the fields of' psychologp (Clark and Clark 1977), the semlntics of programniing languages (Scott 1982), and pragniaties (in particular, tlie scholarship on presupposition). T h e foundational work within linguistics and philosophy was done indcpendentl! by IJans Kanip and Irene Hcim, represented in this volunle bj- Hcim (1983a) and Knmp (1984). An important later framework, dynamic logic, was developed by Jcroen Groenendijk ancl Martin Stokhof (1990, 1991). Throughout the 1980s and I990s, a trcmenclous amount of research on inter- and intra-sentcntial anaphora dt.e\v upon the richncss of the dynamic approach to meaning. We include Heim (1983b), which deals with the particularly natural relation between anaphora and presupposition in a dynamic semantic approach, and simultaneously illustrates the essential integration of scmantics and some parts of pragn~aticson this approach. Another important theoretical framcn-ork de\leloyed during this period uas Situation Semantics (Barwise 1981; Rarwisc ancl Perry 198 1, 1983). 13arwisc and Pcrry's Situation Semantics was in many ways within the formal semantics tradition, and added nevir insights about the value of positing "situations" as a net+ basic type, scrving for instaiicc as the objects of "naked infinitive" perception reports. But it diffcrcd markedly in some of its (onto-)logical underpinnings. In particular, although it is also model-theoretic, it does not make use of the possiblc worlds-based n~odeltheory developed by klontaguc and others; rather it constructs n notion of "possiblc situations" through quite different, set-theorctic, means. Another fran~euorkwhich rcjccts the possible \s-orlds-based model theory, and in Fact rejects model thcory altogether in flvor or' absolute t r ~ ~ tconditions, h is "l>avidsonian Semantics," after Davidson (1967b). This aplwoach produced important scholarship as well (I-ligginbotham 1983a, 1986; Schein 1993; Larson and Segal 1995) and has had a particularly strong follo\ving among a generation of Oxford philosophers (1~;vansand h4cil)owell 1976). Throughout the history of fbrrnal semantics, there has been an incrcasing trend tol~ardsgreater empirical diversity in thc topics which researchers have been interested in. Our collection manages to partiall!- reflect this greater brc;ldth in terms of tlie variety of linguistic constructions treated, but it unfortunately rcflects a historical tendencj- to focus on English. A great deal of recent rcscarch hi~sbegun to erase this bias, but our desire to create a reader of'the essential "classics" of tlie field has meant that none was to be included lierc. (A few early norlis that did address issues in lang~~ages other than English included Cooper (197.5, with a chapter on corrclative

Introduction 7 relative clauses in I-Iittitc), Siege1 (1976a,b), Karttunen and Karttuncn (lO76), Johnson (1977), Link (1979), Gunji (198 l), Miyara (198 11, and Stein (l98l).) Another respect in which the field has developed significantly over the years is in inno\;ations in thc model-theoretic structures in terms of which the semantics is articulated and the increasing sophistication with which the model-theorctic tools of the trade arc deployed. Ladusavv's work on thc role of monotone decreasing functions in the explanation of the distribution of negative polarity items (reprcscntcd hcrc by Ladusaw 1980) was an important step in that direction, as were the innovations in a number of the other papers included here, including Link (1983), Hach (1986), and Kratzer (198 1). A third respect in which thc fielcl has developect significantly since thc ivorlis included here is in tile move away from fully explicit "fiagments" of English or othcr languages to proposals for more general constraints (such as type-driven interpretation; Klcin and Sag 1985) that should interact with a minimal spccification of languageparticular "parametcrs." I11 this respect, developments in semantics have paralleled deirelopments in syntax (although semanticists still tend toward greater explicitness). And it should be noted that within the presently tlourishing area of computational semantics (see Rosncr and Johnson 1992; Nerbonne 199h), explicitness in both syntax and semantics can still be found. All of thcse developments reflect increasing success in blending the perspecti\~esof philosophers and linguists to form a unified field of tornla1 natural language semantics. More 011 the background, historical development, and content of formal semantics in linguistics can be found in o\;ervie\v articles (Janssen 1983; Pastee 1987, 1996, 1097), textbooks (Link 1979; Dowtv et al. 1981; Bach 1989, Chierchia and McConncll-Ginet 1990; Gamut 1991; Heim and Kratzcr 1498), and handbooks (von Stcchow and Wunderlich 1991; Lappin 1996; van Hcnthem and ter Meulen 1997). Major journals in the field are Li~1puistii:~ rind P/tilosop/i,)~ (thrce of our selections, by Karttuncn, L>owty, and Carlson, are fi-om its first volume), and the newer Narurcil Lnr~gutige,J'PIII(IP~/IL.S (too nen- to include "classics", since its first volume was publishcd in 1993).

Selection and Organization of Contents 'The total length of the volume had to be limited so that it could be published at a reasonable cost, and unfortunately this has ineant that we have had to go through the very difficult process of selecting only this small set of papers for inclusion. We have tried to balance a number of competing recluirements and desires, knowing ti-om the beginning that nrc would not be able to include all of the papers that wc ant1 our colleag~~es rccognizc as classics in the field, nor even all of the authors who havc urrittcn articles or dissertations that are considcrcd classics. we started by askins a 131-gc number of colleagues for their suggestions, and then thought and rethought our decisions as we xvorked our way from an initial list of about fifty indisputably classic \vorks down to the present collection of eighteen. We regret every omission, but hope that our colleagues will agree that we have assembled a good collection of outstanding papers that balance the various criteria that are important for a volume like this onc.

8 Introduction Some of the issues we have considered in making our selection arc tlic following: we wantecl papers which have been broadly influential for a number of' ?cars, papers covering a variet~-of empirical domains and formal tools, papers representative of the field's historical development, ant1 ideally papers that are not too long (so as not to displace too many others). Some classics are dissertations or other longer works, but we made a conscious decision to include only complete articles ancl not cscerpts, since it is usually very difficult to react anything in formal semantics without starting at tlic beginning, and the "results" are likely to come near tlic end. Other things being equal, we gave preference to papers that are not already easily accessible (in major journals or in accessible collected voluines such as this one). And because there already exist many fine anthologies of the philosophy of language (see, for instance, Martinich 2000) and very few if any in linguistic semantics, that last consideration eliminated a number of' classic background papers in the philosoplip of language which are alrcady widely anthologized, such as Davidson (1967a), Lewis (1970). and Kaplan (1978, 1979). Tlic volume is therefore unevenly tilted toward the linguistic side of the linguistics and philosophy interaction, and the table of contents does not reprcscnt the degree to which the early development of fornial semantics resulted from the combined efforts and fiuitful interaction of linguists and philosophers. But omission of most philosophical classies also helped us to narrow down the time span of our contributions, increasing their coherence: the earliest paper in the collection is Moiitague (1973), and the latest is Groenendij h and StokhoC ( 1989). Dcspite our efforts to cover a broadly representative range of topics in ous choices of' classics, soine important areas of research remain unrepresentcd or only tangentially represented here. We would have liked to include works on the scnlantics of adjectives and adverbs, such as the classic papers by 'I'homason m d Stalnakcr (1973) and Kainp (1975). Another underrepresented area is intensionality: while i i ~ u c of l ~ thc important work in that area is well represented in collet.tions in the philosophy of langui~gc,it m.ould have bcen nice to be able to include more hcrc, such as Cresswell and von Steclion~(1982), Chierchia (1982, 1985), and Chierchia and T ~ ~ r n (1988). cr Work by formal semanticists on the semantics and pragmatics of f ~ c u s a, major research in more recent Fears, was just getting started near the cnd of the period of our "classics" with the dissertation of Rooth (1985); the later payers by krifka (1992) and Root11 (1992), which are quite accessible, gi1.e a good picture of progress in thc first scvcral years of research on tlic semantics of focus. We have organized our volume more or less bq topic. Of course, ho\vevcr one dividcs 1113 the topics, there will he a great deal of o\-erlap. T h e groupings w-e have come up wit11 are listed below. we have already said a few words ahove about some of thcsc papers and their place in the development of'the field; we add atlditional coinn~entsbcloiv.

1 'The beginnings, representecl by Moiitaguc (1973). All of Montaguc's papers in formal semantics and pragmatics, as well as his work in the dcvclopmciit of intcnsioiial logic, are collected in nllontague (1974), u~liichnlso inclucles an important introductory essaj by Richmond Thoniason. 2 Noun phrase semantics: Carlson (1077), Uarwise and C;ooper (lC)81), and Idink (1983). h4ontaguc (1973) could certainly be included hcrc as well, since it aiid

Introduction 9 Lewis (1970) were the first works to propose that noun phrases should be intcrpretcd as generalized quantiiiers (properties of propcrties, rather than as in a firstordcr sjstcm like predicate logic); Barwise and Coopcr elevate this idea to thc level of claimed universal, suggcsting that all noun phrases denote gcneralizcd quantifiers. Thcy also propose a wide variety of potential applications, thus launching thc sub-field of seniantics known as Generalized Quantifier Thcory, including a first attempt at formalizing the weak/strong distinction among determiners that \vas argued for by hlilsarh (1977) in llis treatment of existential sentenccs in 1.nglish. Carlson's paper focuscs on the interpretation of bare plurals; bare plurals present a variety of semantic puzzles which have long exercised linguists, and these puzzles intersect with many othcr important issues like quantification and genericity. In particular, Carlson's work is important for having introduced, and demonstrated the linguistic relevance of, the distinction between prcdicatcs which express more or less transient properties (stage-level predicates) and those which cxpscss morc or less permanent ones (individual-level predicates). Idink's paper represents an influential step in our ullderstatiding of plurals and mass terms, adding e-type plural entities and mass entities alongside the more familiar singular atoniic entities, and introducing a homomorphism between count and inass domains. At the thcoretical level, it shows the relevance of abstract algebraic structures to thc study of natural language, introducing a mathematical perspective which has rcniained influential. Philosophical pr;~gn~atics: Stalnaker (1978) and Lewis (1979). These papers niahe foundational contributions to the embedding of truth conditional semantics into a broader franicwork of semantics and pragmatics. Stalnahes introduces the notion of "conimon ground" (the set of possible worlds compatible with what speaker and hearer can bc presumed to take for granted at a given point in a conversation), and analyzes the core pragmatic concepts of assertion and presupposition in ternis of tlie wav utterances both depend on and affect the common ground as it evolvcs during a conversation. Lea-is extends this perspective with his "scorekeeping" mctaphos to show that the evolving context on which many aspects of interpretation dcpend includes not only prcs~~pyositions but also indications of such things as the current salience of various entities being talked about, standards of precision, graded modal relations, and perspectives relevant to indexicality. In terms of their iiiipact on linguistics, they la)- tlie groundwork for work in dynamic semnn~ics,thc next section. Dynamic scmantics: Lcwis (1975a), Kamp (1981), and Heim (19$3a, b). We discussed this work, and the further development of dynanlic scmantics in the ~vorkof Groenendijk and Stolthof, and others, above. Thc semantics of thc inflcctional/auxiliary system, i.e. tense, aspect, and modality. This diverse field is quite inadequately represented by Doivt~;(1977), Kratzcr (1981), and Bach (1986). Dowty's paper is onc of the most influential \vol-ks o ~ i an aspectual construction, in this case the English progressive. Bilcli's paper has to do with the progressil-e as well, which is nrl.1~it is includcd in this scction, hut it is also note\\-orthy for its connectioiis to Link's research on bare plurals; it also allows tlie readcr to be introduced to the important role of the notion of "evcnt." Kratzcr's

10 Introduction work from the late 1970s and early 1980s has long been the standard for our understanding of modality. 6 Conjunction and type-shifting: Partee and Rooth (1983) and Partee ( 1986). 'l'ypeshifting is a theoretical tool which allows an explanation of phenomena which indicate that a particular phrase can have a variety of different n~eaningswhich are nevertheless closely related in a logical sense. T h e Partee and Rooth paper, which builds o n earlier w-ork by Gazdar (1980) and others on non-transforn~atio~lal treatments of cross-categorial phrasal conjunction, is usually cited as containing the first systematic proposals for some general principles of type-shifting. Partee (1986) applies the technique to the treatment of noun phrases, and of'fers among other things a w-ay to reconcile gcneralized quantifier theory with the non-uniform NP semantics proposed by lcamp and Heim. 7 Questions: Karttunen (1975) and Groenendijk and Stokhof (1989). Clausc types other than declaratives raise an obvious issue for truth-conditional theories of meaning: how can one think of thc semantics of a question in terms of what would nlake it true? T h e solution to this puzzle involves understanding the meaning of questions in terms of the truth conditions of their answers, and various scholars have worked out this perspective in different ways. Karttunen's papcr, a highly influential early work on the topic, argues that thc meaning of a question is the sct of propositions espressii~gits true anskvers. Groenendijk and Stokhof point out that theories like Karttunen's take answers to be uniformly propositional (Q: I/b'/lo k j t A: May11 lefi.),argue that constituent answers (-4:Mar)?) must be considered as well, and work out a synthesis utilizing type-shifting. (The choice of this particular paper by Groenendijk and Stokhof might be somewhat surprising, as it is not their mostcited work on the topic, but we find it most suitable because it incorporates many of their important ideas while being quite self-contained.) 8 Negative polarity: Ladusaw (1980). Ladusaw's work on negativc polarity was tllc first to show the linguistic significance of a purely model-theorctic property, onc that has no analog as a structural property of a syntactic rcprescntation, even at a level of "logical form." It thus establishes the importance of formal semantic analysis tbr understanding the distribution of linguistic forms.

Acknowledgments Finall!;, we would like to express our thanks to somc of those who contributed LO putting togctlicr this volume. First of ;ill n-e extend our thanks to Stexc Smith, Tanii Kaplan, and Sal-;ih Coleman, as wcll as their collc;igues at Blackwell. for their support and assistance, to all of our many collcagucs who responrled to an e-mail solicitation with liclpful aclvicc about what papcrs they nould lilic to scc in such a book, and to Anna Oxbury for. overseeing the copy-editing and prooii.cading. Wc arc g~.ateli~I to Barbara's research assistants Paul de I,acy for managing and compiling thc responses to our initial c-mails and Ji-yung Kim for co-managing the indexing projcct, and to Paul's rescarch assist.ants Simon Maucli, Matt Baucl., and S h i m Felling for hanclling photocopies and col>! right permissions. Thanks also to tlic indexing team members Luis F. Xlonso Ovallc, Xlaho I lirot;ini, Eva Juarros, hlahoto Kadaffaki, Minjoo Kim, Xlcredith I,andn-ran, and hlil~.cinMorzqcLi. Paul Portner and Barbara 11. Partcc Gcorgeto-I$n and !!nihcl.sr

Introduction 11

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14 Introduction Katz, J e r r ~, and Paul Post:il. 1964. .jri Iii~c~gr.n~eJ T h ~ o r :c!/'I,irig.rruri~ )~ D~siripr~orrs. Canihridgc, hlnss.: h,IIrT Press. Kccnan, Edward. 1971. T w o kinds of presupposition in natural language. In C:, Fillmore and 'r. Lnnpencloen (eds), Sllrili~sin Lirrgclisiir. Setr~nnrrc.~, Ncw E'ork: Holt, Rincliart CSr Winston, 45 54. Klein, En--an and Ivan A. Sag. 1985. Type-driven translation. Li~r~yuls/ii~s clrirl Phrlosnpl~,)~ 8: 163-201. Kratzer. Angelika. 108 I . T h e notional category of modalit!. In FI.-J. Eikme! cr ancl H. Ricser (cds), l h ~ r r l . ~Ilbrlrl.~, , r l t ~ C~ lO ~ I I P I .IV.~'JP I S . ~ ( J ) P ~ o ( ~ L . /toI ( ~I1,h~d s . S C I I I ~ I I IBerlin: I ~ ( . S , Walter tle Gruytcr, -38- 74. Krifk;~,hlanfrcd. 1992. A framework for Lbcus-sensitive quantilication. In Chris Barbcr and David Dontj- (eds), Proi.ec.r/i~igsuf'tlle .'?eroni/ (,'or!fi~lunt.t>on Scrnrr1~11c.smrid I,in,yrirslic 7 ~ 1 i ~ o(.s.iLT r)~ I)), Colunibus, Ohio: Oliio State University, 21 5-36 I.adus;~n,\tTilliam, 1980. On the notion "affective" in the analysis of ncgativc polrlrit!. items. Jo;,rrrtr~/ of'Liir~~rrz.I l~~tticc'-thcor~'tici~I ii1~p1.oach. In 1%. Haucrlc, C. Schwarzc, and "I. von S t e c l ~ o \(eds), ~ /l.l~~(rrrirrg-, lisi. irrril /he l i ~ / t ~ r p t - t ~ / ((!/' //ioi~ Ltrrrgirape, Berlin: Waltcr de Gruyter. Rcpr. in Linb, tiodehard. 1008. :I[yc~hriric .Sei/rirniii~sin Lntlgulrgc' (iirl/ Philo.~oplt)~, CSLI lecture notes no. 74. Stanford, C:i~lif.,:CSI .I I'ublicl~tions, 1 I--34. Rlartinich, 11. P . (ed.). 20O0. Tlir /Jlrr/osoplr,)~o/'i,rrng~rtrgc:,4th cdn. Oxf'orcl: Oxford University Prcss. &la!-, Robert. 1985. Lop.il.rr/ For/rr: Iis S/rvi.iurc lrrzrl Dcri~.tr/ron.C;arnhridgc, Mass.: M I T Press. hqilsark, Gary. 1977. Tornard an explanation of ccrtain peculiarities of tllc existenti;il construction in English. Lirig~~istir .-liicrllvis 3: 1-29. hliyara, Sl~insho. 1081. Complex Preclicatcs, C:asc XIizrhing and Scrambling in J;tyancsc. Ph.U. dissertation. University of ?Vlassachusetts, Amhcrst. klontague, Richard. 1970a. English ss a formal language. In B r ~ ~ nViscntini o 1.1 irl. (cds), Lingrrl~~gi r~cll~r Soc~'e/ric nc,/l~rTcc.irrtn, Milan: Erlizioni di (:omuniti, 180-224. Repr. in hlo11t;lguc 1074, 188--22 1. Montayue, Richard. 1970b. Universal grammar. Tllr~orirr 36: 3'73-98. Rcpr. in Alontaguc 1074, 222-46. hlonraguc, Richard. 1973. T h e proper rrcatmci~tof quantification in ortlinal*!. English. In I(.J. J. Mintihka. J . M. L. hioravusik, and P. Suppcs (eds), .ippioaclii~slo Nu/irr~(r/I , ~ I I I R I ~Dorilrccht: ~I~c>, D . Reidel, 221-42. Repr. in Alontague 1954, 247-70. hfontague, Richard. 1974. F o n ~ i r ~Plrilosopl~)~. I S'tkc./c~rlPapers .vI/'Ric~lrrrrlli \ 1 o t r / 1 1 ~ 1 t i cditcd ~, and u irh ;in introcluction by Richmoncl 13. Thoniason. New I-Iavcn, C:onn.: Yalc Uni\,crsity I'rcss.

Introduction 15 Ncrbonne, John. 1900. C:omputational scn~antics linguistics and processing. In Shalom T,i~ppin (ed.), Tlre H~irrr/l)ookr!/'Corr/enipoia~:)f .S~.rrrrrn/ir.T ~ I L ~ Oxford: ~ I : ) ~ , Hlack~vcll,401-84. Ncwmcycr, Fretlcrick. 1980. Lirigrrrsrl(~Tl7cor:)firr z-17~rrr.ir.~i: Tlie F11:rr Qnrir/er-Cr~~r/vr:)~ I!/' 7j.,rli.~/;)r.rrrlr/rorrul G P I I C ~ ( IG/ ~I I~~' C ~ ~ I NCMI I I ( IYork: ~ . 14cademic Press. I'arsons, Tcrcncc. 1972. An Outline of a Sanantics of English. Ms. thcsis, Lnivcrsit!! of h1assach~1sctts, Amherst. Partec, Barbara. 1073. Some transf'orniational ex~cnsionsot' Montaguc grammar. .?o~irnril I ! / ' Philosoplrinil Logic 2: 509-31. Repr. in Partee 1976, 5 1-76. Partce, Barbara. 197.5a. Comments on C . J. Fillmore's and N. C.hon~sk!;'s papers. In R. Austcrlitz (cd.), Tllr Scopr ( ! / ' , - ~ I N P I . ~ CLtr~guistit.~, ~I~ Lisse: Peter de Ridder Press, 197-209. Partec, Uarbara. l075b. Montaguc grammar and transformation;~l yrammar. I,ingiiis/ir 11rqii11:)l0: 203-300. Partec, Barbara (ect.). 1976. Monl(rfiiie Gn~nrtn(rr.Nccv York: A4cademicPress. Partce, Rurbara. 1984. Compositionality. In Fred Landman and Frank ITcltman (cds), I irrii,/res ( I / Forrtrrrl LSenrontrr~s, I~orclrccht:Foris, 281-312. Partcc, Harbara. 1987. Possible worlds in model-theoretic semantics: a linguistic perspective. In Sture A411en(ed.), P~tssihlc bbi)r/r/.~rrr Hlirnrirr ilits, ijr/s, (r irtl Sr,ier~r.~s. Proi.~rrlr~~gs I!/' oh^/ .5')~111posirii1i 0.7, ljcrlin: Wattcr clc Gru! ter, 93-123. Partcc, Barbara. 1986. Noun phrase intupretation and type-shifting principles. In J. Groencndijk, L). de Jongh, and M. Stokhof (cds), Srul1ie.r iir L ) I S I . ~ I I IR S Pr p w . ~ ~ ~ ~ r r irlreol:)' t i o ~ r rrrrrl 111t T I I I ~ o (!/' I:)~ G~rr~nrli:ci/ Q~rorr~!/ild'rr~, Dordrecht: roris, 115-43. Partcc, Harbara. 1996. T h e detelopment of formill semantics in linguistic theory. In Shalom 1,appin (cd.), Tlic, Hrrndl~~tok ~!/'C~ont~rrrpn~.rtr:)~ ,S~~nrrrn/ir. rlizor:)~,Oxford: Black\\-ell, 1 1-38. Pmtcc, Barbara, and Mats Rooth. 1983. Generalized conjunction ancl type ambiguity. In Raincr Biucrlc, Christoph Schwarzc, ancl Arnim von Stechoa (cds), Mnrairrx, Zlsr, r//i(lIn/erpr.i~tlr/irrrr01' Lri~rgurrp,Rcrtin: Wattel. dc Gruj-tcr, 361-83. Pvtcc, Barbara H. with Herman L. \V. Hendriks. 1997. Montague glamnlar. In Johan van R e ~ l t h c n ~ and Alice ter !vIeulen (eds), Hanr//)ook I ! / Logii. r r ~ r i / Lr~rrgutr~qe,.4mstcrctan1/C:arnbridgc, klass.: I
16 Introduction Stein, .Mark. 1981. Quantification in Thai. Ph.D. dissertation, Department of Linguistics, University of Massachusetts, hmherst (available from GLSA, UMass, L A i i i l ~ ~ r ~ t ) . Subrahmanyam, Ramesh. 1989. Scmantic interprctation in tree adjoining grammar. In Joyce Poncrs and Kenneth de Jong (cds), ESCOL 5: A-oc.et~clirrgsc!/'thi~5th E l ~ s t ~LS/n/es r ~ i Cou/iv-lwr.~ O I I Ll~~gujs~rrs, Columbus, Ohio: Ohio State University. Tarski, i2lfrcd. 1944. The semantic conception of' truth and the foundntions of scmantics. Philnsnph~~ unlf Pl~enom~~nokogrl~rrl' ResearcY~ 4: 341-75. Repr. in Martinich 2000. Tlion~ason,Richmond and Robert Stalnaher. 1973. A semantic theory of advcrhs. I;in~ll~.s/jc l n q i r r ~4:) ~ 195-220. L a n C y u ( ~ Arnsterclam y~. van Benthem, Johan and Alice ter Mculen (cds). 1997. H l ~ ~ ~ d h oofo k Clambridge, Mass.: Elsevier M I T Press. \-on Stechow, - h i m and Dieter Wunderlich (eds). 1991. Sarrrcrtr~iX*:Ein i r ~ ~ e r ~ r a ~ i o iH~ ~u l~r ~ s( ~ (Ier /JIu/I :.til~e?zir,sis~lzen Fonchuirg [Sew~antir.s:.An Itr/em(~tion~rl Hr~ndhook~ ? / ' C o t l / e ~ r ~ p oReserrr.~.lr]. mty Berlin: Walter de Gruyter. Webher, Ronnie. 1979. -4 For,nl~aI,4ppro(r~.hl o Disr.our:~e.41?(rphortr.New Yorh: Garland.

The Proper Treatment of Quantification in Ordinary English Richard Montague

The aim of this paper is to present ill a rigorous way the syntax and seniantics of a certain fragment of a certain dialect of English. For expository purposes the fragment has been made as simple and restricted as it can be while accommodating all the more puzzling cases of quantification and reference with which I am acquaintcd. 1 Patrick Suppes claims, in a paper prepared for the present \vorkshop [the 1970 Stanford Workshop on Graninlar and Semantics], that "at the presenr time the semantics of natural languages are less satisfactorily fbrmulated than the grammars ... [and] a complctc grammar for any significant fragment of natural language is yet to be written." This claim would ofcourse be accurate if restricted in its application to thc attempts emanating from the hlassachusetts Institute of Technology, but Fails to take into account the syntactic and semantic treatments proposed in Montague (1970a, b). T h u s the present paper cannot claim to present the first coinplete syntax (or grammar, in Suppes' terminology) and semantics for a significant fragment of natural language; and it is perhaps not inappropriate to sketch relations between the earlier proposals and the one given bcloiv. Montague (197Ob) contains a general theory of languages, their interpsetations, and the inducing of interpretations by translation. T h e treatment given below, as well as that in hlontague (1970a) and the trcatmcnt of a fragment of Cnglish proposed at the end of Montague (1970b), can all easily be construed as special cases of that general theory. T h e fragment in Montague (1970a) was considerably more restricted in scopc than those in Montagiie (l970b) or the present paper, in that although it admitted indirect discourse, it failed to accommodate a number of more con~plexintensional locutions, for instalice, those involving interz.sionn1 zrerhs (that is, verbs like seelts, worships, conceives). T h e fragment in Montague (1970b) did indeed include intensional verbs but excluded certain intensional locutions involving pronouns (for instance, the sentence John wishes to catch a fish and eat it, to which a number of linguists have recently drawn attention). The present treatment is capable of accounting for such examples, as well as a number of other heretofore unattempted puzzles, for instance, Professor Partee's t h e t e m p e r a t u r e is ninety b u t i t is rising and the problem of intensional prepositions. On the other hand, the present treatment, unlike that in Vontague (1970b), will not directly accommodate such sentences as J. M. E. hloravcsik's a unicorn a p p e a r s to be approaching,

18 Richard Montague in which an indefinite tern1 in subject positioi? would have a nonrcfercntial reading, bur must treat thein indirectly as paraphrases (of, in this casc, it a p p e a r s t h a t a unicorn is approaching or t h a t a uilicorn is approaching a p p e a r s to be true), On thcir common domain of' applicability the three treatments essentially agrce in the troth and entailment conditions imposed on sentences.' Further, when only LC/nr.athlr sentences come into consideration, it is the construction of such conditions that (Suppes notwithstanding) should count as the central concern of syntax and semantics.' Nevcstheless, the details of thc present development possess certain aesthetic merits, of coherence and corlccptual simplicitj-, not to bc found in the treatment of English in Montague ( 1 9'70b). (It is in ordcr to preserve these mcrits that I hcre hrgo il dircct account of such sentences as -Moravcik's.)

1 The Syntax of a Fragment of English Let P and r be two fixed objects (0 and 1, sax) that arc distinct and neither ordered pairs nor ordcred triples. Then Cot, or. the set of l.strgorlcs of English, is to bc the sinallcst set X such that (1) t. and t are in X , and (2) whenever ./I and B arc in X, :I/B and ..4//8 (that is, (0, A, B) and (l,.ii, B) respectively) arc also in ,Y. It should be pointed out that our categories are not sets of expressions but wili instead serve as inrl'icus of such sets. We regard r and I as the categories of entit) expressions (or individual ex;prcssions) and truth valuc expressions (or declarative sentences) respecti\-ely. Wc shall regard the categories A/B and .-I//B as playing tht same semantical but different syntactical I-oles.An cxpression of either category is to bt such that \\-hen it is cornbilled (in some as yet unspecified way, and indeed in differq n~aj-sfix the two categories) with an expression of category 3, an expl-ession of categor! A is produced. (The precise character of the categories l J / B and A//B is unimportant we require only two different kinds of ordered pair.) It will be observed that our syntactic categories diverge from those of Ajdukicwic: (1960) only in our intl-oduction of two compound categories ( d 4 / Band .Ll//B) whcrc Ajdukiewicz would have had just onc. T h c fiact that we necd only two copies is merel: an accident of English or perhaps of our limited hagmcnt; in connection with othc 4 languages it is quite conceivable that il larger number would be requircd. Keeping in mind the intuitive roles described above, we ma!. sin& o ~ as~ follofv t certain traditional syntactic categories.

IV, or the categorv of intransitive verb phrases, is to bc t/e. T , or the category of terms, is to be t/IV. TV, 01- thc category of transitive verb phrases, is to bc IV/T. IAV, or the category of IV-modifying adverbs, is to bc IV/IV. CN, or the category of common noun phrases, is to be t//c. T h c fi)llowil~gcategorics will also be exemplified in our fragment although no special symbol u-ill be introduced for them. t / t is the category of scntcnce-modifying adverbs. IAV/T is the category of IAV-making prcpositioi~s.

The Proper Treatment of Quantification 19 IV/t is the category of sentence-taking verb phrases. IV//IV is the category of IV-taking verb phrases.

By BA4is understood the set of bnsic c.sprt.ssions of the category A; the notion is characterized as follows.

BI\-= {run, walk, talk, rise, change) H..1 - {John, Mary, Bill, ninety, heo, h e l , hez, . . .) B,l-v= {find, Iose, eat, love, date, be, seek, conceive) BI.\\. = {rapidIy, slowIy, voIuntarily, allegedly) BC:h= { m a n , w o m a n , park, fish, pen, unicorn, price, t e m p e r a t u r e } B,:, = {necessarily) BIAI.p-= {in, a b o u t ) BIVlt= {believe t h a t , assert t h a t ) RI~~ll= l \ {try to, wish to) B,4 = L (that is, the empty set) if A is an; category other than those mentioned above. (In particular, thc sets B, of basic entity expressions and Bt of basic declarative sentences are empty .)

BY a bnsic e.t.p~eaionof the present fragment is understood a member of U-i,c,,,,13.1. BY PA is understood the set of phrases of the categorj- A. (We may read "Pc:n", "Pel,\"', and the like as "the set of common noun phrases", "the set of transitive verb phrases", and so on.) These sets are introduced, in a sense to be made precise below, by the following rules, S 1-S 17.

Syntactic rules S1. U 4 PA for every category A. S2. If i E PCN then Fo(<),Fl (i), Fl(<)t P-I., )

ivherc Fo([) = every <, F l ( i ) = t h e i, F2(<)is a ior a n [ according as the first \vord in itakes a or a n . wl~creF3,n(;94) = [ s u c h t h a t SO. If it and 4 E P,, then F3,,(i? 4) E 4'; and d)' comes from (j5 by replacing each occirrrence of he,, or him,, by

{ ) }:{ Fe

or

respectively according as the first B.:.

in

<

is of

neuter

2 E Ptjl\-and 6 E PI\, then F+('r,d) t Pr, where F+(a,d) = rh' and h' is the result of replacing the first wrb (i.e.; member of RlY, Brl.\, HI\./,, or B1\ in 6 by its third person singular present.

S4. If

20 Richard Montague

S5. If (5 E PIl /.I. and /IE PI., thcn F5(S,p) E PI\,, whcre P5(S,[I) = tSP it'P does not have the form he,, and Fs(fS, he,) = 6 him,,. S6. If d E PIrlvI,I'and /? E P.17, then F5(6,P) E PI,l\,. S5. If 6 E PI\ /, and E P,, then F6(ci,/3) E PIv, where F(,(cj,1,') = SP. 58. If 6 E and E PI\., then Fb(O,P ) E PI\;. S9. I f 6 E P,!, and /I E P,, then f.i,(S,P) E P,. S 10. If (S E Pli /n'and fi E PIV,thcn Fi(S, P ) E Ply, where F j ( 6 , /I) = Ph.

Sl I . If 4, Id/ E P,, then F8(4. $), F o ( ~$) , E P,, where F8(C),$) = rf, and !I/, F(,((I,,lI/) = 4)or $. S12. If:l,6 EP,\-, then rx(7j,(5),F9(;~,6)~ PI\-. S13. If 2,/I E Psi,, then FO(a, p) E Pvl..

S14. If x E PT and 4 E P,, then Flcl,,,(a,4) t P,, whcre citlicr (i) a does not have the form hel., and Flo,,,(a,4) comes from 4 by replacing the first occurrence of he,, or him,, by a and all other occurrences of he,, or him, by

}!F{ }::{ or

wspectively, i~ccordingas the gende~of t1.c

}

C

masc. first B~:. or B , in ~ a is {fern. , or neuter (ii) a = heLp, and Flo,,,(a,4) comes fioni ciJ by replacing all occurrences of he,, or him,, by hek,or himk respectively. S15. If a E Prr and i E PCN,then Flo, ,,(a,i)E PCN S16. If z E PT and 0 E Plv, then Fill, ,,(a,6) E Ply.

Rl~lesqf lense and si'qt~ S17. If a E PT and 6 E Ply, then TI (a, d), F12('%, O), F13(a,S), P14(a,d), F15(a,6 ) E P,, where: F1I (a, 0 ) = ah' and 6' is the result of replacing thc first verb in 6 by its negativc third person singular present; Flz(a,6) = adf' and 6" is the result of replacing thc first rcrb in 6 by its third person singular future; Flj(a,d) = adf'' and 6"' is the result of replacing the first verb in 6 by its negative third person singular Suture; FI4(a,8) = xd"" and dJfJfis the result of replacing the first verb in ;5 by its third person singular present perfect; and finally, F15(a,6) = aiS""' and 6""' is the result of replacing the first verb in 6 by its negative third person singular present perfect.

The Proper Treatment of Quantification 21 The precise characterization of the sets P,\, for '4 a category, is accon~plishedils follows. We first define the auxiliary notions occurring in the rules above in an obvious and rlrtir.lr tnkrtr traditional way: the R C I I ~ Cof~ an arbitrary member of BT U BCN,the ~rri/ third person si~~~qrrkrr presewt, the rhird person singulur % f b t u wthe , ncy(itiz1c t h i d penon si?~~qlllr~r firtrrve, the tizi)~ripersonsingukm presetat prt;fir.t, and thc n~>gntjzte tlijri/pcrsot~singrrlarpr~sc~n~ pfi;fict of an arbitrary verb. Then we may regard SlLS17 as constituting a sin~ultaneous inductive definition of the sets P:l. Since, however, inductive definitions of this form are somewhat unusual, it is perhaps in order to state a corresponding explicit definition: the sets P I (for A E Cat) arc the smallest sets satisfying SlLS17; that is to say, (P,l),E(;~,, is the unique fanlily of sets indexed by Cat such that (1) (PA)-4c(:,,satisfies S 1- -S17, and (2) whenecer (P\)A4,c,, is a family of sets indexed by Cat, if ( P i ) satisfies S 1--S17, then P,\ C P i for all A E Cat. (It is easily shown, using an idea I believe to have originated with Dr. Perry Smith, that there is exactly one family of sets satisfying these conditions.) By a rneirrzingfi~Ie.~pressionof the present fragment of English we may understand il member of any of the sets PA for A E Cat. As an example, let us sho~vthat every man loves a woman such that she Ioves h i m is a declarative sentence (that is, member of P,). By S1, love E P,rs and he,, E P.1.. Hence, by S5, love himo E Prv. Therefore, by S1 and S4, hel lovcs himr)E P,. Thus, by S I and S3, woman such that she love hin10 f Therefore, hy 52, a woman such that she loves himo E PT. Hence, by S1 and S5, love a woman such that she loves him* E PIv.Therefore, by S 1 and S4, heo loves a woman such that she loves himo E P,. Also, by S1 and S2, every m a n E P.1.; and hence, by S 14, every man Ioves a woman such that she loves h i m E P,. We nlay indicate the way in which this sentence has just been constructed by means of the following 1rnn1l1si.v trpee: every m a n loves a woman such that she loves him, 10, 0 every man, 0

/

man

he,, loves a woman such that she lovcs him,,, 4

/

love a woman such that she loves him,,, 5

I

woman such that she lovcs him,,, 3, 1 woman love him,,, A

love

22 Richard Montague T o each node we attach a meaningful expression, together, in casc that expression is not basic, tvith the index of that structural operation among I;;,-fi, F3,0,F3, 1, . . . , P4-F9, rlo, o, Flo,1 , . . . ,F1I - F 1 5 (as characterized above, within S l -51.7) which we understand as having been applied in obtaining the expression in question; thc nodes dominated by an? node are to be occupied by the expressions to which the structural operation is uilderstood as having been applied in obtaining the expression occupying the superior nodc. (For example, the nunlbers 10,0 attached to the top node of the tree above indicate that thc expression attached to that node is regarded as the value of the operation as applied to certain arguments; and the nodes beneath indicate that thosc arguments are understood to be the expressions every man and heclloves a woman such that she loves himcl.)A precise characterization of an ~ i ~ ~ n / ) 1rue ~ s i sin the sense of these remarks \vould be routine and will not be given hcre; for such a characterization in an analogous content the reader might consult A4ontague (1970a). Now there are other ways of constructing the sentence under consideration, and hence other analysis trees for it; indeed, it can be shown that eyer? declarati~csentence of our fragment has infinitely many analysis trecs. But in the case considered, the various analysis will differ only inessentially; that is to say, they will all lead to the same scmantical rcsults. There arc other cases, however, of which this cannot be said. 17or instance, thc sentence

rlr,,o

John seeks a unicorn has two essentially different analyses, representcd by the following two trecs:

Tohn sceks a unicorn, 4

4 seek a unicorn, 5

Tohn

a unicorn, 2

seek

I

I

unicorn John seeks a unicorn, 10, 0 /

au

n

I

i

unicorn

c

c

him,,, 4

>eks

A seck him,,, 5

John

A

seek he,,

As we shall see, the first of these trees corresponds to the c/e diltilo (or nonrefcrential) reading of the sentence, and the second to the de re (or referential) reading. Thus our fragment admits genuinely (that is, scmanticallp) ambiguous sentences. If' it were desired to construct a corresponding unambiguous language, it ~vouldbc convcnient to take the analysis trees themselves as the exprcssions of that language; it

,

The Proper Treatment of Quantification

23

would then be obvious how to characterizc (in keeping with MIontague (1970b)) the structural operations of that language and the correspondence relation betneen its expressions and those of ordinary English.' For present purposes, houever, no such construction is neccssary .

2 Intensional Logic We could (as in Montague ( 1970a)) in trod uce the senlan tics of our fragment dircctl j.; but it is probably more perspicuous to proceed indirectly, by ( 1 ) setting up a certain simple artificial language, that of tensed intensional logic, (2) giving the sen~anticsof that language, and (3) interpreting English indirectlj- by showin%in a rigorous way how to translate it into the artificial language. l h i s is the procedure we shall aclopt; accordingly, I shall now present the syntax and semantics of a tensed variant of the intensional logic 1 have discussed on earlier occasions." Let s be a fixed object (2, say) distinct from P and r and not an ordered pair or triplc. The ljyc, or the set of (Ipes, is the smallest set Y such that (1) e,l t Y, (2) whenever cl,h E E',( ~ i , bt ) Y , and (3) whenever a E Y, ( s , ~ E) Y. Uic shall employ denumerably many variables and infinitely many constants of each t y e . In particular, if n is any natural number and a F Type, wc understand hy I),,~,, the ,rr" I-arinblc of type o, and by Con,, the set of constants of type o. (The prccisc cardinality of Coti, need not concern us, providcd only that it hc infinite.) By ME,, is understood the set of metrnin'q-fillr.~pressio?isof type I r ; this notion has the following recursive definition:

1 Evcry variable and constant of type is in NIE,. 2 If a E hlE,, and u is a variable of type 6, then i,rrx E ME(hl,,). 3 If x E and p f hk,, then a@) t ME,,. 4 l f a , / j ME;,, then a = j? E 1\4L<,. !. If Si,,~,!iE ME, and z.r is a variable, then ]4,Irl,r,$J,[&v1[/~l,I~/) 11/.1,[(1) t) 1/11) V nq!!,A u 4 , Oq5, lV41,H4 E ME,. 6 If % E MEI,, then [^lr;l E ME!,,,,. 7 If a E ME(,,,,:, then ['a1 E ME,,. 8 Nothing is in any set ME,, except as required by 1-7.'

