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2058 XXXIII. Formalization Tendencies and Mathematization in 20th-Century

195Ϫ222. New York: Spartan Books. (Repr. in Ϫ. 1999. Pathways of the Brain: The neurocognitive Makkai & Lockwood 1973.60Ϫ83.) basis of language. Amsterdam & Philadelphia: Ϫ. 1971b. “The Crooked Path of Progress in Cog- John Benjamins. nitive Linguistics”. Monograph Series on Lan- Lockwood, David G. 1972. Introduction to Stratifi- guages and Linguistics 24 (Report of the Twenty- cational Linguistics. New York: Harcourt Brace Second Annual Round Table Meeting on Linguis- Jovanovich. tics and Language Studies). Washington DC: Georgetown Univ. Press, 99Ϫ123. (Repr. in Mak- Makkai, Adam & David G. Lockwood, eds. 1973. kai & Lockwood 1973.12Ϫ33.) Readings in Stratificational Linguistics. University, Alabama: Univ. of Alabama Press. Ϫ. 1975. “Mutations and Relations”. The First LACUS Forum 1974, 540Ϫ555. Columbia S. C.: Parret, Herman. 1974. “Dialogue with Sydney Hornbeam Press. (Repr. in Adam Makkai, Valerie M. Lamb”. Discussing Language, 179Ϫ219. The B. Makkai & Luigi Heilmann, eds. 1977. Linguis- Hague: Mouton. tics at the Crossroads. Padova: Livinia Editrice; Reich, Peter A. 1969. “The Finiteness of Natural Ϫ Lake Bluff, Ill.: Jupiter Press, 407 423.) Language”. Language 45: 831Ϫ843. (Repr. in re- Ϫ. 1983. “On Determining the Number of Strata vised version Syntactic Theory I: Structuralist ed. in Linguistic Structure”. The Ninth LACUS Forum by Fred W. Householder, 258Ϫ272. Hardmonds- 1982, 189Ϫ202. Columbia, S. C.: Hornbeam Press. worth, Essex: Penguin. Ϫ . 1984. “Semiotics of Culture and Language: A Schreyer, Rüdiger. 1977. Stratifikationsgrammatik: relational approach”. Semiotics of Culture and Lan- Eine Einführung (ϭ Anglistische Arbeitshefte, 11). guage. Vol. 2: Language and Other Semiotic Sys- Tübingen: Niemeyer. tems of Culture ed. by Robin P. Fawcett, Michael A. K. Halliday, Sydney M. Lamb & Adam Makkai, Schütze, C. T. & Peter A. Reich. 1990. “Language 71Ϫ100. London: Frances Pinter. without a central pushdown stack”. COLING-90: Papers presented to the 13th International Confer- Ϫ. 1985. “Descriptive Process”. The Eleventh ence of Computational Linguistics on the occasion LACUS Forum 1984,5Ϫ20. Columbia, S. C.: Hornbeam Press. of the 25th Anniversary of COLING and the 350th Anniversary of Helsinki University ed. by Hans Ϫ . 1987. “Linguistics, Semiotics, and the Human Karlgren, 64Ϫ69. Helsinki: Yliopistopaino. Information System”. Georgetown University Round Table on Languages and Linguistics 1986, Tai, James H.-Y. 1985. ‘Temporal Sequence and 51Ϫ63. Washington, D.C.: Georgetown Univ. Chinese Word Order”. Iconicity in Syntax ed. by Press. John Haiman, 49Ϫ72. Amsterdam: John Benja- Ϫ. 1994. “Relational Network Linguistics Meets mins. Neural Science”. The Twentieth LACUS Forum 1993, 151Ϫ178. Columbia, S. C.: Hornbeam Press. David C. Bennett, London (Great Britain)

221. The development of Montague

1. Introduction wider sense, however, it has to come to stand 2. Historical Background for a more or less loosely connected set of 3. Montague Original Model views on doing and on integrating 4. Applications and Developments semantics and syntax. 5. Beyond Montague Grammar 6. Bibliography In this article we will first sketch some of the developments in 20th century , phi- losophy and linguistics that form the back- 1. Introduction drop of Montague’s grammar model and briefly relate his model to contemporary con- The term “Montague grammar” denotes, ceptions in both logic and philosophy and in first of all, a particular grammar model de- linguistics. Then we will describe the original veloped by the American logician Richard model of Montague grammar in some detail, Montague in the late 1960s, early 1970s. In a isolating two core principles that are respon- 221. The development of Montague grammar 2059 sible for its remarkable and lasting influence. individual in different ways. Hence a sentence This more systematic section is followed by that asserts the identity of the of an overview of the various ways in which the first description and that of the second Montague’s model has been applied, enriched can be informative. and modified in the period that lasted from Frege extends the senseϪreference distinc- the early 1970s until well into the 1980s. Fi- tion to other categories of expressions: predi- nally, we will briefly review some of the cates and relations, and sentences. His treat- factors that have led to the demise of Mon- ment of of predicates and tague’s original model, and indicate which of relations is somewhat complicated (see his its more general principles still survive in its “Funktion und Begriff” (1891) and “Über successors. Begriff und Gegenstand” (1892), both re- printed in Frege 1962.) He makes an onto- logical distinction between properties as ab- 2. Historical backgrounds stract objects, referred to by complex terms such as “the property of being a horse”, and No history of modern semantics, and cer- the referents of expressions which are actu- tainly no history of Montague grammar, can ally used in a predicative manner, such “is a ignore the momentous influence of the works horse” as a constituent of a sentence such as of Gottlob Frege, the German mathemati- “Pegasus is a horse”. According to Frege cian and philosopher who is also one of the these expressions refer, not to properties, but founding fathers of modern mathematical to what he calls “unsaturated objects”, i.e., to logic. It is no exaggeration to state that much objects which need to combine with another of the work that has been done in linguistic object, in this case an individual, to form a semantics and in philosophical logic since the complete thought. beginning of the 1920s has built, consciously A thought, then, is what constitutes the or unconsciously, on his ideas. sense of a sentence. As such it is constituted Two of Frege’s inventions need to be sin- by the senses of the expressions occurring in gled out in this , viz., the distinction the sentence. This explains, for example, why between ‘sense’ [“Sinn”] and ‘reference’ [“Be- the identity statement expressed by “A = B” deutung”], and the principle of composition- may differ in cognitive value from the one ality. expressed by “A = A”. The latter is tautolo- gous, and hence uninformative, whereas the Fregean sense and reference. In his seminal former, as we just saw, can be informative, paper “Über Sinn und Bedeutung” (1892, re- even when true. The difference between these printed in Frege 1962), Frege argues that a two sentences is explained by Frege in terms proper account of the cognitive of of their respective sense, i.e., in terms of the expressions requires a dis- thoughts they express. The thought expressed tinction between the sense of an expression by “A = B” differs from the one expressed and its reference. Every expression refers to by “A = A” because the former is composed something external to language: an object, a of the sense of “A” and the sense of “B”, property or a relation, an event or a situa- whereas the latter involves only the sense of tion. But its reference does not exhaust an “A”. (And, of course, both contain the sense expression’s meaning, as, for example, reflec- of the identity.) tion on the meaning of identity sentences re- Frege’s senseϪreference distinction re- veals. If such a sentence, say “A = B” is true, instates, to a certain extent, a much older then the reference of A equals that of B. Yet, one, viz., that between the and the Frege notes, a true identity sentence need not of a . However, his analy- be tautological, on the contrary, it can be sis extends the tradition in two important quite informative. This is due to expressions ways. First of all, he does not limit the not just having a reference but also a sense, senseϪreference distinction to predicates, but which Frege characterizes generally as “the applies it across the board, i.e., to expressions way in which the reference is given” [“die Art of all kinds of categories. Of course, most des Gegebenseins”]. Descriptions provide syntactic categories contain, besides lexical suitable examples: “The first man to set foot elements, complex expressions. And some on the surface of the moon” and “the com- categories, e. g., that of sentences, do not mander of the Apollo 11 mission” refer to contain any lexical elements at all. In order one and the same man, yet characterize this to assign senses to complex expressions in a 2060 XXXIII. Formalization Tendencies and Mathematization in 20th-Century systematic way, Frege extends the tradition in Grammatical form versus logical form. In the another way and assumes that complex Begriffsschrift Frege develops a formal nota- senses can be built up from simpler ones in a tion, that is intended as an improvement of step-by-step, syntax driven fashion. (Whether natural language as a tool for scientific Frege actually stated this compositionality analysis, in particular for use in mathematics principle explicitly is subject of debate. For and philosophy. Hence Frege makes a dis- present purposes we may sidestep this discus- tinction between formal languages on the one sion: the compositionality principle