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pdf version of the entry Montague Semantics http://plato.stanford.edu/archives/win2011/entries/montague-semantics/ from the Winter 2011 Edition of the Stanford Encyclopedia of Philosophy Edward N. Zalta Uri Nodelman Colin Allen John Perry Principal Editor Senior Editor Associate Editor Faculty Sponsor Editorial Board http://plato.stanford.edu/board.html Library of Congress Catalog Data ISSN: 1095-5054 Notice: This PDF version was distributed by request to mem- bers of the Friends of the SEP Society and by courtesy to SEP content contributors. It is solely for their fair use. Unauthorized distribution is prohibited. To learn how to join the Friends of the SEP Society and obtain authorized PDF versions of SEP entries, please visit https://leibniz.stanford.edu/friends/ . Stanford Encyclopedia of Philosophy Copyright c 2011 by the publisher The Metaphysics Research Lab Center for the Study of Language and Information Stanford University, Stanford, CA 94305 Montague Semantics Copyright c 2011 by the author Theo M. V. Janssen All rights reserved. Copyright policy: https://leibniz.stanford.edu/friends/info/copyright/

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Page 1: Montague Semantics Sc

pdf version of the entry

Montague Semanticshttp://plato.stanford.edu/archives/win2011/entries/montague-semantics/

from the Winter 2011 Edition of the

Stanford Encyclopedia

of Philosophy

Edward N. Zalta Uri Nodelman Colin Allen John Perry

Principal Editor Senior Editor Associate Editor Faculty Sponsor

Editorial Board

http://plato.stanford.edu/board.html

Library of Congress Catalog Data

ISSN: 1095-5054

Notice: This PDF version was distributed by request to mem-

bers of the Friends of the SEP Society and by courtesy to SEP

content contributors. It is solely for their fair use. Unauthorized

distribution is prohibited. To learn how to join the Friends of the

SEP Society and obtain authorized PDF versions of SEP entries,

please visit https://leibniz.stanford.edu/friends/ .

Stanford Encyclopedia of Philosophy

Copyright c© 2011 by the publisher

The Metaphysics Research Lab

Center for the Study of Language and Information

Stanford University, Stanford, CA 94305

Montague Semantics

Copyright c© 2011 by the author

Theo M. V. Janssen

All rights reserved.

Copyright policy: https://leibniz.stanford.edu/friends/info/copyright/

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Montague SemanticsFirst published Mon Nov 7, 2011

Montague semantics is a theory of natural language semantics and of itsrelation with syntax. It was originally developed by the logician RichardMontague (1930–1971) and subsequently modified and extended bylinguists, philosophers, and logicians. The most important features of thetheory are its use of model theoretic semantics which has commonly beenused for the semantics of logical languages and it adherence to theprinciple of compositionality—that is, the meaning of the whole is afunction of the meanings of its parts and their mode of syntacticcombination. This entry presents the origins of Montague Semantics,summarizes important aspects of the classical theory, and sketches morerecent developments. We conclude with a small example, whichillustrates some modern features.

1. Introduction1.1 Background1.2 Basic Aspects

2. Components of Montague Semantics2.1 Unicorns and Meaning Postulates2.2 Noun Phrases and Generalized Quantifiers2.3 Logic and Translating2.4 Intensionality and Tautologies2.5 Scope and Derivational History

3. Philosophical Aspects3.1 From Frege to Intensions3.2 Compositionality3.3 Syntactic Categories and Semantic Types3.4 Pragmatics3.5 Ontology

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3.6 Psychology4. Concluding Remarks

4.1 Legacy4.2 Example4.3 Further Reading

BibliographyAcademic ToolsOther Internet ResourcesRelated Entries

1. Introduction

1.1 Background

Montague semantics is the approach to the semantics of natural languageintroduced by Richard Montague in the 1970s. He described the aim ofhis enterprise as follows:

The salient points of Montague's approach are a model theoreticsemantics, a systematic relation between syntax and semantics, and a fullyexplicit description of a fragment of natural language. His approachconstituted a revolution: after the Chomskyan revolution that broughtmathematical methods into syntax, now such methods were introduced insemantics.