-

By a nr~~unit~gfi~l e-~plussiclr~ of intensional logic is understood a member of C]ilE7'i#pr, A,IJ<(,. If l i is a variable of type n , then i u o l is understood as denoting that function f1.0111 objects of type N \vl~ichtakes as value, for any such object s, the object denoted bv x nhcn u is understood as denoting .r. T h e expression %(I$) is as usual understood as daloting the value of thc function denoted by a for the arguinent denoted by /I. The equalit!- spil~bol=, the negation synlbol 1,the conjunction symbol 11, thc disjunction s! mbol v, the conditional symbol +, the biconditional symbol ++,the existential quantifier V, and the universal quantilier A are all understood in the usual way. Tllc symbols 0 , Pt', H mily be read "it is necessar! that," "it will be the case that," "it has been the case that," respectively. T h c expression ["a] is regarded as dcnoting (or

24 Richard Montague having as its extension) the I'nln~sionof the expression a. T h e expression ['nJ is meaningful only if a is an expression that denotes an intension or sense; in such a. case ['a] denotes the corresponding extension. We could have donc with a nluch smaller stock of primitive symbols, as in Montague (1970b); but there is no point in considering here the relevant rcductions. In the presentation of actual expressions of intensional logic square brackets will sometimes for pcrspicuity be omitted, and sometiines gratuitously inserted. Let A, 1,J be any sets, which we may for the moment regard ns the set of entities (or individualsx), the set of possible worlds, and the set of moments of time respectively. In addition, let cr. be a type. Then Dl,, , 1 7, or the set of possibkt clenotarions of type a corresponding to A, I,J, may be introduced by the following recursive definition. (If X and Y are any sets, thcn as usual we understand by X' the set of all fu~lctionswith prolllllc~of X and Y (that is, the domain Y and valucs in X, and by ,Y x Y the C(~r/i~silz~~ set of all ordered pairs (x,.)/) such that ,v f X and , ) I E Y). Further, we identify thc truth values falsehood and truth with the numbers 0 and 1 respectively.) %.

U y S,,,,,~,I, J , or the set of senses of type a corresponding to A, I , .7; is understood D;.,,,),.I, r , , ~ that . is, D!:.;)!,!,~. By an interpretation (or intensional model) is understood a quintuple (-4, I,.j', ,I.') such that (1) -4, I , J arc nonempty sets, (2) 5 is a simple (that is, linear) ordering having J as its field, (3) F is a function having as its domain the set of all constants, and (4) whenever n E Type and a E Con,, F ( r ) E S,,, , , J , J . Suppose that A is an interpretation having the for111 (..4, I , J , , F). Suppose also (of values to variables), that is, a function having as its donlain that p is a n A-ns.sig,tz~r~ent the set of all variables and such that g(tr) E D,,, r,r,g w-henever 11 is a variablc of tj-pe [ I . If a is a rncaninpful expression, we shaIl understand by xA,,: the i n ~ ~ n s i o rofr a with respect to A and g; and if (i,,,) E I x J thcn rA*'?l,Lis to be the ~>.r/~vr~io~, of rr with respect to A, i , l , and g-that is, aA,'((i,.j)) or the function value of the intension of '2whcn applied to the point of referenec (i,j ) . Thesc notions may be introduced by the f'ollowing- recursive definition.

<

<

1 If x is a constant, then is r(,x). is g(n). 2 If r is a variable, then nA?J)lr,c 3 If n E ME, and u is a variithle of type b, thcn [halAy'?-''5sthat function /I with domain Dh, ~ , I , -such J that whenever s is in that domain, h ( x ) is & ~ ' ~ f where ) ~ ' , g' is the A-assignment like g except for the possible difference thilt $ ( u ) is .v. 4 If a E M E , , , , and /I E ME,,, then [ X ( ~ ) ] ~ is) ' ~ J ~ ~ ~ (that is, the value of the function aAli,~7"for the argument /jA,',/)'), 5 i f u,/3 E ME,,, then [a = is 1 if and only if a A ~ ' ~is/ , ~ e h If Q, E ME,, then J / ~ ] ~ * ' ) . ' ) %1s if and only if gA7'7'7^ is 0; and similarly for A,b,

+

, w.

The Proper Treatment of Quantification 25

9 If 4 t ME, and rr is a variable of type a, then [ ~ s q 5 ] ~ ? ' 7is~ 1, , ' if and only if thcrc exists r. t D,,,-J,J,.J such that 4A7'*'7i is 1, where g' is as in 3; and sirnibrlg fi)r

a

e

Aucb. 8 If d) E ME,, then [ ~ d ] ~ ~ ' " ) " sI if and only if c,!?7").i'7" is 1 for all 1' E I and j t -7;' 1I V J ~~ * ' , ~is, v1 if and only if ' 3 " 7 ~1sfor some j' such that .j j'; and j # j'; and [ H + ] ~ , ' , J is , L 1 if and only if q"A7'71'7ris 1 for some I' such that j' j and./' # . j . - ' ~ "lo 9 If a E ME,,, then [ ^ x ] ~ ~ 'is~ xA,f. 10 If C( E ME(,,,/),then ["a l A ~ ' ~ ' . y is ~ ~ ~ ~ ' ~ j ~ ~ ( i , , j ) ) .

<

n

<

I1

h e e

If $ is a/i,mruln (that is, member of ME,), then 4 is rrur with respect to A, i , j if mrl only if q5A)i,i,ris I for every A-assignment g. It will be useful to call attention to some particular meaningful expressions of intensional logic. If 11E MEi,,,) and a E h,fE,,, then denotes (that is, has as its estension) a set (or really the characteristic function of a set) of objccts of type a, and we may regard the formula ;'((Y),which denotes truth exactly in case the object dcnoted by r is a mcinber of that set, as assertir~gthat the object denoted by a is a mcinbel- of the set dcnoted by 11.If y E MEi,, (by,) ) ) cz E ME,,, and t ME,,, thcn may be regardcd ~LSdenoting a (two-place) rclntion, and ;?(by a) is to be the expression ;l(r)(fi), which asserts that the objects dcnoted by- /I and x stanif in that relation. If 7 E ME:(,, : and r E ME,; thcn ;: denotes a prfiperty, and y{a) is to bc the expression I'll](a), which asserts that thc object denoted b y r has that pnjperty. If 1. E ME(, (,,, (,,,,) i , n E ME, and (1' E ME,,, then y may be regarded as denoting a rtl~ltion-in-i?lteiuion,and ;I{/$,a$ is to bc the exprcssion [':.](P, a), ~vhichasserts that the objects denoted hg /I imcl n stand in that relation. If tr is a variable of type (1 and 4 a formula then licl, is to bc i.1~14, which denotes the set of all objects of type n that satisfy 6 (with respect to the place rnt~rkedb~ a), and u$ is to be ( iiq!~],which denotes the property of' objects of type a cxprcsscd by (/I. If r E IME,, then a* is to be P f P { n 8 ~ )wherc ], P is r?,, i,,(;, ,a ;I

;I

) )

n

)

r )

,.

3 Translating English into Intensional Logic We first introduce a mapping J from the categories of English to the types of intensional logic. Accordingly, f is to be a function having Cat as its donlain and such that ~ ' ( L= J )e, = L,

l'(if/B) = .[(il//B) = ((s,.[(B)) ,./'(A))whenever A, B E Cat. The intention is that English expressions of any category A arc to translate into expressions of type .Kg/. In all that follows let g be a fixed biunique function such that ( 1 ) the domain ofg is the set of basic expressions of our fragment of English other than be, necessarily, and the members of Br, and (2) whenever .4 E CLU,a E B.,, and r/. is in the domain of g, ~ ( x E) C O ? ! ~ (Let . ~ )j,. m, 6, ?I be particular distinct members of Con,.. (If u-e had

''

26 Richard Montague introduced a definite well-ordering of the constants of intensional logic, we could at this point have explicitly defined g, j, m, h, and r1. Such details would, however, be irrelevant to our present concerns.) Let 21, u be the particular individual wariablcs ~ 0r ,, r l l , ,, respectively; r, y, x,, be the particular individual-concept variables ol,.(., ,,), 03, (s, l,!, 112~, ,( ), respecti~ely(for any natural number n); p be the proposition variable 110,(,., /:;; P, Q b e the variables uo, is, 1, 111, i(s,l,),,) ), which range over properties of individual concepts; 3 be the variable vo,,,( ( (, 1 (, ,), ,) !,,) i , which ranges over propertics of properties of individual concept; M be the variable t)O, (,, (,,,,)), which ranges over properties of individuals; S be the variable oo, (, (,,(,, ,)) ), which ranges over two-place relations-in-intension between individuals; and G be the variable "0, (,, (r,,.u1 1 )) ) . VCTc shall now consider some rules of translation, TlLT17, which will be seen to col-respond to the syntactic rules S 1-S 17 respcctively and to constitute, in a sense to be made precise below, a definition of the translation relation. (,$,

Translation rules Basic rules TI. (a) If x is in the domain of g, then translates into g(a). (b) be translates into i3;1x3{,jl['s = : ) I ) ) . (c) necessarily translates into j l ' p ] . (d) John, Mary, BilI, ninety translate into,;", ni", h", rtQ respectively. ( e ) he,, translates into P P{x,,}. T2. If [ E PCN and i translatcs into [I, thcn every i translates into PA .Y[['(.v) -t P{x)], the i translates into PV,y[A .\.I ['(.z.) -i s = ,111 n P{.y) I, F2(<) translates into P V rl it(s) P{L~).l. TR. If*i t PcN, Q E P,, imd (, Q translate into i t , (bt respectivcly, tllen fi,,,(i, g)) A

A

A

A

translates into .?,,[('(s,,A)4 f 1. 12

Rules qf :firncriorzal application T4.

If 6 E P,,iIy, fi E Ply, and 6, fl translate into h', /I'respectively, then F+((>,P)

translates into 6'("13'). T5. If 0 E PI\;-I., /I E PVr-,and (5, fi translate into (5') /I' rcspccti\lel!-, then P5(6,li) tr;lnslates into St('[?). rcspectivelv, thcn F5(S,Ij) T6. If 6 E PIliv/,I.,/3 E P.I., and 6, /j translate into (it, /I' translates into hl("P'). T7. If 6 E Plvi,, /iE P,, and 6, fl translate into 6')fit respectivelj, thcn F,((j,/l) translates into 6'("pf). TX. I f S E PIv//Iv,11 E PI\^, and 6, /) translatcs into 6'. [I' respectivcly, then F'(,((j,/j) trinslales into 8'("/1'). T9. If 6 E I?,/,, P E P,, and 5, [j translatcs into 6', [j' respecti\elj*, then Fb(6,/i) translates into S 1 ( ^ / j ' ) . TIO. If 6 t PIv,qv, E P I \ , and 6, /I translatcs into (5'. B' respcctively, then F7(0, fi) rans slates into 6'(A/1').

The Proper Treatment of Quantification 27

S 7

e

f s r C

If ( j , t,b E P, and 6 , ~,btranslate into gh', $' respectively, then cf, and I/! translates into [4 A $1, (/, or !,/I translates into [$ v $1. T12. If ;!, (5 E PTvand 71, (5 tra~lslatesinto I?', S' respectively, thcn ;' and 6 translates into .i.kf(.v) A df(s)l,;> or 6 translates into F[-)"(x) fit(x)]. T13. If a, /Ic P r l - and x,3!, translate into x', P' respectively, then or [l translates into i [ d ( v~[I'P ) j].

T11.

\/

1

Rules qf'yua~trJii.aiion 1'14. If a E P.l., d) t P,, and n, 4 translate into a', (11' sespectirely, then Pl0,,,(n,(1)) translates into a'(.?,, (/I/). T15. If a t PT, [ E PC;\,and n, i translate into n', i'respectively, then Flll,,,(r,i) translates into ?,),x'(.i',,[<'(5~)]). 1 1 If y. E Prl., 6 E PI\, and n, A translate into r',6' respectirely, then Flu,,,(n,li) translates into jla'(.~,,[6'(y)l).

Rules

uf'tpi?sr 111111 ~ i g n

1'1 7. If x E Prr.,ci t PlV,and x, S translate into a', Fl (a, 6) translatcs into /a'(^~5~), F1?(a, 0 ) translates into ~ r x ' ( ~ b ' ) , F 13('x,b) translates into / C'C'CX'(' S'), F14(a,6 ) translates into ~ x ' ( ^ h ' j , F1;('%,6) translates into ]13a'(^b1).

1

1

1

1

1

cj'

rcspectivcl\i, thcn

'l'he precise import of the rules T1-TI7 is that the translation relation nlay be defined as thc smallest binary rclation satisfying them; that is to say, an cxpressioo is characterized as tmnslrrting into an expression 4' if the pair {(b, 4') is a membcr of cvcrybinarv relation R such that T1-TI7 hold (with the condition that one expression translates into another replaced by the condition that the rclation R holds between thc tno expressions). The translati011 relation is of course not a function; a meaningful expression of' English may translate into several different expressions of intensional logic. We could, however, speak of rhr 1ra~ulntior7of a given meaningful cxyression of English corrcsponding to any given analysis trce for that expression; the rathcr obvious definition of this notion will be omitted Ilere. The interpretations of intensional logic may, by way of the translation relation, be made to play a second role as interpretations of ~ n g l i s h . ' " Not all interpretations of intensional logic, however, would bc reasonable candidates for interpretations of English. In particular, it would be reasonable in this coi~tcxtto restrict attention to those interpretations of intensional logic in which the following formulas are true (uith respect to all, or equivalently some, worlds and momcnts of time):

28 Richard Montague

1 2 3

4

5 6 7 8

9

V I ~ O I I= I a], where x isJ, rrr, 6, or n, 0[6(.v) 4 Vu.v = ^ u ] , where d translates any member of RCh other than price or temperature, VII/IAw~-O [6(.~) M { " s ) ] ,where 6 translates any-member of ElIb other than rise or change, V.S'A.t.A3Ol6(.r, 3) ++ 3{jS{'s, :)!)}I, where 6 translates find, lose, eat, love, o r date, A3VAMAvrO[d(s, 3) rl4{',x')], where 6 translates seek or conceive, A PV-hfA.rO[G(r,p) * hl{'s)J,where b translates believe t h a t or assert that, A PVMAL~OIG(x, P ) * IZZ{'x}J, where 6 translates t r y t o or wish to, VGA3A QAv~r0[6(3)(Q)(x) 3{,j~[['G](".)1)(Q)(.r)1) J, where 6 translates in, q IseekJ(x,3) try-to' (.v,^[findJ(3)])], where seek', try-to', find' translate seek, t r y to, find respectively.

-

-

-

T h e truth of (1) guarantees that proper nouns rill be "logically dcterminate" according to the interpretations under eonsidcration, that is, will have extensions invariant with respect to possible worlds and moments of time. In view of ( 2 ) , "ordinary" common nouns (for example, horse) will denote sets of COMS~CIIIIindividual concepts (for example, the set of constant functions on woslds and moments having horses as their values; from an intuitive viewpoint, this is no different from the set of horses). It would be unacceptable to impose this condition on such ''extraordinary)' common nouns as price or t e m p e r a t u r e ; the individual concepts in their extensions would in the most natural cases be fi~nctionswhose values vary with their temporal arguments. T h e truth of (3) is the natural requirement of e.v/ensionalitll for intransitive verbs, that of (4) the condition of extensionality (or extensional first-order reducibility) for transitive verbs, and that of (8) the condition of extensionality (or estel~sionalfirstorder reducibility) for prepositions. T h e intrnsionul (or nonextensional) transitive verbs seek and conceive, as well as the verbs believe t h a t , assert: t h a t , t r y to, wish to of other categories, are nevertheless extensional i~~it/l Y L ( S ~ t~oC sllly'ect ~ posilion, and this is expressed by imposing conditions (5)-(7). Condition (9) is thc natural definition of seek as t r y t o find. Several notions of a logically possible interpretation map reasonably come into consideration, depending on whether, and if so how m a n s conditions analogous to (1)-(9), stemming from our intended system of translation, are to be imposed. For prescnt purposes we may perhaps resol\-e the matter as follows: by a /ogical/y possihlr in&erpreratlonunderstand an interpretation of intensional logic in which fornlulas (1)(9) are true (with respect to all worlds and moments of time). Logical truth, logical consequence, and logical equivalence, for formulas of intensional logic, arc to be characterized accordingly. For instance, a formula cj, of intc~lsionallogic is construed as Iogi~.flytrue if it is true in every logically possible interpretation, with respect t o all worlds and moments of time of that interpretation; and two formulas cj, and II/ ol' intensional logic are logic all)^ cyuicalr~ztif and only if the biconditional rkl is logically true. If 6 is an expression of intensional logic of such type as to translate il transitive or intransitive verb, then 6% is to be an expression designating the set of individuals or reliltion between individuals that n a t u ~ l l ycorresponds to the set or relation designated

-

The Proper Treatment of Quantification 29

r r r

:

' 1

; i

hy S. In particular, if S E then S r is to be the expression iiS(lAu]); and if 6 E ME/.(,l.Vl,then 6% is to be iuGG(lAu],[*v*]). Notice that since J'(C:N) = J'(IV), this characterization is also applicable in the case in which 6 translates a common noun. It is a consequence of principles ( 2 ) , (3), (4) that if 6 is among the constants involved in those principles (that is, constants translating "ordinary" common nouns or "extcnsional" transitii-e or intransitive verbs), then 6 is definable in terms of cS%. 12.lorc exactly, the following formulas are logically true: IJ[6(s) * SS('.V)],ifh translates any membcr of Hc5 or RIJ. other than price, t c m perature, rise, or change; IJ[h(x, 3) ++ 3{.1;6s(".v,:y))l, if 0 translates any menlber of other than seek or conceive. Hrl-~-

Notice that although the verb b e (or its translation) is not covered by principle (4), it is by the Iast principle above. T h e reason why the extensionality of be was not cxplicitl\assumed is that it can be proved. (A4ore precisely, the analoguc of (4) in u-hich 0 is the expression translating b e is true in a11 interpretations (with respect to all worlds and moments).)

4 Examples I

;

' i

I I

-

:

1

I

'

Thc virtues of the present treatment can perhaps best he appreciated by considering ~xuticularEnglish sentences and the preciseIy interpreted sentences of intensional logic that translate them. I shall give a list of such examples. It is understood that each English scntcnce listed below translates into some formula logically equivalent to each of the onc or more formulas of intensional logic listed with it, and that every formula into svhich thc English sentence translates is logica11~-equivalent to one of those formulas. It should be emphasized that this is not a matter of vaguc intuition, as in elementary logic courscs, but an assertion to wrhich nrc have assigned exact significance in preceding sections and which can be rigorously prinwl. (The constants of intensional logic that translate various basic expressions of English at-c designated beloii~hy primed variants of those expressions.) The first five exan~plesindicate that in simple extensional cascs symbolizations of the expected forms are obtained.

Bill waIks : walk; (b) a man walks : V~r[mani,(u) n walki(u)] every m a n walks : Au[mank(u) + walkk(u)] the m a n walks : Vv{Au [[mani(u) u = I!] P, walli;(u)] John finds a unicorn : Va [unicorn;(u) n find:(.j, u)]

-

The next sentence, though superficially like thc last, is ambiguous and has two essentially different sy-n~holizationscorresponding to the two analysis trees prescntccf above; thc first gives the AP dicto reading; and the second thc cjc re.

30 Richard Montague John seeks a unicorn :

seek'(:j, ~ ~ u [ u n i c o r n k (neP{^a}I) ) Vu[unicorni(rl) n seekk(j, r,)]

T h e source of the ambiguity of John seeks a unicorn will perhaps be clarified if cvc compare that sentence with thc intuitively sgnonymous John tries t o find a unicorn, which contains no intensional verbs but only the extensional verb find and the "higher-order" verb try to. Here, though perhaps not in John seeks a unicorn, the an~biguitpis clearlq- a matter of scope, and indcccl depends on the possibility of regarding either the conlponcnt find a unicorn or the whole sentence as thc scope of the enistcntial quantification indicated by a unicorn. John tries t o find a u n i c o r n :

try-tot(;j,. j!~lr[unicorn&(u) n findL(Iy, LI)II) Vu[unicornL(rr) A try-tof('j,.i; /I))]

It might be suggested, as in Quine (1960) or Moiltague (1 969), that intensional verbs be allow~edonly as paraphrases of more tractable locutions (such us try t o find).14 Such a proposal, how-ever, would not be naturally applicable, for uant of a paraphrase, to such intcilsional verbs as conceive and such iiitcnsional prepositions as about; and I regard it as one of the principal virtues of the present treatment, as well as the one in Montague (1970b), that it enables us to deal directly with intcnsional locutions. The next cxample accordingly concerns a b o u t and gives us, as intuition demands, one rcading of John talks a b o u t a unicorn that does not entail that tlierc are unicorns. John talks a b o u t a u n i c o r n :

C

aboutf(~Vulunicorn:(,r) n P( ' u } l)('talkl)(''j) Vz~lunicornk(u)n about'(^rr*)(^talkf)(:j) I

T h e next two examples indicate that our uniform symbolixation of b e will adequately cover both the is of identity and the i s of predication; views along this linc, though not the rather complicated analysis of be given here, may be found in (&line (1960). Bill i s M a r y : h = rn Bill i s a Inan : manL(6) T h c next few exainplcs concern an interesting pi~zzledue to 13arl)nr~Hall I'artcc in\-olving a ltind of intensionality not previously ohscr\,cd bq philosophers. From thc ~x-emisest h e t e m p e r a t u r e is ninety and t h e t e m p e r a t u r e rises, the conclusion ninery rises would appear to follow bj- normal principles of logic; yet thcrc are occasions on which both premises are true, but none on which the conclusion is. .According to the following symbolizations, ho~vcvcr,the argument in question turns out not to be valid. (Tlze reason, speaking very loosely, is this. T h e t e m p e r a t u r e "denotes" an iildividual concept, not an individual; i111d rise, utllikc most vcrhs, depends for its applicability on the full behavior of individual concepts, not just on their extensions n-ith respect to the actual worltl and (what is more relevant herc)

The Proper Treatment of Quantification 31 moment of time. Yet the scntencc t h e t e m p e r a t u r e is ninety asserts the identity not of two individual concepts but only of' their exteasions.) the temperature is ninety : V.),[A x[temperaturel(s) ++ .Y = ,111 I, [:I/] = t? I the temperature rises : V,Y[{A x[temperaturel(,r) ++w = A risel(y)l ninety rises : rise'(*n) We thus see thc virtue of having intransitive verbs and common nouns dcnotc sets of individual concepts rather than sets of indil-iduals - a consequence of our general development that might at first appear awkward and unnatural. It would be possible to treat the Partee argument itself without introducing this feature, but not certain analogous arguments involving indefinite rather than definite terms. Notice, for instance, that a price rises and every price i s a n u m b e r must not be allowcd to entail a number rises. Indccd they do not according to our treatment; to scc this, perhaps it is enough to consider the first premise, which, unlike a m a n waIks, requires individual-concept variables (and not simply individual variables) for its symbolization. a price rises :{V.r[price'(.r) A rise' (s)]

The next example shows that ambiguity can arise even when therc is no clcmcnt of' intensionality, simply because quantifying terms may be introduced in more than one ordcr.

-

V I ~ [ W O ~ ~ A~Av[manL(r~) ;(U) lovek(u, u)11

woman loves every m a n :

Au[rnank(o)

+

Vu[womank(u) A lovek(u, v)] 1

The next example indicates the necessity of allowing verb phrascs as well as sentences to be conjoined and quantified. Without such provisions the scntencc John wishes t o find a unicorn a n d e a t it \vould (unacceptably, as scvcral linguists have pointed out in connection with parallel examples) have only a "rcf'crential" reading, thal is, one that entails that there are ilnicorns. John wishes t o find a unicorn a n d e a t it : vu[unicorn:(u) A wish-tol(i,,~[findk("j~, u) n eat;(>:

s)l)l

'The next example is somewhat simpler, in that it docs not involvc conjoining or quantifying verb phrascs; but it also illustrates the possibility of a nonrcferential rcading in the prcsence of a pronoun.

Mary believes t h a t John finds a unicorn a n d h e e a t s i t : ( ~n[unicorn:(u) n believe-thatl(^m, ^[find:(j, u ) A eat;(.j, u)])]

32 Richard Montague O n the other hand, in each of the following examples only one reading is possible, and that the referential: (1) John seeks a unicorn a n d M a r y seeks it, (2) John tries t o find a unicorn a n d wishes t o eat it, ~u[unicorn:(u) A try-to'(:j,.l; [find:(:)/, N) . I r, wish-tof(:j,.l; (Ieata ( ' ' J , u )J)I This is, according to my intuitions (and, if I guess corrcctly from ren-rarks in Partcc (1970), those of Barbara Partee as well), as it should be; but David Kaplan would differ, at least as to (2). Let him, however, and those who might sympathizc with him consider the following variant of (2) and attempt to make nonrefcrontial sense of it:

(2') John wishes t o find a unicorn a n d tries t o eat it. Of course there arc other uses of pronouns than the ones treated in this paper - for instance, their use as what have been called in Geach (1962, 1967) and Partee (1970) pr/)11otm.s u ~ ' ~ ~ ~ z ~ ? zthat P s . sis, , as "standing for" longer terms bearing a somewhat indcfinite relation to other expressions in the sentence in question (or preceding sentenccs within the discourse in question). For instance, it is not impossible to construe it in (2) as standing for t h e unicorn h e finds (that is, the unicorn s u c h that h e finds it), a unicorn he finds, or every unicorn he finds, and in this way to obtain a nonreferential reading of that sentence; but this is not a reading with which David Kaplan would be content.

Notes Much of' the content reported here was supported by Unitcd States National Science k'oundation Grant GS-2785. 1 am indebted to Mr. Michael Bcnnett. Mr. Harry Dcutsch, and Mr. Daniel Gallin for helpfi~lcomments.

1

The mediekal ant1 twentieth-century philosophical literature has pointcd out a nun~berof such difticultics, most of them involving so-called intensional contexts. I am indebted to 13arbalu Hall Partec for pointing out others, both in conversation and in her provocative paper IJartee (1970). (This remark should not, hon-ever, be takcn as implying aprccment with a l l of Professor Partce's conclusions.) 2 iVit11 the exception that in Monraguc (1970b) a numhcr of intuitively pl;iusihlc arnbiguitics &ere for simplicity ruled out. 3 In connection with imperatives and interrogatives truth and entailment conditions arc of course inappropriate, and would be replaccd by fulfilment conditions and n characterization of the semantic content of a correct answer. 4 It was perhaps the failure to pursue the possibilit! of s~ntacticallysplitting categories originally conceivetl in semantic terms that accounts for the fact th;it Ajduliicnicz's pl-oposals have not previousl!: led to a successful s3ntas. They have, ho~vcver,been cmploycd semantically in Mon~aguc(1970a) and, in a modified version, in 1,en.i~(1970). 5 This tva! of constructing ;In underlying unan~biguousIangui~ge,though convcnicnt here, nrould I)c unsuitable in connection with fragments of natural language eshibiting greater syntactical complcsitics of certain sorts.

The Proper Treatment of Quantification 33 0

7

8

9 10

In particular, in talks belore the Southern California 1,ogic C:olloqui~~n~ and the Association li)r Sjnlbolic Logic in April and May of 1969, and in the paper hlontiague (1970b). The :lddition of tenses is rather routine in the light of tlie discussion in Montaguc (1908); and it would he possible to replace the tensc operators by prcdicatcs. thus prescrring exactly tlie language in Montague (1970b), in the manner indicatcd in Montaguc (1970~). Clause (8) is of course vague but can be elinlinated in a familiar way. T o bc exact, the recursi\c definition given above can be replaced by the following explicit definition: ME,, is the set of all objects x such that xRc7, whcrc R is the smallest relation such that clauses (1)-(7) hold (wit11 all parts of the form " p E ME,," replaced by "BRn"). Or possible individuals. If there are individuals that are only possible but not actual, .-l is to contain them; but this is an issue on which it would be unethical for me as a logician (or linguist or grammarian or semanticist, for that matter) to take a stand. [Richmond H. Thomason's note: EIcrc, is interpreted in the sense of "necessarily always."] [Richmond 11. Thomason's note: T h e form of this delinition is not quite correct, since (xA>"s undefined when x is not a conslant. But thc intention is clear; what is to be defined recursively is a A l ! ~ / Clauses ~? (1) and (0) should bc rcviscd to read as follows. s (1) I f r is a constant then r A l ' . l * ~F(x)((i,j)). If Q f ME,, then [-ajA.'>i-x is that function h with domain I x 3 such that whcnever (9) ( i ~ )E I x -7, h((i,,;)) = x A ~ ' ~ l ~ ~ q . The irr/~>nvon xAl!: of a rclativc to A and g is then defined explici~ly: ~ ~ 9 " sthat function Ir with domain I x 3 such that whencvcr (i,.j) E I x 3.Ir((i,,)} = a A ~ ' ~ / l ~ . It then follows as a corollar!l that [Aa]A11317x = xA'? for all (i,j) E I x 3.1

11 The simplicity and uniformity of the present correspondence stands in rcmal.kable contrast to the rill hor character of the type assignment in Montaguc (1970b). 12 [Richmond H. Thomason's note: T o avoit1 collision of varinhles, the translation must be ,i,,j[~'(,v,,l) ,I $1, u-here $ is the result of replacing all occurrences of .v, in cp' by occurrcnccs of .r,,,, wlicrc rn is the least even number such that s,,,has no occurrcnccs in either or (p'.] 13 Alternatives arc possible. For instance, we could instead consider cliwc.1 interpretations of lcnglish induced by interprcvations of intensional logic in conjunction with our translation procedure; the yrccisc gcncral construction is given in Montaguc (1970b). Though this would probably bc the best approach from a general viewpoint, it would introduce slight conlplications that need not be considered in thc present paper. 1-1 Strictly speaking, this ~vouldmean, within tlie fran1cnrol.h of the prescnt paper, introducing ;I syntactic operation r such that, for example, F (John tries to find a unicorn) = Jolin seeks a unicorn, a syntactic rule to the effect that F ( 4 ) t P, whene\-cr Ci, E l',, and a corrcspon~ling translation rille that whenever Q, E P, and 4 translates into (,/I F((/))trallslatcs into cl,'.

c'

Notes 9, 10, and 12 are reprocluced by pcrnlission of Yale University I'ress from Richard klontaguo, Fornlai Pliilosopl~~)~. .Stier.~erlPoprrs (!/Riiknid L140t/ti~C~fi~, edited and with an introduction h j Richmond H. Thomason. New Haven, Conn.: Yale Univcrsitj- Press, 1071.

References r\jduliicwicz, klai'imicrz. 1960. *jf!.z:),k i Po.:ni/~lrt [Langrruge ~ i n dK ~ o ) ~ ~ l byarsari: ~ ~ ~ / g I';IIIS~WOM~ ~~l. U'j;dau n. .Sonic .,I/I~i/irz.(i/(/?/I/:\lor/r,r11 Grach, Peter T . 1962. R~$reni,c. irncj G(~ncnziitj~: -411 E.vnrniria/rotr T11~~oriil.c. Ithaca, N.Y .: Corncll University Press.

34 Richard Montague Geach, Peter T. 1967. Inten~ionalidentity. Journal r!f'Plzilosopl~)t64: 027-32. Ixwis, David. 1970. General seman~ics.Syr~tliesc22: 18-67. Montaguc, Richard. 1968. Pragmatics. In Ra)-mend Klibansky (ed.), C o ~ z ~ c n ~ p 1'111/1isop/~ll: o r . ~ ~ : ~ ~ .-1 S u r z q ~ vol. , 1, Florence: La nuova Italia. Repr. in Montaguc 1071. Montague, Richard. 1969. On the nature of certain philosophical entities. il;lorrisl 53: 161-94. Repr. in Montague 1974. hlontague, Richard. 1970a. Universal grammar. Th~oriu36: 373-98. Repr. in Montague 1974. Montague, Richard. 1970b. English as a formal language. In B. Viscntini et al. (cds), LingriagLqlnclirr Sol-ietri e nclli~Tecnira, Milan. Repr. in Montague 1974. Montague, Richard. 1970c. Pragmatics and intensional logic. S)arrlrc~sc22: 68-94. Rcpr, in _Monii~gl~c 1974. St~ltrrcdPapers (!/Richard Morrlrr~uc,edited and with an Montaguc, Richard. 1974. Forr~znlP/zilo.soph~)~. introduction by Richmond H. Thoniason. New Haven, Conn.: Yalc University Press. Partee, Barbara H. 1970. Opacity, coreference, and pronouns. Synrh~.rc21: 359-85. Quine, Willard Van Orman. 1960. ITord ( ~ n d0hjer.l. Cambridge, Mass.: M I T Press.

A Unified Analysis of the English Bare Plural Greg N. Carlson

ABSTRACT.It is argued that the English "bare plural" (an NP with plural head that lacks a determiner), in spite of its apparently diverse possibilities of interpretation, is optimally represented in the grammar as a unified phenomenon. The chief distinction to be dealt with is that between the "generic" use of the bare plural (as in "Dogs bark") and its existential or "indefinite plural" use (as in "He threw oranges at Alice"). The difference between these uses is not to be accounted for by an ambiguity in the NP itself, but rather by explicating how the context of the sentence acts on the bare plural to give rise to this distinction. A brief analysis is sketched in which bare plurals are treated in all instances as proper names of kinds of things. A subsidiary argument is that the null determiner is not to be regarded as the plural of the indefinite article a.

0 Introduction This study deals with the English "bare plural" construction, by which I mean plural Noun Phrases of English which exhibit no quantifier or determiner before the head noun (like "dogs", "ineffective arguments", or "old white houses that have been painted dozens of times"). For ease of reference, however, I will speak of thesc NP's as containing a null determiner, and leave open the question of whether there is any determiner present at all. This construction has long posed a semantic puzzle for grammarians and philosophers alike, chiefly because of the diversity of its possible interpretations. Although there is no agreed-upon inventory of distinct uses, there seems to be a basic split between the "generic" and "existential" uses, with further subdivisions among the generic uses. The generic is most naturally regarded as something like a universal quantifier, as would seem appropriate for representing the truth-conditions of (la); however, in

36 Greg N. Carlson many cases this "universal" admits of exceptions, and appears to have tlic forcc of "most", as in the examples of (lb).

( I ) (a) Ilorses are mammals/creatures/material objects. (b) Horses are smart/larger than mules/good pets. These uses may be opposed to the use of the generic exemplified in (2), where it is clear that a universal quantifier or the quantifier "niost" tvould simply bc inappropriate.

(2) (a) Horses are widespread. (b) Iiorscs are extinct. (c) Horses are indigenous to eastern Chile. Perhaps other sorts of generic uses can be distinguished, but these exa~iiplesshould suffice to illustrate the variety of generic interpretations that arise. There is another quire distinct use of the bare plural which has been conlmonly referred to as the "indefinite plural", since in many cases it seems to be the semantic plural of the NP's determined by the singular indefinite article a ( n ) . 'l'his use of the bare plural lacks the universal flavor of the generics and seems to be most appropriately modeled by an existential quantifier having essentially the force of sullze. A few exaniples are given in (3).

(3) (a) DOC~OKT tried to save the dying boy. (b) Knute threw rollen p~achesat the library. (c) Mice will come out of that ~ i ~ aifl l you pound on it. It ~ i i l lbe my chief contention here that these apparently distinct uses of the hare plural (henceforth referred to as S N P ) are merely facets of a syntactically and scmantically unified phenomenon, and that in all cases the differing interpretations can bc attributed in an entirely predictable manner to some aspect of the context in which that particular instance of (bNP occurs.' If this hypotllcsis is correct, and the null determiner is in fact unambiguous, then we can gcIlerate thc (/)NP in a rather straightforward manner syntactically, assigning it a constant interpretation in all instances. 2 Though this goal of unification may seem desirable on general esthetic grounds, I wish to argue that a unified analysis is motivated by data unco~-ercdin examining 4 N P ; that is, a unified analysis is not only desirable, but necessary, if we are to havc a complete account of this construction. I will proceed in a rather roundi~bout fashion, first attacking the notion that (/, serves as the plural counterpart of' a, and thereby elucidi~tingsome interesting semantic properties of t/)NP. I will then argue that the indefinite plural use of (bNP is not distinct from the generic uses, and that the generic uses are not distinct from each othcr. I conclude by sketching a ratha. programmatic analysis of the semantics of thc bare plural, one which allo\vs for the seeming variet~-of interpretations but assigns ;I constant interpretation lo ill1 occurrences.

A Unified Analysis of the English Bare Plural 37

1 The Indefinite Plural ?'he notion that the null determiner is the plural counterpart of a is bolstered by certain parallelisms in their distributions. For example, both a and cj~have generic uses, as in

(4).

(4) (a) A nznrnmal bears live young. (b) Mammals bear live young. Both also appear as singular-plural counterparts in predicate-nonlinals, as in (5).

(5) (a)

Gerry is an c~nitrznl. (b) Gerry and Muneie are nrrinznls.

Despite this rather inviting pattern, it has by no means been universally assumed that & is the plural of a. Sweet (1898) and Stockwell et al. (1973), for example, posit the unstressed variant of sornr (often written "sm") as the true indefinite plural. 13ut a number of others have held that c , is the proper candidate." In Chomsky (1965), for example, there is a base rule introducing Articles in the following way:

Art

-+

[ f Definite1

Thc I+ Definite] article is the, which occurs before both singulars and plurals. Tlie I- Definite] article is IL, which is deleted before plurals by transformational rule. Thus, the indefinite plural use of "houses" is derived from the underlying NP "a house".

1.1 Anticipated semantics

I

If in fact 4 serves as the plural counterpart of the indefinite article Q ,we would expect that the two would share all relevant semantic properties except for those attributable to thc presencc or absence of plurality.4 Let us agree to interpret thc indefinite singular as an esistcntial quantifier which also asserts singularity (rcprcsented here by "Esg"), and the plural 4 as an existential quantifier that also asserts plurality ("Epl"). J,ct us assume that "singular" means "one", and that "plural" means "two or more". Hoth of these quantifiers range over the same set of objccts. T h e relationship between semantic intcrpretation and syntactic form is presumed to be of the sort prcscnted in Montague (1952). 'The crucial fcature of this system for us is that quantifier scopc phenomena arc handlcd by syntactic rules "quantifying in" an NP and by associated rulcs of semantic interpretation which assign that NP scope in the semantic representation.' Though the details of this analysis may bc debatable, it gives reasonable semantic representations and should serve our purposcs here for instance, the sentences of (6) cvould be represented senlantically as the corresponding expressions in (7). (h) (a) A clog chased Marvin down the street. (b) Dogs chased Marl-in down the street.

38 Greg N. Carlson (7)

(a) (Esg x) (Dog (x) & x chased M. down the street) (b) (Epl x) (Dog(x) & x chased M. down the street)

We construe (7a) as true just in case there is at least one individual from the domain of objects such that the sentence following the quantifier is true when that individual is assigned as the value of x. (7b) is true just in case there are two or more distinct individuals from the domain such that the sentence Eollou~ingthe quantifier is truc that each of the inctividuals assigned ils the value of x. T h u s (7a) and (7b) tlo appear to represent closely the truth-conditions of (6a) and (6b), respectively.

1.2 Opacity phenomena In the presence of an opacity-inducing operator or predicate, the indefinitc siilg~llar exhibits a rather clear ambiguity. Consider (8) as an example.

(8) Minnie wishes to talk with a young psychiatrist. On one reading, there is some particular young- psychiatrist that Minnic has in mind, and she wishes to speak with him. Let us, following Quine (1960), call this the transparent reading. On the other reading, the opaque reading, Minnie's desires are fulfilled b-! talking to anyone, so long as that person is a youi~gpsychiatrist. The transparent reading is most readily modclcd by having thc existential cli~antifier outside the scope of thc opacitj--inducing predicate "\vish", whilc the opacluc reading is coi~ventionallyrendered by a formula having thc existential quantifier isithin its scope. Thus, (8) may have at least the following two semantic structures associated with it. (8')

(a)

(h)

(Esg x) (young psych. (x) & All. wishes hll. talk with x) bl.wishes (Esg x) (young psych. (x) & h t . talk with x)

We would therefore expect the indefinite plural to show the sanle ambiguity. 'That is, we should find the readings of (9a) and (b), corresponding to (8a) and (8b), for sentence (10). (9) (a) (Epl x) (young psych. (x) & M. wishes M. talk with x) (b) hl. wishes (Epl (s)(young psych. (x) 8i M, talk with x) (10) Minnie wishes to talk with young psychiatrists.

However, (10) does not have both of these readings; the transparent reading represented by (9a) is absent. This is most clearly seen if (10) is compared with (1 I), which does exhibit both readings of (9). (1 1) Minnie wishes to talk with sm young psychiatrists For some reason, the reading of (10) with a wide-scope quantifier is rulcd out, although the parallel reading, (8'a), is allowed for sentence (8) containing the indefinite singular.

A Unified Analysis of the English Bare Plural 39 It is clear that the responsibility for this state of afFairs cannot rcst solely with the plurality marlier, for witness again (1 I), which contains a plural NP. 0 s su\>stitutcfi)r s ~ i rtlic quantifiers rnlrny, all, tme/?le, aiid others that take a plural head noun. All thc resulting sentences exhibit a similar scope ambiguity.

Neither can somc cranky property of the verb i ~ i s hbe held accountable, for ~irtually ;In\.. opacity-inducing operator has the same effect: the singular wit11 a shows an opaque-transparent ambiguity, but the plural with 4 gives rise to only an opilquc reading. Compare the following pairs of sentences, where the opacitj--inducing p1-edicate is italicized. Mas Oeli~i~es a Commie to have robbed Macy's. i'vlax helieces Commies to have robbed Mac y's. A drunk is likely to win the annual potato-sack race. Drunks are lihe/y to win thc annual potato-sack race. hlas is seeking a unicorn. hlax is seeking unicorns. Gerald Tnust talk to a congressman before noon today. Gerald tnust talk to congressmen before iiooii totlay. Ij'a woman were sent to the Supreme Court, busing would end. I/'w-omen were sent to the Supreme Court, busing would end. These fclcts are clearly not predicted by any analysis which analyzes cf, as the plural counterpart of a.

1.3 Narrow scope phenomena ."Zclated set of fiacts comes to light when we examine the relative scope properties of 4 in the presence of negation and other quantified NP's. Consider the followiiig argumen t.

(18) A cat is in this room. ?4 cat is in the next sooii1. Therefore: A cat is in this room and a cat is not in this room. I

The conclusion of (18) is an ambiguous sentence. One reading is a contradiction, being of the form A G -A.