of sense, hand and natural languages on the other. The if not explicitly endorsed by Frege, is cer- former have a rigorously defined syntax and tainly implicitly presupposed by many of his semantics, something that the latter lack. arguments, and in one form or another has Natural languages, in some respects at least, deeply influenced modern theories of mean- do not display the logical forms needed for ing.) unambiguous representation of thoughts and inferential relations. If such is our aim, Frege Fregean compositionality. Here we touch on a holds, we need to distinguish between gram- second element of Frege’s thought that has matical form and logical form and bring out shaped modern thinking and theorizing the latter in a suited for the about meaning and semantics, viz., the idea kinds of representations and inferences we that the meaning of a complex expression is are interested in. determined by that of its constituents parts Thus Frege postulates a fundamental dis- and the particular way in which these con- tinction between natural and formal lan- stituent parts are combined to form the com- guage, one that has deeply influenced the philosophical outlook on language for a con- plex whole. siderable time, in fact well into the 1960s. So The idea of defining the interpretation of here is a Fregean idea that did not serve as a complex expressions in a stepwise fashion, precursor of Montague grammar and other following their syntactic composition, lies at modern semantic theories: obviously, the idea the heart of Frege’s earliest work, the famous of natural language being vague and unsys- Begriffsschrift of 1879. Here Frege applies it, tematic does not invite application of for- not to natural language expressions, but in mal methods. the construction of a formal language. One This idea of a rift between natural and for- of the challenges Frege takes on in the defini- mal language has been a mainstay of both tion of his “conceptual notation” is that of logical positivism and ordinary language phi- giving a proper analysis of relational senten- losophy, the two main branches of 20th cen- ces with multiple quantified arguments, e. g., tury analytical philosophy. Of course, the sentences of the form “for all x there exists a conclusions they derived from it are different. y such that Rxy”. If one allows relations with Logical positivism favored the construction an arbitrary number of arguments, the of formal languages as tools for philosophi- number of combinations is arbi- cal analysis, whereas ordinary language phi- trary as well, and hence cannot be specified losophy concentrated on detailed, but infor- and interpreted in advance. Therefore, the in- mal descriptions of the various ways expres- terpretation of such combinations has to be sions of natural language are used. Both, characterized in some other way. Frege’s fun- however, rejected the idea of a formally ex- damental insight here was to marry inter- plicit analysis of natural language itself, and pretation with syntax: the step-by-step syn- for similar reasons. In philosophical and tactic construction is mirrored in a similar , too, this preconception step-by-step interpretation process. To each of the nature of natural language remained step in the syntactic construction process cor- dominant for a long time. (A notable excep- responds a semantically significant opera- tion was Hans Reichenbach, who as early as tion: the application of a function to its argu- 1947 developed logical analyses of various ments, the introduction of a connective, or classes of natural language expressions, that that of a quantifier. Every constituent expres- are not only detailed and systematic, but also sion hence has a interpretation of its own, very sophisticated. His theory of verb tenses, and the interpretation of the entire formula, for example, is still one of the most influen- however complex, is determined by the inter- tial around.) For example, when Tarski de- pretations of its constituent parts, as re- veloped his theory of truth in the 1930s he quired. quite explicitly stated that such a formal 221. The development of Montague grammar 2061 theory can be applied to formal languages formal, logical approach to semantics of only, and never to natural language, in view natural language to gain a firm foothold in of the fact that the latter’s syntax does not mainstream linguistics, and even then it allow for the required rigorous formal char- caught on mainly outside the Chomskyan acterization (Tarski 1944). tradition, in which it never gained any real All in all Frege’s heritage as accepted by following. However, be that as it may, Chom- his immediate successors is a mixed blessing. sky’s work did pave the way for the logical His ideas on sense and reference are devel- approach in semantics, since it showed the lo- oped directly in conjunction with analysis of gicians and philosophers that their tradi- natural language, but these remain informal, tional assumption that natural language is and the distinction itself, as far as natural too unsystematic and too vague to lend itself language is concerned, is never developed in to formal treatment, is unfounded. a systematic and formal fashion. Frege’s sharp distinction between formal and natural Possible Worlds Semantics. Another impor- language is of course one of the main obsta- tant development that had to take place in cles here. Initially, compositionality is intro- order for Frege’s ideas on sense and reference duced in the context of Frege’s development to become applicable in actual systematic de- of his “conceptual notation”, i.e., with regard scriptive and explanatory theories, was that to the interpretation of formal languages. of semantics. Originally, pos- However, it is also obviously present in his sible worlds semantics was conceived of as a later informal work on natural language and model theory for modal and intuitionistic the senseϪreference distinction. logic, in the pioneering works of Stig Kanger, So other developments, too, were needed , and Saul Kripke. Soon, in order for logical approaches to come to however, its was extended and it be- fruition in the domain of natural language. came the primary analytic tool for dealing First of all, the idea of natural language be- with all kinds of intensional concepts in a ing too vague and unsystematic to lend itself formal way. In particular the Fregean senseϪ to rigorous formal treatment had to be up- reference distinction proved to lend itself to turned. Secondly, Frege’s senseϪreference formal reconstruction in this conceptual distinction needed a formal machinery to be- framework. come applicable in a systematic way. And Assuming a set of possible worlds, refer- thirdly, additional logical tools were needed ences of appropriate kinds are assigned to ex- in order to apply the idea of compositionality pressions in each world. Thus, a term is as- to natural languages, which have a much signed an object as its referent in a world, richer structure than the formal languages a predicate a set of such objects, an n-place of logic. relational expression a set of ordered n-tuples of objects, and sentences are assigned truth The Chomskyan Revolution. Probably the values in worlds. Much of this is familiar most significant development in 20th century from ordinary modeltheoretic semantics. By linguistics, and one without which Montague relativizing reference to possible worlds, Grammar would not have been possible, is however, we also avail ourselves of the func- the rise of the Chomskyan paradigm. The tions that maps worlds to these . idea of describing and explaining natural lan- Thus an individual concept expressed by guages by means of the same kind of formal some term is a function from worlds to ob- that are used in the definition of jects. Similarly, a property is conceived of as formal languages, was a radical change, and a function from worlds to sets of objects, and one that caught on quite rapidly. Already be- an n-place relation as one that links worlds fore the appearance of Chomsky’s Syntactic with a set of n-tuples. A , finally, Structures in 1957, some philosophers pro- is a function from possible worlds to truth posed to wed formal syntax with the ma- values. Such intensional notions as that of an chinery of logic in order to capture both form individual concept, a property and a relation, and meaning. The suggestion did not get a and of a proposition, are sufficiently like favorable reception by the linguists, however Frege’s various types of senses to be used as (Bar-Hillel 1954, Chomsky 1955), and it took formal counterparts thereof. the logicians and philosophers more than a Modal operators are analyzed in terms of decade to develop these ideas into working quantification over possible worlds. To give theories. It took even longer for the idea of a a simple example, a formula of the form ȣf 2062 XXXIII. Formalization Tendencies and Mathematization in 20th-Century