Montague's approach became influential, as many authors began to workin his framework and conferences were devoted to ‘Montague grammar’.Later on, certain aspects of his approach were adapted or changed,

The basic aim of semantics is to characterize the notion of a truesentence (under a given interpretation) and of entailment(Montague 1970c, 223 fn).

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became generally accepted or were entirely abandoned. Nowadays notmany authors would describe their own work as ‘Montague semantics’given the many differences that have taken shape in semantics sinceMontague's own work, but his ideas have left important traces, andchanged the semantic landscape forever. In our presentation of Montaguesemantics the focus will be on these developments.

Richard Montague was a mathematical logician who had specialized inset theory and modal logic. His views on natural language must beunderstood with his mathematical background in mind. Montague heldthe view that natural language was a formal language very much in thesame sense as predicate logic was a formal language. As such, inMontague's view, the study of natural language belonged to mathematics,and not to psychology (Thomason 1974, 2). Montague formulated hisviews:

Sometimes only the first part of the quote is recalled, and that might raisethe question whether he did not notice the great differences: for instancethat natural languages develop without an a priori set of rules whereasartificial languages have an explicit syntax and are designed for a specialpurpose. But the quote as a whole expresses clearly what Montague meantby ‘no important theoretical difference’; the ‘single natural andmathematically precise theory’ which he aimed at, is presented in hispaper ‘Universal Grammar’ Montague (1970c). He became most wellknown by Montague (1973) in which the theory is applied to somephenomena which were discussed intensively in the philosophical

There is in my opinion no important theoretical differencebetween natural languages and the artificial languages of logicians;indeed I consider it possible to comprehend the syntax andsemantics of both kinds of languages with a single natural andmathematically precise theory. (Montague 1970c, 222)

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literature of those days.

Montague's interest in the field arose while teaching introductory logiccourses. Standard in such courses are exercises in which one is asked totranslate natural language sentences into logic. To answer such exercisesrequired a bilingual individual, understanding both the natural languageand the logic. Montague provided, for the first time in history, amechanical method to obtain these logical translations. About this,Montague said:

We next describe the basic ideas of Montague semantics are presented.Section 2 presents several components of Montague semantics in moredetail. Section 3 includes a discussion of philosophically interestingaspects, and Section 4 provides a detailed example and further reading.

1.2 Basic Aspects

To implement his objective, Montague applied the method which isstandard for logical languages: model theoretic semantics. This meansthat, using constructions from set theory, a model is defined, and thatnatural language expressions are interpreted as elements (or sets, orfunctions) in this universe. Such a model should not be conceived of as amodel of reality. On the one hand the model gives more than reality:natural language does not only speak about past, present and future of thereal world, but also about situations that might be the case, or areimaginary, or cannot be the case at all. On the other hand, however, themodel offers less: it merely specifies reality as conceived by language. Anexample: we speak about mass nouns such as water as if every part ofwater is water again, so as if it has no minimal parts, which physically is

It should be emphasized that this is not a matter of vague intuition,as in elementary logic courses, but an assertion to which we haveassigned exact significance. (Montague 1973, 266)

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not correct. For more information on natural language metaphysics, seeBach (1986b).

Montague semantics is not interested in a particular situation (e.g. the realworld) but in semantical properties of language. When formalizing suchproperties, reference to a class of models has to be made, and thereforethe interpretation of a language will be defined with respect to a set of(suitable) models. For example, in the introduction we mentioned that thecharacterization of entailment was a basic goal of semantics. That notionis defined as follows. Sentence A entails sentence B if in all models inwhich the interpretation of A is true, also the interpretation of B is true.Likewise a tautology is true in all models, and a contradiction is true in nomodel.