40 Greg N. Carlson (18')

(Esg x) (cat (x) & x is in this room) & -(Esg x) (cat (x) 131 x is in this room)

This reading arises when the existential quantifier is within the scope of the negative. A more likely way of saying (18') in English would be (19). (19) There is a cat in this room and there isn't a cat in this room. However, (18) on another reading sccms to be a reasonable argument. This reading has the existential quantifier outside thc scope of the negative, ancl the resulting formula is not of the form G -.,4 (nor equivalent to it), and thus is not a contradiction. Its representation would be (18"). ( 1 8 ) (Esg x) (Cat (x) & x is in this room) & (Esg x) (Cat (x) & -(x is in this room) A clearer rendering of ( 1 8") would be (20). (20) There is a cat in this room and there is a cat not in this room. If cf, is the plural counterpart of a, then we would expect the conclusion of (21) to exhibit a similar ambiguity. (2 1) Cats are in this room. Cats are in the next roorn. Therefore: Cats are in this room and cats aren't in this room. T h e conclusion of (21) has only the contradictory reading. Apparently there is no semantic structure that may be associated with the conclusion of (21) that is like that of (18")' where the existential quantifier has wider scope than the negative. Only a narrow-scope reading is allo\ved, which is the contradictory reading parallel to that of (18')." A further example of the scopc restriction on 4 , assuming it to bc an existential, is illustrated by the following facts. (22) is ambiguous with respect to the relativo scope oE the existential and universal quantifiers. (22) Everyone read a book on caterpillars. On the reading where the universal quantifier has wider scope than the existential each indi~idualneed not have read the same particular book. IIowcvcr, on the rcading where the existential has wider scope than the universal, it is thc same hook that every pesson read. These readings nppcar as (22'a) and (22'b) rcspcctivcly. (Rook (y) Si x read y)) (22') (a) ('v'x) (Person (x) -+ (3) (b) (3y) ('v'x) (Hook (y) & (Person (z)-+x reacl y ) )

A Unified Analysis of the English Bare Plural 41

We would therefore cxpcct (23) to be similarly ambiguous.

(23) Everyone read books oil caterpillars. (23) howevcr lias no reading in which the existential quantifier has wide sa)pc with rcspcct to thc unii-ersal. This is a general phenomenon. (24a)-(28a) all contain occurrences of cr and all exhibit a scope ambiguity with respect to some other quantifying expressioil (indicatccl by italics) in thc sentence. T h e corresponcling (b) sentences have the bare plural in place of a but exhibit no such scope ambiguity.

1

(2.1)

I

I

(4 (b)

1

(25) (a)

I

(b)

'

I

(26) (4 (b)

(27) (a)

I

('4 I

i

,

(28) (a) (b)

John saw a dog on his lawn u t 3, 430, 6, and 7 : 15. John saw dogs on his lawn a1 3, 3:30, 6, ~ n 7d : 15. A goat didn 't run across my lawn. Goats didn't run across my lawn. A whale lias attacked this ship ow t m l ~ l eorcnsions. Whales have attacked this ship an tn~elzleoccasions. Max saw an actor in eceql scene. Max sa\17actors in czler-1/ scene. A movie was seen by wrost pzoplc. h4ovies wcre seen by most people.

In none of these is there a reading of the q5NP which would bc appropriately represented by usc of an existential quantifier having wide scope. This is not predicted by- anv - analysis holding that b, is the plural counterpart of a. One could conceivably maintain that q5 is "really" the indefinite plural article, but that some idiosyncratic property of its semantics restricts it to having narrow scope only. It is not clear how a defense of this nature might proceed, but in any case it becomes untenable when we examine the ncxt set of data, where the scope possibilities of i~ and q5 are differentiated, and it is not the case that one exhibits simply a subset of the readings a l l o ~ - e dby the other.

1.4 Differentiated scope phenomena Lncler certain circumstances b,NP can havc narrower scope than the indefinite singular 110ssihl!l can, assuming that we continue to model 4 as an existential. Consider (29).

(29) A dog was everywhere. It is my clear intuition that (29) has only a bizarre reading, in which the same dog pops I

' I

I

up in every location. '.There is no rcadiilg in which the universal quantifier of the predicate has wider scope than thc existential of tlie subject (which rcading is clearly possible in "There was a dog evcrywhere"). Howcver, (30) does l~avc the reading that would be represented by the universal having widc scope. Entirely missing is the reading analogous to (29), in which it is tlie same group of dogs in c\-cry place.

42 Greg N. Carlson (30) Dogs were everywhere. Here is a case where the singular and plural have no readings in common, The plural cannot have narrower scope than is possible for the singular, and this is dit'fcrent from the previous examples, where plurality seemed to restrict the 4NP to a subset of the possibilities already present in the singular. 11 similar phenomenon is seen in the difference perceived betwecn (31) and (32), which serves to raise yet another difficulty for the hypothesis under considcration. (31) An accident happened today at 3 , 1 : 30, and 6. (32) Accidents happened today at 3, 4 : 30, and 6. In (3 l), we are asked to imagine a recurring accident, one which happens three times on the same day. (32), on the other hand, might be used by a local radio announccr to report the happenings of the afternoon. In this latter case, we are not asked to imagine recurring accidents. T h e semantic formulae for (31) and (32) would be of the following approximate forms. (31') (32')

(3) (Accident (x) & x happened at, 3, 4 ~ 3 0 and , 6). At 3, 4:30, and 6 ((3~) (Accident (x) & x happened)).

Here again the plural can have narrower scope than the singular. In this particular example we see further that the notion of plurality fails us. (32) could very easily be used to report the occurrence of one accident at each of the timcs mentioned, although the possibility of more is left open. But this ought not to be a possible state of affairs if 4 carried the information of "two or more", as would scem to follow from any analysis that treats 4 as the plural of (6.' A particularly interesting construction is given much attention in Dowty (1972), wherefor time adverbials are analyzed in some depth. An example of this construction is seen in (33).

-

(33) Marge sat on the coucl~j b v nifze hours. Dowty's analysis treats fir time adverbials as a universal quantification ranging over a given range of time points. Roughly, the semantics of (33) would be that of (33'). (33') 'it: t E 9 hrs ( A T (Marge sat on the couch, t)) (33') asserts that (33) is true just in case "Marge sits on the couch" is truc for each (relevant) time-point in some nine-hour period. One of t-hemore puzzling facts about these adverbials is that they are quite strange with a certain class of verbs ("achievement" verbs), unless the subject or dircct object of thc verb is a bare plural or an unquantified mass noun. For example, (31) appears to describc a rather unusual state of affairs, while (35) has a much more ilatural reading.

A Unified Analysis of the English Bare Plural 43

(34) Max discovered a rabbit in his yard f'or two hours. (35) Max discovered rabbits in his yard for two hours.

If wc use Dowty's analysis, the readings of (31) and (35) could be represented by thc following.

(34') (3x) (Rabbit (x) & Vt: t E 2 hrs ( A T (M. discover x in his yard, t))) (35') Vt: t E 2 hrs. (3x (Rabbit (x) & A T (M. discover x in his yard, t))) (34'), the reading in which the existential has wider scope,' asscrts that the same rabbit is discovered and rediscovered, which would be a strange statc of affairs. In (3Sf), however, we find no such assertion. T h e universal 1x1s wider scopc than the existential, so the same rabbits need not (but, of coursc, may) be discovered time and again. So once more cfi appears to be capable of narrower scope than 0. Thcre are a number of other time adverbials which behave in much thc same fashion, allowing Q, to have narrower scope than is possible for the putative singular countcrpart. The following (a) examples indicate a strange state of affairs, while the (b) esamples need not. In each case, we can attribute the difference to the narrow-scope possibilities of 6. (36) (a) (b) (37) (a) (b) (38) (a) (b)

Kent killed a mouse until Ruidwz~znarriued. Kent killed mice until Raidnratl n ~ r i ~ e d . Chester killed a fly repercteci/y last night. Chester killed flies uepeatedy last night. Leon has killed a cow since before the depression. Leon has killed cows since Iwfi)re the depr.e.ssion.

Aspectual verbs appear to play much the same role, and the exainples pattern similarly. (39) Harvey continued to kill

{

a rabbit i t

1.

(40) The North American Bread Eater tends to eat And evcn the morphologically simple "generic" statements scem to have the same characteristics.

(41) Abner rcpairs

1 1

{ }

for a living.

Similar results are obtained when the NP in question is in subject position. Consider the pairs of sentences in (42)-(44). In the (a) vcrsions, where the subject is the indefinite singular, the existential quantifier is interpreted as being outside the scopc

44 Greg N. Carlson of the time adverbial, and thus it is the same object that is spoken of at all timepoints in that period. In the (b) versions, the "indefinitc plural" is apparently intcrpreted as being within the scope of the quantifying expression implicitly prescnt in the time adverbial, and thus the objects need not remain constant ovcr the period of time. (42) (a) (b) (43) (a) (b) (44) (a) (b)

A dog hung around my valet all last year. Dogs hung around my valet all last year. A cat has been here since Columbus landed. Cats have been here since Colun~buslanded. A11 epileptic ruled Serenia for 200 years. Epileptics ruled Serenia for 200 years.

T h e quantifier need not in all cases arise from a time adverbial. One rather intriguing type of sentence is exemplified in (45); the (a) version has no rcasonaldc reading, but the (b) version has a quite normal interpretation. (45) (a) A wolf gets bigger as you go north from here. (b) Wolves get bigger as you go north from hcre. While (45a) asserts that a wolf in thc back seat of your car uill g 1 . o ~if you head a certain direction, (4%) asserts that northern wolves are larger than southern wolves (and it has an interpretation similar to that of (45a) as well). Though I do not pretend to understand (45), its readings arc clearly not predicted by any analysis that posits (,!I as the plural counterpart of a. One final example of differentiated scopc may be drawn from cleft sentences. (463) lacks the reading of (46b) which allows each pcrson to cat his on-11 tomato. In (46a) the tomato is shared by- all. (46) (a) It was a tomato that everyone ate. (b) Everyone ate a tomato. With ONP matters are different, and (47a) and (47b) are virtually synonymous. (47) (a) It was tomatoes that everyone ate. (b) Everyone ate tomatoes. So far the contrast in differentiated scope has been between and a. But therc is il deeper, and more important, distinction to be inndc herc. As mattcrs turn out, differentiated scope scts apart 4 not only from (1, but from a / / othcr quantifiers and determiners, with the uneven exception of thr and the unstresscd demonstratives in some cases. I cite one example from many to illustrate thc point. Rccall that with f i r time adverbials, a yields a strange reading in sentences like (34), but 4 doesn't. If we substitute other quantifiers and determiners into this context, wc find that they, too, pattern like a in giving old? strange readings. This puts (I!, virtually in a class by itself.

A Unified Analysis of the English Bare Plural 45

(48) Max discovered

se\.cral lots of those many all most twenty few sn~ etc.

rabbits in his yard for two hours.

All the other examples of differentiated scope yield similar results. I further note that (1) is the only quantifier or determiner among the indefinites that fiails to exhibit scopc ambiguities or opaque-transparent distinctions. If $J is an indefinite, it is a special one indeed. The 4 "indefinite plural" thcn, is semantically not the parallel of the singular form 11. Though there is a great deal of semantic overlap between the two, it is clearly not the case that their semantics are coextensive up to differences that can be attributed to the presence or absence of plurality.

1.5 Anaphoric processes A sentence such as (49), as has been mentioned before, is ambiguous between transparent and opaque readings.

(39) Kelly is seeking a unicorn.

In the larger context of (SO), however, the ambiguity disappears, in spite of thc fact that (19) is wholly contained within (50). This lack of ambiguity can be traced to the definite pronominal form found in the second conjunct, here assumed to refer to the objcct NP of the first conjunct."

(50) Kelly is seeking a unicorn, and Millie is seeking it, too. In (.iO), Kelly and Millie must both be seeking the same unicorn. There is no reading in which each is looking for different unicorns, nor is there a reading in which both are engaged in some gcneral activity of unicorn-seeking. Such readings are allowcd in ( 5 I), where thc pro-form ''one" scrves as thc pronoun.

(51) ICelly is seeking a unicorn, and hlillie is seeking one, too. Neither of the readings of (51) are found in (50). We would expect that the I$NP in this position urould not allow any dcfinite pronominalization to take place in a subsequent conjunct, sinee mrc have secn that SNP allows only an opaque reading in sucll contexts. Since only the transparent reading of (49) is found in the context of (N), not the opaque reading, and since cb

46 Greg N. Carlson rules out a transparent rcading, it should be impossible to obtain a reading for the following sentencc (with lhenz meaning "unicorns"). (52) Queenie is seeking unicnrus, and Phil is seeking them, too. Surprisingly, (52) does have a perfectly legitimate rcadi~lg,although it is not thc rcading to he found in (501, but rather one of the readings of (5 I), the opaquc rcading. There is no sense in n-hich Phil and Queenie are seeking the same group of imicorns. It scems to mean only that they are both engaged in some general activity of unicorn-seeking, despite the definite pronominalization in the second conjunct. I must hasten to point out rhat this result is not clue to any difference b e t w e n them and if beyond pluralit!.; rather it is due to the nature of the antecedent. 11s the rei~dermay have noticed, inass nouns with 4 determiners hehave almost identically to QNP with respect to the phenomena notcd." in (53) we find pronominalization resulting in the definite pronoun it, but here, in contrast to (50), the opaque reading of the first conjuilct remains. Cedrick and Hiran1 need not be seeking the same articles of furniture. (53) Cedrick is seeking furniture, and Hiram is seeking ir, too. If i/ can ha\:e this property in (53) then why not in (SO)?T h c answer is simply that it is the nature of the antecedent, and not the form of the pronoun, which gives rise to this property. 1 I It appears that this particular set of facts does not depend on the presence of an opaque context. In (54), the italicized NP's are not in opaque or intensional contexts, but srill Harsiet need not catch the same rabbits as Ozzie, nor must I drink the same beer Dad did. (54) (a) Harriet caught rabbits yesterday, and Ozzie caught thorn today (b) Dad drank heel. slowly, and I drank it fiast. Compare (51a) and the following.

( 5 5 ) Harriet caught

a

rnhhit today, and Ozzie caught it yesterday.

Here again we see Q, behaving in a manner quitc different fiom rr. Similar sorts of results are obtained when deletion occurs in coordinate structures rather than pronominalization. First let us consider the case with the indefinite sin. gulai. a. (56)

A buildi~zgwill collapse in Berlin tomorrow, and u buililirzg will burn down ir Boston the day after.

Clearly, (56) leaves the impression that two different buildings arc being spoken of However, if the subject of the second conjuilct is deleted "on identity with" the subjec of the first, a stranger tale is told in which the same building will collapse and burn i r two different places. This is the only reading of (57).

A Unified Analysis of the English Bare Plural 47 l down in 13oston (57) A building will collapse in Berlin tomorrow, a n d t ~ . i l bur11 the day after. Using the "indefinite plural", we find that (58) means sonlething very close to (56). The difference arises when we remove the subject of the second conjunct, and find that it still means the same as (58). (59) need not denote an extremely unusual statc of affairs, and hence it is like (56) rather than (57).

(58) Buildings will collapse in Berlin tomorrow, and buildings will burn in Boston thc day ilfter. (59) Buildings ~villcollapse in Berlin tomorrow, and day after.

will burn in Boston the

A someivhiit different phenomenon, which hits much the samc flavor as those just discussed, involves reference to the complement of a set. T o illustrate \*-hatI mean, let us examine (60). (60) Jack is hunting h r a unicorn, and Frank is looking for another/some more/some others.

NP's like aizotl~erand sonrf tnol-e, in (60), involve s o n ~ cnotion like "one of the unicorns that Jiick is not looking for", or "some unicosns beyond those that Jack is already

sccliing". There is at least an implicit rcfercnce to the unico~.nsthat Jiick is not seeking. We find that in (60), there is no opaque reading for the first conjunct, in spite of the [act that the first conjunct in isolation exhibits the transpiircnt/opaque distinction quite clearly. One apparently cannot refer to the complelllent set of something that is "down in'' an intensional context, a fact r\-hich seems to inake clear intuitive sense. Since the $NP only to opaque readings in opaque contexts, we would naturally antiuipiitc that sentences such as (61) would be ill-formed. And indeed, (61) lacks an interpretation whcre Jack and Frank are seeking different unicorns. 1

I \

1

(61) Jack is hunting fbr unicorns, and Frank is hunting for ??another/??others/ ??somemore/ ??some others. This result is erpccted. What is unexpected is that similar results ;ire obtained with (/)NP ewn when it appears in extensional contexts. T h e sentences of (02) arc all strange in the sanle way as (61), yet none of the uilderlined NP's are in opaque or intensional contexts. last night, and fed (some) others. (62) (a) ??lMas trapped 1~ecrr~i:r (b) ??Dogsjust ran across my lawn, and some more found their way into 111y kitchen. (c) ??George walhed down the street with kiltcns, and Henry David \valked dojvn the street with (somc) others. Compare (62) with (63), M-herethe NP's differ in their determiners. I vastly pset'er the scntcnces of (63) to those of (62).

48 Greg N. Carlson

(63)

(a) Max trapped (b) (c)

{

dog S m dogs

{

(some) others some more

beaver(s) last night. and k d

}

just ran across my lawn, and (somc) others found their way into my kitchen. sm kittens George walked down the street with , and I-Icnry a kitten David walked down the street with

i i

I

(some) others some more

1.

None of these facts are predicted by any analysis that assumes (/, to be thc plural counterpart of the indefinite singular o.

1.6 Status of the "indefinite plural" T h e facts presented above indicate that 4 simply cannot be the plural of a in any semantically relevant n-ay.12 I therehre conclude that (/) should be stricken from the list of candidates for this position, if in fact there is such a slot in the granlmar. 'l'he unstressed variant of "some" appears to be the most likely candidate, but I will lcave the matter unresolved here as it is not at all germane to the point of this study. I will continue to allotl: myself the liberty of referring to this usc of c/,NP as the "indefinite plul-al", but merely as a convenicilt label n~ithouttheoretical significance. Let us herc sum up some of the properties of (/jNP that havc bcen notcd so far. 1:isst of all, we noted that it had only opaque readings in opilque contexts, ncvcr transyarcnt ones. T h e n it .c~-asshown that (bNP does not participate in quantificr scopc anlbiguitics, but always seems to take narrowest scopc. We then found that c/)hP could actually achieve scnlantically namou-er scope than the other determiners and quantifiers. In matters oS pronominalization and anaphora, we found that thiilgs were also diffcrcnt from what might be expected. Definite pronominalization and coordinate deletion, rules n~hichnormally have quitc strict coreference rcsrsictions, behaved more like "identity of sense" anaphora (o?!c.-pronominali~ltion or VP-11t.letion). I think all thcse properties call be summarized by one short statement about the indcfir~ite plulal use of (/)UP: it fails to pick out a g~oopthat persists tl~roiightinlc and spacc in its membesship. Yet, this doesn't seem quite right, either, since in sentences like (44) it seeins that a group really is in somc scnse, being set 1113 and rcfert-cd to. Othern-ise, n e would simply have no undesstanding of whj- some gives us such a nice paraphrase.

,

(44) Arlene found

sguirn~ls in her attic. XOlllP squil'r~h

2 Generics and the "Indefinite Plural" Now that we have determined that (/I cannot bc the plural of iz, its relationship to thc rest of thc grammar becomes much less clear. 1I:e must ask once ;]gain what it is related I to, and how this relationship is represented in thc grammar. 11

A Unified Analysis of the English Bare Plural 49

A certain amount of evidencc indicates that the indefinite plural use of S N P is not to be distinguished from its generic uses. I,et us for the moment considcr thc hypothesis that there are at least two distinct detcrnliner elements of English, both of which just happen to be pronounced "4".T h e first is like an existential quantifier (but not quite), and accounts for the "indefinite plural"; the second is like a universal (but not quite) and accounts for at lcast one of the "generic" uses of (/,NP (there may hc a number of generic determiners, all pronullced 4, so we let the onc posited represent possibly n whole class of determiner elernents).'"~~ hppothcsis curries with it the claim that 4 N P is systematically ambiguous. I-Iowever, in most cases this is not bornc out bj. the facts (as has been noted previously, for example in Dahl (1975)). Consider the following sentences: (65) Smokers are rude. (66) Dogs bark. (67) Elephants are easily trained. These sentences cshihit the gcneric, or "universal" reading." Hut what is missing is the indefinite plural, or "existential" reading. Why don't (65-67) mean (65'-6'7') as well, if cj, is really systematically ambiguous? (65') Some smokcrs are rude. (66') Some dogs bark. (67') Some elephants are easily trained. Thesc readings are clearly plausible pragmatically, but they are rulcd o ~ l tfol- some reason. Again, why don't we judge the italicized d)NP's of the following sentelices to be ambiguous? Either reading should be possible, but only the "universal" emerges.

(68) Mark really loves puppies. (09) Kris hates smull 1dg1~crentures. (70) The man over thcre believes Te.rans to be friendly And thc following appear to be ilnambiguously existential, even though a universal, or near-universal, would be rcasonable. (71) Sir Snooter slew ~ l r ~ c for ~ o the ~ ~ Baron. s (as an "event"). (72) Pl~rmhersstormed into the convention demanding Ionger lunch breaks. (73) Alice personally knows ilrtresscs. These facts require further explanation uncles an analysis which treats & as an ambiguous determiner. In a number of contexts, an ambiguity does appear. Consider the following ambiguous sentence.

(71) Dinosaurs ate kelp.

50 Greg N. Carlson One reading posits kelp-eating as a characteristic of most, or all, dinosaurs. Another reading of the sentence, one which reports a kelp-eating event of long ago (more readily seen if the sentence is continued " . . . while Grog watched"), refers only to some dinosaurs. So here we have an ambiguity of the type predicted by the hypothesis under consideration. However, these sentences are ambiguous even when the subject of the seiltence is not a $NP, but an N P that is not normally regarded as being ainbiguous in any relevant way. Witness (75):

( Maxwell

(75)

I

Lots of conductors T h e old fireman A few scientists

I

ate kelp.

)

(75) can still be interpreted either ils reporting a past kelp-eating event, or as reporting a past characteristic of the subject of the sentence. These two readings have quite distinct truth-conditions, and thus the distinction drawn coi~stitutcsil genuine ambiguity. Virtuallj any N P in subject position in such a sentence will produce a similar ambiguity, and it is plain that the ambiguity has little to do with the nature of thc subject itself. ate kelp" is already two-ways ; ~ n ~ b i ~ u oregardless os'~ of the Given that "nature of the subject, and nssuining that 6 is itself at least two-ways ambiguous, then (74) ought to be at least four-ways ambiguous. But it is not. T h e existential reading alone appears with the "event" reading, and the universal alone ilppcars with the "characteristic" rcading of the predicate. 'There itre 110 "n~ixti~res"(c.g. it bciilg a past characteristic of some dinosaurs that they ate kelp regul;~rly).Here we see that thc context itself selects certain readings of 4 and disallows others. This is different from the situation with regard to the generic and specific uses of the definite article. An NP such as "the horse" may refer to the species of horses, or to il particular horse (say, Holding Pattern). Sentences such as (76) are ambisuous with ~ h o , but not so with 4,which allows onlj- the "generic" reading. T h e horse \vorks Horses work

quite hard.

'I'he generalization that falls out of this line of inquiry is that the generic and indefinite plural uses of (/, are in complementary distribution. In fact, I wish to rmke the stronger claim that these readings of cjNP arc not only in coinplementarq distrihution, but that their distributions are wholly predictable froin context. cb itself, then, is never ambiguous in a given context. If there is an apparent ambiguity, it can be tsaced to the environment. is in Gct anlbiguous but that thc reasonable counterargument might be that semantic properties of the distinct readings of arc s u c l ~that they- may only rarely, if ever, appear in the same environment. A rather weak ill~illogy might be used to illustrate thc hypothesis. Tlle NP "a crow" is at least two ways anlbiguous, onc reading denoting a large black bird, and thc other denoting the characteristic sound of a rooster at claybreak. Yet the instances of the N P "a crow" in t11c fi)llowing sentences are virtually unambiguous.

A Unified Analysis of the English Bare Plural 51

(77) A '?'Om perched on my doorstep. (78) The rooster rared back and let go with

(1

(lorrd)

CYOII~.

Yet 1 would not wish to claim that the NP "a crow" is unambiguous. 'Thcscforc, complementary distribution of' readings for $NP cannot be used to show conclusivelqthat SNP is unambiguous. However, nonambiguity, though not sufficient argument for a unified analysis, is clearly a necessary one. I nowr turn to other arguments.

2.1 More anaphora Onc of the major differences between the putative ambiguities of QINP and the cxan~plc mentioned above is that different interpretations of 4 may stand in an anaphoric relationship, but not those of "a crow". T h e following sentence cannot be interpreted readily as referring to a large black bird in the first clause, and to thc characteristic noise of a rooster in the second. \

I

I

(79) My rooster lets go with a

1.1'0171

when he sees it near the llouse.

If NP's exhibiting a null determiner were similarly an~bigiious,wc v~ouldnot expect a generic instance of 4 N P to serve as antecedent for an indcfinite plural use, or vicc\-ersa.The use of (/,NP cxempiified in (SO) should not be able to stand in an anaphoric relationship with the use exemplified in (81); the result should bc sotncthing like that of

(79). (80) Le1iz1wit7g.rare protected by law (81) Mick traps k ~ ~ ~ ~ ~ t z i r r ~ s . Hocvet-cr, this state of afhirs can hold, as wc see in (82). (82) (a) NIick traps luwttnir~gseven thoi~ghhe knows full well that r/tqy arc protected by law. are protected by law, but Mick goes ahcad and traps thcnr an!.(b) Lenzvrri~g.~

In (82a), we find an indefinite plural serving as antecedent for a generic use; in (8%) we scc that a generic may serve felicitouslj, as antecedent for an existential. A great number of other such examples can be constructed, and a few are listed here. M y mother hates t-~~crootzs because I ~ Z L stole ; ~ her sweet corn last summer. (b) Rnl.coons have stolen my mothcr's sweet corn every >-eiw,so shc really hates I / ? P ~a Ilot. 4 ) (3) Ally brother thinks s~iclkesare nasty creatures, l>ut that hasn't stopped nic from hai-ing tllrtji as pets m j n;hole life. (b) I've had sncikes as pets my wholc life, but my brother still thinks tl7i;)l're nast creatures. (85) (a) Martha told me that Eetitrs don't grow as well in this ulimatc, but tl7ey grew well for me last year.

(83) (a)

52 Greg N. Carlson B~irnsgrew quite well for me last season in spite of .Martha's warning that thyy can't grow in this climate. (86) (a) I didn't believe that goats liked ti11 l a n s until I actually saw ihenz eating /hem last week. (b) Before I actually saw goats eating /in rairs last week, I didn't believe ~ h t y liked them. (b)

In all these cases, we see a generic (or a universal) serving as antecedent for an indefinite plural (or an existential), and vice-versa. It is not at all clear how this would be possible if (p were at least two-ways ambiguous. There is a complicating Factor here (among a number of others), which I think in the end also argues for a unified analysis. T h e reference of the pronominal form in (87) is ambiguous. (87) -Mark knows ten linguists, and Freddie knows six of them. Ignoring thc rcadings where the pronoun refers to some extra-sentential objects, (87) might have either of the following meanings. (87') (a) (b)

. .. and Freddie lrllows six of the ten linguists lllot ,Mark knows . .. and Freddie knows six (of) lingz4i.st.s.

It appears that the pronoun in (87) may have as its antcccdcnt il subpart of the whole NP tell fi?~,o.u~sts of the form firzguists, or something of the form 4NP. Now this particular antecedent-psonoun relationship can be used to account fix cases of the following sort.

q extinct. ~ (88) Spaceman u-ants to see some gf~/r.sbefore ~ h i1l.e Since the11 stands in a position in this sentence characteristic of generic NP's, its antecedent illust be a generic. Howel-er, examples like (88) can also he constructed in which the pronoun is more like an indefinite plural; e.g.. (89).

(89) Max killed very few r(zBl~it.s,but IHiram killed thcvlr in great abundance. (80) does not require that we imagine some resurrected rabbits. Let us assume that the 4 determiner is ambiguous, that thc NP underlying the pronoun in (88) (or the pronoun's antecedent, as I wish to remain neutml on the deep structure status of pronouns) is something of the form V N o l n , and that in (89) it is solmething of the form ENum (where b' and E are the generic and indefinite pliual markers respectively, and Nom stands for "nominal"). T h e antecedent in (89) then must "contain" an indefinite plural, and the antecedent NP in (88) must "contain" a gencric. This would mean that an NP of the form "three cats" would he at least twoways ambiguous, depending on whether or not it contained ENorn or 'dNora.It is not at all clear that such an ambiguity can be explicated in any reasonable way. Hut our assunlptions would have the further undesirable conscqucnce of allocti-~ga gencric to

A Unified Analysis of the English Bare Plural 53 contain an indefinite plural (be of the form VENorrr) and an indefinite ~>luralto contain a generic (Eb'h7om) in order to account for (83--86). Not only is it hard to make scnsc of the prolific an~biguitiespredicted by such an analysis, but the syntactic slipperiness of' thc invisible elements 'v' and E (pronounced 4 and (/I, respectively) is discon~fortingils well. I know of no othcr determincrs or q~~antifiers that could apycar in all thc positions allowed foi. thcse. However, 4 is a vcry difficult item to find in a sentence, and without spcclfic claiims about its syntactic and semantic properties, it is virtually impossihlc to show that q5 isn't really there. I do not wish to overstate the case that can be inadc from pronominalization, for the processes involved remain very poorly undcrstood, and only in thc framew-ork of solnc definitive analysis can thcse arguments be evaluated adequately. But insof'zir as current theories account fbr these phenorncna, an analysis claiming d'NP to be unambiguous would not suffer from the difficulties of the amhipuity analysis. In all of the cases ;lbove the antecedent of the pronoun is sin~plyof the form (/)NP, and its particular intcrprctation is predictable from context. In any event, an ambiguity analysis of (/) has to face most of' the same problems of' context as il unificd analysis does, so thc rcfcrence to contcxt I rnakc here is not sonlething that can bc avoided cvcn with an ambiguous cb.

2.2 NP's denoting kinds of things I belicve that the strongest argumcnt for a unified analysis of 4 N P conles from the fact that contextual factors that give rise to the gencric and indefinite plural interpretations are independently motivated and are needed elsewhere in the grammar to account tor interpretations of ccrtain constructions that are wholly distinct s!;ntactically fi-om 4 N P . 'I'his class of constructions - NP's that refer overtly to kinds of things - points the path tocvards a correct analysis of the bare plural construction. We might offhand think of kinds of things (I will henceforth simply use thc locution "kinds") as being really quite abstract, as opposed to say, particular individuals. 13ut NP's denoting kinds inny appear as the subject of sentences that predicate vcry concrete things of the subject. They can bt tall, hn7v mings, or even hc sillirzg nrsi 10 w t in rht thrrr/cr.' "

(90) (a) This kind of animal is tall. certain kind of lizard has wings. (b) (c)

Some kind of duck was sitting ncxt to me in the theliter.

If we compiirc the behavior of thcse NI"s denoting kinds to other NP's that are chasacterized as being abstract (iikc c/emol-r.al:y, or 1l7e s p c ~ J(!/' lighl), we find that, grammatically, the NP's making reference to abstract kinds appear cluitc concrete. NP's denoting kinds, such as those in (go), have a number of properties in common with the indefinite plural (/)NP. Let us fix the reference of "this kind of animal" as your f;norite kind of animal, and examine a sentence liLc (91). to have eaten his pet spongc. (91) hlax believes this kind c/J'a~~ili~(il

54 Greg N. Carlson There is no reading of this sentence in which Max believes of any particular indi\~idual that it ate his pet sponge. If Rover ate thc spongc, for example, it does not follow that Max believes that Rover ate his pet sponge, on ail!- reading of the sentence. Conipare this with (911), where there is a reading having this entailment. i

(91')

Adax believes

nlz

animal o$f/zls kird to have eaten his pet sponge.

If we were to entertain the llppothesis that "this kind of animal" refers to certain individual animals, we might be led then to think of (91) as exhibiting only an opaque reading. T h e same follows for the sentences of (92) if we think of this NP as referring to a group of animals. On this false hypotl~csis,we would then think of'(92) as exhibiting only opaque readings.

(92) (a) Max is seeking this kind of animal. Minnie wishes to talk with this kind of animal. This kind of anilnal is likely to win the race. (c) (b)

Ho~~rever, since thcse NP's do not refer to indiiriduals, but to thc kinds then~selves,1 the question of opacity vs. transparent!- wit11 rcspcct to particular ii~di~iduals si~nply does not arise. When such a distinction does arise wit11 N1"s dcnoting kinds, it is a question of transparency vs. opacity with respect to kinds themsclucs, and not wilh respect to individuals. In (93), tl~ereis one reading on which there is a particular kind, e.g. "the kind of animal George has", which is believed by Adax to have eatcn his pet sponge, and there is another reading on which Max's belief is not about any particular kind.

(93) Max belicves

soune

k i d qf'anirtrc~lto have eaten his yet sponge.

If we continue to slippose for the moment that NP's denoting kinds refer to individuals, we find that the NP "this kind of animal" exhibits only narrow scolx. Consider the case of (94), recalling that we agreed to fix the reference of'the subject NP as your favorite kind of animal. (94)

This kind of animal is in this room. This kind of' animal is in the next room. Therefore: This kind of animal is in this room, and this kind of aniinal i s not in this room.

T h e premises of (94) could very well be true, but the conclusion has only a contradictory reading. This is so in spitc of the F~ct that the particulat. indii~iduals in the two roonls are in a11 liltelihood quite distinct. So if we cvcrc thinking of "this kind of animal" as referring to individuals, it ~vouldappear to h~lveonlj narrot+ scope. We also find that the sentences of (95) cxhil~itonly narrow scope l-eadings.

A Unified Analysis of the English Bare Plural 55 (95) (a) Everyone saw this kind of animal. (b) This kind of animal has attacked the ship on tlvclve occasions. (c) John saw this kind of animal in every scene of the movie. In (95,)' h r instance thcre is no reading on which everyone saw thc same particular animals. So we see here "nasrow scope only" alongside "opacity only". Continuing our false notion that NP's denoting kinds refer to particular individuals, we find that they can exhibit differentiated scope as well. Recall that in the context of (96) the appearance of any quantified NP yields a bizarre reading, whcreas 6 N P yields a more natural interpretation. (96) -be everywhere. However, NP's referring overtly to kinds likewise yield quite natural readings. (97) This kind of aniinal was everywhere. In cach place there need be only some animal of this kind; the same particular individuals necd not appear in more than one place. If we put NP's denoting kinds into the contests that were defined in 1 .I as differentiated scopc (or perhaps "to narrow scope"), we find that in all cases a natural interpretation appears. I list a few niorc esamples.

(98) (a) Max discovered this kind of animal in his yarcl for tu-o hours. (b) This kind of animal ruled Serenia for 500 years. (c) It was this kind of animal that cveryone ate.

Thus wc see that these NP's may be thought of as exhibiting differentiated scopc. Given that only an opaque reading seems to occur with NP's denoting kinds, KC see that in (99) definite prol~on~inalization is allowcd, but it docs not yield a reading on which Kelly and Horace arc seeking the same particular individual animals. 17 (99) Kelly is seeking this kind of animal, and Horace is seeking idthen1 as well.

If' we thought of (99) as making reference to particular animals, this ~vouldbe a semantic curiosity indeed. NP's denoting kinds also pattern like indefinite plural (IJNP with respcct to the othcr anaphora phenomena. In ( 100) and ( I 0 1) identity of particular individuals is ilot preserved.

(100) Harriet caught this kind of animal yesterday, and Max caught it/theni earlier today. (101) This kind of structure will burn dourn in Berlin tomorrow, and collapse in Boston the day after.

- will

56 Greg N. Carlson T h e same particular animals need not be caught, nor must thc same particular structures burn and collapse. Reference to the complement is also forbidden with kinds. (102) is strange if one is speaking of the animals that weren't trapped. (102)

Marv trapped this kind of animal last night and fed (some) othcrs/some morc.

((102) is of course fine if this is ref'crence to other kinds. But then so are (61) and (62)). So we sec that with respect to anaphora, NP's dcnoting kinds pattern like the indefinite plural 4NP. NP's denoting kinds also appear to have "generic" and "indefinite plural" intcrpretations. In (103)' for instance, we appear to be speaking of all animals of that kind. (103) This kind of animal is a vertebrate. In (104), howcver, it appears to follow that there are some animals of that kind which Max shot; clearly there is no reference to all animals of that kind. (104) Last night, Max shot this kind of animal. Thus, that NP's denoting kinds also have an "indefinite plural" reading. It would be questionable indeed to account for the existential reading of "this kind of animal" in (104) postulating an ambiguous invisible dcterminer. This becomes even less likely when we note that NP's denoting kinds come in a wide variety of syntactic shapes.'' Evcry one of thc sentences in (10.5) has a perfectly natural reading, provided wc interpret the subject N P as referring to a kind or kinds. (10.5)

This cignretle (yes, thc one 1 am tapping on the table, putting in my mouth, and non- lighting) is made in nine different countries. q~ bird is now extinct. (b) E z ~ e /icclhe~lrss (c) Alo rcfitilcs arc indigenous to the Philippines. (d) Mnn.11 mec.hnnicn1 i/ez:ic-csulcrc invented by mistake. (a)

These, too, may have existential or "indefinite plural" interpretations. (106)

(a) Carter sells this animal in his pet shop (meaning "this kill[/ of animal") in it. (b) This zoo has ez3eqtpal.i~,lrd~v+f?t ( c ) Severcll bircEs were discovered in Spitsbergen by the I>arscn expedition. (d) M.11 (lug has been known to attack leopards.

T h e sentences of (106) are ambiguous as to whether thc italicized NP's denote individuals or kinds. On the "kincl" reading, thc sentence speaks of sotne of that kind, rather than all or most. This is thc "inclefinitc plural". T h u s the hypothesis of an ambigious 4 dcterminer to account for generic vs. indefinite plural interpretations of (/2NP would be difficult to extend to this widc variety of NP's denoting kinds. A l ~ d positing an ambiguous 4 determiner for ~ I N but P some other mechanisnl for the same

A Unified Analysis of the English Bare Plural 57 variation ~ - i t hthe other NP's would be to miss an obvious generalization. l'his all suggests strongly that q5 is not to be represented semantically as an ambiguous determiner.

3 A Brief Excursus on the Diversity of Generic j'NP As mentioned above, a number of "generic" uses of d, might be distinguished. 13ut here, too, positing an ambiguous determiner or quantifier for r j suffers from the same objections that have been raised against the generic/indefinite plural distinctlon. Let us suppose for the momeilt that there are at least three 4 determiners. 'The first would be a strict universal, as in (107). (107) Dogs are mammals. The second kvould be much like a universal but would allow exceptions. (108) Dogs are good pets.

The third as in (109), does not lend itself to interpretation as a quantifier a t all. (109) Plants are \videspreacf/cxtinct/numerous.

The first problem is that these NP's simply do not appear to be ambiguous. (108), for example, does not seeill to be true a n one reading, fialse on another, and true or f'alsc on yet another. It's simply true. I dispense with further examples as the generic scntences presented herein speak for themselves in this respect. We find that thcse various "interpre~ations" of (/,NP can be mixed in antecedentpronoun relationships (and all of these may, in turn, be associated with the indefinite plural). Consider the follo\\ling. (110) (a) Dinosaurs ;are extinct because thqy ate kelp. J difficult to (b) Trzlcks hrrfiling r(ynarnite are illegal in Nevada because I I I ~ are manuever in heavy traffic. (c) W0lz.e~eat only kosher deer, so rbq are less nurnerous than the?; woulcl be if rhey weren't so choosy. (d) Elfphanls are not widespread in spite of the fiict that thy)! are quite largc and strong.

So pronominalization fiacts do not effect a separation among different generic uses. Finally, NP's overtly referring to kinds or types would havc to be distinguished in the same variety of ways. They, too, appear to have the same "readings". 'This kind of animal This animal

is a mammal.

58 Greg N. Carlson This kind of animal This animal

is a good pet/barks.

i

This kind of ani~nal is widespread/extinct/i~umerous. This animal So the ambiguities posited for #NP would have to be allowed in these cases as wcll, if generality is to be preserved at all. There is one further danger inherent in positing an ambiguous (/I determiner to account for the varying truth-conditions associated with the barc plural, namely that one would end up positing a large number of cb's to cover all the desired cases. We have already seen three, but therc would have to be a q!!rc,,,,, to account for "mamma/s givc milk to their young", a #,,,I, to account for L'liop7s have manes", a Onlatu1.e for "birils reproduce annually" a d~,~,,,, for "bees reproduce by laying eggs", and so forth. I do not believe that positing distinct 4's in each of these cases would serve an!. useful purpose, as it seems clear that they would not be modeling anything that iiltuitivelv we would call an ambiguity. These quantifiers would rcflcct more Izow we find out the truth or F~lsit?of generic statements; this is tantamount to building a theory of epistemology into the semantics, something not at all easily done. 'll~erefore,it appears that the various uses of the gencric # arc likenrise contcxtdcterrnined, and that a unified analysis is thercfore desirable. I now turn to a brief description of a program for accomplishing this unified analysis. A great deal of what follows must be termcd speculative, but the general line of' i n q ~ ~ i rappears y to he capable of solving a number of difficulties raised so far.

4 Towards a Solution 4.1 Generic statements about individuals Let 11s bcgin b!; presenting a means of interpreting thc generic uses of c/)NP, which incorporates a unified analysis but ~vhichnevertheless allo\s.s for a wide rangc of different uses. This task is best begun bv drawing somc analogics. Generic statcments can also bc made of individuals. 'l'hcsc statements, too, have notoriously erratic truth-conditions. Consider (1 12).