(read as “It is possible that f”) is true in a ural language without falling back to some possible world w if and only if there is some kind of language reform, this is a problem suitable world wЈ such that f is true in wЈ. that has to be solved. Basically the problem is Depending on the kind of possibilities one how to satisfy Fregean compositionality for takes the set of possible worlds to embody, natural languages with their rich and diverse this gives an analysis of various kinds of mo- syntactic structures: every syntactic constitu- dalities: alethic, epistemic, deontic. The kind ent needs to be assigned a meaning in such a of modality we are dealing with, may put ad- way that it depends on the semantic contri- ditional constraints on the possible worlds bution of its constituent parts. And the stock and the relations between them. For example, of semantic objects to go around (individual if we read ȣ as an alethic modality (i.e., as concepts, properties, ) isn’t large “It is possibly true that …”), then we would enough: we need ways of building other, want f J ȣf to be a valid priniciple: What- complex types of semantic objects from these ever is actually true, is possibly true. But if basic ones. we read it as a (i.e., as “It Other inventions in logic come to the res- is allowed that …”), we would want the same cue here, viz., that of the typed l-calculus, formula not to come out valid: it is not plau- originally developed by Alonzo Church sible that whatever is the case, is therefore al- (Church 1940) and, closely related, that of lowed. categorial systems, pioneered by Kazimierz Ajdukiewicz (Ajdukiewicz 1935). Typed l- Typed l-Calculus and . calculus is an abstract system for studying Possible world semantics was developed in functions and their properties, such as com- the context of modal propositional and putability and decidability. It has connections predicate , systems which employ lan- with the kind of categorial systems, of what guages that from a syntactic perspective are he himself called “semantic categories”, that relatively simple. Natural languages, of Ajdukiewicz developed at more or less the course, are much more complex. same time. Both provide us with tools that First of all, they have more extensive and allow us to treat more complex structures of varied vocabularies, i.e., natural languages a greater variety than the usual logical lan- contain many more types of expressions than guages display, and in such a way that the just predicates, terms, and connectives and syntax and the semantics are treated simul- quantifiers. There are adjectives and adverbs, taneously. Therein lies their usefulness for prepositions, nouns and verbs, personal pro- natural language analysis. nouns and demonstratives, temporal and as- But Church’s typed l-calculus was not de- pectual expressions, singular and plural veloped with application to natural lan- forms, and so on. Moreover, within each cat- guages in mind. (However, in Church 1951 egory there are semantically distinguishable he used similar ideas to give a reconstruction subcategories, such as mass nouns and count Frege’s ideas on sense and reference.) And al- nouns, activity verbs and stative verbs, inter- though Ajdukiewicz’ original work did con- sective and subsective adjectives. cern itself with language, it did so mainly Secondly, natural languages have a dif- from an epistemological perspective, not with ferent, more complex constituent structure. an eye to application in linguistic analysis. One example is provided by the various ways (An early attempt to use categorial grammar in which relations are expressed. Where in to analyze natural language syntax is Lam- logical languages all the arguments in a rela- bek 1958.) So it really took a combination of tion are on equal footing, many natural lan- various insights to bring out the potential for guages distinguish between the subject and natural language semantics. direct object (and indirect object) arguments by assigning them different structural posi- Logical Grammar as a Synthesis. These diver- tions. Frege’s fundamental insight was that gent developments in logic, philosophy and for logical purposes a distinction between linguistics came together in the late 1960s, subject and object is not necessary. Seeing when several philosophers and logicians no way to reconcile that with the fact that started to realize that all the necessary tools natural languages nevertheless treat them dif- for developing a systematic and formal se- ferently, he concluded that logical form and mantics for natural language were available. grammatical form are different. If one wants Thus the idea of a logical grammar for natu- to apply the tools of logical semantics to nat- ral language was born. Of course, different 221. The development of Montague grammar 2063 authors, often starting from diverging philo- ment of English directly, by connecting them sophical principles, conceived of such a with modeltheoretic semantic objects, such as grammar in different ways. Donald David- individual concepts, properties and proposi- son, for example, following Quine eschewed tions. In “The Proper Treatment” paper the use of intensional notions and proceeded Montague takes an indirect route. Here ex- to define an extensional, truth-conditional se- pressions of English are interpreted indirectly, mantic theory (Davidson 1967). But the very by translating them into expressions of a log- idea that such an enterprise was possible, was ical language that is semantically interpreted widely shared. As Davidson and Harman in the usual way. It is the model of the last stated (in their preface to a seminal collec- paper, commonly referred to as “the PTQ- tions of papers, Davidson & Harman 1972), model”, that has become the standard format referring to formal semantics as the emerging of Montague Grammar. And it is therefore interface between logic, linguistics and phi- this model that we will take a closer look at losophy: in the next section. We trust it will be agreed that there is more to this than the usual business of rubbing two or more 3. Montague’s original model disciplines together in the expectation of heat and the hope of light. In the present case, a common General Structure. The PTQ-model consists enterprise already exists; our aim is to make it a of a categorial syntax of a fragment of Eng- cooperative one. lish, a logical language, that of intensional Among the entrepreneurs of logical grammar typed l-calculus, a modeltheoretic semantics was , whose work proved for that language, and a definition of a trans- to have a special impact on the field. This is lation function from the expressions gen- only partly due to its general and rigorously erated by the categorial syntax into expres- systematic character. What really distin- sions of the logical language. As said, the for- guished his work from that of other pioneers, mer expressions are interpreted indirectly, is that Montague was not satisfied with giv- through translation into the interpreted ex- ing either the general outlines of a semantic pressions of the logical language: theory, or a detailed semantic analysis of a particular phenomenon. He wanted to show Natural Logical Models how such a formal semantics could be incor- language translationlanguage interpretation porated in the grammar of a natural lan- guage, i.e., he insisted that a proper account Fig. 221.1: Indirect interpretation in the PTQ- of the relationship betweens semantics and model syntax was an essential ingredient of the en- terprise. And Montague illustrated his gen- The principle of compositionality is imple- eral ideas with actual, detailed analyses of mented in this model at two points. The in- important aspects of natural language syntax terpretation of the logical language transpires and semantics, such as quantification and in the usual manner. First the notion of a structural , adjectival modification, model for the language is defined. Models intensional verb phrase constructions. contain the semantic objects that can be ex- He did so in a series of papers, among pressed by the language, and an interpreta- which “Universal Grammar” (1970), “Eng- tion function that assigns appropriate objects lish as a Formal Language” (1970), and “The to the basic expressions. Secondly, the inter- Proper Treatment of Quantification in Ordi- pretations of complex expressions are defined nary English” (1973) were the most influen- in a way that parallels the rules of the syntax tial. (Along with several others, these papers that generate them. For every syntactic rule have been collected in Montague 1974.) The that operates on expressions a1 … an, to yield first paper contains a very abstract, algebraic an expression g there is a clause that defines formulation of his ideas. The latter two are the interpretation of g in terms of the inter- concerned with applications to fragments of pretations of a1 … an. If the syntactic rule in- natural language, each giving a specific troduces syncategorematic expressions, such model of how such an application may pro- as connectives, quantifiers, or sentential op- ceed. The main difference is that in “English erators, their interpretation is implicitly de- as a Formal Language” Montague defines an fined in the corresponding clause. This is the interpretation for the expressions in the frag- usual way of specifying the semantics of a 2064 XXXIII. Formalization Tendencies and Mathematization in 