An essential feature of Montague semantics is the systematic relationbetween syntax and semantics. This relation is described by the Principleof Compositionality which reads, in a formulation that is standardnowadays:

An example. Suppose that the meaning of walk, or sing is (for each modelin the class) defined as the set of individuals who share respectively theproperty of walking or the property of singing. By appealing to theprinciple of compositionality, if there is a rule that combines these twoexpressions to the verb phrase walk and sing, there must be acorresponding rule that determines the meaning of that verb phrase. Inthis case, the resulting meaning will be the intersection of the two sets.Consequently, in all models the meaning of walk and sing is a subset ofthe meaning of walk. Furthermore we have a rule that combines the nounphrase John with a verb phrase. The resulting sentence John walks and

The meaning of a compound expression is a function of themeanings of its parts and of the way they are syntacticallycombined. (Partee 1984, 281)

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sings means that John is an element of the set denoted by the verb phrase.Note that in any model in which John is element of the intersection ofwalkers and singers, he is an element of the set of walkers. So John walksand sings entails John walks.

An important consequence of the principle of compositionality is that allthe parts which play a role in the syntactic composition of a sentence,must also have a meaning. And furthermore, each syntactic rule must beaccompanied by a semantic rule which says how the meaning of thecompound is obtained. Thus the meaning of an expression is determinedby the way in which the expression is formed, and as such thederivational history plays a role in determining the meaning. For furtherdiscussion, see Section 2.5.

The formulation of the aim of Montague semantics mentioned in theintroduction (‘to characterize truth and entailment of sentences’) suggeststhat the method is restricted to declarative sentences. But this is need notbe the case. In Montague (1973, 248 fn) we already find suggestions forhow to deal with imperatives and questions. Hamblin (1973) andKarttunen (1977) have given a semantics for questions by consideringthem as sentences in disguise (I ask you whether …), with a meaning thatis based upon sentences (viz. sets of propositions). Groenendijk andStokhof (1989) consider questions as expressions with meanings of theirown nature (namely partitions).

Since Montague only considered sentences in isolation, certaincommentators pointed out that the sentence boundary was a seriouslimitation for the approach. But what about discourse? An obviousrequirement is that the sentences from a discourse are interpreted one byone. How then to treat coreferentiality of anaphora over sentenceboundaries? The solution which was proposed first, was DiscourseRepresentation Theory Kamp (1981). On the one hand that was an

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offspring of Montague's approach because it used model theoreticsemantics, on the other hand it was a deviation because (discourse)representations were an essential ingredient. Nowadays there are severalreformulations of DRT that fit into Montague's framework (see van Eijckand Kamp (1997)). A later solution was based upon a change of the logic;dynamic Montage semantics was developed and that gave a procedure forbinding free variables in logic which has an effect on subsequent formulasGroenendijk and Stokhof (1991). Hence the sentence boundary is not afundamental obstacle for Montague semantics.

2. Components of Montague Semantics

2.1 Unicorns and Meaning Postulates

Montague's most influential article was ‘The proper treatment ofQuantification in ordinary English’ Montague (1973). It presented afragment of English that covered several phenomena which were in thosedays discussed extensively. One of the examples gave rise to thetrademark of Montague grammar: the unicorn (several publications onMontague grammar are illustrated with unicorns).

Consider the two sentences John finds a unicorn and John seeks aunicorn. These are syntactically alike (subject-verb-object), but aresemantically very different. From the first sentence follows that thereexists at least one unicorn, whereas the second sentence is ambiguousbetween the so called de dicto reading which does not imply the existenceof unicorns, and the de re reading from which existence of unicornsfollows.

The two sentences are examples of a traditional problem called‘quantification into intensional contexts’. Traditionally the secondsentence as a whole was seen as an intensional context, and the novelty ofMontague's solution was that he considered the object position of seek as

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the source of the phenomenon. He formalized seek not as a relationbetween two individuals, but as a relation between an individual and amore abstract entity, see Section 2.2. Under this analysis the existence ofa unicorn does not follow. The de re reading is obtained in a differentway, see Section 2.5.

It was Montague's strategy to apply to all expressions of a category themost general approach, and narrow this down, when required, by meaningpostulates. So initially, find is also considered to be a relation between anindividual and such an abstract entity, but some meaning postulaterestricts the class of models in which we interpret the fragment to onlythose models in which the relation for find is the (classical) relationbetween individuals.