( 1 12) Jake mows his neighbor's lawn. This clearly does not mean that Jake's days and nights arc spent mowing. \Vc might hypothesize that (1 12) is truc just in case it is Jake, most of the time, who mows the lawn. T h e lawn must furthermore be mowed regularly (onc mowing cvcry tivc years wrould not do). Jake is allolved to be sick occasionally, or to be on vacation, etc. Comparc this line of thought with thc one that emerges from an examination of (1 13).

(1 13) Kenney bcats small children.

A Unified Analysis of the English Bare Plural 59 -

-

A great deal of regularity is not required for (1 13) to count as true; nor must Kcnney be the one who beats children largely to the exclusion of other maniacs. He necd not beat children a t every opportunity, nor every time a child needs a beating. A very few childbcating instances would suffice for (1 13) to be true. 'I'he reader may take issue with certain aspects of these remarks, but this is not entirely germane to the point at hand. These generic statements about individuals clearlv vary greatly in truth-conditions. Consider the following, asking for each h o ~ many times Jake must do whi~tif the sentence is to be true.

(I 14) (a) Jake wears contact lenses. (b) (c) (d) (c)

Jake runs to school. Jake runs the mile in 3:58.2. Jake is a drunk. Jake is a failure.

(f)

Jake writ-es

Careful examination will reveal a maze of factors to be taken into account in cases such as these. 'I'herc is a means at our disposal of allowing for all this variation while retaining a cohcrent scinantics. If we fi>llowa semantic theory of the sort proposcd in Montagiie (1972), the truth of hlsity o f a sentence is detern~inedby finding out whcther or not the property attributed to the subject of the sentence is in the set of propertics that the subject of the sentence has. (We might also talk equivalently in terms of the predicate naming a set, and finding out if the subject of thc sentcncc is in that set.) So, for example, (1 1 -#a) is true just in case the property of wearing contact lenses is in the sct of properties associated ~vithJake, and (1 14d) is true if being a drunk is in that property set. How do we know whether or not these set membership relations hold? In a modeltheorctic semantics, the nlodel kt-ill tell you, so the determination of truth or thlsity i s simply a matter of consulting the model. If we think of thc real world as being the model m-e consult, matters become a good deal more complicated. We no longel- just need to be able to read off some information that is given to us; wc ~ ~ c also c d to bc ablc to perocive, to compute, remember, make i~lductiorrs and deductions of startling complexity, and go through a host of othcr cognitive processes to tell if sonlcone is a drunk, or \\;ears contact lenses, \ V h t is suggested, then, is that thc apparent variation in the truth-conditions of'(114) can be attributed to our strategies of investigation and not to any inherent semantic marker in the sentence (in particular, a qui~ntificr). Let us return now to the question of the proper interpretation of the bitre plural. Wc have already noted the semantic relationship that holds between 4 N P and kinds. T h e suggestion herc is that w-e treat the bare plural in all cases as denoting il kind of thing. In particular, we suppose that the bare plural acts as the proper name of a kind, and that kinds are to be construed as individuals. Of course, these individuals arc a little different from more normal individuals in that kinds can hc hcrc and there, n-hereas normal individuals are generally confined to one location (though it might bc il big location) at a given time. 'That is, while hlark Spitz at a given time is spatially quite

60 Greg N. Carlson confined (he can only be in one place, roughly), bees can be in 111ilny locations (wherever there are one or more bees). (Zemach (1975) makes a siinilar point.) Postal (1969) notes a striking similarity between bare plurals and proper names with respect to the "so-called" construction. Consider (1 15).

(1 15) (a) Slim is so-called because of his slender build. (b) Cnr(1itzuls are so-called because of their eolor. "Those cardinals 'All cardinals are so-called because of their color. *No cardinals "The cardinals ctc. Quantified or determinccl NP's are cxcludcd, leaving $NP, proper names (and the generic definite "the cardinal"). TActus agree then to treat 4 N P as a proper name of a kind, and let us think of kinds as being abstract individuals. Tn this treatnlent, 4NP's arc treated semantically as if they were unanalyzable wholes. This assuinption is clearly incorrect in many cases, but this does not affect the point of the anal?-sis sketcl~edhere. Generic s ~ a t e n ~ e nfor t s bare plurals are then handled cxaetly as they arc for regular proper names. ( 1 16a) is true just in case the individual Bossie has in her property set the propcrty "eats hay", and (1 16b) is true just in case "cats ha!-" is in the property set of the individual Cows. ( 1 16) (a) Hossie eats hay. (b) Cows eat hay.

HOJDwe go abo~itdeciding whether a given property is in a property set is not a semantic issue.'" In this way we avoid dealing with thc extremely recalcitrant problem of the widely-varying truth conditions of sentences like (I lob) in the same way that we do in the case of generic statements about particillar individuals. This, then, is how we go about accounting for the various uses of the generic 4 N P . It is not ambiguous, but it may take on the appearance of ambiguity when we assign different properties to the individual in question. Tf we assign the predicate "lives in caves" to the individual "bats", our strategies for determining whether or not that property is in their property set let us tolerate the exceptions. If we assign "reproduces by giving live birth" to LLrabbit~", our strategies deterininc that wc nced not take into account the male rabbits. Thcse strategies are not so different from those we nced for determining the truthvalue of generic statements about individuals, as in (1 14) above. Those for kinds may bc a bit more complex, but it is clear that the proeesscs involved are closely related to one another. T h e relationship between particular individuals and these "kind-level" individuals is, I believe, a tighter one than might be imagined. For example, I know of no predicates that can be assigned to particular individuals that cannot also be assigned to kinds. And the predicates that cannot be assigned to particular individuals (or groups of particular individuals) but which may be assigned to kinds arc not numerous.

(I 17)

A Unified Analysis of the English Bare Plural 61

1

*Fred "All goats Goats This kind of animal

} { Ze )

~ ~ ? i d e s j r ~c lt /u ~ n e r uTs , I ; mre/ ~.ommon/irzcligenous to . .

The predicates of (1 17) represent a saml~leof what appears to me to be a rather exclusive class. I now leave the problem of the generic use of (I,NP and turn my attention to its indefinite plural use.

4.2 The indefinite plural \Ve still must account for the indefinite plural interpretation of $NP. In light of previous discussion, one may wonder why we single out this particular interpretation for analysis. Why shouldn't it be treated in the same simplistic fashion as the generic interpretations? That is, could (1 18) be construed as being true just in case "he sitting on my lawn" is one of thc properties of the individual "dogs"? (I 18) Dogs are sitting on nly law-n.

It would then follow that the "indefinite plural" interpretation would be, in a certain sense, illusory, and not really an existential statement at all. I do not believe, however, that such an approach is entirely justified. Fos one thing, there seeins intuitively to be a rather clear distinction bctween the generic and indefinite plul-a1 uses of 4NP. T h e generic secms to speak of tendencies, dispositions, characteristics, and thc like; thc indefinite plural does not have this flavol- at all. There is a more important reason for wishing to split off the indefinite plural scnsc from the other senses of $NP and treat it as a semantically distinct phenomenon. Onc of the chief aims of semantic theory is to represent correctly the entailment relations that hold between sentences. As matters turn out sentences with the indefinite plural md sentences with the generic sense of $NP have quite distinct ent;~ilrnents."' Consider the argunlent presented in (1 19). (119) Dogs arc sitting on my lawn. All dogs arc n~an-~mals. Therefore: Mammals are sitting on my lawn. The infcrence of (1 19) appears to be valid, and could easily be shown to he correct if thc indefinite plural wcre represented by an existential quantifier. Contrast this with (1 20), nherc the bare plural has the generic scnsc. (120) Dogs are good pets. All dogs are rnainmals Therefore: Mammals arc good pets.

62 Greg N. Carlson This invalid argument is a case of overgcneralization, as would he clearly demonstrable were thc generic to be construed here as a universal quantifies. Clearly the indefinite plural and generic senses of Q)NP give rise to different entailments and are thercfol-e distinct. But here I seem to be arguing the contrary of what I havc argued fils at lenglh a bit earlier - that the generic and the indefinite plural arc not to bc differentiated syntactically or semantically. I seem to be sitting atop a paradox. 1 am not really; the remainder of this work is devoted to thc resolution of this contradictor)- state of affairs. I11 thc following discussion I will consider only Q)NP in subject position, as matters arc clearest there. No doubt the reader has noted that therc is another diffcrcnce betwecn (1 19) and (120), namely in the tense, or aspect of the scntcnce. This diffcrcnce cr)uld be the ultimate source of the difference in entailment rclations between (110) and (120). 1h ~ i s h in the end to claim that essentially this is so. Let US begin by asking ourselves about the relationship between the two sentences of (121). (121) (a) Max is being clei-cr. (b) Max is clever. Both appear to be predications concerning thc same individual. (121a) says son~cthing about ~24ax'scurrent actions, m.hereas (121b) says i ery little about his current actions (he ma; ill fact be making an utter fool of himself when this is uttered) hut speaks more of a disposition or characteristic. T h e generic is, in a sense, timeless, whilc the present progressive refers to a particular period of timc. We have seen that thc simple past tense may yield eithcr of these interpretations. A sentence suc11 as (122) is ambiguous, one reading being akin to that of (l2la), and the other being like that of (12 1b).

(122) Jake ate kelp. There is one reading that refers to a particular stretch of time, and another which attributes morc or less timeless characteristics to Jake. T h e same holds for the "filturc" tense. (123) Jake will eat kelp.2' In all these cases, the reading that is "timeless" and sycaks of characteristics and the likc is the one that unambiguously sclects the "universal" reading of ONP. tind that reading of (1 19)-(123) which l ~ a referencc s to a particular stsetch of timc, and intuitively secms to be reporting events, unambiguously sclects thc "existential" reading of the hare plural. For instance, (121a) selects the existential and (124b) selects the universal or generic. (124) (a) Dogs are running around in circles. (b) Dogs run around in circlcs.

A similar sort of phenomenon can be observcd in the casc of English adjectives. Somc adjectives select the indefinite plural existential reading, and others select only

,

,

I

1

1

A Unified Analysis of the English Bare Plural 63 -

-

the genenc. In A4ilsark (1974) and in Siegcl (1976), two classes of adjectives are isolated, the chief diagnostic being whether 01. not a given adjective will fit into the types of content cited in (125). (125) (a) Jules caught the girls . (b) There were five dalmatians Into these contexts may go only those predicates that Milsark calls "states" (which may be rozlglily characterized as being fairly temporary), as opposed to those predicates he calls "properties" (which are roughly more permanen t sorts of things). Among the states are adjectives such as "hungry", "sleeping", "awake", "drunk", "availal~le" and the like. _4mong the properties wc find adjectives such as "fat", "tall", "clever", "obnoxious", etc. As Milsark noted, when the subject is a 4 N P the "propcrties" select the generic or universal reading, while the "states" unambiguously select the indefinite plural reading. Compare the "states" of (126) to the "properties" of (127)."

(126) (a) (b) (c) (127) (a) (b) (c)

Soldiers were available. Dentists \\-ere drunk. Frogs are awake. Soldiers are brave. Dentists were tall. Frogs are clever.

In (126), the only possible interpretation of the subject is the indefinite plural, whereas in (127) only the generic readii~gis possible. We find one particularly interesting contrast in the case of the adjective "sick". This has two senses; the first, a state, is physical illness, and the second, a "property", indicates mental instability. Notc that in the context of (125), only the physically ill reading is to be found.

(125') (a) Jules caught the girls sick. (b) There were five dalmatians sick. In (128), however, this predicatc is ambiguous, but here the physically ill reading sclccts the indefinite plural reading of bNP, while the mentally ill reading selects only thc generic.

(128) Girls are sick. Among the other predicates, we find that predicate nominals unambiguously refer to "properties", while most prepositional phrases (especially those of location) refer to thc "~tates".~" And wc find, as expected, that predicate nominals select the generic reading \vhilc prepositional phrases select the indefinite plural. (129) (a) Dogs are sivect animals. (b) Dentists are book collectors.

64 Greg N. Carlson (130) (a) Dogs are in the next room. (b) Children were without pasents. (129) is generic, while (130) is existential. In all of these cases, one might hypothesize roughly that the predicates selecting the "indefinite plural" arc predicating something of an individual for a short period of time, while the predicates selecting the generic leave the implication that what is predicated of the individual is of a more permanent nature. Though this is most assuredly on the right path, time as the crucial fiactor does not satisfactorily distinguish c4 states'' from "properties". For exan~ple,one can be physicallj ill for several years, and mentally ill for only a few w-eeks. O r one can be "in the next room" for a lot longer than one is "a butcher". I wish to look at things in a slightly different wrap. Suppose that the "states" and "properties" are being predicated of dzfirent sorts qf'things. Suppose we take an individual, Jake, and look at him as being composed of a set of Jake-stages, or temporallybounded portions of Jake's existence. T l ~ e r eis more to Jake, howcvcr, than a set of stages. There is whatever it is that ties all these stages together to make them stages of the same tl~ing.Let us call this whatever-it-is the individual Jake. Those predicates we have been calling "states" then are not predicated of individuals, but of stages of individuals; and those we have been calling "properties" (in the sensc of Milsnrk) are predicated of the individual, or the thing that ties all the stagcs together. Now thcso "stages" can be short or long in duration, but they are nonetheless perceived as parts of a whole. Thus the apparently temporary nature of such predication. It is not at all clear that anything of a temporal nature falls out of the characterization proposed for thc "properties", but since they are predicated of the individual, no doubt the permanence of the "properties" arise from this notion. Perhaps a cautionary note on the intuitive idea of "stages" is in order here. I do not see them simply as clips of film of an individual's lifetime that are taken out and examined, w7ith the sum of the clips of film being the individual. T h e individual is more than the sum of the parts, and the stages are not static sorts of things. T h e stages aren't simply things that are; they are more akin to things that hr~ppen.That is, stages are conceived of as being much more closely related to ejrenrs than to objects. I think this characterization can be taken quite scriousl?;, but rather than try to mect possible objections the readcr may have at this point, I will leave matters quitc open regarding the ontological aspects of this proposal and move on to the formalism. Let us take the individual as basic, and define "stages" in terms of an individual. An individual's set of stages is denoted by the follon-ing formula (exemplified here for Jake):

This may be read as "the set of all things, x, such that .x bears the relation R to Jake". (1 henceforth assume the reader to be familiar with the notation used in such places as Montaguc (1972)). 'rhc predicate R ma! bc thought of, roughly, as "rcalizes". 'l'ho stages then may bc callcd "rcalizations" of an individual. When one prcdicatcs a "state" of an individual, intuitively I urish to say that one claims that that state is in

A Unified Analysis of the English Bare Plural 65 the property set (in the scnse of Montague (1972)) of a realization, or stage, of that individual, rather than in the individual's property set directly. The "properties" (in Milsark's sense) are asserted to be in the property set of an individuaI, rather than in the property set of one of that individual's realizations. Let us exe~nplifythis with somc formulae. First I will present "Jake is intelligent", ignoring tense. I will treat be as semantically null; it won't show up directly in any of the translations. "Jake" translates as: ),PP{j) "be intelligent": I' 'tJake is intelligent" is: APP{ j)(^J1) This formula reduces to the following: I(j) is predicated of the individual Jake ( j ) . I,et us compare this with the tral~sla~ion of the sentence "Jake is sick" in the physically ill sense, which is a "state".

Here, we find that I ("intelligent")

"be sick" translates as: Il,r3y[R(y,s) & sick'(y)] "Jake is sick" ivould then have the following semantic representation: /ZPP{j)(^A.viI)! [R(,y,.r) &- sickJ(,)/)] This formula reduces to: j)11R( ,-j) 8: s i c k l ( j ~ ) ~ This illustrates formally what was said in words above. Being "intelligent" is a property of Jake, but being "sick" (physically) is a property not of Jake but of one of his realizations. This invites a characterization of the function of the English progressive marker. No doubt a far more sophisticated treatment is ultimately required,'4 but among other things t l ~ eprogressive seems to have the function of predicating a verb of a stagc, but not of an individual. Let us give the following translation of thc progressii~emarker, which is of the category IV/IV, or something that takes IV-phrases (or Verb Phrases) and turns them into other IV's. T h e progressive, then, turns a "property" into a LL~tate".

I then compare "Jake runs" with "Jake is running". "Jake runs": RPP{ j ) (^Axrunl(s)) Which is equivalent to: run1(j ) "Jake is running": ; I P P { j ) ( A L P 1 l . x ~ y [ R (& . YP'{y)](^),z Y~) runl(.z))) This reduces to: ?ylR(.)/,j) & run'(.y)] In the case where "runs" is predicated directly of Jake, it may be intcsprcted v:~riablp as a habit, or a disposition, or an occupation of Jake's. Thcsc various characterizations arc not distinguished under this analysis, though closer examination may rcvcal that ccrtain distinctions will have to be made.

66 Greg N. Carlson Another problem raised here is that individuals and stages appear to be of the same type, as run' may be predicated of either. These should be distinguished at some level, and can be, but to do so would require a certain amount of additional notation (introduced by Terry Parsons in class lectures, spring 1976), so I leave the matter unresolved here. Intuitively, if run' is predicated of something that realizes an individual, it means something like: "running is a characteristic of this event-like thing, il realization of an individual". I now turn to the matter of 4 N P and the indefinite plural interpretation. Formally it is a rather simple matter to incorporate 6 N P into this framework if we treat it as a proper name of abstract individuals. T h e translation of "dogs" would be very much like that of "Jake". (We ignore the obvious internal structure which the N P dogs exhibits.) (a) "Dogs" translates as: ilPP(d}. If we construct the proposed translations of the following sentences, we can see how the "indefinite plural" reading arises. T h e translations are given in their reduced forms. "Dogs "Dogs "Dogs "Dogs

are intelligent": I(d) are sick" (physically): 3x[.R(.r,d ) & sick' (x)] run": run' (d) are running": 3x[R(s,8 ) & run' (x)]

T h e indefinite plural reading- arises whenever it is a dog-stagc that something is predicated of. A dog-stage, or a realization of the hind dogs, thcn, is whatever realizes the kind dogs at a time and a place. That is, it is a temporally and spatially bounded appearance of a kindVz5Particular individuals are by definition spatially bounded (i.e. can only be in one place at a time) but not temporally bounded (can exist at different times), so the main difference between kinds and individuals is that kinds are not sparially bounded, but individuals are. A realization of a kind, appearing at il time and place, would be simply one or inore of that kind. As an individual may be thought of as whatever it is that ties a bunch of stages of an individual together, so might a kind be thought of as whatever it is that ties a buncl~of things of that liind together, maliing them realizations of the same thing.2" T h e notion that a realization of a kind should be a subset of' the set of individuals of that bind might run counter to the feeling of some that a realization of a kind should instead be a11 of the individuals of that kind. But we simply do not speak that way. If we say "Marvin owns that kind of do%'' we clearly do not mean that he has a monopoly on the ownership of that kind of dog, but only that he owns at least some of that kind. Or, if we say "that hind of animal is found in India and is also found in Pakistan", we do not mean that all of that kind of animal are found in each place, only that s o m 2 are fbund in one place, and some in the other. If there are some of a kind present, then this counts as the presence of that kind. With this in mind, we see that the indefinite plural does indeed have an existential quantifier associated with it, but that the source of the existential quantifier is not the determiner of t11c 4 N P , hut rather what is being predicated of it at the time. Thus, the

A Unified Analysis of the English Bare Plural 67 existential quantifier itself vvill have constant scope, and in fact will have "narrowest" scope. This clearly accounts for the lack of interaction between the existential quantifier and other predicates in the sentence. We will conclude by showing how this way of looking at things can account for the opacity, narrow scope, and differentiated scope phenomena discussed earlier. Let us first look at what is proposed to bc the difference between the sentences in (131) and (132).

(131) Max believcs some dogs are here. (132) Max believes dogs are here.

1

'

The question we are concerned with is why (131) exhibits a transparent reading, but (132) does not. T h e transparent reading of (13 1) would be derived by introducing the NP "somc dogs" outside the scope of "believes". This structure would bc represented as follows. '[be here" translates as: ;lx3y[R(y,x) & Here1(j~)] "belie~es" is: Bell "Max" is: iLPP{m} "Some dogs" is: ; ~ 3 s ( D o ~ ' ( x&) " ~ ( x )(I] ignore plurality) "Max believes some dogs are here" translates as: (131') 3r[Dog1(x)& [Bell(cjJ~[R(.y, x) & He~-e'(,~/)l)(uw)]] Substituting the NP "dogs" for "somc dogs" in this structure and translating "dogs" as ~.PP{IJ),u7e arrive at the follon-ing representation:

(132') A~[Bel'('3,~[R(~~t, x) & He~-e'(.~!)])(rn)](d) What this says is that the individual denoted by d is believed by Max to have a stage that is here. Since d is a kind of thing, there is no reference M-hatsoever in this formula to any particular dogs. Hence, Max's belief has nothing at all to do with particular canines. Our impression that sentence (132) has only the narrow scope reading derives from the fact (which in a full formalism would be explicitly stated) that any stage of 11 (the kind) also has to be a stage of some particular individual of that kind. In making this inference, though, we find that the expression denotes thc stages in question appears in an intensional context, being part of the predicate itself, and not part of the NP. As no particular stages are referred to here due to the intensional context, these stages need not be associated with any particular dogs. I11 extensional contexts, as for example in "Dogs are here" the specific stages may be associated with specific dogs. In this way the notion arises that (132) and he like exhibit only opaque readings. It should also be noted that if we consider d to be a rigid designator (as Montaguc considered proper names to he) the formula in (132') would be equivalent to the following.

68 Greg N. Carlson So the logic makes the claim that sentences like (132') are not ambiguous with respect to scope possibilities. As is well-known, this might not be exactly correct. In any casc, the claim made here is that whatever ambiguity proper names may exhibit in intcnsional contexts, bare plurals will exhibit the same sort of ambiguity. I think a littlc reflection will show that this is reasonable. Compare (13Ba) and (133b). (133)

(a) Max believes that Bossit has horns. (b) Max believes that conls have horns.

It sce~nswhatever anlbiguity can be attributed to (133a) may also be attributed to (133b), though the judgments here are notoriously subtle. T h e treatment of barc plurals as proper names also leads us to an account of thc narrow scope phenomena. Recall that a sentence like (134) has only a contradictory reading.

(134) Cats are here and cats are not here, If we look at the second conjunct alone and introduce the NP "oats" outside the scope of the negative, we come to the following representation:

But because c is treated as being a proper name, this formula is equivalent to thc follow-ing.

Thus, the representation of sentence (134) will always be equivalent to somcthing ofthc forin A & -A, or a contradiction. T h e same equivalence will account for all the othcr cases of narrow scope. T h e other quantifiers in the sentence will always, in the cascs exemplified, have wider scope than thc existential cluantifier that is a part of thc predicate itself, and the relationship between the other quantifiers and thc bare plural is irrelevant since proper names do not show scope behavior. T h e sentence ( 1 31) is a contradiction for the same reason that a sentence like "Fred is here and Fred is not here" is a contradiction. And a sentence like "everyone saw movies" fails to exhibit relative scope ambiguity for the same rcason that "everyone saw Fred" fails to exhibit that sort of ambiguity. We will conclude this section by presenting a brief analysis of differentiated scope, which once again cxploits the analysis of barc plural NP's as proper names. An examination of the sentences of (135) reveals that (a) and (b) cannot receive ilorrnal literal interpretations (though a hyperbolic usage of (1351)is heard on occasion, a matter I disregard for the time being). T h e proposed translations of these sentences r e ~ c a lwhy it is that the first two are strange in a way that (135c) is not. (13 5 )

Jake is everywhere. (b) Some dog is everywhcrc. (c) Dogs are everywhere. (a)

A Unified Analysis of the English Bare Plural 69 "Jake" translates as: %PP{ j) "Some dog" translates as: IP3.lel~Dog1(x)& P { x ) ] "Dogs" translates as: 'PP(cl) "be evcrywhere'' transIates as:

"Jake is everywhere":

"Some dog is ever~where":

"Dogs are everywhere"

In these cases, we see that a felicitous transIation results with $NP in spite of the fact that the universal in the predicate is always restrictcd to being within the scope of any quantifiers present in the subject NP. In the case of (135a), the assertion is that in cvery relevant placc there is a Jake-stage. Since Jake is an individual of the type that can be in only one placc at a time, this sentence, taken literally, speaks of'a worlcl we simply don't live in (but if "Jake" were the name of a god, for example, the sentence would make a bit more sense). Likewise, w-ith "some dog" in (135b), it is realizations of the samc animal that must appear ever~rwhereif the sentence is to bc true (supposing that the phrase means some particular individual dog, and ignoring thc well-formcd "kind" rcading for now). This sentence encounters the samc problems as "Jake is everywhere". Note that I am not claiming that these sentences are either syntactically or scmantieally ill-formed, only that they are strangc in our ~ o r l dIn . the case of ( 1 3 5 ~we ) exploit the notion that 1-ealizationsof a kind consist of sofile of that kind, appeasing at a time and a place. It is asserted that the same individual is everywhere, just as with Jake, but this individual is not of the type that can be in only one place at a timc. Whatever it is that ties all individual dogs together as a kind - thc abstract individual "dogs" has the propert! of having sonic realization in every relevant placc. 'This, then, is a means of accounting for differentiated scope. Ilue to thc programmatic nature of this formalization, I leave undiscussed a number of other difficulties with 4NP that have been raised here. T h e analysis proposcd can handle some of these quite readily; others at this time remain unresolved. A number of othcr issues raised by the proposecl analysis have not been notcd here, and I do not wish to pretend that it is u-ithout its difficulties.

70 Greg N. Carlson

5 Conclusion T began by noting that the analysis of the semantics of the English bare plural was full of difficulties because of its apparently diverse uses. Hen-ever, there was reason to believe that the divisions noted were not so clearly distinct after all, and that a unified analysis LWSfound to be desirable, if not necessary. A unified analysis was then psoyosed which allows a constant translation of (bNP in all cascs, existential and generic, and which seems to be able to account for some of the surpsising semantic characteristics of the "indefinite plural" use of this construction. Ally number of directly related matters have been left untouched, but tbs which this analysis of the bare plural, if adequate, would have direct consequences. For example, a striking similarity was noted between bare pli~ralsancl Illass nouns that have no cletermi~lerassoeated with them (see Cartwright (1975) for similar observations). In addition, the singular generics with rr and the were left untouchcd, though the relationship bctween a, the, and d, is a mast interesting (and difficult) one. I hope tliat this analysis of the bare plural will be able to shed some more light on such matters.

Notes This paper represents a major revision and extension of my h.3.L4.thesis written at the Univcrsit! of Iowa in the fall of 1973 under the direction of Larry Martin. I nish to thank Lisa Sclkirk, E n ~ n ~ o n Rach, Edwin Williains, 1,arry Martin, and Barbara H. Partcc For reatling and criticizing earlier versions of this paper, and for their constant interest and encouragcnicnt. T h e quality of the contents has been enhanced reniarkablp by their comments; of coursc nonc ncccssarily bclicvc anything contained herein, and I am alonc responsible for errors. In addition, I rcccived man! fine coninlcnts from the anonvmous rekree which contributed grc;ltly to the tinislicd product. 1

This hypothesis appcnretl in t\vo publications in the coursc of' niy work on this topic. See Schachtcr (1976) and Burton-Roberts (1976). 2 E:xcluded from consideration arc the predicate noniinals, though an extension of the suggested analysis may be able to cover them as well. 3 T o mcntion but 3 f~ 5 - ,Dowty (1972), Gough (l069), a11d Do~lgI~crt!.and 13elonnc (1072) worked under this assuniption. I do not intend to claim that thcrc ~ i ~ c c s ~ n1s.r;1i lplural ~ counterpart for the indefinite article: only that if there is one, it is not the null determiner. 4 That is, the differences we find in other singular-plural pairs such as /hrw-/hr.\., he sp ancl 1Itc pl, that-lhose, r r ~ y sg, and nr!)! pl (thc quantilier, not the polarity itcni), and I - sg and some -t pl (the quantifier). 5 T h c gcncral prediction is tliat any NP in any contcrt mily gct t1icl.e cithcr b y direct introduction or by quantifj-ing in, and if it makes a difference in the ultiniate semantic rcprescntation, tlierc will be an ambiguity. However, thcrc are many cascs \\~llerepredicted ambiguities clo not ilplJf3di". Sometin~esit depends on the nature of the YP itself:

+

+

+

+

, ,

I

A Unified Analysis of the English Bare Plural 71 (i) Bart wants to slio\v Jcnnie ( I gaoi/ ?trozq~~. (ii) Bart wants to show Jcnnie rr A-oocl tiirrr. (ii), unlihc (i) docs not appear to bc ambiguous with respect to relative scopc of' thc italicized NP. In other cases, the nature of ~ h predicate c is responsible. (iii) Bart wants to have LI I I I I S I ~ C S S . Note that (iii) is not nmhiguous with any other NP's in place of the italicized expression either.

(iv) Bart wants to have

6

7

8

0 10

I 1 lots of twenty set-cral

1?:: 1

mistresses.

I assunic that certain predicates like "have" reyuirc a lcsical direct object, h;~nning.rluantjfving in, and that certain NP's, like "a good time" cannot be quantified in. It is not clelir to me how to sure thcso restrictions formally. The sentence "there aren't cats in this room" is really quite awkward, if not ungramma~ical.I believe [his stcnls from the fact that the ncgative here negates the quantifier on the subject N1' and not the \\-hole sentencc. If there is no quantifier present, as may he thc case \\ith dNP, this result is expected. This lict conflicts with our feeling that "cats are here" means that Inore than onc cat is here. At this time, I think it rrwans one or more, but inlpljes morc than one; if ?-ou knew there was just onc there, you'd say so. Thc fact that the existcnti;il must hold wider scopc is nor predicted by Ilo\~-ty'sanalysis, and an! ,~tlccluateanalysis of.fi,r tinic adverbials should reflect this observation. Pcrl~aps/ i ) r time advcrbiills are verb modifiers and not VY or sentence modifiers. This is discussed in hlnntague (1972), Partcc (1970) and references citccl. Unquantified Inass nouns exhibit unambiguousl~opaclue readings, narrow scope, and dillkrcntiatcd scope as wcll. (i) Jack belicvcs that .furniture is kept in Ncll's attic. (ii) Ever>-onctll-nnk water thri~~r,os.fi~arir/~ited. (iii) Chloritrc gas was e\.eryw here.

11 This problem is discussed in Cartmright (1965). 12 One readcr notcd that it cvould be possible to assign scope after the translation from the object language into the language that is to be intcrprctcd, and that is r~rouldbc possible thcn for the singul~rand plural ( I and 0, to participate in diffcrcnt scope-assignment rulcs, bur remilin singular ilnd plural counterparts nonetheless. I mean to exclude this logical possibilitl as violating the very assuniption I started out with, that singular and plural shoultl bcha\-e ;dike up to thosc differences that can he attributed to plurality. Attributing differential scope beha\-ioi to pluralit! per se is not possible, sincc so t"ir as I know it has no inrlcpendcnt motivation clsewhere in the grammar. T h e differential scope behavior \voiild then be retluccd ro sonic ,~rbitrarj-property of thc bare plural, and not simply the h c t that il is n plural. generic quantifier or article, though I clon't know 13 I know of no languages that have an e~i.lrr.uz~e/,~ whether this is universall\; so. Smith (1964) notes that generic NP's of ICnglish arc gcncmtcd s!-ntacticall!- just lihc non-generic NP's, requiring no special rulcs ; ~ tall. I suspect that all languages pattern liken-ise. 14 I assilnlc some notion of "normal intonation", and T ;111i 11ot responsible for M-hath~lppenswhen additional stress is added to some constituent. For examylc, "smokcrs .4RL rudc" can apparently

72 Greg N. Carlson mean that some smokers are rude. I doubt that it can bc maintained that such scntcnccs in\-olve a simple existential claim howevcr, for the following seem cstre~nclc.strange to rncb, even if existentially true.

(i) (??)Smokers .4REChincse. (ii) (??)Trccs ARE 350 feet tall. (iii) (??)Babies ARE six-toed. It seems thc stress has to do with disposition or some similar notion in this case, having some sorts of behavioral implications. 15 T h c p~.escnceof the mass noun "kelp" cannot be held rcsponsiblc, for "chirpcd", "walked without stumbling" and other VP's having no dircct object cxhibit the same iunbiguit!.. 16 This is similar to referring to. say, all the members of a team by calling them "the team". In (a)(c), the predicate is true of each team member individually. (a) T h c team is m-caring red shirts. (b) T h e tcaln died in the plane crash. (c) T h e tcam is quite tall. Hov.cver, "the team" can be used to rcfcr to something more than thc sun1 of'its parts: (d) Thc tcam has ~ v o n22. pennants over the last 40 years. (c) T h c team has becn in continuous cxistcncc for 100 years.

I do not wish to push the analogy too far, but kinds arc a bit like tcanls in this respect. I thank Barbara Partee for his obser~ation. 17 I am not sure rvhy this variation of pronominal fbrm is roleratccl, but sometimes one, and then thc other, seems preferable. In fact, "sm" alone appears to disallow any reference to hinds, while all the rest allow it. 18 (a) ""Sin birds are \videspread.

19

I do not mean to discount as a possible line ofliguistic inquiry an investigation of ~ h relationship c belween binds and individuals of that kind. For example it scems to me that thc fiict that (a) below can mean that linguists, collectively, have .30.000 books in print but th;~t(h) cannot mean t h a ~linguists, collectively, have 62,344 legs (even though (b), on a collccti~ebasis, m i ~ h bc t true) is a fact that needs to be accounlcd for within a semantic aniilysis.

j

7

;; 1

, I

(a) 1,inguisls lla\-e over 30,000 books in print. (b) Linguists have 62,.744 legs.

20 21

I do mcan to excludc from a semantic analysis hen-c\cr, the q ~ ~ c s t i oofn whcthcr 200 out of 000 birds born without wings t~louldfalsify "bircls have wings". This is a different cluestion li-om the above, thoug1:ll I grant that the dividing line is not al~vaysclear it1 any given casc. I wish to separate linguistic knowledge from the act of recognizi~lgficts about the world. This sort of analysis might also bc uscd to investigate abstract individuals such ;IS "honestj", bL democracy", and the like. These arc csscntially mass nouns, lihc "water" and "tire". Noted in Lawler (1972). Will also give rise to a very natural third reading, indicating present clisposition. as in: (a) Water r~v'llboil at 100°C.

22

kIilsiirk also notes that his "states" are predictablc of NP's detcr~nincclhy ~ h ~~nstresscd c "sm" and b! the non-generic "a", but that the "propcrtics" only sclect thc generic "a" (assuming

1

: I

j

A Unified Analysis of the English Bare Plural 73 there to be a distinction, which there may or ma>-not be, bctaeen generic ant1 existential "a") and ;ire not acceptable when predicated of an N P determined by "sm".

(a) A soldier nas available Sm soldiers were a\;ailablc (a "state") (b) ,4 soldier was tall, (generic only) "Sm soldiers were tall (a "property") Though I have no account of thcsc distributional facts at this time, they may be uscd (at least in subject position) to test \diether the predicate is a "state" or a "property". 2.1 1 exclude those cascs whcrc it is thc hr of identity that precedes the predicate nominal. These are quite different, for they allow an indefinite plural reading: (a) Children were thr z?rr.tims(![/he nssliult. (b) Horses were /he p n n ~ sin /tic parnr.

24 For example, we ~vouldwant it to entail that john hadn't finished crossing the street in (a): (1)

John was crossing the strcct (when he was flattened by a truck)

See Bennett and Partee (1972) for some problems and suggested solutions. of translation 2.5 This sort of notion may be what Quine (1960) had in mind in his clisc~~ssion procedures, but whether or not this is true, it served as the sourcc of thc line of thought l~ursucd herein. 26 We night try, then, to define a "gcncric sentence" as any sentence that attributes a property to thc individual that s e r ~ e sas the subject of the sentence, and not to one of that individual's realizations.

References Iknnett, Michacl. 1975. Some extensions of a Montaguc fragment of English. Ph.11. dissertation, UCLA. Rcpr, by the Indiana Linguistics Club. llcnnett, Michael and Barbara H. Partee. 1972. Toward the Logic of Tensc and Aspect in Ilnglish, unpublished manuscript. Ilurton-Roberts, Noel. 1976. On the generic indefinite article. Llrlrgt/n~r52(2): 427-48. C;arlson, Greg N . 1973. Superficially Unquantificd Plural Count Nowl Ph1-ases in llnglish. M . 4 . thesis, University of Iowa. Cartwight, Elelen. 1965. Heraclitus and the bathwatcr. Philosopliir~rrlR P Z ~ C71: J ) 406-85. ~ (;artwright, Helen. 1975. Some remarlcs on mass nouns and plurality. . S ' ) ~ t ~ / h31: ~ ~ s39.5-410. e (:honisky, Noam. 1945. .-lsj)i~i.tsoJ/lre T11ror:yr![S')~nta.\.,Cambridge, Mass.: M1T Press. Chonisk~~, Noarn. 1975. Questions of form and interpretation. Lrng~ris~tc .41z~!j~sI.s 1: I. Ilahl, &ten. 1975. O n generics. In Edward Keenan (cd.), Fonnirl Se~rrtrr~tr~..~ q/',V(~~trnrl /,an,qr,rrg(~, (:ambridge: Cambridge University Press. Uelorme, ICvclyn and Ray Dougherty. 1972. ,4ppositive N P constructions. F'or4nrlnlioi1s ( ? / ' I , l i ~ i , y ~ ~ o , y ~ ~ 8: I . Dowt!, David R. 1972. Stttdrrs irr the Logic. of' I'i.rb -4spi>c/~rnd 7 i m i ~Rq/i,rerne ill EtrK/islr. I'h.11. dissertation, L ~ n i v c r s i tof ~ Texas, Austin. Rcpr. in Stztdirs in Li~zpuis/ic.s,University of Texas, Austin. Gough, James, Jr. 1969. T h e syntax-based semantics of the English determiners a, /he, and 0. Pcrptr:\. itr Ling 14is1il.s1: 1 . Lawler, John. 1972. Generic to a h u h . In Paul h4. I'eranteau, Judith N. Levi, and Gloria (:. l'liarcs (eds), Prlpers#i-om /hc Eigh~hRiyio,lwtrl i2.lrrtin.~c!f /lzr Clril.npo Lin,y~rtslit.S o r i i ~ i )Chicago, ~, Ill.: (:I ,S.

74 Greg N. Carlson J,ewis, David. 1975. "Adverbs of Quantification", in Edward Keenan (ed.). firr~jal S1~nrrrnrli.sr!/' ;Virlurul Licnguage, Cambridge: Cambridge University Press. McCawley. James D . 1970. Where do noun phrases conic from?. In Roderick 11.Jacobs and Pctcr S . Rosenbaum (eds), Rua(litrgs in Englzsh T ~ ~ I ~ ~ I MGrrrrnnrur, I I I I J OWallham, ?? Mass.: Ginn and Co. Milsarli, Gary. 1974. Cxistcntial senlcnces in English. P11.D. thesis, MIT. Montague. Richard. 1972. T h e proper treatment of quantification in ordinary Cnglish. In Richard 11. Thomason (ed.j, (1974) for~ri~cl Pfzi/ost~ph.)f. Stlerted P(zpers r!/'Rirhrlr-d.Mon/~lgrr~, New I Iat en: Yalc University Press. Parsons, Tercnce. 1970. An analysis of mass and amount terms. Fourtd~r~ions c![Lrrlaguugi~6: 363-88. Partee, Barbara H. 1970. Opacity, coreference, and pronouns. Slln~ki~se 21. Partee. Barbara H. 1975. Deletion and variable binding. In Edward Keenan (ed.) Forrttnl Sem(mlics I!/' Naturul Language, Cambridge: Cambridge University Press. Perlmutter, David. 1970. On the article in English. In hlanfrcd Rierm-isch and karl Erich Heidolph (eds), Progress 212 Lirtguistic-s. The Hague: Moutton. Poslal, Paul. 1969. Anaphoric islands. In Robert I. Rinnick et al. (cds), P(ipe,:rJ h m the I;i/ih Rqionrrl Mertrng of ~ h Chicago t Linguistil- Socirt.11. Chicago, Ill.: CLS. Q ~ ~ i nWillard e, van Orman. 1960. IVoril irl~dOl!;ect, Cambridge, hlass.: MIT Press. Schachter, Paul. 1976. A nontransformational account of gerundive nominals in English. Lin~rris~ic. Inquit:)/7: 2 . Siegcl, hlufft E. A. 1976. Capturing the adjective. Ph.D. dissertation, University of Massachusetts, ,4mhcrst. Smith, Carlotta. 1964. Determiners and relative clauses in a generative grammar of English. Langmii,~e 40: 1. S~ockwell,Robert P., P. Schachtm, and B. H. Parlcc. 1973. The /Mr!;or- .Synr~rc.ll'lSlruclur~,sc!f'Erzg/ish. Ncw '170rk: Holt, Rinehart, and Winston. S~veel,Henry. 1898. -4 ~ V e mE n ~ j i s hGnlmnrur. Oxford. Zemach, Eddq-. 1975. On the adequacy of a type ontolog!.. Sy~rrhesc31: 509-15.