20th-Century formal language, that actually is not strictly Ϫ a set of categories, in terms of basic cat- compositional in view of the way in which egories and derived categories; syncategorematic expressions are introduced Ϫ a set of syntactic rules, that determine and interpreted, but which is compositional how expressions of various categories can in spirit, and can be made to comply with be combined; strict compositionality. Ϫ an assignment of categories to the lexical The second point at which composition- expressions of the language. ality comes into play is with the translation of the natural language into the logical lan- PTQ’s categorial syntax has three basic cat- guage. In order to transfer the compositional egories: S, for sentences, CN, for common interpretation of the latter to the former, the noun phrases, and IV, for intransitive verb translation process has to meet certain phrases. Complex categories are defined as requirements as well. Given a translation of follows: lexical elements of the natural language into if A and B are categories, then so are simple or complex logical expressions, every A/B and A\B syntactic rule that derives complex expres- sions is matched with a translation rule that with A/B the category of expressions that defines the translation of the derived, com- combine with an expression of category B to plex expression in terms of the translations of their right to form an expression of category the constituent parts. In the natural language A, and A\B that of expressions which take an syntax, too, Montague allows for rules which A to their left to form a B. This type of cat- introduce syncategorematic material, but, egorial system is called “bi-directional”. Ex- again, such rules could in principle be elimi- amples of derived categories are: T = S/IV, of nated. Notice also that there may be syntactic terms; TV = IV/T, of transitive verbs, IV/S, rules which do not affect the meaning, here of sentential complement verbs; CN/CN, of the corresponding translation rule is simply pronominal adjectives. an identity map. In a simple categorial syntax there is only In his “Universal Grammar” Montague one syntactic rule, that of concatenation of analyzes the syntax of the two languages and expressions of matching categories. Thus a the models used in the semantics as algebras. sentence S is the result of combing a T = The translation and interpretation compo- S/IV with an IV. And an IV may be either nents in this model then show up as homo- lexical, or derived, e. g., the result of combin- morphic mappings. Translation is a homo- ingaTV= IV/T with a T. morphism from the syntactic algebra of the In the categorial syntax of PTQ syntactic natural language into the syntactic algebra of rules do more than concatenate. For exam- the logical language, and interpretation is a ple, there are rules that introduce determiners homomorphism from the latter into the alge- syncategorematically. There are lexical el- bra of semantic objects. Since the composi- ements of category T, viz., proper names and tion of any two homomorphisms is again a pronouns, but also complex expressions con- homomorphism, the two-step process of fig- sisting of a determiner and a (lexical or com- ure 1 indeed defines an interpretation for the plex) common noun. In the PTQ-model these natural language expressions. Actually, it T’s are not derived by combining a lexical de- shows that the indirect and the direct ap- terminer, that would be of category T/CN, proach are equivalent, and that the interme- with a CN, but by a special rule, that intro- diate translation step is there for conve- duces the determiner in question syncate- nience’s sake, not because it is necessary. gorematically. Let us now take a closer look at the three Other examples of syntactic rules that do components of a PTQ-style Montage gram- more than simply concatenate, are Mon- mar: the categorial syntax, the language of tague’s “quantification rules”. These rules intensional , and its semantics. combine terms with sentences that contain an (The following presentation is slightly simpli- occurrence of a special, variable expression of fied at some points. See Dowty et al. 1981 category T to form new sentences, and are and Gamut 1991 for extensive introductions.) introduced to deal with scope . Thus a sentence such as “everyone admires Categorial Syntax. A categorial syntax for a someone” has two derivations corresponding particular (fragment of) a language consists to the two relative scopes of the quantified of a specification of: terms “everyone” and “someone”. The first is 221. The development of Montague grammar 2065

the so-called ‘direct derivation’, that results into one of type . The cup-operator from a straightforward combination of “ad- only takes expressions of type and mires”, of category TV = IV/T, with “some- turns them into expressions of type a. one”, a T, to form the complex IV “admires Another essential ingredient of the lan- someone”, that combines with the T = S/IV guage is l-abstraction. From any expression “everyone” to form the sentence. As will be- a of type b and variable x of type a,wecan come apparent shortly, the interpretation of form the expression: lx a, of type . this derivation is the one in which “everyone” So, l-abstracts are also functional expres- has scope over “someone”. The other deriva- sions. As will become clear shortly, l-abstrac- tion, called the “indirect derivation” first de- tion is an important tool for translating, and rives the S “everyone admires him0”, with hence interpreting, complex expressions of “him0” a variable T, and then combines this various categories. with the T “someone”. The semantic result is that “someone” now has scope over “every- Intensional Type Theory: Semantics. The se- one”. The same mechanism is used to ac- mantics of the language of intensional type count for so-called “de dicto Ϫ de re” ambi- theories consists, as usual, of a definition of guities, such as occur in a sentences like suitable model structures and of a definition “John is looking for an assistant”. of the interpretation of the expressions of the The occurrence of this type of syntactic language in these models. rule makes clear that translation (and, hence, Unlike more standard languages, the lan- interpretation) operates not on strings, or on guage of intensional type theory contains ex- constituent trees, i.e., labeled strings adorned pressions of a great many types, each of with categories, but on so-called ‘derivation which is to be interpreted in a suitable do- trees’, i.e., strings with their derivational his- main. The basic types e, t, and s are associ- tory, including category information. ated with a set D of individuals, a set {0,1} of truth values and a set W of possible worlds. Intensional Type Theory: Language. The sec- A functional type is associated with ond component of a PTQ-style Montague the set of functions from the domain Da to < > grammar is the logical language into which the domain Db. For example, an e,t ex- the natural language expressions are trans- pression, i.e., one-place predicate, denotes a { } lated. This is the language of intensional type function from De to Dt, i.e., from D to 0,1 : theory, with l-abstraction. (the characteristic function of) a set of indi- All expressions belong to a type. Basic viduals. And a predicate modifier, i.e., an types are e, for expressions which denote ob- <, > type expression, denotes a jects; t, for expressions which denote a truth function from functions from De to Dt,to value, i. e. for sentences; and s, a special type functions from De to Dt. The intensional used to account for intensional expressions. types are linked to functions from W From this set of basic types complex types to Da, i.e., functions from possible worlds to are derived: if a and b are types, than objects of type a. For example, an expres- is a type, and if a is a type, then so is . sions of type denotes a function from Complex types are functional: an expression possible worlds to truth values, i.e., a propo- of type is one that combines with an sition. Similarly, an expression of type expression of type a to form one of type b. > denotes a function from worlds Examples are , of one-place predicates; to sets of individuals, i.e., a property. >, of two place predicates; A model M consists of some hierarchy of <, >, of predicate modifiers; domains generated by sets D, W, and {0, 1}, and so on. Expressions of types are and an interpretation function I that assigns also functional, but in a special way, since suitable to the basic expressions: ෈ there are no expressions of type s that could if a is a basic expression of type a,I(a) Da. be their argument. Likewise, an assignment g takes variables x The language contains constants and vari- of type a into the corresponding domain. The ables of the various types, and the usual con- definition of the interpretation of the lan- nectives, quantifiers and intensional opera- guage then follows the definition of the syn- tors, such as modal and tense operators. In tax. Every expression is interpreted relative addition it features two operators, ∧ (often to some world w and assignment g. called “cap”) and ∨ (called “cup”). The cap- Concatenation of expressions is interpre- operator turns an expression of any type a ted as function application. If a is of type 2066 XXXIII. Formalization Tendencies and Mathematization in 20th-Century