As a consequence of this strategy, Montague's paper has many meaningpostulates. Nowadays semanticists often prefer to express the semanticproperties of individual lexical items directly in their lexical meaning.Then find is directly interpreted as a relation between individuals.Nowadays meaning postulates are mainly used to express structuralproperties of the models (for instance the structure of time axis), and toexpress relations between the meanings of words. For a discussion of therole of meaning postulates, see Zimmermann (1999).

2.2 Noun Phrases and Generalized Quantifiers

Noun phrases like a pig, every pig, and Babe, behave in many respectssyntactically alike: they can occur in the same positions, can beconjoined, etc. But a uniform semantics seems problematic. There wereproposals which said that every pig denotes the universally generic pig,and a pig an arbitrary pig. Such proposals were rejected by Lewis (1970),who raised, for instance, the question which would be the color of theuniversal pig, all colors, or would it be colorless?

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Montague proposed the denotation of a descriptive phrase to be a set ofproperties. For instance, the denotation of John is the set consisting ofproperties which hold for him, and of every man the set of propertieswhich hold for every man. Thus they are semantically uniform, and thenconjunction and/or disjunction of arbitrary quantifier phrases (includinge.g. most but not all) can be dealt with in a uniform way.

This abstract approach has led to generalized quantifier theory, seeBarwise and Cooper (1981) and Peters and Westerståhl (2006). By usinggeneralized quantifier theory a remarkable result has been achieved. Itconcerns ‘negative polarity items’: words like yet and ever. Theiroccurrence can be licensed by negation: The 6:05 has arrived yet is out,whereas The 6:05 hasn't arrived yet is OK. But there are more contexts inwhich negative polarity items may occur, and syntacticians did notsucceed in characterizing them. Ladusaw (1980) did so by using acharacterization from generalized quantifier theory. This was a greatsuccess for formal semantics! His proposal roughly was as follows.Downward entailing expressions are expressions that license inferencesfrom supersets to subsets. No is downward entailing because from Noman walks it follows that No father walks. A negative polarity item isacceptable only if it is interpreted in the scope of a downward entailingexpression, e.g. No man ever walks. Further research showed that theanalysis needed refining, and that a hierarchy of negative polarity itemsshould be used see Ladusaw (1996).

2.3 Logic and Translating

An expression may directly be associated with some element from themodel. For instance, walk with some set of individuals. Then also theoperations on meanings have to be specified directly, and that leads toformulations such as:

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Such descriptions are not easy to understand, nor convenient to workwith. Montague (1973, 256) said: ‘it is probably more perspicuous toproceed indirectly’. For this purpose he introduced a language, called‘intensional logic’. The operation described above is then represented by∧λtλu[t = u]. The λt says that it is a function that takes t as argument,likewise for λu. So λtλu[t = u] is a function which takes two arguments,and yields true if the arguments are equal, and otherwise false. Thepreceding ∧ says that we consider a function from possible worlds andmoments of time to the thus defined function.

Two features of the Montague's ‘intensional logic’ attracted attention.

1. It is a higher order logic. In those days, linguists, philosophers andmathematicians were only familiar with first order logic (the logic inwhich there are only variables for basic entities). Since in Montaguesemantics the parts of expressions must have meaning too, a higherorder logic was needed (we have already seen that every mandenotes a set of properties).

2. The logic has lambda abstraction, which in Montague's days was nota standard ingredient of logic. The lambda operator makes it possibleto express with higher order functions, and the operator made itpossible to cope differences between between syntax and semantics.For instance, in John walks and he talks there is only one occurrenceof John, whereas in the logic John should occur with the predicatewalk and with the predicate talk. The use of lambda-operatorsenables us to plug in the meaning of John at several positions. Theimportance of lambdas is expressed by Partee at a talk on ‘The firstdecade of Montague Grammar’: ‘Lambdas really changed my life’

G3 is that function f ∈ ((2I)A×A)Aω such that, for all x ∈ Aω, allu,t ∈ A and all i ∈ I : f(x)(t,u)(i) = 1 if and only if t = u.(Montague 1970a, 194)

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(Partee 1996, 24). Nowadays lambdas are a standard tool in allpapers in semantics. In section 4.1 an example will be given thatillustrates the power of lambdas.