Generalized Quantifiers and Natural Language Jon Barwise and Robin Cooper

0 Introduction In 1957, the Polish logician Andrej Mostowski pointed out that there arc many mathematically interesting quantifiers that are not definable in terms of the firstorder ti, 3 and initiated study of so-called generalized quantifiers (cf. Mostowski 1957). Since then logicians have discovered and studied a large number of gencralizcd quantifiers. At last counl there were well over 200 research papers in this area. Most of this work has been directed toward cardinality quantifiers (e.g. Keisler 1969) and topological quantifiers (e.g. Sgro 1977) which are not particularlj? relevant to natural language, but even so, it has fbrced logicians to rethink the traditional theory of quantification. The quantifiers of standard first-order logic (as presented in elementary logic textbooks) arc inadequate to treat the quantified sentences of natural languages in at least two respects. First, there are sentences which simply cannot be symbolized in a logic which is restricted to the first-order quantifiers b' and 3. Second, the syntactic sti-uctureof quantified sentences in predicate calculus is completely different from the syntactic structul.e of quantified sentences in natural language. T h e work on generalized quantifiers referred to above has led to net+-insights into the nature of quantifiers, insights which permit logical syntax to correspond inore closely to natural language syntax. These insights, we argue, may also make a significant contribution to linguistic theory. Section 1 discusses the nature of generalized quantifiers and thcir relationship to the syntax of English in general terms. Section 2 develops a logic containing generalized quantifiers. Section 3 shows how this logic may be formally related to a fragment of a syntax for English. Section 4 is the main section of the paper. In it we discuss some of the general implications of the notion of generalized quantifier for a theory of natural language of the kind that is interesting to linguists. Our conclusion, in secrion 5, attempts to draw some general conclusions about thc relationship between syntax, semantics and logic.

76 Jon Barwise and Robin Cooper 'Thc papcr has four appendices. Appendix A contains additions to the fragn~entin section 3 which are suggested by the results in $ 4 . Appendix 13 contains sol~lcpossible semantic postulates on the meaning of non-logical dctcrminers. Appendix C contains the proofs of the Facts about quantificrs asserted in the body of the paper. Appendix D consists of a chart classifying English determiners according to the semantic categorics introduced in $ 4. Some (but not all) of the points made in section 1-3 of this paper are implicit or explicit in A4ontaguc (1 974), especially in PTQ, "The Proper Treatment of Qyantification in Ordinary English". (Some of the suggestions in 1-3 are also similar to suggestions in other papers: e.g. Fenstad (1978); Peacocke (1 979)). Our hope is to develop h4ontague's treatment of noun phrases further in a straightforward way (without lambdas), and to show some of its implications for a theory of natural language.

1 Generalized Quantifiers and Noun Phrases 1.1 Some examples of generalized quantifiers Viewed from a modern perspective, the familiar b' and 3 are extremely atypical quantifiers. They have special properties which arc entirely misleading when one is concerned with quantifiers in gcncral. JVc begin this paper by discussing some simple examples of generalized quantifiers froin mathematics to draw out some of the general features of quantifiers. Consider the following examples. (1) (a) (b) (2) (a) (b) (3) (a) (b)

There are only a finite number of stars. No one's heart will beat an infinite number of times. More than half of John's arrows hit the targct. More than half the people voted for Carter. Most of John's arrows hit the target. Most people voted for Carter.

1.2 Many quantifiers are not definable using first-order 'd and 3 There is no doubt that in any human language in which modern science can bc formulated, sentences like (1) and (2) can be expressed. We suspect that sentences with quantifiers like those in (2) and (3) can be cxpresscd in any human language. But tlie quantifiel-s in (1)-(3) cannot be expressed in terms of tlie first-order quantifiers b',r(. . . ,v . . . ) and 3x( . . . .t. . . . ). It is not just that we do not see how to express them in terms of b' and 3; it simply- cannot be done. Thus, a semantic theory for natural language cannot be based on the predicate calculus alone. First, before sceing just what thc problcms arc, Ict us abstract out thc quantifiers at \vorli in (1)-(3) as follows. (1') (2') (3')

Finitely many things s satisfy q ( s ) , or., Inore .\:ymho/ic.a/l)/,1;inite xlcp(.v)l. More than half the x such that $(r) satisfy cp(.v), or, (more than t~b).rlq~(.v)]. A/lost ,I? such that II/(.v)satisfy cp(x), or (most rl/).~[rp(.v)l.

Generalized Quantifiers and Natural Language 77

Let E be an arbitrary non-empty set of things (individuals, entities, call tlicm what !.ou will) over which our variables range. First-order logic only a l l o ~ squantification over objects in E, not over arbitrary sets of things, functions from things to things or other sorts of abstract objects not in E. Within this framework, it is easy to prove that none of the quantifiers used in (1)-(3) is definable in terms of the ordinary 'v' and 3. Consider the case of "more than half'. It is a routine application of familiar tcchniques in first-order logic to prove that this cannot be defined from 'v' and 3; that is, that there is no fixed definition that works even in all finite domains. 'This is proved in Appendix C (C12). One has to leave traditional first-order logic in oiie of two ways. One possibility is to expand he domain E of quantification to a bigger domain E U .#I,where .4 includes numbers and funetions fiom subsets of E to numbers. That is, one might mirror the high-order set-theoretic definition of "more than half' in thc semantics by forcing every domain E to contain all of the abstract apparatus of modern set-theory. A different approach, oiie that model-theorists have found more profitable, is to keep the formal definition as part of the metalanguage, and treat generalized quantifiers without bringing all the problems of set theory into the syntax and semantics of the logicper se. We'll see just how this is done in a moment. T h e point to make here is that, once we make this move, it also gives us a way to treat determiners like "niost", "many", "few" and others.

1.3 Quantifiers correspond to noun-phrases, not'to determiners We have bccn at some pains no/ to call "inost" ancl "more than half" quantifiers. T o see why, note for example that there is no way to define "morc than half of John's arrows" from "more than half of all things", i.e., it cannot be formalized as sonietl~inglike "More than half s( . . . x . . . ),' . This is why, in (2')' we symbolized the quantifier with (I/ built into thc quantifier prefix. What this means, semantically, is that "more than half" is not acting like a quantifier, but like a determiner.l It combines with a set expression to produce a quantifier. On this view, the structure of the quantifier may be represented as below. Quantifier

/"',

Determiner

,

, 1

Set expression

If we compare this structure with the syntactically simple sentence (313) we can see that the structure of the logical quantifier corresponds in a precise n-av to the structure of the English noun-phrase (NP) as represented in:

Det

ANoun

78 Jon Barwise and Robin Cooper For exactly the same reason, "most" must be treated as a determiner, not as a quantifier. It is the NP "most people" that is the quantifier. There is no way to paraphrase a sentence like (3b) that begins "most things are such that if they are people then. .. ". This can be proved, given reasonable assumptions about tlie meaning of L< most", in the same ways as for "more than half".

1.4 Quantifiers are not necessarily logical symbols 'There is a mistaken notion that the meaning of the quantifiers nlr~stbe built into the logic, and hence that it cannot vary from one model to another. This is mistaken on several counts even for mathematical examples. Unfortunately, the most convincing examples of this are outside the scope of this papcr. For example, the rncaning of tlie quantifier Q~cp(x)wl~icliasserts that {x 1 (p(.~))contilins a 11011-empty open set (studied by Sgro 1977) is determined not by logic, but by somc underljring notion of distance, or, more precisely, by an underlying "topology". T o interpret such a quantifier, me need not just an ordinary model, but also a topology to make the quantifier precise. 'The same idea can be applied to the determiner "more than half" when onc turns to infinite sets. hleasures have been developed in which (4) :u~d( 5 ) makes perfectly good sense. More than half the integers are not prime. ( 5 ) More than half the real numbers bctn-een 0 and 1, expressed in dccilnal notation, do not begin with 7.

(4)

However, the truth or falsity of (4), ( 5 ) will depend not on rr priori logic but on which underlying measure of infinite sets one is using.-his measure must be includccl as part of the model before the sentences have any truth value whatsoever. One of the simplifying assumptions often made in the model theory is that one hiis a fixed context whieh determines the meaning of the basic csprcssions. We can think of this context as providing an intcrpretation for non-logical dcterrniners in the above examples. In this paper we shall assume throughout that thcrc is a rich context held fixcd that determines the precise meaning for basic expressions, even those like "most", "niany" and "few". We refer to this as the.fised ~'orttr.vtassu~ilplz'or~. It should bc pointed out, however, that even with this assumption the interpretation of quantifiers, even those like "every man", will vary fro111 model to model since the interpretation of "man" is clctcrmined by the model. T h e difference between "cvcry man" and "nlost men" is this. T h e interpretation of both "most" and "man" depend on t l ~ cmodel whereas the interpretation of "ever!;" is the same for e\~erj-model. "Every", unlike "open", "niorc than half" and "most", is a logical quantifier. T h e fixed context assumption is our way of finessing the vagueness of non-logical determiners. We think that a thcory of vagueness like that given by Kamp (1975) for other kinds of basic expressions could be superimposed on our theory.4 We do not do this here, to keep things managcable.

1.5 Quantifiers denote families of sets Quantifiers are used to assert that a set has some property. 3.\-(p(.z)asserts that the set of things which satisfy q(.r) (informally {,vlq(s)} or, in our formal notation .i'[(p(.z')l) is a

Generalized Quantifiers and Natural Language 79 nonempty set. That is, the set of individuals having property cp contains at least onc member. b'.vcp(.v) asserts that the set contains all individuals. Finite ,v(p(.~) asserts that the set its finite. It is clear that a quantifier map be seen as dividing up or partitioning the fimily of sets provided by the model. When combined with sonlc scts i t will produce the value "true" and when combined with others it will produce the value "false". In ordcr to capture this idca formally, quantifiers are taken to denote thc fhmily ofscts for which they yield the value "true". T h e truth of a sentence QPvlcp(s.)jis the11 determined by whether or not the set .i[cp(,v)-Iis a member of the quantifier denotation. The denotation IlQII, of a quantifier symbol can be specified informally as follows for some of the quantifiers we have discussed. (Wc Ict E represent the set of entities provided by the model.) 11311 = { X It'7'II

=

L E I -x# 4 )

{El

J(Finitel1= {X C E X is finite} contains more than half of the Ns} lMore than half of NII = {X 2 E 1 Ilhlost NII = {X & E I X contains most Ns} To e~nphasizcthe role of the set, we will write Q,i.Lcp(,v)Jrather than just Qr[rp(.\:) 1 in the logic developed in section 2. If cp is a simple set expression we may write QW

1.6 Proper names and other noun-phrases are natural language quantifiers

I

Wc are now in a position to examine the notorious mismatch between the syntas of noun phrases in a natural language like English and their usual representations in traditional predicate logic. To review the mismatch, notice that thc sentcnccs in (6) arc all to be analyzed as consisting of a noun phrasc followcd hy il verb-phrase as represerlted by the labelled brackets.

(6) (a) [HarryJkp[sneezed]\ (b) [Some person],p[si~eezed]\.l, (c) [Every man],,[sneezed]\ (cl) [Most babies]Np[~ncc~e]\:l., There is strong evidence that the phrases labelled as NP's here belong to a singlc syntactic category. For examplc, they may occur not only as the subjccts of intransitive verbs (as in (6)) but also as the objects of transitive verbs (7) and of prepositions

(8) Harry some pel-son (7) Susan kissed every man ( most babies

80 Jon Barwise and Robin Cooper

( Harry (8) I saw Susan with

some person every man ( most babies

This constituent structure is not reflected in the translation of sentenccs containing NP's into predicate calculus. (6a-c) might be represented, ignoring tense, as (9a~c) respectively. (9) (a) (b) (c) (d)

sneeze (h) 3.r[person (,r) sneeze (,x)] 'v'.r[n~ai~ (x) + sneeze (x)] (There is no predicate calculus reprcsentation for ((id))

While (9a) contains a representatioi~of the English NP Hurq), (9b) and (9c) do not eontain constituents representing the NP's some person and e o e ~ n2an. : ~ Furthermore these two expressio~lscontain open sentences joined by two place connectives which do not correspond to constituents of thc English sentences. T h e corrcct choicc of thc connective depends on the quantifier which is to be prefixed to the open sentence. From our discussion of generalized quantifiers we can see that the mismatch bctwcen (ha-d) and (9a-d) is not necessary. (9b) is not really a translation of (6b), but of the logically equivalent, but linguistically quite different, sentence: (10) Something was a person and sneezed. What is wanted, to translate (6b-d), is (in our notation): (1 1 )

(a) ( S o m e person) .i- [sneeze (s)] (c) (Every m a n ) ,i.[sneeze (s)] [sneeze * (.v)]. (d) (Most babies) ,i

Or, more simply, (12) (b) ( S o m e person) (sneeze) (c) (Every man) (sneeze) (d) (Most babies) (sneeze). These sentences will be true just in case the set of sneczcrs (~+epresented either by .i, [sneeze (s)] or by sneeze) contains some p c r s o ~ ~every , man, or most b~bies, respectively. All that is left to make the treatment of NP's ;IS quantifiers uniform is the observation that even proper nanles can be treated as quantifiers. In our logic, (13) may bc translated as (14), or rather, something like (14) in structure.

(13) Harry knew he had a cold. (13)

I-Iarry i[x knew s had a cold].

Generalized Quantifiers and Natural Language 81 - -

-

~p

(14) must be truc just in case Harry is a illember of the set. Hence the quantifies represented by the NP H~i1.1.y can be taken as denoting the family of sets which eontain I-Iarry. T o hare our cake and cat it too (preserving the intuition that proper narnes denote individuals, rather than scts of sets) we will let the lexical item or word Hur-?)I denote an individual. However, the NP containing just this word, represented by [HarryjNp,will denote the family of sets containing Harry.

1.7 Quantifiers can degenerate in some models As mentioned above, we can think of a noun phrase as dividing the sets corresponding to verb phrases into t\vo classes correspoi~dingto those which make it truc and those which make it hlse. As a denotation of the noun phrase, we choose the set of those whicl~ make it true. It seems the most natural way to formalize the intuitions. Thus noun phrases act, semantically, like tlte logician's generalized quantifiers. In some interpretations (models) however, thesc NP denotations ma>-degenerate in one of three ways. The) may denote the cmpty set, the set of all sets, or, the worst case, hi1 to denote any set at all. T h e first two types of degeneracies are discussed in S 4.5. To see how a noun phrase can fail to denote, notice that determiners will be interpreted as functions from common noun deilotations (scts of things) to noun phrase denotations (sets of sets). Ho~vever,functions have domains and a set may fail to be in the do~nainof function u-hich serves as the denotation of a given detel-miner. In particular, the determiners the, boll1 and rreitlrer havc domains which are special. The l~londtncin, for example, does not denote anything at all unless there is a unique blond man in the state of affairs represented bj- the model. Any attempt to assign it a11 (lr/ ~ I J L denotation is bound to give rise to some incorrect infcrcnces. Thus, we treat the determiner Ithell as a function wit11 domain the set of scts with esactly one element. llbothll and llneitherl are dcfined on those sets with exactly two elements. (?'his treatment is siinilar to some presuppositional treatments that havc been proposed in the literature.) We now turn to spelling out the ideas of section 1 formally. Some readers might prcfer to turn directly to section 4 to see tRe kind of applications we have in n ~ i n d . -

2 A Logic with Generalized Quantifiers: L (GQ) The logic developed here has no basic quantifier symbols. All quantifiers are built up by applying some basic determiner sj.mbol D to some set tcrm 17.

2.1 Logical symbols Thc logical symbols of L ( G Q ) include: propositional connectives: A, v, (b) variables: s , y, Z , xu, . . . (c) n distinguished srl / e m : thing ( d ) parenthesis: (, ), brackets: [, 1, and a c r ~ psymbol:

(a)

^

82 Jon Barwise and Robin Cooper (e) an equality symbol: = (f) sonic of the following logical i1~7/ernzirzers: some, every, no, both, neither, 1,2, 3, ... , !l, !2, !3, . .. ,the 1, the 2, the 3, ...

The semantics of L ( G Q will be defined so that thing alw-ays denotes the set E of things in our model, i.e., the set of individuals or objects. The semantics of the numerical determiners will be defined so that 3 m e n run will mean that at least three men run; !3 men r u n will nlean that exactly three men run; the 3 men run, following I .7, will only have a meaning in those models where thcrc are cxactly three men. In such models it will be true if they- all run.

2.2 Nonlogical symbols Thesc include: some sct (possibly empty) of constan/ s)nnbols, say c, d , .. . (b) for each 12 = I , 2, . . ., sonie set (possibly empty) of n-ary ~.clu/ion.synzj,ols, say R, S, .. . The I -ary relation symbols are also called pl-errlilatc .\:lrmi~ois. say D l , Dz,. . . . These may (c) some set (possibly empty) of now-/ogil.sl ~l~~terrnine~s? include most, m a n y , few, a few, etc. (a)

'Thus L ( G Q ) is not just one language, but is rathcr a whole fin~ilyof languages, dcpcnding on the choiccs made in 2. If and 2.2.

2.3 Syntactic formation rules There are six syiltactic fbrn~ationrules which, together, provide an inductive definition of thc three kinds of expressions of L ( G Q ) , namely set terms, r/ul~?7t?fiers, and/i)rn~zr/ris. These rules are given in Rl-R6 bclo\v-. R1. Atzy preu'ic.ilte synbol is B sel 1er.w. R2. I f q is a Jorv~ulnarrd 14 I s rr zlnriahle tllelr i1.qJ zs a set trrni. R3. I[ D is a rleter-nliilrr rln~l11 is n set tertll the72 D (11) is rn quuntifi~~r. R4. I[ R is an tz-rir:y relnrion sy??1bol and tl, . . . ,t, (11-c conslunls or- i>(/~-i(~bits //IP~ R(tl, . . . ,t,) is a jbrmuln. Sirnilur/)/,!/'11 is 11 srt irrtrr irrrtl I 15 (1 z*~rrirrl?Ic. or. rbonstrzn/ tllerr q(t) is aJi)rrnrd/a. R5. I/' Q Is a qunrntijiri- l~nliq Is a sel tel-rrz Q(II) is rr /i)n?1rr/~r.Wc lcave off' thc parentheses if no confusion is likely. R6. The ~ ; ) T N I L I / L ELII'P S c/o.\-ed untier t/ie pl,opnsilionr// ~.orlr.rcl.tii~es ,A (i~nd),v (or) and (not). N

Some remarks and then some examplcs. 1;ormulas are built up by R4-R6. Set tcrms arc built up by R l and R2. Quantifiers are built by R3. In particulas, RS givcs 11sthe quantifiers: every (thing), (dcnotcd in accordance with tradition by V), some(t11ing) (denoted by 31,and no(thing). Gi\-cn a set term rl we u-rite t h e (11) for the quantifier the 1 (11). In R1, t ~ ( t is ) used rather than thc more customary ( t E I!), just becausc it

Generalized Quantifiers and Natural Language 83 makes the formulas neater. We will abbreviate the formula (1,

= (11, t l )

(given by R4)bl;

= h).

2.4 Some examples In the examples below we assume that our language L ( G Q has the determincrs displayed in addition to the obvious stock of relation symbols. Below each sentcncc of L ( G Q ) we write an appropriate rendering in English and, whcre possible, a predicate calculus equivalent.

(18)

(a) (b)

(4 [19)

(4 (b)

(4 (20) (a) (b)

(4

Some (thing) run. Something runs. 3x[run (.E^)]. Every (man) sneeze. Every nlan sneezes. b'x[man (s)4 sneeze (,r)]. 5(woman)i [the (man).?[kiss (s,.)l)]l. Five (or more) women kiss the man. 3.~1 3 . ~ ~ 3 ~. r~~3 . x - # ~ [xz x ~ # " ~ 3 . . . r\ woman n woman (x2)fi . . . 3y[man (y) n b'zlman ( z ) + , ) I = z.1 A kiss (.q, y) n kiss (sz,.p) r\ . . . n kiss (.t'j,,)r)] No (woman) .? [run (x) r\ sneeze (x)]. No woman runs and sneczes. 3,r[woman (x) n run (x) n sneeze (s)]. Some (woman)j j [most (men) .i.[kiss (s?y )1). Most men kiss a (particular) won~an. (No predicate calculus equivalent for rrrost.) Many (men) i [ see (.v, h)]. -Many men don't see Iiarry. (No predicate calculus equivalent for ~rrlrn.y.)

(XI )

I.

N

2.5 The semantics of L(GQ) A model for L ( G Q is a function &I which assigns interpretations to expressions of tho language that need interpretations. It assigns to thing some nun-empty set E and it assigns to each basic symbol S an interpretation (ISI( satisfying S1-S6 below. (To cxhibit the important parts of M separately, we sometimes identify M with the ordered pair (E, I1 I\}.)

SI. IJti.sa l~onstarrtorvnriablr., tlzrn lltll E E. S2. ((thing// = E. / = (1 = ( ( a , ~ ) l aE E ) (i.e., thc equality rclation on E). S3. S4. If'R is un n-uqy rr/c~tion.y~rnbolthert I(R(I C E x . . . x E (n-1it11r.s).Sivrilorl)/. !/' U is n basil set PI-nz (2.2b) then 1 I U / 1 C E.

84 Jon Barwise and Robin Cooper \ ) S o m e ( ) is the #bnction ~ ~ h i c assigns h to crrch & L i11t IurnIl~ llsornell (A) = { X E I X n l / l # 0). (b) IIEveryll i.s the j i ~ n ~ . t i o tI~ N S S J ~ ~ tS u erlch ..I C E I/U ,/i~~ijljt IIEvery)((A) = {X C E124 X ) . (c) llnoll i.s the ./irrzclion L P ~ I ' C ~assigns t u tach - . I C E the /iinlily llnoll (A) = {X 2 E I A n X = 4 ) . (d) For erich natural r~ulnber n, Inll, Il!nll, cznd 11 t h e rill arc /i{t~r.tinn.sorz sels defined by:

S5. (a)

I > 71) I X n24 I = n)

llnll (A) = (X

g El I X

Iln!ll ( 4 = (X

El

nil

\\every11 (A) undefined llbothll (14) = llthc2)1 ( A )

/ ) t h en J /(_,I) =

llneitherll (A) =

if 1-41= 11 otherwise

{l~noll(-4)

if/.,,=?

undefined

otherwise

rvhere I Y I is /he i.nrd?nu/il.~lqf'tlze set Y. Note that for each of these determiners D , llDl[ (-4)is a fimily of sets Q with the property that X E Q if and only if (X n L 4 )t Q. That is, wlicthcr or not ,Y is a member of I D I (-4) depcnds only on -Y n This property is described by saying that the quantificr ID11 (A) /ive.s on A. It is a universal semantic feature of determiners that thcj- assign to any set -4 a quantifier (i.e. family of sets) that lives on A. When nre turn to non-logical determiners, it is the only condition we impose as part of the /ogt'c. T h e other properties will bc determined by the meaning of thc determiner in question in a given context, just like with other non-logical symbols. Just as with other non-logical symbols, we may place non-logical semantic constraints on their interpretations. We will discuss somc of their constraints in Appendix A.

S6. [f D Is a non-loLgica/rlt~erminersyn~holthe71 ID11 ~ ~ ~ j g 1101~~(1l.h s .(Pi (!/'sets that lizqes o r z A.

sorrre /i,nzillt

If more than one model is around, we can keep track of the modcl M by writing [ I S1"' for the denotation of S with respect to M . Givcn a model M = ( E , 11 11)) a variable 11 and an n f E, we let M(::)= (E, 11 11' ) ) be the model n~liich is just like M except that llull' = a . We use this notation belo\v to assign interpretations to all expressions of our logic by extending the function 11 1 . We use 1 for "true", 0 for "false" below. (Formally speaking, rules S7-Sll constitute a definition of I ~ S I I by " ' recursion on expressions S of I.(GQ), simultaneously for all models A I . )

Generalized Quantifiers and Natural Language 85

Similarly, if rl is a set term then

D is I / ~leleriniratr1twl1 q a set /rl.~nrhe~zthe qlrnntz/ier D(q) ~/etlot~.sthc rcslrlt (!/' S9. If ~~pp/jling the ~kt~zot~ltiofl ?/'D to file l/enotcbtion 1!/')7; i.e. lID(v)lI = IlDllcllvlIl).

It is a family of sets that lives on (Irjll 017 S10. I f Q i s (1 quantlfie~mad I./( is a set tern thcrz Q$ dttzotcs t r u ~or~/ill.wcI'i'~,enr/i~g i~hrrhrror mot the denoration of'll/ is one qj'the sets in the ~lt>notation of'Q i.e.,

T l l e u . s u ~ ~ / ~ r l i t h ~ r c h k ~ ~ . z ~ / e s-., / i )L.g. rn,v,

Sll.

if II(PII = lli4l = 1 otherwise. We arc only interested in models where our quantifiers turn out to he defined. However, to make things definite, we might use the conventions of Klcene (1952) 111s logic of "true", "false" and "undefined" on p. 344, extended in thc usual ways. A formula g is said to be tnie in A4 !/ I I ~ I ( ~=" 1. -

3 Application to English Syntax The similarity of the structure of the logical expression in the examples (15)(20) at the end of 2.4 to thc strucrure of their corresponding English expressions should be evident. To makc this relationship explicit we characterize a small fragment of English 2nd define a translation relation between the English fragn~cntand L(GQ), thereby inducing a semantics onto the fragment. T h e fragment will be extendcd in Appendix A.

FRAGMENT 1

3.1 Lexicon NP N

{.John, Harry, Sus(iir, so~rzcfhirzg,e v ~ ~ ) f t k i nlg~, e ( I~I L, J ~., . .) - {person, miln, woman, hook, thing) -

86 Jon Barwise and Robin Cooper ITP- {sneeze, vun) IT- {kiss, see, read}

3.2 Syntactic rules SD of structurlif ~/escri'tions (phrase structure trees) by

IVe define the set means of an inductive (i.e., recursive) definition. We say jr is an S D rather than the more accurate: a is a member of SD.

3.2. I

Le.ril.al inserrion

SDO. If a is a word listed in the lexicon under A (where -4 is NP, N , etc.) then I a]A4 is .%

an S D . [a], may be identified with the tree / . X

3.2.2 Phlzlse structure mlts SD1. N P - + D e t N. (In words, if a, /3 are SDs of forms [S],,, [illr\; respectively then [a/31KPmay be identified with the tree NP.)

is an S1>.

A

(NP

SD3.

SD.1. SD5.

S-{

N R

VP

N P do not VP S and S

-+

S or S NR

-+

that I'P

SD1 and SD5 are used to generate rudinienrary relative clauses, as in ez3tr:)l7rzutz /hu/ runs. Of course, these two rules do not present anything like a coinpletc treatment- of' English relative clauscs.

Generalized Quantifiers and Natural Language 87

3.2.3 Q~rm~tification rule

C(NP, hc/she/ it)

n

This is shorthand for: If [[INl, and [t],are SDs and if'c contains at least one occurrence of hr; then thc result of replacing the first in by and subsecluent occurrences of I I ~ by , he, she, or it (depending on the gender of [ ( l N p ] is ) a n SI).

<

3.3 Morphological rules We need additional morphological rules to obtain the correct forms of pronouns and verbs. We shall not specifjr these here. See Cooper (1978) for details.

3.4 Examples

I

We give structural descriptions of the English esn~nplesin 2.4. (15')

(a) [[S0?t~elh27?g']~~ [uunIvpIs ( b ) [[Sor?~e]~,,,[thing]~

PIS.

The two SD's obtained for sonzethifzg may be regarded as alternatives for the analysis of this n~ordor as making the claim that English contains a word sorn~thi~rg with stress on the first svllable in addition to the phrase S V V I P thing with stress oil the second syllable. T h e truth co~~ditional semantics of the tv*o a]-e sho\vn to bc the same by our translation procedure. Similar remarlis can bc made about ~z~e?:y~//ir/g and eolJql rking.

To get the SD's for 2.1 (19) and (20) whose derivation rcflects the desired scope dependencies we must usc the quantification rule.' (19')

[ [ a ],], , [ ~ ~ 0 1 ? ~ ~ l ? -k ' Z L[[ml~st]~),,[~l(/nl~]~~,l.[k~.\.l~,[~l~~)~~~*1\ ]~]~~~ PIS

(20!)

+[ [ l l ~ o s / ][~m, n, n ] ~ ] ~ l ~ [ [ k i s s ] ~ ~ [ [ ~ ~ ] l ~ c ~ [ ~ ~ ~ o ~ ~ ~ n n ] N ] N P ] \ ~ P ~ S [ [ r n i ~ ~ ~ ~ ] ~ )lN]ll, , , [ n ~+l r[n[ h ~ l do ] ~not ~ ) L[sc.e]\ [ H ~ r l - r : )IS~ I ~ ~ ] ~ ~ ~

0

I

ih lZot [ [ s P Y ] ~ [ H L c Y ~ ~ ~ ] ~ ~ ] ~

[ [ Y ? I Q ? Z ) ~ ] [ ) , , [ ~ ~ ~ ] ~ ] ~ ~ ,

88 Jon Barwise and Robin Cooper

3.5 The translation of fragment 1 into L(GQ) Wc define a relation a' is n tra~zslution? f a by induction on derivations of structural descriptions a by means of rules T g T 4 below. We will use d to vary over translations of a, keepii~gin mind that a' is not necessarily uniquely determined by a (due to rule SD6). z' is uniquely determined by a derivation of a . The Lexicon. On thc left we list lexical items x of Fragment 1, on the right their translation a' in L ( G Q )

NP

VP V Det

John, fIarvy, Susan something, eceglt hing hc, person, man, momarz, book, thing sneeze, run kiss\., see. rend IL, somr t're4y. emh, a1/ the no, neither one, tlvo, LJZWP hot h rtzost, ! I I C ~ ~ )fin), ~, (i fl.111

j, h, s (constant sj-mbols) some(thing), every(thing) person, m a n , w o m a n , book, t h i n g (predicate symhols) sneeze, r u n (predicate symbols) kiss, see, r e a d (relation s ~ m b o l s ) some every t h e (i.c. t h c 1) 110, neither 1, 2, 3 both m o s t , many, few, a few

TO. If' a is an SD of the form [ q ] , , u-here q is in the lexicon, then a' is t ~ 'as given in the above table u~z/es.sX is NP and g is a proper namc or pronoun, in ~vhichcase a' is the quantifier t h e .i,[ll = I?']. Let us explain the exception in the above rule. 'The denotation (in a model M = ( E , I I ) ) of the lexical item Hur?)~is the denotation of its translation h , namely Ilhll or Harry. However, the nounphrase [Harry lK, dcnotes {X El IlhI t X ) . To see this we sinlply compute:

-

I t h e .Yf[.)) = hlll = Ilt~~ell(ll.i.ll.l' hill) IIthell({ llhlll) = {X C El{llhll} =

= {X

C El

CX )

since I(II11II)I

=

1

llhli E X}.

This set of sets is called the prim.ip~/z~/trr~filte~generated bp Ilhll. T h c coinputatio~i sholvs the logical validity of thc following: t h e (?[]I = [I) .? [ c p ( x ) )

cp(r)

(as long as t is not a bound variable of cp(,r)). We could have used a special notation, say h*, fix such quantifiers in our logic. T h e present treatment has the virlue of pointing

Generalized Quantifiers and Natural Language 89 out the relationship of proper names to definite dcscriptions. Many languages cmplo) definite determiners with propcr namcs (e.g., German L r Hetzs, Spanish r l . p ~ o n ) . Pronouns are vanslated similarly: Ire; and are translated .Y,and as the .711 .11 = .r, 1, respectivclp. translated as S'(ql), a quantifier. (We suppress the labelled brackets in translations for ease of reading.)

translates as .+[pl(x)A ql(x)l

If the cluantifiel- Q is a translation of lplNr,then translates as .i[Q(
There would be a closer correspondencc between the structure of this kind of phrase and its translation if we were to adopt Montague's treatment in P T Q , of intensional verbs like seek. Under his treatment the translation would be (" " Q)where " " " is a11 intensional operator). We avoid this here because we are not presently concerned with the semantics of intensional contexts. T h e translation we have provided corresponds to Montague's rule of VP-quantification. 11'

translates as if(<')

translates as

[UNPc/o

not

rV

[jr(<')l

[51vr translates as

(['A

tl)

^

90 Jon Barwise and Robin Cooper translates as (('A <')

translates i [ q ' ( s )A <'(b4)]

translates as

['

that

T6. Suppose a is an SD that comes via the quantification rulc

Then x' is (['.?,[<']). We lcave it to the reader to check that if a is an! of (15')-(20') above, tlicn a' is the corresponding formula of 2.4. This is true except in thc cases of (19') and (20')) kvhcre thc quantifier rule is used. The actual translation 01 (?Ot), for example, turns out to be

many (men),i[ N the

(illy

= h]),? [see

I

i'

1

I

(X,J,)~].

T o get the formula (20). you need to use the logical validity of the scheme t h e (.?[.I, = tl) .i I (~(.1'>1 ( ~ ( f ) pointed out in the cliscussion of translation rule TO.

4 Generalized Quantifiers and Linguistic Theory Linguistic theory is concerned, in part, with nntural l ~ r ~ ~ ' q r aeirerrr~is, ~igc ficts \vhicli llold for all naturally occurring liilman languages and which disringuish them from other logically possible languages In this section we suggest some potential univcrslls which relate to the theory of generalized quantifiers. In discussing these universals we

1

1 11 i

Generalized Quantifiers and Natural Language 91 shall restrict ourselves largely to siwlple NP's of English: Proper nouns, a singlc determiner clement followed by a common count noun and basic count words like (1 n2eH and ez'eyybody .

4.1 The universality of noun-phrases There is a strong intuition that every natural language has a syntactic category which should be labelled NP. This category includes proper names and is also the locus of detcrmincrs such as errgl, most, one. If the language has pronouns and dcfinite or indefinite articles (tkr or a), they also occul* insicle NP's. IIowever, thcrc is no simple way to give a universal syntactic definition of NP. NP's in the world's languages having varying internal structure and the positions in which they can occur in a sentence vary from language to language. This is just the sort of situation rvherc semantics can contl.ibutc to syntactic theory. The kind of semailtics we arc suggesting allows us to proposc thc follorr-ing univcrsal for consideration:

U1. NP-Quantifier universal: Every narural l u n ~ u c ~ ghas e .qvrtac./il. c.unsii~u~vrr.v (called noun-phrases) 1s7hosc serjzantii- -fi~nctir,nis to e.rpress gcn~nrlizedyum~~/~/icl:s o iwr tile dorm/iw of dis~.nzrt-j~>. It would probably be wrong to claim that NP's are thc only quantifiers in natural language. (It seems possible, for cxample, that tcmporal adverbs should cxpress quantifiers over moments or intervals of time as has been suggested by Partee (1073); Dow-ty (1979) and others). It docs seem reasonable, however, to claim that the noiutiyhrascs of a language are all and only the quantifiers over the domain of discourse, i.c. the set E of things provided by the model. The quantifier universal not only allows us to consider something which mil! be true of all natural languages but also scrr-es to distinguish natural languages from some other languages like the standard formulation of first-order prcdicatc calculus. -

4.2 Scope involves whole NP's, not just determiners In readings for natural language sentences, it is alcvays the interpretations associated with whole NP's that cnter into scope relationships, rznt simply the detcrniincr interpretation. This is one of the mismatches between standard predicate calculus and natural languages. O n a generalized quantifier analysis, rvherc NP's function as genesalized quantifiers, this fact about natural language would follow from thc univcrsal fict that quantifiers ma; be given different scope interpretations.

4.3 Dislocated phrases It has been proposcd recently (e.g Karttunen 197'7; Ciwper 1978; Gazdar 1970; Chomsky 1980) that somc phetlonicna (\vhich in a tranditional transformatiotial grammar would bc accounted for bv means of mor7ement rules) arc associated with

92 Jon Barwise and Robin Cooper seinailtic rules having to do with the binding of variables by quantifiers. An example is the rule of wh-movement, which woulcl be involved in dcriving the sentence who (lid Jo/rn .stJe. O n the traclitioi~al analysis who has been moved from its deep structure position after see. It seems significant that many such rules involve the movement of NP's. In order not to prejudice the issue of whether such sentences are to bc accounted for by a nlovemcnt rule IYC will adopt the term p l ~ n ~ sinc il2slocat~)d position (due to Stanley Peters) and proposc the following as a candidate for a univcrsal:

U2. Dislocated phrase universal. I f IL litnzguuge i ~ l / o p~ l~~ ns ~ s etos occtrr in rr. disfucntcdposiiion a.ssol.iated zuith a I.u/~ql'zlari(rbli>birding, then rrt /cast NP's (i.c. the syntactic category corresponding to quantifiers over the domain of discourse) r ~ i /occur / in this position. We would not expect to find a natural language in which adjectives or prepositional phrases, but not noun-phrases, OCCLLI. in dislocated positions associated with variable binding. In particular we would not espect a language to allow dislocated determiners if it did not also allow dislocated NP's. If determiners were regarded as qilantifiers one inight expect the opposite to bc true.

4.4 The property "Lives on" Quantifiers denote hmilies of subsets of the domain E of discourse. In this and future sections we will often refer to thc families as yurrnt!ficrs-, rather than as quantifier or NP rlenoiatio~zs,and use Q a s a variable over such quantificrs, It should always be clcar from the context what is meant by the term yuan~zfier. In a modcl /tl = ( E , 11 II), a quantifier Qfizwon a set C h' if Q is il set of subsets of E \vitlz thc property that, for any ,XC E,

X

E Q iff

(X n -4)E Q.

English examples u-hich illustrate this notion are the following equivalences Many men run tt Many men are men who run Few U ~ O ~ I C Isneeze I +-+ Fcw wonien are women who sneeze John loves hilary --t John is John and loves -Mary. T h e quantifiers represented hp the subjccts of the sentcnccs livc on the set of mcn, women and the singleton set containing John, respectively. ?'he peculiarity of the sentences on the right hand side of the biconditioi~alsis presumably due to thc fict that they arc oh\-iously I-edundant. LVc know of no counterexamples in the world's languages to the fbllowing requirement.

contrri~~s husic e.rprcssions, (called U3. Determiner universal. Ere?))nrrtui~alItrn~l~ugr determiners) ruhose senzaniic.fik~z~~tinn is to N S S Z ~ I ?to co?n?r~olrcount noun r/enotatinns (i.c., sets) 2-1 n gzlnnt~fiprtlzrst /ires on -4.

Generalized Quantifiers and Natural Language 93

4.5

Proper quantifiers as "sieves"

We think of a quantifier Q on a model IM = ( E , (1 (1) as separating ("sifting") thc VP denotations into those that do and those that do not combine with it to makc il true sentence. There are situations, I~owevcr,wherc this sifting process is degenerate: \chcn it lets ~'z'etyset through (i.e. Q = {X I X C E ) , called Pow(E)) and when Qdoesn't let an)! sct through (i.e. Q = 4).We call Q a prtjpe?. quantiji~>r denotation or skoe if neither of thcse hay yen, i.e., if Q i s a non-empty proper subset of Pow(E). Notice that this is a property that applies to cluantifiers as NP denotations, not to NP's themselves. I.e., it is a semantic property. For example, 11 inany inell [I= q5 in exactly those models where there aren't many men. 11 Every man I = Pow(E) in exactly those models wherc thcre aren't any men. Table 1 shows conditions under which various NP's can fail to denote proper quantifiers. We work in a fixed model 11/1 = (E, I I ) . T h e first column indicates by ayes" or uno" whether the quantifier can ever be 4. T h c second column indicates whether it can ever be Pow(E). T h e ncxt two columns give equivalent conditions for the quantifier to bc proper; column three a formal description, column four an informal description. It is often assumed in normal conversation that noun-phrases denote sieves. For example, given the utterance (2 1) we could naturally assume thc truth of (22). I Iotve\*er,this assumption may be explicitly contradicted as in (23). (21) No boy at the party kissed Mary. (22) T11ere were boys at thc psrty. (23) No boy at the party kissed Mary since therc \verenlt any boys at the part!. With some NP's it is harder to contradict the assumption that the denotation is a sicvc. For examl~lc,(24a) invites the assumption (24b) and the explicit denial in (24c) soiul~ds quitc odd. (24) (a) Every nlan at the party kissed Mar!(b) There were men at the party. (c) ?Every man at the party kissed Mary, but only because thcre weren't any men at the party. In column 5 of our chart we indicate our judgement as to whether it is easy or hard to interpret the NP as denoting an improper cluantifier (i.c., a non-sieve). 111 thc next section we find an independent semantic characterization of these classes. Column 6 has t o do with that characterization. Note that, according to our analysis, the n 17, both 17, and neither v1 will always denote a proper quantifier when they denote an! thing at all. \Vc suggest that these arc special cases of a general phenomenon which we attempt to capture with U4.

U4. Constraint on determiners that call create undefined NP's. scrrt u simple J t t e m i n e r such //?at 11 D 11 (A) is so?ne/ii?zesundcjinrrl.

/,LV L) fillre-

Table 1 --

-

-

(5)

When does it denote a sieve?

(4) Informal description of 3

Easy or hard to interpret as a non-sieve?

(6) Strong or weak determiner

hard easy hard easy easy easy

strong weak strong weak weak weak

easy easy impossible

weak weak strong

impossible

strong

impossible

strong (-1

always

There is some 11 There is some 1 There is some rl There are many 11's (There are few 11's) There is an 17, i.e. (There are no 11's) There are 317's There are 311's Undefined unless there is exactly one 11 Undefined unless there are exactly two 11's Undefined unless there are exactly two 11's always

impossible

strong (the y [ y = j l )

(1) Can NP denote 47

(2)

(3)

Can NP denote Pow(E)?

1. Every r l 2. Some 11 3. Most 4. Many rl 5. Few 11 6, No 11

No Yes No Yes No No

Yes No Yes No Yes Yes

7. 3 1 ~ 8. !3 17 9. The 1 tl

Yes Yes No

-

10. Both I/

11. Neither 11 12. John

NO

Generalized Quantifiers and Natural Language 95 1.

2.

R'lzerterc~r(1 D (1 (A) is c/
This suggests that the partial determiners D function semantically just likc their completions D' with the added import that they clenotc sieves. A completion of the n and hot12 is ez~er1t.The completion of neither is no. By U4, we would not expect to find a language with a word for nrithrr without one for no. Similarly, we predict that no natural language has a simple NP D men with the same meaning as ez5ri:ymnri except that it would be undefined only if there were more than 2 men, D men would be defined but improper if there were no men. This contrasts with The 2 men which is undefined unless there are exactly two men. Finally, note that no determiner has a "yes" in both column (1) and column (2). It would bc logically possible to have a determiner D which sometimes fiailed to be a sieve by having D(g) denote 4 for some q and Pow(E) for other q. It appears to be a rather trivial universal that ~zonaturr~llrrngiinge delev?niner mill' h ~ i v ethis propert)f. An even more trivial universal is that no (letrr?iziner is rrlr~~rqts tri-~~i~l.