and b of type a, then a(b) is of type S is paired with type t. The other basic b. The interpretation of a(b) is obtained by categories, IV and CN, are linked to the applying the interpretation of a, a function same type, >. Complex categories from Da to Db, to that of b, an object in Da, A/B and A\B are mapped onto types < < >> < < >> the result being an object in Db. The inter- s, f (B), f (A) and s, f (A), f (B) , pretations of the connectives, quantifiers and respectively, where f (A) and f (B) are the intensional operators follows the usual pat- types associated with categories A and B. terns. The operation of l-abstraction seman- Notice the use of the intensional types: Ex- tically amounts to functional abstraction: if a pressions are taken to denote senses. Thus is of type b and x of type a, lxa is interpreted the category T = S/IV is associated with the < << < >> > as that function from Da to Db that, applied type s, s, e,t ,t . ෈ to some d Da, returns the interpretation of Most lexical elements of category A are a with d assigned to x. The semantics of the translated into constants of the type corre- cap-operator ∧, too, is functionial abstrac- sponding to A. Some, however, receive spe- tion: ∧a, with a of type a, denotes that func- cial treatment: they are translated into either tion from W to Da that applied to some logical constants or into complex expres- w ෈ W returns the interpretation of a with sions. A case in point are proper names. A respect to that world. I. e., the cap-operator name “John” translates, not into some con- abstracts over the world parameter in inter- stant of the appropriate type, but into the pretation, and hence forms a function that complex expression lP∨P (j), where j is a con- gives the interpretation in each world. Thus stant of type e and P a variable of type expressions of the form ∧a denote Fregean >. In general, terms are taken to senses. Correspondingly, the cup-operator ∨ denote functions from worlds to sets of prop- semantically amounts to application of such erties, and the proper names “John” is associ- a sense-function to a world: in a world w: ∨a ated with a specific function of this type, viz., denotes the value of the function a in w. the function from worlds to the set of proper- Given the semantics, a logic results, i.e., a ties which the individual denoted by j has set of entailments that hold among the for- in that world. Other lexical elements that mulae of the language. One of these merits may receive special treatment, i.e., a distinct special attention in the present context, viz., translation, that results in a distinct meaning, the principle of l-conversion. Given certain are modal auxiliaries, temporal expressions, restrictions on b, viz., that it be “semantically and so on. closed”, and provided no variable clashes oc- The translation of complex expressions is cur, it holds that: defined in tandem with the syntactic rules. This is necessary for two reasons: first all, lx a(b) = [b/x]a such rules may introduce syncategorematic This says that whenever we form an expres- material, that needs to be interpreted; and sion by applying a l-abstract to an appropri- secondly, structural properties, such as scope ate argument, the result is equivalent to the properties, are coded in the derivational his- expression abstracted over with the variable tories. Hence, every syntactic rule, that de- in question interpreted as denoting whatever rives a complex expression g from a number the argument denotes. This allows us to sim- of expressions a1 … an, is paired with a trans- plify formulae. lation rule, that defines the translation of g in terms of the translations of a1 … an. Rules The Translation Component. The actual natu- which involve only concatenation are ral language semantics in a PTQ-style Mon- straightforward: the corresponding transla- tague grammar resides in the translation tion consists of functional application of the from the natural language into the logical translations of the component parts. Given language. This translation proceeds in famil- the strict association of categories with types iar fashion: there is a specification of the this is guaranteed to result in a well-formed translations of the lexical elements, and on logical expression. the basis of that the translations of complex An example of a rule that introduces a expressions are defined. Translation presup- syncategorematic expression is one that poses a relation between the categories in forms a quantified T from a CN by introduc- the syntax of the natural languages and ing the determiner “a”. The corresponding the types assigned to expressions of the logi- translation rule defines the meaning of the cal language. As is to be expected category determiner in terms of the existential quanti- 221. The development of Montague grammar 2067 fier as follows: ∧lP∃x(∨a(x) ∧ ∨P (x)), where meaning postulates are formulae of the logi- a is the translation of the CN. The expression cal language that serve as restrictions on the as a whole denotes the function from worlds models that are taken into consideration as to the set of properties P such that there is proper models for the natural language ex- some x that has both the property expressed pressions. Only such models as make these by the CN and P. formulae valid are to be taken into account. Another type of translation rule that de- The meaning postulates express a variety of serves to be mentioned is the one correspond- semantic properties of expressions, or classes ing to Montague’s quantification rules. The of such. For example, the synonymy of two latter combine a T with an S containing a lexical expressions is not accounted if they special pronoun into an S, by substituting the are translated into non-logical constants, T for the pronoun. This mechanism is intro- since the interpretation function of a model duced to account for the relative scope of two only says something about the semantic type, quantifiers, or of a quantifier and a modal not about the actual semantic object that is expression. The translation rule involves ab- the interpretation of such constants. By add- straction over the variable x in the trans- ing a that expresses that lation of the sentential input f: lx f, result- the respective constants are co-extensional in ing in an expression of type , that de- all worlds, we effectively limit ourselves to notes the set of objects for which f holds. models in which the two lexical expressions Application of the cap operator gives an are interpreted as synonymous. Other postu- >, i.e., an expression that denotes lates concern properties of classes of expres- the property of being an object for which f sions, such as the intersectivity of certain ad- holds. The translation of the T is applied to jectives. Used in this way, meaning postulates this expression, the result being a formula permit us to blur the sharp distinction be- again, one that expresses that the said prop- tween logical and non-logical constants that erty belongs to the set denoted by the T is inherent in most logical formalisms. translation. Thus, to refer back to our initial example, if we derive “Everybody admires Core Principles. From this sketch of the main someone” indirectly, i.e., by means of the ingredients of Montague’s original PTQ- quantifying-in process, we first derive the model, we can derive the following general sentence “Everybody admires him ”, with 0 features, which are characteristic of the enter- “him0” a variable. The translation is a for- ∀ prise of Montague grammar in general: mula with a free variable, x(admire(x, x0)), that by l-abstraction and application of the A. Semantics is syntax-driven, syntax is se- ∧ ∀ cap-operator is turned into: lx0 x(ad- mantically motivated mire(x, x0)), which denotes the property of B. Semantics is modeltheoretic being admired by everyone. From the trans- lation of the T “someone” we get: lP∃x ∨P (x), The first characteristic is, of course, closely which denotes the set of properties someone related to the principle of compositionality. has, and the translation of the entire con- It comes to the fore at several points, among struction is functional application of these which the correspondence between syntactic ∃ ∨ ∧ ∀ two expressions: lP x P (x)(lx0 x(ad- categories and semantic types, and the mire(x, x0))). This says that the property of grounding of structural ambiguities in syn- being admired by everyone is a property that tactic derivations are the most prominent. It someone has, which is the intended reading. clearly demonstrates that compositionality The formula in question can be simplified by constrains both syntax and semantics. Any performing l-conversion (twice) and taking semantic object or operation on such objects alphabetic variants, resulting in the more fa- has to have a correlate in the syntax, an ex- miliar, first-order representation ∃y∀x(ad- pression or operation that triggers it. And mire(x, y)). In a similar fashion de dicto and