This motivation for using translations (a tool for obtaining perspicuousrepresentations of meanings) has certain consequences.

1. Translation is a tool to obtain formulas which represent meanings.Different, but equivalent formulas are equally acceptable. In theintroduction of this article it was said that Montague grammarprovided a mechanical procedure for obtaining the logicaltranslation. As a matter of fact, the result of Montague's translationof Every man runs is not identical with the traditional translation,although equivalent with it, see the example in Section 4.1.

2. The translation into logic should be dispensable. So in Montaguesemantics there is nothing like ‘logical form’, which plays such animportant role in the tradition of Chomsky.

3. For each syntactic rule which combines one or more expressionsthere is a corresponding semantic rule that combines thecorresponding representations of the meanings. This connection isbaptized the rule-to-rule hypothesis Bach (1976). Maybe it is usefulto emphasize that (for rules with one argument) it is not forbiddenthat the corresponding semantic rule is the identity mapping (in casethe syntactic operation is meaning preserving).

4. Operations depending on specific features of formulas are notallowed. Janssen (1997) criticized several proposals on this aspect.He showed that proposals that are deficient in this respect are eitherincorrect (make wrong predictions for closely related sentences), orcan be corrected and generalized, and thus improved.

The method of using a logic for representing meanings has a long history.One might point to philosophers such as Dalgarno and Leibniz who

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developed formal languages in order to express philosophy clearly. In the19th century there were several proposals for artificial languages in orderto make mathematical argumentation more transparent, for instance byFrege and by Peano. Frege's ‘Begriffsschrift’ Frege (1879) can be seen asthe birth of predicate logic: he introduced quantifiers. His motivationcame from mathematical needs; he did not use his Begriffsschrift in hispapers on natural language. Russell (1905) used logic to represent themeanings of natural language. A classical example in his paper is theanalysis of The king of France is bald. Syntactically it has the formsubject-predicate, but if it would be constructed logically as a subject-predicate, then the king of France, which denotes nothing, cannot be thesubject. So there is a difference between the syntactic form, and thelogical form: natural language obscures the view of the real meaning. Thisbecame known as the ‘misleading form thesis’. Therefore philosophers oflanguage saw, in those days, the role of logic as a tool to improve naturallanguage. An interesting overview of the history of translating is given inStokhof (2007).

Note, however, that Montague semantics has nothing to do with the aimof improving natural language or providing its logical form.

2.4 Intensionality and Tautologies

Montague defined the denotation of a sentence as a function from possibleworlds and moments of time to truth values. Such a function is called an‘intension’. As he said (Montague 1970a, 218), this made it possible todeal with the semantics of common phenomena such as modifiers, e.g. inNecessarily the father of Cain is Adam. Its denotation cannot be obtainedfrom the truth value of The father of Cain is Adam : one has to know thetruth value for other possible worlds and moments of time. Theintensional approach also made it possible to deal with several classicalpuzzles. Two examples from Montague (1973) are: The temperature is

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rising, which should not be analyzed as stating that some number isrising. And John wishes to catch a fish and eat it should not should beanalyzed as stating that John has a particular fish in mind, but that hewants to eat the fish he will catch.

Intensional semantics has been criticized for the fact that all tautologiesget the same meaning (are synonymous). Indeed, a tautology as John is illor he is not ill gets as intension the function that constantly yields true,and the same for other tautologies. If one is interested in discriminatingsemantically between tautologies, then a refinement of the notions‘meaning’ and ‘equivalence’ is needed: ‘meaning’ should see distinctionsbetween tautologies, and ‘equivalence’ should be sensitive for the thusrefined notion of meaning.