4.6 Weak, strong and definite determiners In this section we dcfinc, semantically, a division of the determiners into "weak" ancl strong". We then make a hrther division by defining thc definite determiners as a subset of the strong determiners. LC

PROPOSITION. [f'n quantiJier Q o n M = (E, 11 (Cf. Appcndix C , C1 for the proof.)

I

11)

l t ' ~ )on ~ sA then .A E IQ ~ [ f E' E Q.

DEFINITION. A determiner D I,\.posilizle strong (or rregurk~i~ strong, resp.) if for evcry model M = ( E , 11 11) ancl every A E, if the quantifier IIDII(A) is defined then rl E IIDII(A). (Or A $ IIDJl(rl),resp.). If D is not (positive or negativc) strong then 13 is wei~k. To classify a determiner D as (1) positive strong, (2) negativc strong or (3) weah, you form a simple sentencc of the form

ancl sec if it is judged (1) automatically valid, (2) contradictory or (3) contingent on the interpretation. For examples, ez!eql gnu is (1 gnu is true in every model, ~ e i l h e g?zu r is n p i 4 is false in every model in which it is definecl and ??tanygnus ore gizr4s will bc true just in case there are many gnus. These judgemeilts classify rrpyy, l~cilherand riuztqt as positive strong, negative strong, and weak, 1-espectil-cly.'Table 2 presents our classification of the determiners we are considering. The terms "weak" ancl "strong" (though not the dcfinitiuns) are borrowed from Milsark (1977). Weak determiners for _Milsark are those which create noun-phrases which sound good after tlzCri~zs or there arc. (Such NP's are called indefinite in earlier literature.)

96 Jon Barwise and Robin Cooper Table 2 Weak

Strong

a some one, two, three many a few few no

the 1, the 2 , ... both all every each most neither (negative strong)

Note that a theory of demonstratives (this, that, thue, those) should work out so that they are strong determiners, since they sound odd in chere is contexts. Note also that the weak determiners are exactly the ones marked "easy" in colun~n5 of Table 1. We can use our definition to explain tvhy noun-phrascs with strong determiners sound strange in tlzew-sentences:

A sentence of the form there is/nw NP can be intcrpretcd as meaning that thc set of individuals in the model (E) is a member of thc quantifier denoted by the NP. For any positive strong determ~nerthe result will be a tautology, since to say that E is in the quantifier is the same as to say that A is in the quantifier. For negative strong determiners, the result will be a contradiction. While tautologies and contradictions are not ungrammatical they are not very informative and are normally restrictccl to use in special situations construcd as set phrases. For example, to say llzrre'.v John (in the existential, not the locative sense) is to say something that could not possibly be false sincc our semantics will require that John has a denotation, whatever the model.7 T h e sentence is therefore used in a special kind of situation with special intonation as in the following dialogue: Who coulcl possibly play Hamlet? Well, there's John. T h e speaker is using a tautology here to avoid making the direct assertion that John could play Hamlet but nevertheless implicating that this is a possibility. We can also gain some insight into the dichoton~yobserved in the previous section. NP's constructed m-it11 strong determiners sound much inore peculiar than those with weak determiners when they do not denote a proper quantifier (sieve). For an NP Dy \vhei.e D is weak, the "sieve-hood" of IDilll is contingent on whether I(rlJI E II/hlI or not. On the other hand, with strong determiners, Iqll f llDqll is always true (except for the negative strong, ~lihercit is always false). Thus in the case of weak determiners wc arc ablc to cancel the implicature that Dq is a sievc by saying soincthing like r h ~ wis (n't)/ilvr ( n ' ~ )Dq whereas this is not possible in the case of strong determiners. This hardly- constitutes an cxplanation of the dichotomy, but the meak/strong distinction is

Generalized Quantifiers and Natural Language 97 clearly relevant to any explanation of this phenomenon. 'l'he exact match between columns 5 and 6 of Table 1 ean hardly be an accident. Wc now turn to definite drtc~mine~s. Of the determiners we are considcring, the definite ones are thc n and both. DEFINITION. A determiner D is defhile if for every model .M = (E, 11 1 ) and every A for which j(D(((_.l)is dcfined, there is a non-empty set B, so that ((D(I(;f)is the sievc {X El B C _XI.(Hence, I Dl1(A) is what is usually called the principal filter gcncrated by B.) PROPOSITION. If.D is d
(We suppose that in the case of demonstratives, one would have only nllDyl( 2 IIqII.) We suspect that it is this ability to ~ ~ n i q u e deterrninc ly the generator from the NP that allows the N P to play the role of a common noun and recombine ivith a determiner. The additional information being supplied by the definite determiner is just that the set being quantified ~ \ ~isenon-null. r We shall interpret ofNI' in the abovc construction as the intersection of the quantifier denoted by the N P and apply certain detcrn~incrsto the result. An implemeneation of these suggestions for the treatment of /heve scntenccs and definite determiners can be found in Fragment 2 in Appendix A. Note that wc have no explanation of the contrast between ortr of tlzr two rnerl and "ol-16 c!f'both nwn since we arc treating the ~ J T ~and O both as equivalent.

4.7 Monotone quantifiers In this section we discuss two subclasses of quantifiers suggested by work in moclcl theory and recursion theory. T h e classes seem equally important for linguistic theory.

1

I

r)

DEFINITION. A quantifier Q is nronorowr inrreasitig (mon if X E Q and -X Y C E implies Y E Q (i.e. for any set € Q, Q also contains all the superscts of X.) Q is ~noncitnnedecrrasing (n-lon 1.) if X E Q and Y C X C E implies Y E Q (i.c. for any set X E Q,Q also contains all the subsets of X).A determiner D is moi/otoizr increasing (or decreasing) if it al\vays gives rise to monotonc increasing (or decreasing) quantifiers IlDl((_4).T o test an N P for monotonicity we take two verb-phrases, Vl'] and

98 Jon Barwise and Robin Cooper VP2 such that the denotation of VPI is a subset of the denotation of VP2 and then check whether either of the following seem logically valid.*

I f ' N P VPI, then N P VP2. ( N P is mon IJNP VP2, thew NP VP1. (NP is moll

1')

1)

LXAMPLES. Take VPI to be entered the race eady and VP2 to bc entered the race. T h e following are valid:

(30)

some Republican every linguist o n most peanut farmers many inen

If

)entered

the race early,

some Republican every linguist entered thc race most peanut farmers many men Notice that the reverse implications do not hold, since these clearly could be people who entered the race but did not enter early. T h e validity of these implications follow from the fact that the NP's are mon '/. T o exhibit somc mon 1 NP's, we note the validity of:

l3 If

{

no plumber few linguists }entered the race, neither Democrat no plumber entered thc race early. neither Dcmocrar

By considering such examples one comes to the following fiairly clcar judgements of monotonici ty: hlonotone increa.sing: he, John, men, a man, some man, some men, somcbody, the 0 man/men, these/those men, most men, many men, several men, either man, at least two men. Monotone &creasirig: no m a d m e n , few men, neither man, nobody, nonc, nothing, at most two men. Not monolone: exactly two men, exactly half thc men. There are some NP's which could arguably be regardcd as bcing able to denote both monotone and non monotone quantifiers. If ir .fern is used to mean some but no1 mnn,y, then n.)m men is not monotone. If it is used to mean ul 1erl.s~r z j i ~ itl ~is mon 1'. It is likely that the man T reading is the only one that should be accountecl for by the semantics, conversational implicature explaining the illusion of a non-monotone

I

,

1.

II II

Generalized Quantifiers and Natural Language 99 I.'

reading. (Cf. Grice 1975; Horn 1976.) Similar remarks apply to sez'rrml, qlrile a Ji)i7 and tmo. The first thing that strikes one about the above list is that there are Car fewer rnon 1 NP's than rnon T. What decreasing ones there are have traditionally- been trcatcd ils negations of increasing quantifiers (no man of n man,j>m ?Tienof mnaql ?nenor perhaps of sezleral men). We can state a general relationship between rnon T and rnon 1 quantifiers, once we define Q and Q -.

-

DEFINITION:

Given a quantifier Q on E, define nenr quantifiers

-- -

Note that Q and Q are sieves just in case Q is a sieve and that if Q i s not a S~CVC, then Q = Q . Q corresponds to negating a sentence beginning with (c.g. no/ one nrnn r~zn).Q corresponds to negating the VP following Q (e.g. onc man didn't run). N

a

N

--

then Q and PROPOSlTlON. Negation rez~erstsmmol2otonicity. 1. I j - Q is won Q ure mon 1. 2. If Q is nzvn .1 then Q and Q arc Inon T. Further Q = Q = Q --, (See C9 in Appendix C.) It follows that we can think of any nlonotone decreasing quantifier as Q for some moil .T Q. This together with our discussion of weak determiners in section 4.6 allows us to consider the following potential language universal. N

N

-

US. -Monotonicity correspondence universal. There is a sirnple iClP n7lril.h expresses the rnon 1 yuant$fier Q af nlzd onZy $t/~ercis a sitnple N P with a nwak non-cardinal dtterwliner whic.h expresses the rnon T yuantzfier Q. This potential universal suggests the following relationship between English determiners. Corresponding mon no man/men neither man

some man (men) a man

fen- men

many men several men

nobody

TQ

some person etc.

This proposal would predict that no language would have basic determiners meaning no/ nrost, not eveq, or not the since most, czlery and the are strong. It would also predict that no language would have a basic determiner nleaning not (at lt.as/) tnlo since two is a cardinal determiner. Thus, such a proposal, if correct, puts real constraints on the set of basic determiners in a human language. Another significant aspect of this kind of

I!

100 Jon Barwise and Robin Cooper universal is that we can talk in semantic (i.e., model theorctic) tcsn~s.We do not havc to assumc, for example, that.f?n. is the same as not macy at any syntactic lcvcl.

4.8 A Monotonicity constraint on simple NP's

1 1

There do nor seem to be any simple NP's in English which could not conceivably be 1 analyzed a s monotone quantifiers or as a conjunction of monotone quantiticrs. Rr example, if we claim that a fin, ,nm has a non-monotone reading, wc a ) ~ ~say l d that it 1 expresses the same quantifier as the conjunction some men hui no1 n m q lmen. Sinlili~rly,I a non-monotone reading of t r ~ omen could be the same as at Icasl tn?o inell hut fit IIIOSI two 1 mrn. (Semantically, conjoining NP's is simply intersection of quantifiers and will he taken up in section 4.) These observations suggest the following candidate for a ! universal.

1

1

e U6. Monotonici ty constraint. The sin~pltl NP s! c! f' an.)! r~llfzrrnlI ( ~ n g u a ~il,rpnss nlonolone quant?fiers or rcoi~junrrionsof nzou~otonequtznt!/ier-.s. This proposed universal has the effect of ruling out many logically possible quantifiers as simple NP denotations. Examples arc the denotations of [ I N ~ ~ r enurnher ll ~!f??zcn, e.vnctly t h ~ e eor esuctly . .live men, all but olre mun. It seems unlikcly that any natural languagc would havc a basic cletern~incrmeaning u n ereu nlcrnber o/,. L J X ~ L ' I ~I ~h r e ror e.ractly.fi~)eor 011 hul one. If the monotonicity constraint is true, it seems to be more than an arbitrary restriction on the quantifiers found in human languages. Rather, it seems to be related to the way people understand quantified sentences. We take this up in the next section. Recall our discussion of strong and weak determines from section 4.6. D is positive strong just in case A E IIDII(A) is always true. It does not follow from this that D is monotone increasing. For example, we could define an artificial determiner D which was strong but not mon 1. by

11 D1((,4)= {X

Card(X - A) is finite and even).

A E IlDII(A) and if n , b ~ ; 4 , a # I ? then - { I , E ( 4 ) but (A {u)) $! 1D(1(,4). However, there do not seem to bc an>- such determiners that arise in natural language. This leads us to propose anothcr possible ~~niversal.

Then -

U7.

Strong determiner constraint. In n~riurizllr~ir'q-~~agn, po.sitiz:r slrovz'q ~/~~lermin~.rs ar-r morrnlonr ~zcreasing.Nrg~z~irt. slrong ~/eterrni?zersarc t~Jonolo?lcr/eec.r-casing.

This proposal makes some predictions as to rhc logical behavior of st]-ong determiners. T o see just what the>-are, we note the following proposition, PROPOSITION. [f D zs pusitit:e strong aird inonotonr inrreasirr'y then ,fill. un)t wiorlcl IZI = (E, (1 (1) and nn.y sets A, B in M :

B

E 11D11(-r2 n B).

Generalized Quantifiers and Natural Language 101 Ij'D is n e g ~ t i c estrong und nzonotane decr,ensirig then nw huz~cB 6 IIDII(A n B). (C;f'. C8 in Appendix C.) Thus, U7 predicts that if D is a (natural language) positive strong determiner then any sentence of the form

Dek

N

D a

A

shoulcl be judged valid (since its translation is true in M just in case

which is equivalent to

IIP'II

l l ~ l l ~ l l ~ ' nl l ll/j'll).)

This prediction is borne out by the following examplcs.

men that love Mary, love Mary. T h e three

("I

{ :::'

}man that loves Mary, loves Mary.

T h e corresponding prediction for negative strong determiners is that such scntcnccs are judged contradictory, as in (34). (34) Neither man that loves Mary, loves Mar?-. When one carries out the same test with weak determiners the results ilre odd and certainly not universally valid as the examples in (35) and (36) show. (a) No nlan (c) (At least) three inen (d) Exactly threc lncn

that love(s) Mary, love(s) Mary.

102 Jon Barwise and Robin Cooper (36)

{

(a) Many men (b) Few men (c) A fccw men

}

that love Mary, love Mary.

We judge the examples in (35) to be logicall? equivalent to those in (37) but it is not I clear that the same holds fbr (36) and (38).

(37)

( (a) No

man ome man (c) At least three men ( (d) Exactly three men

I

lovc(s) Mary.

)

Accepting these equivalences amounts to asserting that weak determiners satisfy the intefiection con~izr'ilion.

i I

DEFINITION. = (E,

)I 11)

D satisfies the intevse~.tion~~on(iition if for all models

and all X,-,.l

E,

x E 11011(.4) iff x E 11011(--l n s). do i n ~ satisfit i ti7c inlrnecliorr l.orrditior?. (Cf (26.) PROPOSITION. Strotzg ck'tet.m~~zt>rs

T h e second author is inclined to think that weak determiners rrll satisfy the intersection condition but it violates the first author's intuitions for mar^)^ a11d.fl.n). It would say, for example, that 117arl.)/could not mean something approximately like the following in a model h l containing one thousand men: (

4

)={

E

1Y

4

1

1A

and I X n -4

I>

30)

Here the nurnber that counts as "many" gets smaller the sinaller / I is, but nothing smallcr than 30 ever counting as "man!;" men. If rlznny is interpreted in this way in somc model, and if 35 men love Mary ill the model, then Mi~n,yinen l o z ~M[zry is hlse, but il/l[rrr,lmer~[hilt love A.1211:)1, 107'~MLII;)~ is true. T h e issue hangs on the one's interpretation of the fixed context constraint in rclation to relative clausc constructions. It can probably only bc resol\.ed by working out a (Kamp-like?) theory of the vagueness to superimpose on o u r tl-catmcnt, and seeing \vhich interpretation of the constraint provides the smoother theory. (This samc issue coines up in deciding whether ~ r ~ c l nisy persistent, as defined in thc next section.) Sentences involving determiners that satisfi- the intersection conditioil can be expressed, up to logical equivalence, in a number of waj-s, due to the follo~vingfacts.

Generalized Quantifiers and Natural Language 103

(CE Appendix C, C4 and C5 for proofs.) The right hand of the first biconditional corresponds to the semantics for the there is/are sentences, as in (37'). T h e right hand side of (2) corresponds to switching the verb and noun as in (37").

( (a) no man Thcre is/are

(37")

some man (c) at lcast thrce mcn ( (d) exactly three men

I

that love(s) ~Marg

)

(a) No one that lovcs Mary is a man. (b) Sorneonc that loves Mary (c) At lcast thrce pcople that love Mary are men. (d) Exactly three people that love h4ary

The proposition predicts that corresponding sentences in (37), (37') and (37") i~rc equivalent. And, as above, the equivalence of the sentences in (38), (38') and (38") is much less clear.

(38')

(38'')

There are

{

(a) manj- men (b) few men (c) a few- men

}

that love Mary.

i

(a) Man]- people that love Mary (b) Feu. people that love NIary (c) R few people that love h4al.y

4.9 Processing quantified statements _An objection that could be leveled against Montague's treatment of NP's is that it would seem to make cl~eckingthe truth of a simple sentence like John rlnns wcll nigh impossible. For, the argument might go, one tvould first have to "calculate" thc denotation of namely, the family of all sets X to which John belongs, and then see if the set of runners is one of these sets. 13ut this clcarly corresponds in no way to the reasoning process actual1~-used by a native speaker of English. Using the monotonicity constraint, we wish to show that something very much like an intuitive checking procedure is always possible for simple NP's. The procedure rests on thc notion of mitness sets. DEFINITION. A witness sct for a quantifier such that m E D(24).

D (A) living on ,/i is any subset a, of '4

EXAMPLES. T h e only ivitncss set for 113ulznl is {John). A witness set for 1111 woman1 is any nonempty set of women. R witness set for Ili~zosl)~,orlz~i?(( is any set

104 Jon Barwise and Robin Cooper of women which contains most women. A witness set for I[/ir~n?omen)lconsists of any set of women which contains only few women. A witness sct for Il(c.zac.tl)l)ljoo a~o~nerall is any set of exactly tw-o women. PROPOSITION.

Let m range over a~ktrresssets /or the qutzntqier

I f D(A) is mon T then .fhr nny X, X E D(14) ?ff3w[m C .X] (ii) I f L44) is mon the??- f i r ntz.), iY, E D(A) $f ~ P [ ( xn '1)

lizing o ~ A. t

(i)

x

cII~.

(Cf. C11 in Appendix C for the proof.) We can paraphrase this Proposition as follows: To e z ~ a l l ~ nX f c E D(.4) do tlw ~7)/lomi~1g:

1 Take some subset r1) of A which you ltnow to be in D (A). 2 (i) For mon D(/4), check rv C X. (ii) For mon J, D(A), check (X n A) 2 113. 3 If there is such a m, the sentence is true. Otherwise it is false. These procedures are not totally unlike some computational models for the verification of quantified sentences which have been suggested in the psychological literature. (See Clark 1976.) We imagine it might bc possible to design experiments which check the predic~ionsof a psychological theory based on witness sets. For example, we predict that response latencies for verification tasks involving decreasing cluantitiers would be somewhat greater than for increasing quantifiers, and that fix the nun-monotonc it would be still greater. These predictions are based on the complexity of the checking procedure we have suggested above.

EXAMPLE. Imagine a yard full of animals, including some dogs. 1,et us imagine a dog Fido that looks like a spaniel, but we're not sure if it is a spaniel. Iinaginc deciding which of the followiilg are true. (a)

(b) (c) (d) (c)

Fido is in the yard. Some spaniels are in the yard. No spaniel is in the yard. hxactly one spaniels is in the yard. An even number of spaniels are in the yard.

In all of these the set X = {,)/I.)1 is in the yard) is the set .Ydenoted by the VP. For (a), the only witness set is {Fido). M7e check to scc if Fido E AY.For (b), we need to find some non-empty set n? of spaniels, 1~ C_ _X.For (c), the only \1-itncss set is 0.We must see if A- n llspanielsll = 4.This will causc us no problems if there is a clear cut spanicl in the yard, for then clearly S n I)spanicls)l# (b, so (c) is fillst w~l~ctl~cr Fido is in the vard or not. If Fido E X but no clcar cut spaniel is in .Y then ivc won't be able to compute the truth or falsity of (c) without deciding wl~etherFiclo is a spaniel. For (d),

Generalized Quantifiers and Natural Language

105

we must do two things to scc that it is true; find some spailiel in A- and show tliat tllerc is at most one spaniel in X. This corresponds to breaking down e.t.ar*t/)~ O I I P sp~~?/icI into sorrzc sprtniel and nr tnosr otle spnrri~~l. For (e), we niust decide whether X n I(spaniclsl( contains an even number of things. We will not be able to do this without deciding whether Fido is a spaniel. There is another distinction, related to monotonicity, that seems to affect processing of quantifiers, and is bound to interact with processing requirements by nionotonieity.

DEFINITION. A determiner D is ~ C V . S Z S ~ P ifI I ~ for all M = ( E , 11 I ) , and all A i B i E, if E IlDll(A) thenX E IlDll(B). (On the other hand, D is crtzti-pcr:vis/e,~/ if nl 2 B C E and X E IlDll(B) implies X E 11011(.4).) T h e idea here is tliat if D is persistent then once you see that X E 11D11(-4)then you know that X E (IDII(B) for any set B that contains '4. For example, if B = {x I x is a man that left the party before IOPM) and .?l = {x I x a man that left the party beforc 9PM) then A C_ B so that for persistent determines D. (39) D men

that

lcfi the p~krtyhefire 9Ph4 n m t hotlae

will imply

(10)

D met? that Iefl /he port]/ befire IOPM went konze.

Logical (and mathematical) examples of persistent determillers are solize, ai /(>as/I?, (infilile/y ITULI!~~, uizcoun/nh()l IT/LE~)I). Other determiners that seem to function as persistent dctermiliers are .sever[~l, and rnnr/j/ (??). For anti-persistent deterillincrs D,the inlplication goes the other way, from (40) to (39). Thesc include PZIPP)I, IIU, f r 1 7 7 (?), C L ~ most n, .fini/t>lyununy. Other determiners are neither persistent nor antipersistent. A glance at the table in appendix 2) table suggests another proposition for consideration as a univessal. (See also C7 in Appendix C.)

U8. Persistent d e t e r m i n e r universal. Ez!ery pvv.si.steitt ~/tternzine~c!J' Izzinlcrrl Inrrgullge i s lnon

und m m k .

Since it is not difficult to construct artificial determiners which Fail U8, (ct: Appendix C ) , this would, if truc, provide another constraint on the class of human languages among the class of all possible languages. In terms of witness sets, pel-sistence works as follows. If D is persistent and if n? is a witness set for D (A) then 1~ will be a witness set for any B that contains .-l (,-I C B). It seems clear that between monotone quantifiers D l , D2, which arc otherwise comparable, if D l is persistent and D2 isn't, then Dl should be easier to process, especially when the universe is too large to pcrceii-e all a t once, since a \$-itnussset for D l ( @ may bc able to be found on the basis of some manageable A C B. Continuing the above examples, it should be easier to verify (f) than (g), since any witness set for (g) must contain most dogs in the yard, whereas for (f) it might suffice to have a witness sct for .scz:~r~~l dogs in /he .vnrd t~/~il-lr are closc enough / o see.

I

106 Jon Barwise and Robin Cooper (f) Several dogs in the yard are spaniels. (g) Most dogs in the yard are spaniels.

Persistent determiners were introduced in Barwise (1978). Ladusaw (1979) has put them to excellent use in his discussions of polarity items. They turn out to be important for the logic of perception (Barwise 1980).

4.10 Monotonicity and NP-conjunction An advantage of treating natural language NP's as generalized quantifiers is that we can treat NP-conjunction (instances of N P and NP, N P or NP, NP hut NP, etc.) directly. NPlorNP2 denotes the intersection of IJNP111 and IJNP211, NPI or NP2 denotes the union of the two quantifiers. We may similarly extend the logic l,(Ci(B to L(GQ,) by adding a formation rule R8.

R8. If Q 1 nlzd Q 2 are qutrntjfifiersso are

(a a ) , (av a ) . A

T h e corresponding semantic rule is S13:

This logic allows us to represent new quantifiers but provides no real strengthening of the logic, since

are logically valid. That is, we cannot express any sentences that were not already represented, up to logicnl equivalence. Not all instances of NP-conjunction are acceptable in English. In general, it seems to be difficult to use a.nd or or between two NP's if they represent quantifiers of different monotonicity . Examples are given in (32). (32)

+

(a) inc~eczsing increasing: a man and thrcc women, several men and a few women. the professor or some student, most men and any woman (could lift this piano) (b) de~-reasi?zg decrecrsing: no man and fcw wonien (could lift this piano), no violas or few violins (are playing in tune) (c) mixed: *John and no woman, *few women and a few mcn (could lift this piano), *two violas ancl few violins (are playing in tune).'"

+

T h e unacceptability of the mixed conjunctions is not simply due to the peculiarity of the message \vliich would be expressed by sentences containing them, l'here are acceptable sentential conjunctions which would express the samc proposition.

Generalized Quantifiers and Natural Language 107

(33) (a) John was invited and no woman was, so he went home alonc again. *John and no woman was invited, so he went home alone again. (b) Few mathematicians have worked on natural language conjunction and a few linguists have - so I don't think you have the right to illakc thcse unfounded statements. *Few mathematicians and a few linguists have worked on natural language conjunction .. . (c) Whcn two violas are playing in tune and few violins are, Berlioz begins to sound like Penderecki. riVhen two violas and few violins are playing in tune, . . . This restriction on NP-conjunction could be related to the preservation of properties of monotonicity. T h e conjunction or disjunction of two increasing quantifiers will be another increasing quantifier and sin~ilarlyfor the decreasing quantifiers. T h c conjunction or disjunction of an increasing and decreasing quantifier will normally not bc a monotone quantifier. For example, the putative conjunction John lznd no wornan would have the denotation represented in (34).

(.34)

{X I John E

X and 171 n {woman)

=

4).

This would for example contain the set {John) but not all of its superscts or subsets. Not all instances of NP-conjunction demand monotonicit!- in the way we have suggested. It is possible to conjoin an increasing and a decreasing quantifier wit11 hut as illustrated in (35).

(35) (a) John but no woman was invited. (b) Few mathen~aticiansbut

{

linguists have worked on natural lin-

guage conjunction. (c) T w o violas but

{ }

violins ase playing in tune.

In fact, in order to use hut in this way it seems necessary or at least prefcrable to mix increasing and decreasing quantifiers. Comparc the sentences in (36). (36) (a) (b)

(c)

*John but a woman

{ze }

invited.

*Few mathematicians but no linguists have worked on natural language conjunction. ?Two violas but three violins arc playing in tune.

We assume that the interpretation of hut is the samc as that of und for the purposc of truth conditional semantics. However, there are important ways in which it behaves differently from and. Conjunction with and can be repeated indefinitely many times. This is not possible with But no matter how one mixes thc quantifiers.

108 Jon Barwise and Robin Cooper (35) (a) John and a woman and three children were invited. (b) *John but no woman but three children were invited. (c) *Few inathematicians but many linguists but no physicists have workcd on natural language conjunction. This lack of iteration might be related to the fact that monotonicity is not guaranteed for a mixed conjunction and hence that the verification procedure we have discussed might apply separately to each conjunct. It is interesting to note that similar pcculiarities are true of more c o i ~ ~ p l econjunctions s that might be considered as mixed

(38) (a) John and not Mary is invited to the party.'' nobody else (b) John and can kcep the party golng. no other man As with but these kinds of NP's cannot be further conjoined with other NY's. (39) (a) *John and not Mary and not Helen is invited to thc party. (b) *John and no other man and Helen can keep the party going. An extension of fragment 1 to include the basic cases of conjunction of simple NP's can be found in Appendix A.

4.1 1 Negation of noun phrases and duals Certain NP's in Lnglish may be prcceeded by not when they occur in suhject position, while others can't. Here are some data: Not every inan left. Not all men lcft. Not a (single) man left. Not one man left. Not many men left. *Not each man left. *Not some man left. *Not John left. *Not the man left. (*) Not fcw men lcft. *Not no man left. ?*Not most men left. Notice that this distl-ibution cannot be explained pure/), in tcrins of the semantics of quantifiers, as comparison of (40a, h) with (412) and (40c) with (51b) shows. (One might try to explain the unacceptability of ( I l a , b) as having sonlething to do with the preference of so17~cand ecrr.h for wide scope reading.) Therc are some semantic gencrilizations to be captured, ho\vcver.

Generalized Quantifiers and Natural Language 109 The tirst observation is that only mon T quantifiers can be negated in this way. Recall our universal that to every mon simple NP a denoting a quantifier Q there corresponds a nlon 7 simple NP a' denoting the mon 'f Q. Thus, to negate [ [ C I ] ~ ~ [ P ]one \ ' ~could ]~ simply use [[a1][P]Js.For example, instead of saying ( 4 1 ~ or ) (4lf) one could say rna91.y men l ~ f or j some ??le?zIrSf. T o see what is going on in (41c, d), we use the model-theol-etic notion of the dual of a quantifier.

-

-

DEFINITION. T h e ' h a / of a quantifier Q on E is the quantifier =- (Q ) = ( Q ) -. If Q = = {X EI(E - X) @ Q, i.e., called self-clual.

0

0

N

0 defined by 0 then Q is

EXAMPLES. 'The dual of' llsome man11 is llevery man11 and vice versa. On a finite set A 2 E of odd cardinality, {X C_ EIX contains more than half A ) is self-dual. For any iz E E, {_X C Elu E X ) is self-dual. Hence / the I rlI is always self-dual, when

0

-

defined For any the dual of is the original Q. Also, if Q is nlon " Q =- Q and two minuses niake a plus, so to speak). T h e following is clearly valid:

1. so is 0(since

As special cases of this we have the usual

-

-

vxcp 3X(P

-

3s

t .

VX

2 . r

cp (P.

If Q i s self-dual, then the above simplifies to

That is, wc can push negations back and forth across self-dual quantifiers. Hence there is no need to use any syntactic construction to show that negation has wide scope ovcs quantification when the quantifier is self-dual. Thcse observations lead us to proposc the following as a candidate for a languagc

U9.

Constraint on negating seIf-dual and mon 1 quantifiers. I f a l(lngr~iige has a syntnclic constrmc~ior~ll~hosrsemnntic Junc.tion is to negate a quantz/iL.v, t/z~'n this c.onstruction mill nor he usell i ~ i l h-W'se.vpressing mon 1 or s e ~ / ~ l l u ayunrzli/ Jieev.

Of the unacceptable determiners in (41), aside from (a), (h) \vhicli we haw alrclldy discussed, this constraint leaves only (g) unexplained:

(41) (g) "Not most men left.

110 Jon Barwise and Robin Cooper T h e odd thing herc is that there just isn't any way to express the intended sentence without using sentencc negation:

1, ~

1'

j

(42) It is not true that most men left. (If most meant cxaetly the same as more tllarl hid[ then (42) could be paraphrased by

(43) At least half the Inen didn't leave since the dual of ) ( More than half the q'sll is 1) ,4t least half the rl'sll.) ,If 1 Dl is a determiner interpretation, nre can define Ilbll(-d) = IlDll (-4). For example, IlsoXlell = Ileveryl and IIe~G-yll= IIsomell. We do not know exactly why the following should be true but we can find no counter-examples to it, so propose it for consideration:

UlO. Dual quantifier universal. I f n ~iatural/~inguwgrklrs u basil. i/~.teriiliircrf i r zacl~ f D liirr/ D rlwn ~ b r s 61t re .st~iz(lnti~d/y ~yuiz~~i/(v?i' 111 ' ?onlr9' NIU/ " ~ > z w ] r 'I,

,4n apparent exception to U10 as stated is the pair ~ h rI , the I. In other words, the 1 is self-dual when defined. But then this is not rcall!. an exception, since whcil fhc 1is defined it is semantically equivalent to both soine and cz'r'i-y. In connection with UlO, we ~vouldpoint out the follou-ing simple fi~ct.It may have something to do with tlic reason U10 is true (if' it is true). PROPOSITION. If Q is mvnoLvar inrreosiq thcn Q i [ g ( s ) ] n o i l ) l ( , r ) 1 implies 3.ifq4.v) r. $(s)l.(Cf. C10 in Appendix C.) An example of a pair of dual determiners from mathematics is "more than half" and "at least half". Some people consider "most" as synonymous with "more than half" but therc is no l~asicdeterminer in English synonymous with "at least 1ia.lf". Another dual pair is "more than 75%". and "at least 2jC?4o". U10 would predict that no hurnan language would have basic determiners for each eleinent of such pairs. T h c proposed universal also predicts that of the sentences below, only (#a, b) could be paraphrased as D Inen /
1

which amounts to using "quite a few" as thc dual of "many". However, faced with (46) they are not usually willing to paraphrase it by (47), which suggests that they arc not 1 consistent in treating "quite a few'' and "ma11!7" as dual.

Generalized Quantifiers and Natural Language 111

(46) Not many men left. (47) Quite a few men didn't leave. Also, faced with the fidlowing argument, which would be valid if "many" and "quite a fen-" w-ere dual, (by the above proposition) they judge it highly dubious. Many men voted for Carter. Quite a few inen voted for Ford. Therefore, some mail voted for both Ford and Carter Thus, we see that "quite a few" and ''man>" arc not consistently used as duals of each othcr. And, even if they were, it is unlikely that "quite a few" should be considered a basic determiner element.

5 Conclusion In this paper we have focused attention on the semantics of English determiners and r l o u ~phrases, ~ defined and illustrated a numbcr of semantic propertics of them and proposed a number of possible universals. Ultimately, however, we are less concerned with the fate of these proposals, or eve11 of the details of our senlantic treatment of nounphrases and determiners, than with illustrating sonlc genesal points about the analysis of natural language - points not always fully appreciated by linguists or logicians.

5.1 Semantics is part of a linguistic theory Linguists often feel that a model-theoretic semantics is an appendage to a linguistic theory and that it will not further the linguist's aim of characterizing thc class of possible h u n ~ a nlanguages. Sucll a view suggests that the relationship between Ianguages and models of the world has more to do with the world than with thc structure of language. It might even claim that facts elucidated by a model-theoretic scinantics arc logically necessary facts and thus cannot possibly serve to separate the class of natural languages from the class of logically possible languages. We believe that the t-csults of section 4 show some ways in which this is mistaken. 'The confirnlation of any of the universals presented there (or more refined versions of them) would invalidate such a view. None of the proposed universals is logically necessary, and scveral of them effect sharp reductions in the class of possible human languages. Fut.thermore, it seems that such universals could be related to a psychological theory of language. T h e drastic reduction in the available interpretations of natural language determiners suggcsted by our proposed universals hints at a theory of acquisition following Chomsky, in that children faced with the task of learning language need only consider a restricted set of possiblc determiner interpretations. Another relationship to a psychological theory is discussed in section 4.9, where it is suggested that the nature of' determiner interpretations gilarantces the availability of certain verification procedures.

112 Jon Barwise and Robin Cooper ~-~ -

We should emphasize that the psychological considerations have emerged from examination of certain fornlal set-theoretic propcrtios of the interpretations of natural language determiners. We feel that this illustrates the possibility of basing psychological theories on research in model-theoretic semantics. It is a mistakc to reject such research as irrelevant to psychological theories just because its relationship to a thcory of language learning or use is not apparent on the surfiacc. For example, it has been suggested that taking NP-denotations to be families of sets runs counter to any reasonable psyohological theory. Wc believe that our further investigations of the structure of these families suggests otherwise. Our discussion has concentrated on purely semantic distinctions among vai.ious kinds of dctcrminers and noun-phrases. While these semantic distinctions are often reflected in thc syntax of sentences, there are no syntactic correlates to these distinctions in the structure of the noun phrases thcmselves. For example, the semantic u ~ a k / ,, strong distinction is closely reflcctcd in the acceptability of the "Thcrc is construction, but it is not reflected in the syntactic structure of the associated NP's (6 many men", "most men", etc. Similarly, the fact that an NP corresponds to a rnonotonc increasing or decreasing quantifier is not reflected in rhe s j ntactic structure of "many men" or "few men" (although it is rcflccted in the arbitrar~choice of basic Icxical i tems). T h e importance of this type of semantic analysis for a linguistic theory has rarcly been emphasized - even in the literaturo which has adopted a n~odcl-theoreticapproach r has sl~owil toward semantics, such as hlontague Grammar. Previous work, f i ~ example, that the use of model-theoretic semantics allows us to capture semantic relationshil)~ between sentences bithout making the relationships explicit in the syntax. It is clcu that, in the long run, ui~lerstandingthe relationship between syntax and sclnantics will be at least as important as, say, that b e t w e n syntax and phonology. Howevel-, important as this is, we believe it is only a part of the role that semantics can play in linguistic theory, and we suggest that the study of semantics in its own right will be as important as the study of phonology or syntax. -

5.2 Semantic intuitions AThileit is seldom made explicit, it is sometimes assumed that thcre is some system of that an inf'crencc in natural axioms and rules of logic engraved on stone tablets language is valid only if it can be formalized by means of these axioms and rules. In actuality, thc situation is quite the reverse. T h e nati\.e spcalicr's judgcinents as to whether a certain inference is correct, whether the truth of'the hypothesis irnplics the truth of the conclusion, is the primary evidence for a semantic theory in just the way that gran~maticalityjudgcments are used as primary evidence for a syntactic theory. We have used such judgeri~entsconcerning inference in order to detcrniinc many aspects of the model-theoretic treatment we have provided. In particular, all o E the properties of determiners and NP's w-e have used rest on such evidence. Just as a syntactic theory must draw the boundaries around grainmaticality in some way, so too with semantic thcory and infcrcncc. For example, just which inferences -

Generalized Quantifiers and Natural Language 113 involving "most" count as "logical" depends just whcrc one draws the lines. 'l'hc raw data of speaker judgements may be represented diflcrent1)- within different theories. Nevertheless, our clues to the meaning of the string of souilds rcprescntccl by "n~ost", and the inferential uses to which it may be put, are not determined by ally logic \\;sit in stone, but come only from the intuitions of native speakers. We have built one semantic intuition about determiners (the one captured by the determiner universal) directly into the semantics. Thus, "hilost men run" is /ugj~.nllJ~ equizllllent to "Most men are ~ n c n and run" in our semantics. And, of course, just as with grammaticalit? judgements, there are some very clear cases and some for which it is difficult to get a clefinitivc answer - as we saw at the end of 4.8.

5.3 The Role of translation and complexity of fragments Like htontague in PTQ, wc have used translation into a logic to induce a scina~ltieson our formalization of n fragment of English. It has been pointed out m a n j times in the literature, initially by Montague himself, that the intermediate language is a convenience, not a necessary stage in the interpretation of English. It \vould be easy enougli to define the semantics directly on the syntactic component of the English fragment. In studying Montague's fragment in P T Q and subsequent work in thc same tradition, however, it is easy to get the in~pressionthat the model-theory per sc is essentially trivial. It often appears k o m this work thal the translation procedure must be more contentful since it is so complex. 'There are taro reasons for this imprehsion. A theory may be trivial because it says so little that it is easy to understand it completely, or because it is so complicated that there is little known to say about it. T h e model theory that goes along with Montague's logic I L is trivial for the second reason. Montague packed so much into the semantics of his logic that it is extrcmcly difficult to discover any very general facts about it. His reasons for doing this were, presumably, two-fold. In the first place, he wanted his logic to be sufficiently expressive to use ils a tool for showing that very large portions of English could be given a model-theoretic semiultics. Secondlj-, Montague had an unflinching Platonistic attitude toward set tlieorj- which is deeply imprinted on his logic. The reason Montague's translation procedure is more complex than necessary, is that the syntax of his logic 11, very directly reflects the semantic intcryrctation, Once one has mastered his symbolism, the model-theoretic interpretation can be read directly off the formulas of the logic in a straightforward manner. T h e situation can be visualized as a scale of 1 to 10, with the syntax of Montague's English fragment at 1, the model-theory at 10. One could imagine interpolating a formal language anywhere in between. Montaguc chose to put his I L at about 9.5. His decision has allowed researcl~ersextending his work to concentrate their efforts on the translation relation of larger English fragments into 11,, leaving the model theory largely untouched. This has been a fiuitful approach, but it has had the unfortunate sideeffect of diverting attention away from the properties of the model theory. A major lesson learned fro111over 60 years of work in model-theory is that there are great insights to be gained by (temporarily) limiting the cspressive power of your

114 Jon Barwise and Robin Cooper formal language (as with first-order logic) so that one has tools for studying the resultant models. By making the language less expressive one obtains a non-trivial model theory with applications to those areas which happen to lie within the rcalm of the logic. Now that we have learned that it is possible to give a model-theoretic semantics for large portions of English, the timc seems ripe to apply this lesson as a research strategy for natural language semantics. T h e strategy has guided us in the work reported 11cre. We havc deliberately restricted ourselves to a very si~lzplefragment of extensional English, concentrating on determiners and NP's. We have set up our logic so that the translation procedure is essentially trivial, concentrating our efforts on genuinely scmantic issucs. The and noun-phrascs potential contributions to the linguistic theory of dcterrnine~.~ suggested in section 4 result from this attention to model-theoretic semantics. Thcy would certainly not have been apparent had we studied translation - eithcr ours or Montague7s. I

5.4 Logic as a part of linguistics

/

If our claims in 5.1 and 5.2 are correct, then the traditional logical notions of i validi~yand inference are a part of linguistics a conclusion not likely to comfort , many logicians or linguists. T h e phenomenal succcss of first-order logic within mathematics has obscured, indeed, nearly severed, its tics with its origins in language. Except for tense and modal logic, rescarch in model theory in thc past twentv five years has taken irs problelns almost entirely from pure mathematics, becoming ever more specialized and renlotc from languagc. 1;vcn thc work in generalized quantifiers mentioned in the introduction is devoted almost cxclusively to mathematical quantifiers, going out of its way to avoid mentioning possible applications to natural language. This same success of first-ordes logic within mathematics also fostered the mistaken idca, discussed in 5.2, that the "laws of logic" are autonomous, perhaps part of mathematics, but not a property of language and language use. It is here that kIontaguc nlade his biggest contribution. I o most logicians (like thc first author) trained in model-theoretic semantics, niltural language was an aiiathema, impossibly vague and incoherent. 1'0 us, thr re\-01utio1iai.yidca in Montilgue'~ paper P T Q ( a n d earlier papers) is the claim that natural languagc is not in~possibly incoherent, as his teacher Tarski had lcd us to hclicvc, but that large portions of its semantics can be treated by combining known tools from logic, tools like functions of finite type, the A-calculus, generalized quantifiers, tensc ancl modal logic, and all the rcst. R~fontaguehad a certain job that hc wanted to do ancl used whatever tools he had at hand to do it. If the product he built looks a bit libc a Rube Goldberg machine, well, at least it works pretty well. It proved its point and should lead others to explore natural language semantics further, while at the same timc paying rigorous attention to its syntax. It is an exciting possibility, onc that could lead ro a revitalization of model theory and open up ncw domains for thc construction of linguistic theories. r-

7

Generalized Quantifiers and Natural Language 115

Appendix A. Additions to fragment 1 Fragment 2 incorporates ilzeru sentences and definite determiners following the iclcas in section

4.6. /!/'~h~re-serzt~r.rlc~>s. T h e s!-ntactic analysis we present is culled from FRAGMENT 2. I~ll.o~-pcr~atiorz Gazdar (19'79)' and Jcnkins (1975). .Add to the rules of Fragment 1:

SD1.1

SD 2.1 SD3.1

NP [there] VP [there] S-NP [thcrel

+

there

+

be NP

STP [thcre]

T o the translation rulcs we add: translates as thing

NP [therc] there VP [thcrel

translates as j'

[il\~

14;1\T

[there]

[therc]

It is important for our seman~icanalysis that only an NP follon- bc in the /here-VI' and it is llopcd that this fcature of thc analysis could be niaint.ainecl in larger fragments which includc si~chscntc~lccs as 1hei.e u o I?JLL?It n I ~ ~PL I Y I / ~ t?/ i~t ,> ~~lP-r,fi,~li, .(~ C J O P / CS ~ L I T I ( / I ~ ?/?)I R ~ / ~ ~ . / ; , L ~ N ~ The N I I I . plausibility of such an anal~sisis argued for n-ith a lilrgc nulnber of csnmples by Jenkins (1975). Even Jenkins ho\j,c\cr clocs not consider the unclerlincd strings in (1) to be NP's. (1) (a) Thcre are Inlo pcoplt ril.k/drunk (b) There are,fire i c~okieslt;/i. Indeed, the!. do not occur in all the pti~ccs~ l i e r cone ~ ~ ~ o ucxpect lcl NP's to be, as shown in the ungi.arnmilticat sentences in (2).