conversely, all expressions and all structural de re readings of sentences with intensional operations in the syntax have to have a se- verbs and quantified objects are derived. mantic correlate. Thus the autonomy of syn- One element that strictly speaking does tax is limited. That categories are mapped not belong to the translation component, but onto semantic types illustrates this. In so far that is involved in getting the semantics of as the categorial labeling of the expressions in the natural language right, is Montague’s use a string is akin to more familiar constituent of so-called ‘meaning postulates’. These structure, it implies that all of constituent 2068 XXXIII. Formalization Tendencies and Mathematization in 20th-Century structure is semantically interpretable. And of the data, no other constraints, of a philo- the requirement that any syntactic rule has sophical nature for example, are accepted. to be matched by a corresponding translation However, this does not mean that the ontol- rule, in effect, says that there can be no syn- ogy of the models is completely fixed by the tactic operations which are not somehow in- semantic facts: there is, in most cases, more terpretable in terms of operations on seman- than one type of model that fits the data. tic objects. This explains, to some extent at least, that in The strong coupling of syntax and seman- some cases more or less the same range of tics implies that for every semantic phenome- semantic phenomena is accounted for in the non there has to be an analogue in the syn- literature in a number of divergent ways. Es- tax. The way in which scope is handled is, pecially those aspects of natural language se- of course, a perfect illustration. There is no mantics which involve expressions and con- independent syntactic motivation for some- cepts that are loaded with philosophical im- thing like Montague’s quantifying in rules: plications and , display this. they are part of the syntax simply and solely A case in point is the semantics of attitude for semantic reasons. verbs, such as believe and know, or that of the In order to implement compositionality in temporal and aspectual system. The analysis this particular way, the use of a logical repre- of plurality, too, is rife with philosophical sentation language with a type structure and and logical controversies. the means to form new complex expressions An immediate correlate of the model- by means of l-abstraction is imperative. Only theoretic perspective on meaning is that there with such expressive means at our disposal is a strict division between semantics and can compositionality be implemented as . Semantics deals with meaning, strictly as is done in Montague grammar. i.e., with modeltheoretic interpretation, and An interesting question is whether this pragmatics deals with use. Meaning and use strong form of compositionality actually can are hierarchically related in this sense that be empirically motivated, or whether it is an meaning is supposed to be independent of empirical hypothesis in the first place. It use, but not vice versa. Hence, semantics can seems that without independently motivated proceed without pragmatics, but the converse constraints on either syntax or semantics, the does not hold. Since pragmatics is about how latter question has to be answered in the we use the expressions of our language given negative. If there are no such constraints, we the meaning they have, pragmatics presup- can always set up things in such a way that poses semantics. compositionality is met. In that sense it is not Here Montague grammar abides with what so much an empirical hypothesis as well as a for a very long time was accepted wisdom, methodological principle. However, once we viz., the Gricean division of labor between se- impose certain restrictions on either the or- mantics and pragmatics. Despite early dissi- ganization of the syntax or on that of the se- dents, such as Stalnaker (Stalnaker 1979), the mantics, it may very well be that the compo- idea that the tasks and domains of semantics sitionality requirement can not be met. and pragmatics can be strictly separated, was The second characteristic, that semantics dominant until well into the 1990s, certainly is modeltheoretic, places Montague grammar among semanticists of a modeltheoretic incli- squarely in the dominant logical tradition. As nation. It was only due to a change in seman- far as meaning is concerned, it seems, entail- tics itself that this picture began to change ment relations between (classes of) expres- as well. sions are the primary data of semantics. It is Concomitant with the second characteris- in terms of entailment that all the important tic is a feature that Montague grammar semantic properties of an expression can be shares with almost all linguistic frameworks explained, including ambiguity, synonymy, up until quite recently, viz., an exclusive fo- special properties such as factivity, and so on. cus on the sentence as the primary unit of These data, acquired by means of introspec- analysis: grammar is sentence grammar. That tion, are accounted for by associating the ex- was the case in the Chomskyan framework, pressions involved with the proper semantic and in almost all of its offshoots and alterna- objects, viz., those that have corresponding tives, and Montague grammar is no excep- relations among themselves. Consequently, tion. It actually inherits this feature from models are constructed, in principle, solely both its ideological sources. The first of these on the basis of what is needed for an account is the Chomskyan tradition, that Montague 221. The development of Montague grammar 2069 himself considered as an ally in the fight Extensions of Montague grammar were against the then still prevailing opinion often triggered by a felt need to use different among philosophers and logicians that natu- semantic concepts than were available in the ral language is too vague and unsystematic original formulation. A good example is the to lend itself to formal treatment. (Of course, analysis of tense and aspect. In due time it for the reasons sketched above, Montague became clear that the original tense logic in- adopted a different outlook on syntax than corporated in Montague’s intensional type Chomsky did.) And secondly there is the theory was rather limited. Other options were logical tradition itself, which throughout its explored, such as analyses in terms of events long tradition has focused on relations be- and their properties (Hinrichs 1986), or tween judgments, propositions, statements, analyses that used intervals (Dowty 1982). or whatever they were called, but all “sen- Another example of an extension of this kind tence like” objects. involves the use of a strictly typed logical sys- tem, another feature of Montague’s model. 4. Applications and developments Every expression here can be assigned to only one type, fixing both its logical syntax and its Montague grammar, in particular the PTQ- semantics. Thus, for example, properties and model, was the dominant paradigm for for- objects are strictly separated. In the analysis mal semantics of natural language during the of nominalizations this was felt as a short- 1970s and much of the 1980s. To review the coming, and some proposed to (partly) re- various ways in which it has been applied is place Montague’s typed system with a form beyond the scope of this article. We will have of property theory (Chierchia & Turner to confine ourselves to a brief indication of 1988). Yet another example concerns the some remarkable specimens. For more de- analysis of plurality and quantification in- tails, and an extensive bibliography, the reader volving plurals. Montague’s set-theoretic, in- is referred to the survey in Partee/Hendriks dividual based semantics was found lacking, (1997). descriptively and explanatorily, and other It may be useful to distinguish between ap- analyses were proposed in terms of algebraic plications, extensions, and amendments of structure (Link 1983) and plural objects Montague’s original model. Applications of (Landman 1989). the PTQ-model stay, more or less, within the All these developments extended Mon- confines of Montague’s formulation and ex- tague grammar by replacing some elements tend its range of application by analyzing of the semantic theory by different ones, thus new phenomena, using the tools already at increasing its descriptive and explanatory hand. Extensions broaden the scope of the range. Another type of extension is exempli- theory, not just by bringing it to bear on new fied by work that explores the original model phenomena, but also by extending its theo- in new directions. One of the most successful retical apparatus, i.e., by introducing new examples of this type of development is pro- concepts and techniques. Amendments actu- vided by theory. Mon- ally change the model in some way, by re- tague’s original analysis of terms, i.e., noun placing certain tools or techniques by dif- phrases, treats them, extensionally speaking, ferent, more adequate ones. An amendment as sets of sets. (And determiners, as a distinct may be triggered by both empirical or con- syntactic category, are analyzed as relations