(2) (a) *Two pcople sick/drunh entcrecl the room (b) ?*The cookies left were mould\;. Howcvcr. they do show up in gran~rnalicalsentences after analysis 1%-ouldshow thcm to be NP's.

where tlic on14 rcasonablc

116 Jon Barwise and Robin Cooper (3) (a) “bong the people sicli/drunli, were 13ob and his wife (b) Of tlie cookies left, two were mouldy. While we do not u~iderstandtlic distribution of thcsc strings it clocs not sccni unlibcly that thcj arc NP's which are somehow restricted ill their distribution to ~lr~~rc-sen tcnces and certain othcr contexts. 011 illr r.onIcsr(often rcfcrrcd to in the Other problematic examples i~ivolvcph~ascssuch as s~~rlr~lirig transformational litcraturc as reduced relative clauses) when they occur aftcs a full rclativc clause. It is not normally possible to have such a scqucncc occurring within a single NP. This is shown b j the ungrammatical example (4a) which is contrasted with tlic grammatical example (4b) with the same string in a tlrere-sentence. -I [n~ho~ I I O I P S.you] Is/ri~irIin~o r ) 111~ c~onic*r.l~+-l;avcd to nic as I went by in the hus. (4) (a) *,girl (b) There is a girl who knows !-ou standiiig on the corner.

,iZ possible

w7i1y out of this problem is to say that s~trlrilrrrgorr ilrr r.m,aur is a scntcntial nioclificr i ~ n d hence not uithin the \'P at all. This is supported b!. the fact that sl~r~r~lit~g or/ 111(.(~1171~1.in also occur at the beginning of the sentence: Stul~(/i~rg on tlre i.or~rrrllrerc is tr grr1 i i 4 o k11/~n7s yolr. Iio\vcecr, not all such "reduced relati\ cs" may be explained away in this fi~sl~ion. (5a) suggcsts {hat 11 girl w h o Avlnr~?v,)!a~~ artr.r-~,.
( 5 ) ( I ) *I rnet a girl who knows you intcrcstcd in this proble~ii. (b) There is a girl who knows you intcrcstcd in this problem. ( c ) *Interested in this problem, there is a girl who lino\vs you. It is not clear, howc\er, whether there Ilia!. not bc otlicr special NP positions in which these strings may nevertheless occur as NP's. Consider: O[tkt, girls m)lio o ~ zA ~ r r r ~ Ii I~ Il I P ~ ~ ~ .111 < I /lrr.i C ~ proh/t111,.I/Ior:yIS /!)I /i/r //if ~ r i o ~likely */ 10 ./i~id (1 ,co/uttott. Finall!-, arc would expect that in a larger fiagment tlie feature /IIL'~LJ on the l'-P would percolate up to higher VP-nodes in thc rnaniicr suggcstcd by Gazdar (I 970) in order to account for sentences s ~ ~ as ch ~ h r r rupperrr. l o he./ji~~ m ~ i~r ~ t h p(/rk. ~

I?uorporntion of Det uf'NP Acld to the lexicon:

Xcld to the s!;ntactic rules:

SD1.2 If is a N P constructed with a ~lcfinitcdctermincr, then [of (1, is an SI>. SD1.3 N P d D c t N I"( I lof] [ofj In order to translate this cxtendetl fragment we must sliglitl~enlarge the logic (thcrcby creating L(GQ),) by adding the Sollowing syntactic and semantic 1.~11~s: Syntax: if Q i s a quantifier, then Semantics: Ir;Qll is n 11-(211.

AQ

is a set term.

Generalized Quantifiers and Natural Language 117 l'hc trailslatio~lof the syntactically special determiners in tlie lexicon is thc same as their traoslation in Fragment 1. ranslates

as A

translates as

<'


~ / ~to( .the I~O syntax ? I . oC fragment 1: FRAGMENT 3. X P - ~ ~ I ~ ! ~ I.4dd ,%-P-~.o,!/~mnc.tio~~ ilnd the ( a ) If x, {j are of the form [[lN1>and [jlNPwhere neither or cj is oF the form [qINr,bur I~%jNI,, denotations of r , fi induced by translation arc mo~iotonequantifiax of the same kind, then [ x ant1 PIYP and [ m or- flJNP arc. mcinbess US SD. (b) If x, 11 arc of the form specified and denote monotone quantifiers of dil'fcrcnt kinds, then [ a hut IjINPis ;1 rncn~bcro~ SD". 5

<

Wc specif) that tlie translation is into the bnguage L((;Q)2 defined at thc beginning 01'4.10 and acld the following translation rule:

translates as j'

translates as

A

5'

6' v 5'

Appendix B: Semantic postulates for few, most and many In the course of'section 4 wc discussed 1-ariousscniniitic properties of the non-logical clctermincrsjh, rnos/ and I I I N I I ) I , propcrtics which are not insured hy our definition oF modcl in section 1 . j . To gual-antee that thc fornial scniantics rellects the intuitions from English we must restrict the class of ;ill models to those which satisSy thesc properties. Wc refer to tlie Sornial versions :IS seman~ic

118 Jon Barwise and Robin Cooper postulatcs. Thep are not quite of the same character as thc restrictions on moclcls introduced by M o n t a ~ u ein P T Q for thcy arc not expressed as sentences in our lornlal language, but rather as scttheoretic condi~ionson the models thcrnselves. The extent to which such semantic postulates can bc capti~rcdsyntacticall! ma! be discussed in a latcr paper. Let $1 = { 1 I ) . From section 4.2 we ha\.e the follon-ing:

SP1. Most i.~n po.dizt~s/rorzg Iletzrrrzincpr. That is, for every A

C E, A E

llrnostl(-d).

From 4.3 n-c have

SP2. Most und many ore nzon [',f'cw is mot/

L. That is, for all X, Y,.4 c E:

.Yc L', X t Ilmost ll(A) implies L' E Iinust 1(?4) -YC I',A- E (Imanyll(.4) implies Y E Ilmany ll(.A) Y & X,.Y E Ilfew1(-,4) implies Y E Ifewl/(.-i).

If most people do X and most people do Y then someone does hoth. This I I I L I C ~S C C I ~ Sclcar from the meaning of "most". JVc can express this by:

If one wants to demand that ,/in)mzn bc ccluivalcnt to no/ nlnqt r~ren(or postulate one of:

7101

stzvrrll m t i i ) onc cdn

IFewll =- (manyll. 1.e. for ail!- -4, SP4. (op~~onlrl): l(17ewll(-A)=

(1\111anyIl(-i))

=

or :( 1 fewI/(.i)=

{-Ji C E ( X # Imanyl1(..4)) Iscv~raIl((~-l).

T h e persisteilcc of nrurzy and anti-persistcncc of/>n)discussed in 4.7 nas less clcir than most of'the above. many SPS. (oplio?ln/):few is pti.sasr~t~t,

1s

~1~7tipcrxrst~~11t. That is for a11 -4

c B C E,

SP6. I / ' X E llmany1((.4)thcn X f @. This guarmtees that if man^- men (lo something thcn some man docs it. These are by no mcatls all of thc semantic properties that sccm to bc cnjoj-ctl h!- the dctcmincrs iuost, nr~rtt)r,~/;'~r, but they are the ones which seem most clearll- rcilectcd in tllc scnlantic judgements of iiati\;c Ihglish speakers.

Appendix C. Some simple results about quantifiers and determiners LJntil C12, wc k t hf = (E, 11 I ) be a fixed model. A Y ~ ( ~ ~ ~Qon t ~ / :1/f ; ~is~an) r sct ofsuhscts o l ' l . Q i s a proper qnontz/ier.or .vi~.i.tif Qis nun-empty and is not the sct of ;ill subsets oi' I?. Q/izle.v nt? -4, where :i is some subsct of L, if for a e r y A- C I?, X E Q iff (X n : I ) t Q . ('Lit't"' is all abbreviation for "if and only if.")

Generalized Quantifiers and Natural Language

PROOF. Since Q lives on . i , E E Q iff (I7n A) E Q but E proof'.) A quantifier Q is called the principtrl filter gencriltcd by B if

n.4 = .-I.

119

(amarks the cnd of a

Notc that the principal filter generated by H is a sieve unless B = 1/1.

C2. PROPOSITION. PR001:.

If R

!f'Q Is the prini.~pol~fille~gcize,z~le(l /!I, B rlreir Q 1ii.t~1111 .-I ! / ] ' R C : I .

-I, thcn for ;my X' the following ilre ecluiralcnt:

?i E

B g .AB & (A- 1-1 - 4 ) (since B C '4)

. \ n . -E~Q . Thus Qli\ cs on -4. Notv supl>oseQ l i \ cs on i and Ict us show B (:I. T h u s B C_ .-I hy the definition of princip;ll filter.

C -4. Siilcc 11 cII E, /1 E Q so .-l E Q h!

.4 cletcrn~incrD is n i l e / i i ~ ideterminer /~ if tbr all . I E, D(.4) is ii such that, lor ;in! .-I f tloiiiain(D), I)(.-!) is a quantifier that liws on -4. :I clcterinincr D ispi.opc1- if D(.-1) is dcfined tor all .I C_ II and, for some .-I & E, D( -I) is proper. Uni\crsal UJ in 5 4.4 illlplics that ever! naturnl languilgc clctcrmincr is thc restriction of a proper natural language determiner. r i dctcrmincr D is ;I tlc/jn~recletcrniincr if for all -4 2 E, D ( 4 ) is ;t principi~lfiltcl,. I ) is ~ ( ~ . < i /s/rnri,:r ~i'l' if for c\ cry A, -4 t D(.-I).

1 I

I'ROOE'. Sincc D is clcfinite, D(.-I) is the filter gcncratcd by solnc R. Since D(.-I)lihcs o n . f , B b! C:2. But then .-I E D(.4) by the definition of principal filter. [7

5 :I

il determiner D is .glln~~telric if f i ~ rall -4, B. B E D(-4) iff .i E D(U). D satisfies the ~n/c~r;sc~~~/~orr rn/~rli/ronif for all . i , B, R t D(.4) iff R E D(.i n R). iVc will shon that thcsc two conclitions itrc eqitiralcnt in C5"

1 i

U E: DCi) iff E E D(/i n B). JJROOF. T h c follot+inp are cquiv;llcnt:

,

R E D(I) ( i n B ) E n(-4) (-4 n B ) E />(.i r~ (.i (-1 11)) ( 4 n R) E I>(.-1 11 B) E E 4 4 i?B )

(since D(. 1 ) lives on -4) (the i~~tersectiol~ condition) (-4 n R = ,i n (-4 n R ) ) ((11 applicd to f n B),

120 Jon Barwise and Robin Cooper PROOF. Assul-tle I) satisfies the interscctian condition. Thcn .-l E D(R) iff 1; :' lI)(il n R) by C4 hut B E D(A) iff.,!?E D(*4n R) also by C1. Thus .,.IE D(B) iff /1 E D( -1). For the converse, assume I1 is symmetric. The follo~~:ing arc then equivalent:

(by symmetry) (since D(H) lives on B) (by symmetry). [7 C6. THEOREM. 1,er D be rr: p r o p ~ rmont clrrrrnrrnc*r. Tjren D ron(/iiiotr (rrnd hewre j.< 11ofS J ~ I I I N I ~ ~ / ~ ~ ( ~ .

( ~ I ) P S rror

so/is[jl /lip irilerrc~/to~i

PROOF. T o rccall the definition of strong, D is posrrrre S I ~ U I Iif~ for it11 . I, >.I E /)(--I). D I.? I I C , ' C ( ~ ~ ~ M strorz.q if for all 4,,4 $ D(,-1). D is stro112 if D is positi\c strong or nc#ati\c strong. .lssurne that I1 satisfies tllo intersection conclition. tiye claim that if D is positive strong then Li~rcvcry -4, lJ(.i) is the set of a11 s~ibsctsof E whereas if D is negati~estrong thcn for every .4, D(.f) is empty. Thus if D is either positive or ncgati~estrong, thcn L> is not propur. To prove our claim, note thc tidlowing equivalences:

R E L)(A n B) R E D(-4) iff iff(..l n B) E D(B) iff(A ilB) E D(.4 n B)

(the n-condition) (s_vmmctr~-, C5) (the n-conclition again).

Thus if D is positive strong, R E D(.i), for all ,i, B wliercas if D is negative strong thcn B $! D(:.l),J i r ull .,.I,B. 17

,4 quantifier Q o n M is monotone I Y J ~ . ~ P I I (mon S I ~ ~ T) if for all Y, 1' C E, .Y t Q and . I C'Y implies Y E Q. determinu I> is wnrl f if for all .4, if I)(-1) is defined then D(:l) is mon T. 'I'his should not bc conf~~sed with thc notion of a persis/ewi determiner, one such that for ill1 4,11 E, if . A C B then D(A4) lJ(R). Here is an examplc of'a determiner which is persistent but not mon 1 or syn>inctric,Let E h;lw at least two elen~cntsand define I) by D(.i)

=

{.Y C E.4 n X # O and -4 X f 8). -

1

to scc that .-l C B implies 1'11at is, Dc4) mcans "some but not ill1 of the things in .-I". 11 is D(--1) C D(B) so that I) is pcrsistcnt. H o ~ ~ ~ e vDc ris, not mon 1 since I: $! /J( 4) for :ill :f. Sincc, For I 0 # 4 # E,.i E D(E), this also slio\vs that D is not symmctric. 111 our persistent drtcrminer universal (U8 in 4.0) \vc proposcd that 1111 siiuplc persistent determiners of human languages arc nion '[ and n-cak. In view of the following proposition (and C b ) , a stronger universal would he to assert that the simplc pel-sis~cntdctcr1ninel.s of' I i ~ u m ; ~ nI languiigcs arc all s!n~metric, i.c.. satisfy the intersection conclition. Our lacl, ol' clear c u ~intuitions " ahout \vhicll non-logical weak determiners satisl'!. the intcrsc.ction corldition prcvcnts us from ruiiking , this proposal. I

C7. PROPOSITION.

[/.Di . ~ptrsis/(vi/ (!11,1: ) ~ ~ I / ~ I I L //li'7/ ~ / ~ I ~ IJ , is

II~IIII

1.

PROOF. Suppose .y E D(.-1) arid .Y C I.'. b'c nccd to show I - E 13( 4). I3ut .X E D(.i) implies 4 E D ( X ) by symmetry so -4t D( I.') by pcrsistcnce and hcncc ).' E I)(--f) hv symmctr>.

Generalized Quantifiers and Natural Language 121 (1) (.-I n B) E D(a-In 4) since D is positive strong so R E I)(.-!fl 8 ) sincc .f is similar. Recall the definitions nf'Q and Q from section 4.7 and from 4.1 1.

PIi00F.

PROOF.

- -

(1) ,lZssuine Q i s mon f . First, suppose Y

( E - Y) c ( E - X) so ( E - X) E Q s o .Y t (Q then I' 4 Q so X @ Q so X E ( Q ) .

-

( 2 ) This is similar to (1). (3) This fllovis from ( I ) and ( 2 ) sincc C10.

PROPOSITION.

I f ' Q i s nrorr

r,

zN

-

n R g L3. ( 2 )

E (Q -- ) and X C_ 1'. Then ( E - I.') E Q and ). Now suppose J.- f ( -- Q )and .Y C 1'. Rut

(Q

E Q nrwl

).

O l h m -i n 4 f 0.

BE

-

PROOF. Suppose A TI B = G?. Then .4i (E - B ) so, by monotonicity, ( K - B) E Q .13ut thcn U E (Q ) SO 4 @- ( Q ), a contradiction.

A n7irnrs,( set fur a quantilier Qliving on A is any subset n ) of.2 that is an clement 01'Q. C11.

PROPOSITION.

LC/ rrrrige oT7er~ ~ i t t z zsets,fi~r a the rl~inf~ttfin. Q fhur lii.~.r011 .-I.

/ i ) ~X, X Q $jfsoii~eIP 1s (1 S U / ) S ~ J(!JAY. / (i) I1.Q i.q 1 1 1 0 ~ 1 ~ / 0 1 i1 ~t ~ o . ~ v z . ct lt '~f (t ~~ n -((?Z)I J ~ ~ ~ ~ ~rblw < ~ ./ij~. L z .LII?.)) s ~ -X, I z ~.\- E Q I[[ 3-n .-I ix ~/tttlr1?7c(l sonic (ii) r f Q Is r~~oi~ototrc

131.

PROOF. (i) Assumc that -A' E Q. Then .Y n .4 is in Q since Qlives on -4, so wc may tillic .X n .-1 lib]. 111. Conversely, if IT, ,Y, thcn since 11. E Q and since Q is monotone increasing-, Y E 4. (ii) It' E Q, then X n ,4 is suitable. Conversely, if (.Y n .I) I?), thcn sincc n? f Q ;lnd Q is monotone decreasing, X n -4 is ill Q ancl hence .X- E Q, since Q lives on -1.

.x

T o conclude this appendix we return to some af thc pointsmadc in 5 I, cspccially in 1.2 and 1.3. Wc want to prove that, in our terminology, "niost" and "more than half" must be treated as dcterniincrs, nor ;IS qu;~ntificrs.In other aiords, we u-ant to prove that there is no uay to clcfinc wlosl I tr urt. li's in tcrrns of i ~ o s it l ~ i ~ l ~,v(g .s . . La'. . . C' . . . I.. . . ). T o a ~ o i dproble~nsof vagueness, we treat "niorc ha11 hall". be convenient to first protc a weakcr result, namely, that ~trorc/htr?l ktr//'/he 1':t urc3 For the proof; i t li:r cannot be defined in first-order logic. This result is probably somewhere in the litcraturc, bui wc ha\ cn't been able to find it so present a proof. It is a routine application of t l "Vraissi: ~ letho hod". T o motivatc the cnmplcxity of tlic yrool; note that for ail!- fixed uppcr bound k' on thc sizc of tlic unikerse E, there ts a sentence c p , that "works'' Ibr models of sizc 5 K , a giant disjunction of k formulas.

C12. TIIEOREM. CIINSI(/CY 11 jiu/-or(ftr I L I I ~ ~ L I U XLP nlifh qutl1i1.1.rlrru' t n ~ oi r ~ r t r r : )predr'culc ~ s)rrrlhols L, V. Tl~erei\. 1111 xrnirnc.cJ cp ( ! f L so i h ~ l iin ei.cr:l~-/i~ri~e n~odiv'.I4 = ( E , I ', 1'),

PROOF. Wc will prove morc for thc purposcs oi'thc ncxt proof. Namely, for all n;~turalnunlbers nl and k with k > 3n1 we construct ttvo mndcls MI = ( E , 1/', If) ancl .Mz = (E, Liz, 1 -) with thc same domain E and same interpretation {'of Y, such that

.

122 Jon Barwise and Robin Cooper (1) til g Ij2 k. (2) 2 . Card (lTr)> Card(L7) = 3n1, hence, ,+Izk "hlorc than hall' the j'.'s are I:'s" (3) Card (1,;) = 2 . Card(Cil), hence, ,+I1 "More than half the C.'s are Ir's." (4)

(5)

Chrd (I?) = k For an! sentcncc cp of L \\-it11 less ihnn

HI

q ~ ~ a l ~ t i f j c.+Il rs,

y iff MZ

(p.

Ignoring condition (4), this \ d l prove thc theorem, for given a purported definition rp of "More than half the I/"s are i;'s", we apply this to some nl greater than the number of'cluantifiers in rp and some k 2 3 7n. T o construct M I and satisfying (1)-(5) we let E be anv set of 1, objects, b ' a s~thsetof /?of size 2 nr, C'? a subset ot' I/' of size nr 1 and LTl a subset of 1:? of size rr1. Only (5) needs to be proved. Notice that if n(nl then Card ( E - V ) 2 r1, Card ( L - / I I ) n, Czard ( I ' - [ I r ) 2 /I as nell as Card (I;,) n and Card (l??) n. Tliis fact allo\ts us to prove (5) b!- proving the following strongcr (6). Define for an!? formtila ip, r.(cp) = number of cluantifiers in cp+ number of free variables in (p.

+

>

>

>

) a formula with ~.(ip) < HI and if wc h a w an?;. one-one corr~.sponclcncc ( 6 ) If (p(.ri . . . . I . ~ is

hl

'71 &+

.

bctnccn clcnlents o f E satisfying ir, E lil iff b, E L.'2, and '1, E 1,- iffh, E I , l i ~ rall i = 1, . . . I,, then k tp((li . . . l r i ) iff /El2 q(bl . . . h,). ( ( 5 ) is the special case of (0) \there t = 0.)

+

.MI

Stated this explicitly, the proof of (6) is cluile eas! - by induction o n I-(cp).'I'hc point is tlint therc is always enough room to cstcnd the onc-one corrcspondenec onc more step c\licn !.OLI comc to a quantifier. (Draw a picture.)

PROOF. hlorc explicitly, \\ hat \t-c prove is the follon ing. I.ct I , be 11ie fi~.st-orctcrmonaclic langi~age of C12 and introduce a ncn quantifier symbol Q. Lcr I,(Q hc the Ii~nguagcwhich nllo\\s ;ill thc sj-ntactic constructions of' 1, plus, for each tbr~nulacp(.\.)of L(Q ), Q.\,[q~(.v)) i ~ ~ic\v i f'ornlula of L ( Q ). T h e semilntics for Q i s defined on finitc moclcls !Vf by I%f

k Q.~[rp(.v)]

iff Card

111.24

t=

cp(rl)) > $ard

(E).

(a

What w-c prove is that there is no scntencc (p of L so that :I1 cp iff more tliiln half the Ins arc Lrs. l'lie intuitil-c idea is that i f E is very large con1p;ircd to [,.and I -111cni t \till swamp out Iliitid 1.' in the language L ( D . T o mahe it prccisc Ne will dciine 1' fi~nction* fiolti for~iiulasI/! of' L(Q) to formulas (1% of L so that II/ is cclui~alentto I//" on moclcls it3 nhcrc thc gap bcl\vccn the size ol' I',~IIJ that of E is great enough. Namel!-:

(P) For an!. formula 11/(.\.~. . . .rr.)of L ( Q ) and an! lnodcl 44 (E) 2 (Card ( V ) i.(~,!!))

>

+

=

( E , I:, 1 ) uhcrc I 1 C I '

C E and Card

Generalized Quantifiers and Natural Language 123 From (P) and the proof of C12 we can easily concludu the proof of C13. For supposc tliat cp is ;I sentence of ~vhichis tl-uc in a modcl /l4just in casc illore than half the I"s and (j's. IAct ill > r(cp) and let k > 2(2 111 r.(cp)). For this rra and let .All, M2 be models sarisljing conditions (1)(5) in the proof of C12. Thus &IL /=- (p but /MI q , b\- (2) and (3). B L I ~since (lard ( E ) = k > 2(2 m r.(cp)) = 2 (Cart1 (F) r.(q)), Condition (P) implies that :Vll (cp -+ cp"), M2 ((p '-' cp"). But b) ( 5 ) , M I rp* iff ,%Ir (p", since cp" is a first-order sentence with r.(cp*) < IN. This is a cont~adiction (since All (p implics M I cp" iniplics Mi rp* implics & .. /= (p but /t12 k- q). T h u s u-e need only define I//*, show- that r.(~//)= C(I/I"),and prow (P). ?'he definition of I//* is h!. recursion on 11and only docs things to the quantifier Q. Thus:

/,(a +

+

17

+

If I// is atomic then I)* is 11. If Gi/ is 10, (01 ~ 1 1or ~ )b'.v[R] rcspectivcly, then $* is ](H"), (l17,t0;) or V.rl O"], rcspecti~.ely If i// is Q~.r(I(.v,.]1~ . . ..ill ), then ~b* is ti.v[V(.r) ,v = ,111 1, . . . , v = .),k v O*(X,.),~ . . .,yk)l. ,J

(l.c., I/I" says that ever! .Y # I - C! ( y I . . satisfies ~"(.Y,,J,~ . . .yi.).) 'I'o prorc (P) onc ;irgucs b!. induction on the length of 11).T h e onl!- nontrivial casc is \vhcrc $ is of the for111Qd)(.r.,l~~ .. So supposc that (P) holds for 0, by induction, ancl Ict M = (E, I.', I.') be ;I nioclcl t\ it11 . , ) j b )

.,)ln.).

First assume $(nl . . . i l k ) holcls in .l4, i.c., that more than half the h's in E satisf! O(h, 111 . . . u k ) . Since X, 5 r(0). and since f- C:;~rd ( E ' ) > Card ( V ) r(t1), at least one such h is not in 1 C! { o 1 . . . ( / A , } . But a trikial autoniorpliisn~ argument then shows tllat any h ' # I - U { a l . . . ak} satislics O ( / j ,r r l . . . I I ~ . ) . Il! our inductive assun~ption,O*(hl, rtl . . . irk) holds I'or an! such h'. In othcr words th'(rtl . . . (11) holds in .*I, T h e othcr half of thc equit-alcnce is easier. This proves (1') ;IIICI hc-ncc tllc thcoreni.

+

'

After finishing this papcr, n r learned that ;I tlieorcni relatcd to Thcoreni C;13 J V ; I ~pro~cclb! l.);~i;iJ Laplan in 1065 hut v-as ncvcr puhlished. Extend the yual-rtifier "\,lore than 1ii11fof all things" into thc infinite following Rcschcr (1062) by defining JM k Qa.4 to mean that the set of'n such thilr .Ill k r/)(n) !>;IS greater cardinality than its complement. Kaplan showeel tliat the rclatijizcd notion could not be defined from the unrelati\izcd. This follow-s from C13. Kaplan's proof nlahes esscnti:~lilsc ol'inlinitc structures. His proof is unpublished, but some othcr intcresti~~g results fill- this clu;lntificr ilrc contuincd in his nhstracts Kaplan (1966). 3lonotonc iluantificrs hat c been stuelied in model thcorj- ;ind gc11cr;lliz~drecursion theor! fix somc tinlt'. Sce l3ar\vise (19'79) for refcrcnccs. l ' h c notion of persistence \\as ititroduccd in Har\visc (1078). 'The othcr notions are nee hcre. b'c 1ia1.e incluclcd 3 Lin of the classic pi111e1-s on gcnercllizcd quantifiers in the rcfcrenccs: hlosto\vski (10.57); Lindstroln (1900); Lcislcr (1060). Other relkrcnccs can be lounil in I 3 a r n . i ~(1078, ~ 1970).

Appendix D

NP's containing simple Det's 1, some/a 2. every/each/all 3. no 4. (at least) 1, 2, 3, ... 5. the 1 , 2, 3, ... 6. both 7. neither 8. most 9, many/several 10. few 1 1. th~s/that Other Det's 12,a few 13. exactly 1, 2, 3, ... 14. at most 1, 2, 3, ... 15. more than half 16. at least half 17, fin~telymany 18. infinitely many 19. open Proper names and

logical?

always defined?

always sieve when defined?

Yes Yes Yes Yes Yes yes Yes no no no no

Yes Yes Yes Yes no no no Yes Yes Yes ?

no no no no Yes Yes Yes no no no Yes

no yes yes no no yes(?) yes(?) no

Yes Yes Yes Yes Yes yes yes Yes

-

Yes

no

no no no no no no no

man 1‘,

& strong

or weak (w)

w +S

w w +s s -s

+

is

w w SI

w w w

+s i s w w w

definite? no no no no Yes Yes no no no no Yes

1 or neither

simple dual?

self-dual?

pers~stent?

antipers~stent?

intersection condition (= symmetric)

1

(2) (1 no no the 1 (11 no no no no (11)

no no no no the 1 no no no no no Yes

Yes no no Yes no no no no ? no no

no Yes Yes no no no no no no yes(?) no

Yes no Yes Yes no no no no ? ? no

no no no (16) (15) no no

? no no no no no Yes

no no Yes no no Yes no

no

no no no no no no no no

(20)

Yes

-

-

T 1 I

r

i

no no no no no no no no

pronouns 20, John, he

Yes

A

s

Yes

T

-

-

-

Generalized Quantifiers and Natural Language 125

Notes 1 This is proted formally in Theorcni C13 of Appendix C. 2 'I'hroughout this paper we use "determiner" to refer to a wide class of s!-iitactic clcnlcnts which also include what are solnetinies called predetermincrs, postdctermincrs, numerals, ctc. -4 more detailed investigation may well show that some of these liner distinctions arc ncccss;uy. 3 For esample, a number theorist interested in prinlc numbcrs will usc a measure which "livcs on" the set of primes so that (4) would be false. More common ineasurcs tvliicli do not give special weight to primes will make (4) true. T h e notion of "lives on" will be clcfined below. 4 Karnp's proposal is basically to evaluate a sentence with respect to a class of nlodcls, rather than a single model. 5 For each of (19) and (20) there are alternate derivations of thc same SL) bthich d o not use the quantification rule. In thc translalion dctined bclon, these alternilre deritations \\.ill translate, respectivel\;, as: m o s t ( m e n ) ,i[ s o m e (woman),j/[kiss [ m a n y ( m e n ) .i [see (s,h ) ) ] .

-

6 7

8 9

10

II

(s..)l)]l

T h e unlikelj-hood of interpreting (6) with this particular scope relation between no/ ant1 I I I ~ I is I ~ discussed in section 4.11 below. T h e notion of simple N P is well-defincd only !$-ithi11 tlie contest of a given synt;lctic analysis. For example, not ctcry analysis s i l l treat u.f,?u~as a single determiner clement. Notc that this is different from claiming that John cuists. U'e might assuiiie that the n~otlvl includes sonic lhings whicli do not actually exist. T h e set of things that exist is a subset of tlic sel of things that there are (in the model). I11 applying this tcst, we must niakc sure that we do not violate our assuniption of t i ~ c dcontcul. za1./'1 docs not cntail I ~ I O Rrpriblil.rit~s .YI It n i i g h ~be objcctcd that I I Z O S ~ Repub/zc.llns r~rlrrellrhe rlllr~c~rl I ~ NI(.P Z since thc former is often used in contest to mciln I I I O S I R e p ~ i h l ~ ~ u I Vi/ IiO. ~ r.r~r~r.r~i/ ~ltr, rille e)l/rrerl it c a r l ~ This ~ . is an esaniplc of the violation of thc fised contcst assumption. Clcnrly the N P mosl Repub/ic.titis corresponds to different q u i ~ n t i f i ~ins diffcrcnt contests. We arc intercstcd at prescnl in isolating the properties of the quantifiers thcmsclvcs ratl-icr lhiui t l ~ c coniples relationship bctwccn NP's, quantifiers and context. Some spcakcrs scein to feel thilt not :\I1 ]nixed conjunctions with irntl eleser\,c a l'ull star, particularly if the conjuncts are not simple NP's: John hirs rlii'i~ed or /~u.\.t J;ZY I P ~ ) I I I ~ [~ II I I ~111 most f k r nwll lo r h ~~( z I - / nwn,)! y. ~fthr I ~ Z C I IZ I M I~ I ~uost ~ s i . O~/ ' I / I P I P O ? ? I I ~ ~ IIra~.c./iii/i~d 1/11. e . ~ , t ~lrv~ lici . have no explanation for why there should be variation on the juclgcments in these cases. -Ipl>arcntl>-11ot A I N I should ~ not be considered as an N P since i t cannot occur ~ l o n cin 211 NP position. "Not Mar! is invited to the parly.

13 Tllcsc rules do not meet tlic rcquircmcnts of autonomous syntax, i.e. thcy use information about the semantic intcrprctation of constituents in order to define sj~ntacticwell-for~lictl~.rcss. 7'hei.c are, of course, ccluivalcnt fol.mulations in which the category NP is subclividccl in the syntax according to tlie kind of quantifier dcnotcd (a species of autononl>-preserving tlutment often h u n d in Montaguc Grammar). Wc beliebe that it might also bc possible to allow frcc gcncl.atioii of conjoined NP's and design a semantic filter (cf. Chon~sl;!~)which would rulc ouc ccrrain of thc conjunctions as semantically unacceptable.

126 Jon Barwise and Robin Cooper

References Barwise. J ~ I I .1978. Monotone quantifiers and admissible sets. Ln J.-L. Fcnstad, R. 0. CGanJ!., and G.E. Sacks (cds), C ~ ~ n ~ r n l ~Rn.rrr-siojl zcd ir'l~eir~l 1I: Proi.rerIrrrgs (!/' 1hl8 1977 O.vlo ,5'~~111/)1~srunr, Amsterdam: Nort h-Holland, 1-38. B:~r~visc, Jon. 1079. On branching quantifiers in Fhglish. -~orrrrrtr/~~/'Plrilosophii~tII I,o~rr.8: 47-80. Barwise, Jon. 1980. Scenes and othcr situations. Jo~irrrrrlr~/'l'l~ik~soplr,)~ 5 0 : 300-90. Chon~sky,Noani. 1080. On binding. /,irigur.\./I't.Inqrlir)~1 l(1): 1-40. C:lark, I-lerbcrt 11. 1976. Ser17a~rlti.sonrl Cr~~i~pr-~/z~~tzsion. The I laguc: Aloutot~. C;ooper, Robin. 1978. .-\ Fragment of English with Qucstions and Rclativc C;l;~uscs.University of T\'isconsin, Madison. Dowty. David K. 1979. Il/>~tlMeanitr,y u t d Ml/n/ugur Grir~tirzlar:771~~ Se/!rr~rr/ic.s I!/' I >r/)srrrrrl TIIIIPS III C&r~er-ci/i;.e.9cw1onlic.~(atlil i ~ iVlon/cicyuc's i PTQ. Dordrccht: D. lieidel. Fensrad, J.-1. 1978. Models for natural languages. In Jnahko IIintilika, Ilkha Niiniluoto, and Esa Saarinen (eds), E.c.sc)ls 011 iMiirh~nrrt~r,rrlclrrd Pliilr~soplrI'r.lrl I,o<~rt~: Proiec~ilirrgs(!/'I 11i l.'orrr/l/ .St riacli17i/i'r(/r1 Lo$i(. S ~ ~ n r p ~ ~ rr~il s i r(!/'/hr~ ~ t ~ ~P ~ I . S.Eorr~~r-Finnrsh I 1,ogii. Ctr~!/bre~tii,, JyiisLyli, Finland, Junc 20-July 6 1976. Dordrecht: 13. Reidcl. C;;lzdar, Gerald. 1970. Erz~li.v/( 1 s a (;'ontr.v/-Fr~~~~ La~tg-rrugc.Univcrsitj. o f Susseu. Gricc, Paul. 1975. Logic and conversation. In Petcr (;ole and Jcrry ;\lorg;un (ctls), kYpc~r~i.l~ .-I( 1s (.S)vr/a+v trtrrl S ~ ' P I Z ~ I IYO]. I ~ ~.3), ' L .NS ,e w York: Acadcnlic I'rcss. Jenkins, Lyle. 197.5. T I I E ~ I I ~ / IL S. ~~i F t ~ ~ ~ Tiibingcn: /iirl. Nicmcycl-. l,t~.y11~~11 Op1~~(11ors 111 L'tzg-lislt. I3loonii1igton,Ind.: Horn, I .aurcncc. 1976. 011111rS t ~ ~ i ~ zPr~~prlrt~c>s n~ic Indiana University 1,inguistics Club. Kanlp, J.X.W. 1975. T n o theories about adjectives. In b;clwnrd Kceni~n(ccl.), firmer/ S'rvr1r11111i.s o/' Mrrtnral 1,trt1gurrtqr: Prrptrs ./ion1 u C~ol/oq~~i~rrn .\'l,on.vor~d h y IIIP h;rr,y's Collrgr~ Rase~trrr.lr C'errlrr, Grnibri&c, Cambridge: C:ambridpc University ['rcss. Kaplan, D. 1066. Rescher's plurality-quantificatioli and gcncralizcd plurality quan~iiication,abstracts. Journrrl c!~S)~nrholr'i L ( < ~3I1:C 1.5.34. Karttunen, Lauri. 1977. Syntax and se~nanticsof questions, l,ityii~!il.,s( I N ( / P / I I / I I . ~l(1): I ~ ~ 3-44. ~I)~ Keislcr. 11. Jerome. 1969. Logic t ~ i t hthc quantifier "thcrc cvist c~ncountabljnian~". :l~rrnr/si!/' 2 4 ~ l ~ r / h ~ ~ ~ldosqic t z i ~ /1i:~I r-93, ~/ Klccne, S.C. 1953. In/ror//irlion I I I :2/relr1111a1h~n1~11i~~s. Amstrril;lni: North-T IolI;111cl. Ladusaw, William, 1079. Polarit? Scnsititity as Inlicrent Scope lielation. Pli.11. dissertation, University of' Texas, Austin. l,indstrom, P. 1906. First-ordcr logic and gencralizcd quantificrs. 7hcwr.1'1133: 187-95. Vlilsark, Gal-!. 1977. To\+-ardand cspian,ltion of ccrtain peculiarities ol'thc. cuistcriti:ll constrcrction in 13:nglish. l,irlglrt~./icAntr/lls,\< 3(1): 1-30. hlontague, liichard. 1974. For-rrrr(lPlrilosoplr~~. Srll~c.rtJI'iipriu ~~/'Rrrlrrr~.rl ~~/1oll/~i,~/lr~, ccli~ccliind with an introduction b!- Richmond H. Thomason. Nev fla\.cn. C:crnn.: Yale Uni\crsit! Press. Zlostc;.rr.sbi, A. 1057. On a generalization of cluantilicrs. l'rrrrt/trnr~rr/ir44rr/lr~11rn/rt~cr~ 44: 12-36. l';~rtcc., H;lrb;lrll. 1953. Some structural analogies hctu ccn ~cnscsand pronouns in Lnglish. J'ojl)rr~rrtl I!/ 1'11i/osop11~)~ 1X X ( I 8 ) : 601--0. Pcacocbc, C. 1979. Game-thcoretic scrn,lntics, cluan ti liers ancl triltli: conltiiellts on ProI'Cssos Hintikha's p'ipcr. 111 Lsa Saarincn (cd.), C ~ a 1 ~ 1 r ~ - ? 7 1 c o r c.Sc'trlrri~/iis: ~/i~~ril ~ ; : s . Y N1/11 ~ ~ sJ'(JIIIIIII/II..T, 1101.drccht: 1). Reidcl, 1 19-34, Rcschcr, N. 1962. Pluralit!- qu,lntif?cstion, abstract. Jorrr-trol ~!/'.Sllrnho/ic.1,1q1e.27: 373- 4. Sgro, J. 1977. C:omplcteness tlreorerns lor topological models. ,.ltrrr~~/.s 4' /LIu/l~c~~nir/ic.rrI I,O%I(I(. 1I : 173-93.

The Logical Analysis of Plurals and Mass Terms: A Lattice-theoretical Approach Godehard Link

1 Introduction The weekl!~ Mnga.zirtc. of the Gernlan newspaper Fra~lkjiurtcrA i l l ~ e t i z ~ ~Ztirung ine rcgulasly issues R/Ial.cel Proust's famous questionnaire which is answered each t i n ~ eby a different personality of West Gcrman public life. One of those reccntly qucstioncd was Rudolf Augstcin, editor of Der Spiegel; his reply to the question: "Which property do you appreciate most with your friends?" was ( I ) "that they are fcv ." Clearly, this is not a property of any one of Augstein's fiiends; yet, cven apart from the espyit it was designed to display the answer has a straightfbrw-ard interpretation. The phrase ( I ) predicates something co/lective()~of a gro-olrp of objects, here: Augstein's friends. As it is well known, coIlcctivc predication is a rather pervasive phenomenon in natural language, as the following sanlple of sentenccs shows:

(2) (3) (4) (5) (6) (7) (8)

T h e children built the raft. T h e Romans built the bridge. Tom, Dick, and Harry carricd the piano upstairs.' T h e playing cards are scattered all over the floor. T h e members of the committee will come together today Mary and Suc are room-mates. T h e girls hated each other.

There is a striking similarity betwcen collective predication and predication involving mass nouns.

(9) (a) The.children gather around their teacher. (b) T h e water rathers in big pools.

128 Godehard Link Moreover, a characteristic feature of mass terms, their c.11112rrIirtizlr~ ~ ' ~ ; / i r ~p~rioipcec~ I y2, can be imitated by plurals. (10)

If n is water and /I is water then the sum of I/ and /I is water. (b) If the animals in this camp arc horses, and the animals in that cilnll> are horses, then the animals in both &Imps arc horses. (a)

All this has been observed and discussed in the literature although the notcd pi~rallclism has perhaps not been stressed too much.3 As it can be seen fi-om Pcllcticr's 1079 volume, l~owcver,there is much disagreement about the proper bvi1y 01' attacking the logical problems posed by plurals and inass terms. 1;rom a scn~anticpoint of view the basic question is: what do mass terms and plural expressions denote? Some havc thouglit that in order to be able to give a satisfactory anscver to this question it is necessary to givc up or at last extend the underlying set theory and to clefine new kinds of objects, for instancc ~nsc~nlhkes (Bunt 1979) or k ' o l l t . k t i o ~ ~(Blau ~ n 1079). 1 think, l ~ o ~ ~ c vthat e r , cvc can retain the usual set-theoretic metalanguage and sinlpl~lenricl~the structure of our models as to ~ ~ c c o i ih~rl tproperties like cumulative reference. On 111y view, such properties arc also not secured by ik/inirrg some plural or mass term denotations out of others through settheoretic n~anipulations;they all should be recognized as simply being thcrc. What wc rather should try to discover, then, is the nctworli of thc various relations which thcy enter and through which they are tied together. In the case of group and mass objects this picture naturally leads to the notion of lattice structi~rc,'an iclca which is, again, not new: it is inherent in mereolog-ical predicate logic and the (:aIculi~sof Individuals as developed by Leonard and Goodman ( 1940) and Goodman and (luiric (1947). Iiowcver, its possible use in the present context has perhaps been ohscured b! reductionist ontological considerations which are, in my opinion, qilitc alien to the purpose of logically analyzing the inference structures of natural language.' Our gnidc in ont~logicillmatters has ro he language itself, it seems to me. So if we havc, for instance, two expressions ir and h that refer to entitics occupying the same place at the samc time but have cliFferent sets of predicates applying t o them, then the entitics referred to arc simply not thc samc. From this it fol1ou.s that my ring-and the gold making up my ring arc different entities; they arc, however, connected by what I shall call the c.ons/itzl~ionrulrrtioi~:'I'licrc is exactly onc portion of matter making up my ring at a time. A co~~stitution 1.clilti0110' has bee11 explicitly introduced into the discussion by Parsons (1079). Sharing his intuitions on this point 1 shall provide a similar 2-place I-elation, "/,", for ~.ons/it~rtrs or nrr~krs~ l p Its . semantic counterpal*tin the theory to be presented below, the "materialization" function h, lies at the heart of my reconstruction of the ontology of plurals and Inass tc~-nls: individuals are created by linguistic expressions involving dift'ercnt structiircs cvcn if thc portion of n~attcrnlaking them L I is~ the samc. C:onsidcr thc esitmple from I3lau (1079) (imagine that there is a cleclc of playing cards on the table):

(1 1) (a) the cards (h) the deck of cards 1Vhile the portions of matter denoted by ( 1 la) ilnd ( I 1h) arc the same, I consiclcr the individuals as being distii~ct.~' ( 1 l a ) refers to thc pure collection of objects, and in many

The Logical Analysis of Plurals and Mass Terms 129 contexts (1 lb), too, refers just to this collcction. In general, however, ~ h introduction c of a collective term like (1lb) is indicative of connotations being added enough for it to refer to a different individual; for instance, a committee is not just the collection of its members, etc. Note, by the way, that the transition to an intension function would be of no help here. There might be two different committees which necessarily consist of exactiy the same members.' It might bc tllougllt, then, that collective predication is just the context in whidl pairs of expressions like ( l l a , b) refer to collcctions and thus are corefi'rential. 'I'his is not so, however, as can be seen from the following cxample. Imagine that thcrc are several decks of cards, a blue one, a green one, etc., lying on a table.' Then the two is a collective following sentences do not mean the same although number con.s~cu~iz~eIj~ predicate (the German ~ ~ is du~hnumerier~n). r d (12) (a) T h e cards on tllc table arc numbered consecuti\rcly. (b) T h e decks of cards on thc tablc are numbered consecutively.

By contrast, spntio-ten~por~ck collective predicates do refer to the pure collection, or, as 1 conceivc it, the portion of matter making u p the individual in question. Examples are bc-on-llzu-lahlu, occupy, etc. I call such predicates inz'nrinnt. So the following (a) sentences are indccd equivalent to their corresponding (b) scntcnccs.

(13) (a) T h e cards are on the tablc. (b) T h e decks of cards are on thc table. (14) (a) T h e stars that presently make up the Plciades galactic cluster occupy an area that measures 700 cubic light years. 0 (b) T h e Plciades galactic cluster occupies an area ctc. 111the following I shall distirlguish between pure plural ilzdividuals involved in (1 2) and collections in the portions-of-matter sense referred to in (13) and (14). 'I'he I'ormcr 1 call (individual) sums or plzlral ohjecrs; thcy respect levels of "linguistic cornprchcnsion" as shown by (12). Ry contsast, collections do not, they typicallj- merge those Ievcls. Sums and collections are similar, howevcr, in that they both are just individuals, as concrete as thc individuals which serve to dcfinc them, and of the same logical typc as these. T h e latter feature is important because therc is no systematic type ambiguity inherent in predicates like ( . a n y , b~ii/d,derr~o/ish,ri
130 Godehard Link to refer to the concrete "scattered individual" that you just find everywhere, hcnce Quine's analysis in terms of mereology. In this sense the sentence should be synonymous to the water (on earth) is mirJespread. Of the samc type is the use of~qcp/,lcl in ifmericn,i gold is stored in Fort Knos. Here, again, a concrete object is refcrrcd to by :lmtlua \gold, namely the material fusion of all quantities of the US gold reserve. So there can be no doubt that some notion of fusion is needed to account for definite descriptions involving predicative mass terms (the poperator dcfined below docs just this). Gcnuine stuff names, however, arc something else. Substances are abstract entitics and cannot be defined in ternls of their concrcte manifestations, Thc question, then, is of what kind the connections are that are intuitively felt between substances and their quantities. Take water, for instance. A quantity is water if it displays thc internal structure of water, that is H20. Hut this relation is not a logical one. Or elsc we might look for substance properties which carry over to the quantities of the substance in question. Water is a liquid and yet, all concrcte water might be fsozen. So we have to go over to dispositional properties, getting morc and more involved into our knowledge of the physical world . . . What I am getting at is that laominat w i ~ ~ sterms s do no1 stew to lzilve a proper logic. Be this as it may, this issue is completely independent of' the lattice structure that governs the beltaviour of predicative mass tcrms ancl plural expressions; it is only this structure that I want to address myself to in the prescnt paper." Now what I am going to propose, then, is basically thc Sollowing. First of all, lct us take seriously the morphological change in pluralization, which is present in many matural languages, and introduce an operator, """, working on 1-placc predicates P, which generates all the individual sums of members of the extensions of P. Such a starred predicate now has the same cumulative refercncc property as a mass predicate, it is closed under sum formation: any sum of parts which arc " P is again *P. This property gives rise to the introduction of a Boolean structure on the domain of discourse, E: technically, E becomes an atomic Boolean algebra which is taketi to be complete so that every subset of E possesses a sum. Now let 11. (1 be the denotation function in a model, and ( ( P (the ( extension of P. Then I"PII, the extension of "P,can be defined in terms of IlPll as the coinplete join-subsemilattice" in E generated by 11 PI(. This construction is the mathematical expression of the closure propcrty referred to above. T h e set A of atoms of E consists of the "singular objects", like this card, that deck of cards, etc. Among them are all the different portions of matter, like the water making up this ice cube. Now, let a and h denote two atoms in A. Then there are two more individuals to be called below a 6 and a @ /I, a b is still a singular object in ,4, the nzatel-kiil./i~.sionofcz and b; n $ b is the irdi~kduul.turn or pturrd objt>rtof o aod b. l ' h e theory is such that a b constitutes, but is not identical with, a .'f; b. This loohs like a wild platonistic caprice strongly calling for Occam's Razor. Language, however, seems to function that way. Take for a, b two rings recently made out of some old Egyptian gold. The1 /he N $ 6, are new, tht stutf; n + / I , is old. Sunls are partially ordered through the intrinsic ordering relation on L' which is expressed in the object language by the 2-place predicate "11". It is callcd thc in(l,'.z'id~~nl part relation (i-purt relation, for short) and satisfies thc biconditional

+

+

+

I ~ ~ I ~ J J ,

"<;"

The Logical Analysis of Plurals and Mass Terms

131

The semantic countel-part of (15 ) is the well-known Boolean rclation (wherc U, denotes the join operation on E):

Now if P is a 1-place predicate and *P the corresponding plural predicate let (,'P (the proper plural predi~'atrof P ) be true of exactly the non-atomic sums in the extension of "P. Then we can define the sum and the proper sun1 of the P's, 0sP.v and a'.rP-~:, respectively, as the supremum of all objects that are 'P and 'P, respectively ("i" is the description operator): (17) (a) oxPx : = IX("P-YA A .y(*Py -+ ~1rI.r)). (b) a3xPx : - rx("px P, Ajf(*Py -+ ~ I l x ) ) . a*xPx carries the presupposition that there arc at least two P's; in this case a.z.Px and a"sP.~*coincide. The material~fi~sion of the P's, denoted by pxPx, becomcs that object which constitutes the sum of the P's:

The poperator has the effect of merging levels of comprehension, as I called it above. Thus we M-illhave p.z.Px = /isQx, if P stands for Is a curd of v~teorzhe card decks, and Q for is r i i-nrd rrleik. cr.rP.t.and a,xQz' are not the same, however. If we have two decks of Bridge cards, for instance, axPx contains 104 atoms whereas axQv contains only 2 atoms. There is a second ordering relation, called the material part (FFZ-picrt) rCln/ion and denoted by "T". It establishes a partial order on portions of matter, but only a on , the , whole ,domain of individuals. Objects which are m-parts preorder, called I, of one another are materially ~gunlal~tlt in that they have the same portion of matter constituting them. If a is an i-part of 11 then a is an m-part of h; syn~bolicallg:

In order to explain the meaning of "T" Inore precisely let me supply the remaining concepts of the model structure to be defined below. In addition to the domain of indivictuals, E, thcre is a set D which is endowed with a join operation "U" making D into a complete, but not necessarily atomic, join-semilattice. D is partially ol-dered by its intrinsic ordering relation, defined by

<,

(20) x 5 v

iff

.r U ,v = .y

D represents the set of all individual portions of matter in the model, and as such, it is considered as a subset of A, the set of atoms of E. T h e relation (20) stands for a material part-whole relation." It can be used to order the individuals of E materially. Rlr this I postulatc for everjLmodel the existence of a scinilattice homomorphism h from E' \ (0)

132 Godehard Link to D such that h is the idcntity function on L) and conlmutes with the supremum operator. Then h is order preserving, i.e. we have (2 1)

.Y

5 ,31

+

iz(.~)5 h ( , l ~ )

for all x,y E E with x (22)

Y

,

y

iff

# 0. I can now

define the relation

h(x) 5 h(31)

<,,, mentioned

above by

( ~ x )E . ~E~ \ (0)).

I,,, is the semantic counterpart of "T".

Ob\~ioi~sIy, 5 , is coarser that1 <,,,, a Fact that is syntactically expressed by (19). In its intended interpretation, then, which is formally inlplementcd into thc nod el structure, a T b can be read as follows: "the portion of matter constituting a is m-part of the portion of mattcr constituting 6". One-place predicates are interpreted as subsets of E \ (0). (Herc, thc null clcmcnt 0 is assigned the role of a dummy object to which no predicate applies and which can take care of denotation gaps arising in our description theory; for details see 1,ink (1979, 5.2).) If P is a mass term the extension [IPII of P in a given model should be a set of portions of matter which is closed under joins. So IlPll becomes a subsemilattice of D. This completes the informal description of the Boolea??moikd simiiure @! = (E, '-1, D, h) and of the LP-M-mode/ /)/I = 1 . 1 1 ) . For precisc definitions scc bclom-. If P is a predicate we can introduce its mass term cowr.spondt?~t,called "'P; if llPll is the extension of P in a model, define

(a,

If P is already a predicative mass term we have, by this definition, IlPll term correspondents occur in natural language.14 Considcr

C IItl'l'II.Mass

(24) (a) There is apple in the salad. (b) Vx(" P.r A

ar)

(24 b) is a formalization of (24a), with P for is an applt, "'P for is uppic, and Q for is in /he s a l ~ d From . this we can infer, intuitively as well as formally:

(25)

(a) There are apple parts in the salad. (b) VXVJI("P.YA,,ITX/\@)

We have, in fiact, the following theorem (cf. (T. 14) lxlow): (26) "'Pa

-

V.y("Py~aT.11)

Returning now to the model structure consider once again the extension of a 1-place predicate P. It can contain atoms as well as proper i-sums. This general case of a ii71.vrd e.~~e?zsion is exemplified by predicates like can-)! rkepillno: think of Tom, Dick, and (together), Obelix (all by himself), ctc. Common nouns and intransitive verbs likc die, however, seem to admit on\y atoms to their cstcnsion. 1 call such predicates iIis/rihu/ive:

I

The Logical Analysis of Plurals and Mass Terms

133

(here the prcdicatc -At stands for the property of being an atom in the model). To illustrate take the intuitively valid inference from (a) to (b) in (28).

(28) (a) John, Paul, George, and Ringo are pop stars. (b) Paul is a pop star. This inference can be fornlally represented if we consider pop star as a distributive predicate P in the sense of (27). In this case the extension of "P is closed under nonzero i-parts (see theorem (T. 10) below), so every atom of an i-sum which is "P is itself "P, hence it is a P . In symbolic form the inference (28) looks likc this. (29) (a) (b) (c) (d) (e)

*P(a (3h ( 7 > c @ d ) Distr(P)

bnu@h@~.@d "Ph Ph

In a similar way we have the following valid inference with the distributive predicate Q for dip ( P stands for nninznl):

(30) (a) T h e animals dicd. So every animal died. (b) *Q(a*xPx) A.v(Px + Qx) For non-distributive predicates we have, of course, no such result, witness ravq!: n CP h might be in the extension of cnrqr while a or h alone is not. Noticc that these predicates enter formalizations unstarred. T o illustrate,

(31) (a) T h e children gather around their teacher (b) Q(c~*xP.v) (with Q for gnther. etc.). I think there is no harm in accepting this systematic difference: i.e. distributive predicates working on plural terms have to be starred, all the other predicates must not bc. For distributivity seems to be a lexical feature. If we have a formal translation procedure the predicates have to be subcategorized accordingly. There is one morc operator, "T", in the logic LPM to be presented below. ' I ' P , for a predicate P , is to be read as part~ikesin P . I introduce this operator in order to be able to distinguish betwccn the plural terms tht ~.kriWrtnand LL// the ~.hilrlren.It seems to me that in ill1 the chikdren /?uilt the rufi it is claimed that every child took part in the action whereas in tire chil~/vpnbuilt thp r.4it is only said that the children somehow managed ro build thc raft collectively without presupposing an active role in the action for cvcrjsiilglc child. This intuition enters thc fi~rmalizationof the two phrases givcil bclow. I Y ~ only bc partially characterized in want to stress, however, that the operator C ~ ~ I ' can view of the essentially pragmatic nature of its intended interpretation. So Ict me formulatc just thc following lneaning postulates for "'l'" which seem to be plausible:

134 Godehard Link

@);

(-32) (a) A.V('PX + V,y(sn)!A b)) (b) Distr(P) + A.Y("'P,Y+ PX)

I conclude the informal part of the paper with some more formalizations of natural language sentcnces involving plurals and mass terms. Most of thc principles governing thcir logic that have bccii mentioned in the literature come out valid in the system L P M below. Some of them are instantiated here, like Masse!,'s pluralitv principles of symmetry, expansion, and contraction (scc _Masscy (1976)). What I did not treat in LPIL'I, however, are any "downward" closure properties that are somel~owfelt to bc present in the behavior of mass terms ("a part of water is still watel-"). Such principles can be added when a careful linguistic analysis has succeeded in giving them a form that takes care, in particular, of the problivz (!f'nrinimal p~irts(is every m-part of lenlonade really lemonade again?). For some discussion of downward c1osul.c propcrtics scc, c.g., Hunt (1979) and Hocpclinan and Kohrcr (1980). (33) (a) (b) (34) (a) (b) ( 3 5 ) (a) (b) (-36) (a) (b) (37) (a) (b)

child built the raft Vs(Px ,Y @); Children built the raft. VX("P.~A @); T h e child built the raft. V,JI(]I= i.rP.r A QJ); The children built the raft. V.9(11 = n*vsP.v11 Q.1)); Every child saw the raft. Ax(Px

-

(38) (a) ,411 rhe children built tlze raft. (b) /\y(y = o " s P . ~+ Q.1)); (39) (a) (b)

(10) (a) (b) (41) (a) (b) (42) (a) (b) (43) (a) (b) (41) (a) (b)

P.v: .r is i~ child Q.Y:.I. built the raft P.r, Q-r: dto. J'.Y, Q-x: dto. Ps, Q v : dto. Px: dto. IQt.: s saw the raft

+

P s , Q.1, as in (33) = i"Y(QX. A A.:(."rI.V +"Qz)). Tom and Dick carried the piano upstairs. P(n @ /)); a: Tom, 6: Dicli, P.v: .x carried the piano upstairs Tom and Dick carried the piano upstairs, so Dick and Tom carried it upstairs. P(a B b) + P(h @ a) (S~~III~~IC~I;~) John and Paul are pop stars and George is a pop star, so John, Paul, and George are pop stars. ' ; ~ ( a
0:

-

i '1

@Y:

The Logical Analysis of Plurals and Mass Terms

(45) (a) This ice cube is water. (b) Pi.z(s L a);

135

a: this ice cube Px: as in (43) (46) (a) T h e gold in Smith's ring is old, but Smith's ring is not old. (b) Qis(Px .Y L a) A I@; P.Y:,v is gold .T is old : Smith's ring

2 The Logic of Plurals and Mass Terms (LPM) LP%T is a first ordcr predicate calci~luswith the usual logical constants "I", "v", "A", , H", "V", (LA",the description operator denoted by "i", and the abstraction operator "A" .1" T h e syntactic variables are, for forn~ulasof LPM, "d)", "tk", "x"; for (<+v

kL

individual terms, "a", "B", "I."; for variables, "Lv", ")I", "z"; for (1-place) predicates, "P","Q". These symbols can also appear with primes and indices. For metalinguistic (definitional) identity the s!-inbol "A", (": =") d l be used. Thc set of I-place predicate constants contains two specified subclasses: the set M T of prtdii.i~lizlr' nrass t~rmsand the set DP of 1listribz4tieltpredicates. h4'T and DP are taken to bc disjoint sets. As specialprimilii:~.\:l~nlbolsof 1,PM we have a I-place predicate symbol "E", thrcc 2place predicate constants, "II", "T", "L", and two operators on 1-placc predicates, cc*?, ~('1'" , The intended interpretations arc the following. En stands for " n e.~is~s"; all/) for "a is an in~lizli~llirzlpav~ (i-part) of h"; r ~ T hfor "11 is a ma teri~zlpr~rl (m-part) of I)", rr 1,I!. for "o ~onsti/u/esor mnkts up h"; *P for "thc pllrml prnlnutr of P";"'P for " p ~ ~ n o hinc ~ P". No\\- I introduce a number of defined e,zpri~ssion.s. Let P, Q be I-place predicates, R a 2-place 1 Gl~neralfirst o~prler i/blr~ezlin~kons. predicate, and n an individual term.

Furthermore, the usual definitions for the following formulas involving R arc assumed: R ~ f (R) l ("R is reflexive"), Trans (R) ("R is transitive"), S I ~(R) I ~ I("R is symmetric"), -Antis),vu (H) ("R is antisylnmctric"). We then have, with P r o (R) for "R is a preordering rclation", P O (R) for "R is a partial ordering relation", and Eqzr (R) for "R is an equivalence relation": (D.5 ) PrO(R) (D.6) PO(R) (D.5) Equ(R)

--

H

Kefl(R) 4 Trans(R) R
136 Godehard Link

-

2 Defined preriicate con.stnnts. With two individual ternls L I , h, let a = b stand for " a equals b", a b for "a is m-equival~nlto b", ,-l, n for " ~ is i an L L ~ O ~ I I " , and Mpa for "a is a portion o f matter". Define

3

&find prcilicate oper.rztors. Let P be a 1-place predicate and rl an individual term. T h e proper plz~rnlprrrlicale CP of P and the nlass term (.orrespond~ntto P, "'P, itrc then defined as follows.

(D.12) ' ~ r r ( D .13) "'Prr 4

t t

t-t

Defined lndicir/uu/terms. With I-place predicates P and individual terms a, b, definc the i-sum of the P's, the proper i-sum of the P's, the i-suttz of N and h, thc ~?zatericll .fusion of the P's, and the rtiateritcl JLsiun of N and 11, respectively:

( D .14) axPx : = ( D .15 ) aS.vPx : = (D.16) a i s b .. ( D.17) psP,r : = ( D .18) n b : =

+

5

"Pur,r~~$t~~ Vjl(*Pll A &Ti.+ L.y)).

ix("P.1.A Ay("Plt ia("~,nA A y(*Py abr(Ii,@ I/Jbr ix(s L crxPP~) ~ , V (3 I (I/,)&r ~

-+ +

.I)

rZ s ) ) . r[s ) ) .

Sp~cirrlabbrczliali~wfuvtn~rlrcs.Let P be a 1-place predicate.

( D .19) ( D .20) (D.21 )

--

Dlstr(P) tf A.v(P;t. A&,\') M ( P ) tt As(P.t lVpvr) Itiz!(P) t-t /'\.rA~!(x ,)I + (Ps ++ PI,)).

Dlslr ( P )is to be read as " P is ~listributiz~e" (i.e. Pis true of atomic individuals only), -44( P )as " P i s a r?tnterialpredicntc"(i.e. it is true of portions of matter only); In;r (P) means: " P is an in~lariu~~t predicate" (i.e, it is closed uncler substitution of mequivalent terms). 6

Mermirrg postulates fnr.".'"'. v,]f(xl-I]! A. PI!)). A . ~ ( ~ Pt.. YP,.) i.c. for all .r, if .v partakcs in P, thcn .t is an i-part of a -11 which is P (VIP.1); if P is distributive, then for all .r, .\- partakes in P just in case .z. is P (MP.2).

The Logical Analysis of Plurals and Mass Terms 137 'I'l1e sevnantil:~c!f LPM. I shall en~ploythe usual set-theoretic terminologj (scc, for for set iilclusion and 't/'[YJ" for the/: instance, Eberle (l970), Link (1979), with image { J'(.Y)1 .T E 1') of Y under the function J': _X + Z. Moreover, some elementary notions of lattice theory are needed (see, e.g., Griitzer (1971). Sikorski (1969) for their definition) pr~rtircladeritzg replnlion, pnrtial!j/ or(l~>r~~rl sr/ ("pos~t"), 1elemenl, 0-ele~tent,atom, sewzi-lattice, Join-semil~ittir~,lattice, ~~{blntlice, iill>al,prin~.ipnl illral, cov~pleteI~ittit.e,Boolc~znlattice (ulgrhm), irtomll. ldttice, (serizi) llittice /~orrzomo~pllism. The supremum of a subset k;of a poset X, relative to its ordering rclation if it exists, will be denoted by sup,- Y , the principal ideal generated by an clement .v E X,whcrc X is a lattice, by (x]. So we have (x] = E X Ij t 5 s). If X is a complete (semijlattice. and Y C X, then [E'] denotes the complete sub(semij1attice generated by Y. 1 now provide a model-theorctie interpretation of the formal system LPIL1.

"c"

<,

-

(D.22) 1 2

3

4

Boolenn rnodel structure [pith homogeneous kernel ("hoosk") is a quadruple @, (E,A, D, h ) such that E is a complete atomic (c.a.) Boolean algebra, with join operation U, and the intrinsic ordering relation 5,; A E is the set of atoms in E; D C is a complete join-semilattice with join U and ordering relation 5; h : E\{O) + D is a semilattice homomorphism such that (i) h2D idD, i.e. h(x) A .t. for all .r E D; (ii) / ~ ( s u= Ip B) . ~ = suphi B] for all B C E\{O).

It follows from this definition that we have, in particular, for all s,,I!

r: E\{O):

The homomorphism h induces a second ordering relation on E\{O) which will be called the material part relation on E\{O) and denoted by <_,,,: I

(D.23)

.tT

<,,,

iff

/z(x)

5 /I()!)

I,,, is only a preordering relation. It gives rise to an equivalence relation, defined I

(D.24) s N ,,,11

iff

by

xxl,,,.11 and . ) ! S m . ~ ( b ~ , . ~ ~ ~ E \ { o } ? ,

I

and thus to a partition of E\{O} into equivaleilce classes each containing all the indivicluals that are made up by the same portion of matter. In the following definition of a model for 1,PM I am going to interpret only I-placc predicates, for simplicity (except for the special 2-place predicate constants). All the other many-place predicates receive their usual first order interpretation.

I

(D.25) A moi/el.fiv LPhl is an ordered pair .MA(@, (1 - (I) such that if1 hl, 24the srt oj'nlonrs in 1 @ (E,A, D, h ) is a boosli (E is the domai~rq/'indiz~ir/~inl.s M, D the S P qf'portions ~ c!rmatter in A4, and h the "nznterir/licn /it111Ji~n(.~ior~" in Aq);

1

138 Godehard Link 2

11.II is a first order assignment of denotations to the primitive expressions of LPM such that if a is an individual constant; (i) lll~llE E if P is a I-place predicate constant; (ii) 11 Pll E\{O) if P E DP; (iii) 11 PI1 (iv) IlPll is a complete subsernilattice of D if P f MT; (v) I " ~ P I I{r t Elx # 0 & 3.~1E IlPll such that x 5;, y ) ; (vi) II"PII-IIPII ifP E DP.

T h e model is such that predicates are interpreted in the non-zero elements of E (condition (ii)). Conditions (iii), (iv) guarantee the special properties of the distributive and mass predicates, respectively. (v) and (vi) are designed to validate the meaning postulates (MP.1) and (MP.2) above, respecti\rely. Let h f= I /I) bc a model for 1,PM. The usual first order sernantical reci~rsion rules are assumed, where quantification is taken to run over the non-zero elements of the domain of individuals only. T h e truth and (/cnota/ion cortditiun.~for the special primitive symbols of 1,PM are defined as follo\irs (" 1" stands for "truc"; bivalence is assumed):

(a,

(D.26) (D.27) (D.28) (D.29) (D.30)

~ ~ P a [ ~ ~iff l 11i.sP-~l=d iff IIEa,ll 1 iff Iln.llhI=I iff IlaTbllAl iff

(D.31) (D.32)

Ila L lill -1 iff l\*pll=rllpll~,

IlaIl E IIPlI; d # 0 and 11 PI = { d ) , and 0 otherwise; IIa11 # 0 Ilnll # and llffll 5,Ilhll; Ilall, Ilbll # 0 m d h(llnll> 5 h(llbll); 1611 # 0 and IlallAh(llbI); (the complete U, -subsemilattice generated by 11 PI[);

,4 nu~nbcrof scmantical facts can now. be derived that give truth and denotation conditions for all the defined s~mbolsof LPM.

llp c QII-1 iff Ilpll l l Q I IIP = Q l l A I iff IIPIIAIIQII IIP @ QII=IPI u IIQII I I O ll={IInlI) [la = bll=l iff I l ~ l l ?Ibll # 0 and II~ll-llbll iff /Iff 11, 1611 # 0 and h(ld)-h(llhll) 110 bll-1 (IA4ta II 1 ifT EA ~ ~ M p ~ i ~ [ = l iff llal E D ll*PII A {X E E13X C lIPI & X # 0 s.t. vv-supiX} IICPII = II*PII\A IIU1PII A (suPih[llP11]]=(x E D ( x 5 sup5h[llPII])= .r E 0133' E [*PI\such that ,v 5 k ( ) l ) )

-

The Logical Analysis of Plurals and Mass Terms 139

(60) 1o.t.P.rII = sup, IIPII, with sup,@'O (61) Ia*sP.vI lon.Pxll if llPll has 2 2 elements, and 0 otherwise (62) In @I hll 11~111U, llhll (63) 11j1.~P;t.(I = sup,hlllPII]if llPll#@, andootherwise (64) /la + bll = h(llall) h(llbll) if llall # 0 # 11BIl Let the notion of ~rutlrin a given model be defined in the usual way. A formula is italirl if it is true in every model. I now give, in a loosc order, a list of theorems of L P M that come out valid undcl- the above interpretation. I may mention, in particular, the honrogenrous refi.renl,e properties (T. 1l), ( T . 12), and the eslsierice and idunlil.)~~,rilerinfor the a- and ![-terms, (T.16)(T.21). (T.1) (T.2) (T.3) (T-4) (T.5) (T.6) (T.7) (T.8) (T.9) (T. 10) (T. 11) (T. 12) (T.13) (T.14) (T.15) (T.16) (T.17) (T.18) (T.19) (T.20) (T.21) (T.22) (T.23) (T.24) (T.25) (T.26) (T.27) (T.28) ( T .29) ('r.30) (T.31) (T.32)

rt. = h

aHbr\hIIa aIIb + aTh n =B nTb~bTn Equ( = ) A Equ( ) PO(17) fi PrO(T) Dlstr (P)+ As(P.y -+ J ~ P ~ Y ) P c "P A.v (-Atx 4 (P,Y c-1 "P.1.)). Ax(*Px xlIaxPx r\V,y(P)l~ . ) l n . ~ ) ) . Distr(P) + AsA.y('Py A x l l ~ + l "Px) AxAy(" P.r r,"Py + "P.r 91,) for P E k I T A.x A-]l(P,r A P)!+ P,Y+ y) M (P) + P C "'P M P ~ ("lm4 v.)I(*P~ b, ~LT-II)). VwP*t.A P c Q 4 *Q(a.rP.v) V.vP.t. 4 EaxP.r VxP.v E~J'P,z' Ea'xP-r +t VxV.y(Px A P y A x # .I/) P = Q + a x P . ~= cr.~Qx Dislr(P) A Distr(Q) --; ((TXPX = a,rQr + P = Q ) P = Q 4 p.~P,z.= 1rxJQr jltd 4 a = a.r(.r = a ) A4pn n = p,x(,r = n ) ~ i x P x= is(Px A Ajl(l3, + yTx)) for P E M T a h p.v(x = a) = /i.r(x = b) /1xPs = ix.(.VC /lLxPAY) Mp(px.Px) a L h c-1 Mpo n (1 -. 17 /lbvPx-. CTXP.T 0xP.z. = a.vQ~+ /~xP,t^= [L~VQY A.vA.y (x"y * .Y CE = -11) A.v/ly(sTy .r +.)I = y) c-1

-

-

-

-

-

140 Godehard Link

3 Applications to Montague Grammar I.et the basic synta.r be as in PTQ'" T h e category CN of common noun phrases h a to be subcategorized into MCN (mass noun phrases), SCN (singular count noun phrases), and PCN (plural count noun phrases). 'The quantifiers disc~~ssed are a, some, ihe, a// the, roer)r, all; they are of category T / C N with suitable subcategorizations (term formation is done compositionall?;). have the obvious plural rule

a,,,

cpl=

cpl&

where [A hand becomes /lands7 5; child, c/zil(/ren, ctc. T h e logic of P T Q TITL,, is extended as follows. TIT'L', or esten(/e(l Y'ITL, contains as new symbols all the special symbols of LPM. Additional rnenni?rg/i/ c,rprts.sions are: E ME,, for t = e(t E Type, p-tt or ST/,[ E .ME,,); (i) *<, (:(, .l'< t MEi,, and (ii) aucb, ~ " u d ) , luqb E ME,, and ,urn4 E ME,. (z f Type, u E V~zr.,,gS € ME,); (iii) all/?E IZIE, ancl a % fl E ME, ( t E Type, a, E /tlET); (iv) a T P , a j, a L / 3 E ME,, and ct+p E ,441/IE, (x,B E )'llE,.). T h e sevlanlir.s for 'TI'I'L' has to specify the "boolification" of thc sets of possible denotations since the i-sum operation applies to expressions of any type (m-sums are formed on level e only). Let E be a c.a. Boolean algebra, I , 3 sets, and K = I x J. A Boolean hiererch~y?fpossib/c (knotnlions over E with respe1.t to K is a family of sets (DT,E , K ) i t l j l r defined as in P T Q , but cndo\r.cd with a c.a. Boolean structure for every D, as f o l l o ~ ~(i) s : T h e sets Dl, = E and D, = 2 are already c.a. Boolean algebras; (ii)l; g E D,,; then: (f' U,, g)(k) : = J'(k) U, g(k)(k E K), similarl!? for the other operations; (iii) J'; g E D,,; then (f U,, g-)(.r): = /'(x) U, g(x) ( s E D,), similarly for the other operations. T h e resulting structures are c.a. Boolean. An in!erpretation for TITI,' is a 7-tuple 8=(E, A, D, h, I , 3 , I l . I I ) such that (i) (E,.4, D,h) is a boosk, (ii) K = I x 3 is as in P T Q and (iii) I l . j J is the interprctation function that naturally emerges from a combination of T I T L and LPM. As rults o/' truth ntzd dtnotntion provided by an appropriate recursive definition let me mention the the characteristic function of L{x t DTlllill(x)= I}], ,,([ E MEn); following: (i) (ii) *
"'c

//'
x,,,:

<

<,

(c

The Logical Analysis of Plurals and Mass Terms 141 Most of the basic count nouns like child are taken as distributive, similarly IV phrascs like die or see. In accordance with what I did above the translation rule T4 involving distributive expressions has to be such that their translations al\v\lays entcr this rulc under the star operator. While this seems to me the correct move from a technical point of view 1 am aware that it carries the empirical prediction that d i s t r i b ~ t i ~ i is t ya lexical property. I now give transl~~tions for the quantifiers. Let U be the translation relation, and P: ) , ~ I [ P (A ~Aslxnl1 ) -'V~(X)]].

-

1 2 3 4

a),,

a, some the a// /he emry, nN

U i @ ~ r [ ~ A( Pr ()x ) ] u ~ Q P lv ~~ b A, )AX[Q(X)

u U

I~QFA.~[.,,= OYQ(+) I~QFAX[Q(X)+ P ( s ) l

--

x n y l PC?)]

P(.~?)I

a applies only to singular count noun phrases and GP1to PCN phrases. 'The quantifiers some and the, with one and the same translation, apply to both singular and plural phrases. As can be seen from 5 through 8 below it is only the incoming CN phrase which differentiates between the appropriate singular and plural readings. While this is obvious with some, a comment on the transformations under 7. is in order. T h e conjunct A s [ Q ( x ) + xIIJJ-] in the translation of the asserts maximality which is needcd for the intended sum formation in the extension of*Q. At the same time, however, it gcncralizes the usual uniqueness condition for definite dcscriptions. This comes out when /he is applied to a singular count noun like chili/: T h e i-part s of.)! in 7. cannot bc 0, and.y is an aton? because of DISTR (i,hildl);so x equals .y. 1 think this is a nice instance oC strict compositionality in times in which this principle has come under heavy fire in vicw of all sorts of recalcitrant data (see below). Finally, 9-1 1 show that universal q~lantifiuationis as in LPhl where, again, the difference in meaning between the c h i l ~ / ~ eand n 1111 thi~ a/rildrr.n is only partly characterized ( P being defined in terms of "l'").

5

a child some child some water 6 (sonze) children 7 the chill

8 the childrepi

1 may mentioil here that the incorporation of nuincrals like oue, III~O, thr'e, into the fragment is straightforward. T h e extension of t h r e ~men, for instance, is the set of 1111 isums in l/rnen'll~llcmsn'llwhich contain exactly three atoms. Sentences like rlrrut nrm went t o .mom a rnearlolz?receive a translation of the form V:z[(lhreeyylen)'(z), A Q ( z ) ] .

142 Godehard Link ?'he final topic 1 want to say something about is the vexing pl-oblem of relativc clauses with more than one head noun.'' Call those structures I!)u/~l'ilc.Let me consider restrictive relative clauses only. First of all, there is a rather fi-iendly type of hydra, like the following. (67) the German or Austrian John met yestcrda!~ (68) the cabinet-member and mafioso who was deeply involved in the scandal P T Q style relativization, i.e. C N modification, takes care of these pets yielding the obvious representations for (67) and (68), respectively:

-

( 6 7 ) the ( (German or Austrian) such that John met him yesterday) Bv.]~A.r [Gemran' @ Ar,strirrnl(.r) n @[.r] .r = ,)I]A P(y)] (68') the ((Cabinet-member and mafioso) such that he was decply in~olvcdin the scandal PV.)~[A r [Cnbinet-menrb~r~ G n~~/io.so'(.v) n i,hlsl * r. = ,)I 1 n P(y)J

-

Here, @ and rl/ are the translations of the relative clauses of (67)and (68), respectively; furthermore, i€3 q ;Is[r(.~)v rl(s)] is thc Boolean join and ( ci,~17 h[[(x) ,\ ~(.r))l.' the Boolean meet of iand 11. This case is simple because CN wnjunction does not lead I to plural structures, and there is only one determiner present.'X 1 Next, let us conjoin t\vo singular count nouns to form a plural phrase. 13san1plcs arc ! 77alin l ~ r t l n?ot?latl, l h q f crnd girl, hushand arivl )PI/;>,I~rnlllovJ( I r1(1tennllt. Such phrascs can bc ; true of a sum of two individuals one being of the first and the othcr bcing of thc sccond kind. An appropriate translation, therefore, is of the form (69) ([ and 4)

U

IL.zV.YV,)![~'(S) n )11()1)A c = s $,)I]

\%Tiththis sentences like J'oltn art(/ Mar)! nrt hu.chanrl N I I 1711/i~ ~ can hc handled in a first approximation wl~ilcthe non-symmetric features of this sentence have to be taken care of by other clues. This is because the predicate (69). by itself, is ncccssarily symmetric reflecting the fact that pluralization has the force of group formation which typically gives rise to symmetric construction^.^^) Such is the following CN phrase: (70) boy and girl who dated each other Wit11 (69), the standard P T Q r u l e yields for this:

Now term formation with one and the same determiner for both head nouns is again standard if we decide to have it distributed over the two nouns in a kind of copying process. So, for instance, n ho)! and il girl n~homecptbccon~es (72) an((buy c l t d girl) sltch tllirt the)! meet) U PVn[VsV.)~[bo~)~(,v) ~ g i r l ' ( j )A s = s B ,y A YIIPP~'(.C)] A P(.s)]

The Logical Analysis of Plurals and Mass Terms 143 In a further step we admit conjunction of plural nouns. This move doesn't really add any ncvl difficulty so I can write down an example immediately. Let (1 be the translation of thc phrase J I I I ~ C M ~ EM^ S profi~sorsm/lo IlorJ ~ I I Cin~ srrrrt:

Adding a determiner as before, say the definite article, cve get thc term

(74) the students and the professors who had met in secret u P V z [ z = fi*.I"O(x) A P(z).I h

T h e present conception of conjunction of two CN phrases [ and 17 is such that the extension of (( nntl y) contains i-sums of objects which are of kind [ and r l , respcctively. Unlike the Boolean join this carries the presupposition that both sets of objects be non-empty. I think this is as it should be. Notice that in this case the overall sum is the same as the one u-hich is formed in terms of the Boolean join, i.e. wc have a.z.(i'@ ~j')(w) = ax([ n,zd vl)'(.v). What is not captured in either approach, howcvcl-, is thc proper pair reading which is dominant in sentences like Inncllorri.~ilrrd tr~r~lnts 1~/7o l ~ n t eerrc./l other rvili ai~r~nysfifinil so~lrrhingto argzic r~lrout.I find these constructions hart1 to treat at the moment though I an1 confident that they will finally lend thcn~selvcsto an analysis which is compatible with thc present framework. What is nccclccl here is somc notion or ordered pair in the objec~language, apart horn the symmetrical sum operator."' It is only with such an additional instrument, it seems to me, that 1r.c can. in a close-to-language treatment, attack the most dangerous hydra lurking in the reillin of pair reading and hrmching quantification.21 T h e last type of hydra I want to address myself to shows up in a scntencc like thc followiing who hacl met in secret \vest (75) all of the students and some of the professo~.~ arrested after tlle coup d'ittat. From a purely syntactic point of view it might look natural to try a T-S-analysis 11erc: first conjoin the t n o unrclativized terms ( I / / the .stud~'?clsand som~.cd'the p~(!fis.sonand then modify the conjunct by the relative clause. Iiowever, I agree with Jansscn (1981) in his sceptical attitude towards the semantical soundness of this approach in