ceptual shortcomings. Of course, in reality between sets.) Such sets of sets are general- most contributions to the field are mixtures. ized quantifiers, and by exploring their un- The early years were devoted mainly to ap- derlying logic, a whole new branch of seman- plications of the PTQ-model, on a variety of tics grew out of Montague’s proposals. Initi- empirical data, such as tense and aspect ated by Barwise and Cooper (1981), van Ben- (Dowty 1979), plurals (Bartsch 1973), mass them (1983), and explored by many others, nouns (Bennett 1979), lexical meaning this turned out to be a very fruitful area of (Dowty 1976), adverbs and adjectives (Siegel investigation, with ramifications far beyond 1976), control verbs (Bach 1979), and so on. the original analysis of terms. Another area Most of these analyses stayed fairly close explored in this way is the analysis of non- to Montague’s original formulations, but indicative sentences, such as interrogatives brought out its descriptive and explanatory (Karttunen 1977, Groenendijk & Stokhof potential by incorporating more empirical 1982). The fact that Montague’s model can phenomena. be extended to apply also to interrogatives, 2070 XXXIII. Formalization Tendencies and Mathematization in 20th-Century shows that the idea that modeltheoretic se- ticular, it was the category Ϫ type agreement mantics is biased towards, or even restricted that made categorial grammar something of to, declarative language is a misconception. a natural choice. As we saw, for Montague Important amendments of Montague’s syntax was not autonomous in this sense that original model are connected with his analy- he did not hesitate to add rules to the syntax sis of structural ambiguities involving quanti- just to get the semantics right, without any fied expressions, intensional verbs, and so on. independent syntactic motivation. Further- As we saw above, Montague postulated addi- more, the kind of categorial syntax Mon- tional syntactic rules in order to provide a tague used, provided a rather poor descrip- derivational basis for such ambiguities. For tive framework. various reasons, a number of authors were From a linguistic point of view these were dissatisfied with this approach and developed definite weaknesses in Montague’s original alternative ways of dealing with these seman- model, and over the years a lot of work has tic facts. Cooper provided one such an alter- been done to overcome this. Several people native, using so-called “storing” of quantifi- have combined a semantics in Montague’s ers and variables (Cooper 1979). Another ap- style with different syntactic frameworks. proach was pioneered by Partee (Partee There were some early attempts to combine 1987) and worked out in detail by Hendriks a Montague semantics with a classical trans- (Hendriks 1993). It involves so-called “flex- formational grammar (Partee 1976). Later ef- ible type assignment” and does away with forts were concerned with syntactic theories Montague’s strict correlation of syntactic cat- which were closer to the original ideas of cat- egories and semantic types by using a calcu- egorial syntax, but were motivated by consid- lus of so-called ‘type-shifting rules’ that allow erations internal to syntactic theory. Here expressions to be analyzed in different types work on unification grammar (Shieber 1986) and with different scope relations. needs to be mentioned as well as modifica- The upshot of the various extensions and tions that use generalized phrase structure amendments was that certain basic aspects of grammar (Gazdar, Klein et al. 1985) and Montague’s original model were gradually head driven phrase structure grammar (Pol- beginning to change as well. For example, lard & Sag 1994), as well as lexical-functional compositionality becomes an issue that has grammar (Kaplan & Bresnan 1982). Some- to be re-thought once the strict category- what separately, categorial grammar itself type association is loosened (see Janssen 1997 was explored and expanded in various direc- for an extensive overview). And another ef- tions (Moortgat 1988, Steedman 1993). fect of the proliferation of extensions and Many of the developments in this area changes of the original semantic apparatus have also received an impetus from the use was that the original unity of the various of derivatives of Montague grammar in com- analyses carried out within the framework putational linguistics. Since Montague gram- was lost. “Montague grammar” became a mar was the first grammar model that fea- way of doing semantics, still modeltheoretic tured a formal semantics, it was a natural semantics, but using different logical theories choice for those tasks in computational lin- for different domains. In this sense the very guistics which required semantics, or some character of the enterprise of Montague kind of understanding on part of the users grammar changed. of a system, such as translation and dialogue modeling. In these contexts extensive descrip- tive coverage is a necessary feature of the 5. Beyond Montague grammar model, something that Montague’s original syntax did not supply, hence the use of dif- There are a number of reasons why formal ferent syntactic frameworks. semantics eventually abandoned some basic With the incorporation of internally moti- tenets of the Montagovian framework. We vated syntactic theories, Montague’s original will briefly sketch two of them, each con- conception of the syntaxϪsemantics interface cerned with one of the core principles that was modified. First, other ways of satisfying were identified in section 3. compositionality were explored that did not rely on syntactically unmotivated rules in the The syntax-semantics interface. Montague’s syntax to get the semantics right. Here such use of categorial syntax was motivated techniques as mentioned above, Cooper mainly from a semantic point of view. In par- stores and flexible type assignment, were 221. The development of Montague grammar 2071 used to loosen the demands placed on syntax them with respect to a suitable model. Dy- by semantics. Later on the outlook on the namic semantics (Groenendijk & Stokhof syntaxϪsemantics interface itself changed, 1991, Groenendijk, Stokhof & Veltman 1996) and syntax regained a more autonomous takes the idea of dynamic interpretation one status. step further and locates the dynamics in the concept of meaning itself. The basic starting Modeltheoretic semantics. An important de- point of can be formu- velopment in the 1980s and 1990s was the lated in a slogan: “Meaning is context-change move away from the traditional sentence potential”. Unlike DRT dynamic semantics grammar to systems which are able to deal does not rely on representations, but assigns with larger units of , such as texts expressions a dynamic meaning in a composi- and dialogue, a move that also involved tional way. radical changes of the Montague grammar The development of these dynamic theo- framework. ries also paved the way for a major reorienta- An external impetus came from natural tion of the relationship between semantics language research within artificial intelli- and pragmatics. More and more aspects of gence, where interpretation is viewed as the language use, such as embodies in Grice’s execution of procedures which change the original theory of conversational maxims state of a system as it proceeds. Another in- (Grice 1975), or in theories of presupposi- fluence came from computational linguistics, tion, are incorporated into semantics. Mean- that had to deal with larger units than senten- ing no longer is identified with Fregean sense ces in actual implementations. And there was (and reference), but rather with the way in an internal stimulus, that concerned certain which utterances, i.e., uses of sentences in persistent empirical problems that could not contexts, change that very context. And the be solved in a sentence oriented context includes information states of lan- such as Montague grammar. A simple but il- guage users, about the world and about lustrative example is provided by cross-sen- themselves and each other, their intentions tential . In a discourse such as “A and expectations, and so on. These develop- woman walked into the room. She was wear- ments have given rise to new perspectives ing a black velvet hat.”, the pronoun “she” is such as the use of game theory in the analysis naturally interpreted as bound by the indefi- of language, in which the borderline between nite noun phrase “a woman”. If interpreta- semantics and pragmatics is drawn in quite tion proceeds on a sentence by sentence basis, different ways, if at all (Dekker & van Rooy this can not be accounted for, since the ante- cedent is processed before the anaphor is en- 2000). countered. And delayed interpretation, i.e., Finally, another way in which Montague’s linking antecedent and anaphor only when original proposals are being changed and the discourse is finished, makes wrong pre- transformed is connected with the increasing dictions in other cases, such as “The restau- interest in the cognitive reality of linguistic rant was empty except for one man, who was modeling. Montague stood in a tradition that sitting by the window. He was wearing blue was shaped by Frege’s antipsychologism in suede shoes.” logic. And his grammar model bore witness It was phenomena such as these that led to that: meaning is analyzed in a purely for- to a view on interpretation as a dynamic pro- mal way, that does not refer to any cognitive cess, one that proceeds incrementally as a dis- embedding or realization of the concepts and cource or text unfolds. In the beginning of principles used. The increasing interest in the the eighties such a dynamic view was formu- way in which meaning and use interact, that lated explicitly in Discourse Representation is characteristic for the developments in the Theory (DRT; Kamp 1981, Kamp & Reyle late 1990s and that is still continuing, is 1993) and File Change Semantics (Heim changing this, too, and is bringing, if not 1983). DRT is a dynamic theory of inter- Montague grammar itself, then at least his pretation, locates the dynamics in the process successors closer to cognitive research on lan- of building representational structures. These guage and meaning. structures are initiated by incoming utter- For that, despite all the changes, there is ances and added to, or modified by, subse- still a living tradition in semantics that owes quent utterances. The structures themselves a great debt to Montague’s work, is evident are interpreted in a static way by evaluating from the high standards of rigor and exact- 2072 XXXIII. Formalization Tendencies and Mathematization in 20th-Century ness that were once revolutionary, but are Dowty, David. 1976. “Montague grammar and the now accepted as defining characteristics of lexical decomposition of causative verbs”. Mon- what this field of inquiry is all about. tague Grammar ed. by Barbara Partee, 201Ϫ246. New York: Academic Press. Ϫ. 1979. Word Meaning and Montague Grammar. 6. Bibliography Dordrecht: Reidel. Ϫ Ajdukiewicz, Kazimierz. 1935. “Die syntaktische . 1982. “Tenses, time adverbials, and composi- Konnexität”. Studia Philosophica 1.1Ϫ27. tional semantic theory”. Linguistics and Philosophy 5.23Ϫ33. Bach, Emmon. 1979. “Control in Montague gram- mar”. Linguistic Inquiry 10.515Ϫ531. Dowty, David, Robert Wall & Stanley Peters. 1981. Introduction to Montague Semantics. Dordrecht: Bar-Hillel, Yehoshua. 1954. “Logical syntax and Seidel. semantics”. Language 30.230Ϫ237. Frege, Gottlob. 1962. Funktion, Begriff, Bedeutung: Bartsch, Renate. 1973. “The semantics and syntax Fünf logische Studien. Ed. by Günter Patzig. of number and numbers”. Syntax and Semantics, Göttingen: Vandenhoeck & Ruprecht. Vol. 2 ed. by James Kimball, 51Ϫ93. New York: Seminar Press. Gamut, L.T.F. 1991. Logic, Language and Mean- ing. Volume II: and Logical Barwise, Jon & Robin Cooper. 1981. “Generalized Grammar. Chicago: University of Chicago Press. quantifiers and natural language”. Linguistics and Philosophy 4.159Ϫ219. Gazdar, Gerald, Ewan Klein, Geoff Pullum & Ivan Sag. 1985. Generalized Phrase Structure Grammar. Bennett, Michael. 1979. “Mass nouns and mass Oxford: Blackwell. terms in Montague grammar”. Linguistics, Philos- ophy and Montague Grammar ed. by Steven Grice, H. Paul. 1975. “Logic and conversation”. Davis & Marian Mithun, 263Ϫ285. Austin: Uni- Syntax and Semantics, Vol. 3 ed. by Peter Cole & Ϫ versity of Texas Press. Jerry Morgan, 41 58. New York: Academic Press. Benthem, Johan van. 1984. “Questions about Groenendijk, Jeroen & Martin Stokhof. 1982. “Se- wh Linguistics quantifiers”. Journal of Symbolic Logic 49.443Ϫ mantic analysis of -complements”. and Philosophy 5.175Ϫ233. 466. Ϫ. 1991. “Dynamic predicate logic”. Linguistics Chierchia, Gennaro & Ray Turner. 1988. “Seman- and Philosophy 14.39Ϫ100. tics and property theory”. Linguistics and Philoso- phy 11.261Ϫ302. Groenendijk, Jeroen, Martin Stokhof & Frank Veltman. 1996. “ and modality”. Hand- Chomsky, Noam. 1955. “Logical syntax and book of Contemporary Semantic Theory ed. by Sha- semantics: Their linguistic relevance”. Language lom Lappin, 179Ϫ213. Oxford: Blackwell. 31.36Ϫ45. Heim, Irene. 1983. “File change semantics and the Church, Alonzo. 1940. “A formulation of the sim- familiarity theory of ”. Meaning, Use, ple theory of types”. Journal of Symbolic Logic and Interpretation of Language ed. by Rainer 5.56Ϫ68. Bäuerle, Christoph Schwarze & Arnim von Ϫ. 1951. “A formulation of the logic of sense and Stechow, 164Ϫ189. Berlin: de Gruyter. ”. Structure, Method and Meaning ed. by Hendriks, Herman. 1993. Studied Flexibility: Cat- Paul Henle, Horace Kallen & Susanne Langer, egories and types in syntax and semantics. Amster- 3Ϫ24. New York: Liberal Arts Press. dam: ILLC. Cooper, Robin. 1979. “The interpretation of pro- Hinrichs, Erhard. 1986. “Temporal anaphora in nouns”. Syntax and Semantics, Vol. 10 ed. by of English”. Linguistics and Philosophy Frank Heny & Helmut Schnelle, 61Ϫ92. New 9.63Ϫ82. York: Academic Press. Janssen, Theo. 1997. “Compositionality”. Hand- Davidson, Donald. 1967. “Truth and meaning”. book of Logic and Language ed. by Johan van Ben- Ϫ Synthese 17.304 323. them & , 417Ϫ474. Amsterdam: Davidson, Donald & Gilbert Harman, eds. 1972. Elsevier. Semantics of Natural Language. Dordrecht: Reidel. Kamp, Hans. 1981. “A theory of truth and seman- Dekker, Paul & Robert van Rooy. 2000. “Bi-direc- tic representation”. Formal Methods in the Study of tional optimality theory: An application of game Language, 277Ϫ322. Amsterdam: Mathematical theory”. Journal of Semantics 17. 217Ϫ242. Centre. 222. Le de´veloppement des grammaires casuelles au XXe sie`cle 2073

Kamp, Hans & Uwe Reyle. 1993. From Discourse tation Theory and the Theory of Generalized Quan- to Logic. Dordrecht: Kluwer. tifiers ed. by Jeroen Groenendijk, Dick de Jongh & Ϫ Kaplan, Ron & Joan Bresnan. 1982. “Lexical-func- Martin Stokhof, 115 143. Dordrecht: Foris. tional grammar”. The Mental Representation of Ϫ, ed. 1976. Montague Grammar. New York: Aca- Grammatical Relations, 173Ϫ281. Cambridge, demic Press. Mass.: MIT Press. Partee, Barbara & Herman Hendriks. 1997. “Mon- Karttunen, Lauri. 1977. “Syntax and semantics of tague Grammar”. Handbook of Logic and Lan- questions”. Linguistics and Philosophy 1.1Ϫ82. guage ed. by Johan van Benthem & Alice ter Meu- Lambek, Joachim. 1958. “The mathematics of sen- len, 5Ϫ92. Amsterdam: Elsevier. tence structure”. American Mathematical Monthly Pollard, Carl & Ivan Sag. 1994. Head-Driven Ϫ 65.154 170. Phrase Structure Grammar. Stanford: CSLI. Landmann, Fred. 1989. “Groups I & II”. Linguis- Reichenbach, Hans. 1947. Elements of Symbolic Ϫ tics and Philosophy 12.559 605. Logic. New York: Macmillan. Link, Godehard. 1983. “The logical analysis of Shieber, Stuart. 1986. An Introduction to Unifica- plurals and mass terms: A lattice-theoretical ap- tion-Based Approaches to Grammar. Stanford: proach”. Meaning, Use, and Interpretation of Lan- CSLI. guage ed. by Rainer Bäuerle, Christoph Schwarze & Arnim von Stechow, 303Ϫ323. Berlin: de Gruyter. Siegel, Muffy. 1976. “Capturing the Russian adjec- tive”. Partee, ed. 1976, 293Ϫ309. Montague, Richard. 1974. Formal Philosophy. Ed. by Richmond H. Thomason. New Haven: Yale Stalnaker, Robert. 1979. “Assertion”. Syntax and University Press. Semantics, Vol. 9 ed. by Peter Cole, 315Ϫ332. New York: Academic Press. Moortgat, Michael. 1988. Categorical Investiga- tions: Logical and linguistic aspects of the Lambek Steedman, Mark. 1993. “Categorical grammar. Tu- calculus. Dordrecht: Foris. torial overview”. Lingua 90.221Ϫ258. Partee, Barbara. 1976. “Some transformational ex- Tarksi, Alfred. 1944. “The semantic Conception of tensions of Montague grammar”. Montague Gram- Truth”. Philosophy and Phenomenological Research mar ed. by Barbara Partee, 51Ϫ76. New York: Ac- 4.341Ϫ375. ademic Press. Ϫ. 1987. “Noun phrase interpretation and type- Martin Stokhof, Amsterdam shifting principles”. Studies in Discourse Represen- (The Netherlands)

222. Le de´veloppement des grammaires casuelles au XXe sie`cle

1. Le paradoxe casuel langues. Ambivalente parce qu’a` la fois 2. L’approche paradigmatique syntaxique Ϫ les cas grammaticaux expri- 3. Les roˆles se´mantiques (Fillmore) mant voix et valence Ϫ et se´mantique, les cas 4. Le cas dans la grammaire ge´ne´rative des concrets exprimant les coordonne´es spatio- anne´es 1980Ϫ90 5. De´finition et typologie casuelle selon temporelles et adverbiales de l’e´nonce´.En Mel’cˇuk outre, ses inventaires et son expression, qui 6. L’approche conceptualiste (Jackendoff et al.) concernent la morphologie, sont a` la fois 7. Conclusion distribue´s dans les paradigmes lexicaux 8. Bibliographie (noms et pronoms) et fonctionnels (les de´ter- minants, les adjectifs), et contraints par le 1. Le paradoxe casuel filtre phonologique, responsable de phe´no- me`nes de distribution comple´mentaire et de Le cas en grammaire est une cate´gorie para- syncre´tisme. doxale et ambivalente. Paradoxale parce que La the´orie du cas se trouve tiraille´e entre re´gie par le verbe mais marque´e sur le nom toutes les composantes de la langue: se´manti- et suppose´e comme universelle en tant que que, syntaxe, morphologie, phonologie. La cate´gorie re´gie, mais exprime´e par ze´ro ou au tentation est donc grande d’en faire un obser- contraire par une ple´thore de formes dans les vatoire privile´gie´ en faveur de the´ories plus