There are several proposals to account for this problem. The oldest is byLewis (1970): propositions are structured by including in their meaningthe meanings of their parts. Then indeed Green grass is green and Whitesnow is white have different meanings. However, lexical synonyms pose aproblem for this approach. Since woodchuck and groundhog are namesfor the same species, John believes that Phil is a groundhog is, under thisview, equivalent with John believes that Phil is a woodchuck. However,since John might not be aware of the synonymy between woodchuck andgroundhog, the two sentences should not be equivalent. One couldconsider belief contexts a separate problem, but most authors see it as partof the problem of equivalence of all tautologies. The literature providesseveral proposals for dealing with this. Bäuerle and Cresswell (2003) givean overview of the older proposals, and Fox and Lappin (2005) of morerecent ones. The latter authors explain that there are two strategies: thefirst is to introduce impossible worlds in which woodchuck andgroundhog are not equivalent, and the second to introduce entailmentrelation with the property that identity does not follow from reciprocalentailment. Their own proposal follows the second strategy.

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2.5 Scope and Derivational History

A well known example of scope ambiguity is Every man loves a woman.Is there only one woman involved (e.g. mother Mary), or does every manlove a different woman? The sentence has no lexically ambiguous words,and there are no syntactic arguments to assign them more than oneconstituent structure. How to account for the ambiguity?

In Montague (1973) the scope ambiguity is dealt with by providing forthe sentence two different derivations. On the reading that every has widescope, the sentence is produced from every man and loves a woman. Onthe reading that only one woman is involved, the sentence is obtainedfrom Every man loves him1. The him1 is an artifact, a placeholder, or, onemight say, a syntactic variable. A special kind of rule, called a‘quantifying-in rule’, will replace this him1 by a noun phrase or a pronoun(in case there are more occurrences of this placeholder). The placeholdercorresponds with a logical variable that becomes bound by the semanticcounterpart of the quantifying-in rule. For the sentence under discussion,the effect of the application of the quantifying-in rule to a woman andEvery man loves him1 is that the desired sentence is produced and that thequantifier corresponding with a woman gets wide scope. When we woulddepict its derivation as a tree, this tree would be larger than the constituentstructure of the sentence due to the introduction and later removal ofhim1.

This quantifying-in rule is used by Montague for other phenomena aswell. An example is coreferentiality: Mary loves the man whom she kissedis obtained from He1 loves the man whom he1 kissed. And the de rereading of John seeks a unicorn is obtained from a unicorn and Johnseeks him1.

Many researchers did not like this analysis in which powerful rules and

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artificial symbols (him1) are used. Two strategies to remedy this can bediscriminated.

The first strategy was to eliminate the ambiguity. Some linguists haveargued that the scope order is the same as the surface order; this is knownas ‘Jackendoff's principle’ (Jackendoff 1972). Others said that suchsentences have only one reading, viz. its weakest reading (every widescope), and that the stronger reading is inferred when additionalinformation is available. These two approaches work well for simplesentences, but they are challenged by more complicated sentences inwhich the surface order is not a possible reading, or where the differentscope readings are logically independent.

The second strategy was to capture the ambiguity in another way than bythe quantifying-in rules. Historically the first method was to put theinterpretations of the noun phrases in a store from which theseinterpretations could be retrieved when needed: different stages ofretrieving correspond with differences in scope. One might see this as agrammar in which the direct correspondence between syntax andsemantics has been relaxed. The method is called ‘Cooper Store’, after theauthor who proposed this Cooper (1983). A later proposal is DRT(=discourse representation theory), where representations are used toaccount for such ambiguities van Eijck and Kamp (1997). A recentmethod is by means of ‘lifting rules’ (see Sect. 3.3): the meaning of anoun-phrase is ‘lifted’ to a more abstract level, and different levels yielddifferent scope readings Hendriks (2001). A radical position is taken byJacobson (1999) who does not use free variables at all: a pronoun like heis interpreted as the identity function on individuals, and sentences withunbound pronouns as functions from entities to truth values. This workswell for pronoun binding, but for the treatment of scope one of the othermethods has to be used.

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Even if the role of derivational history can be avoided for scope andcoreferentiality, other phenomena remain for which derivational historieshave a role. An example is John wondered when Alice said she wouldleave. This is ambiguous between John asking for the time of leaving, orfor the time of saying. So the sentence is ambiguous, even though thereare no arguments for assigning to it more than one constituent structure.Pelletier (1993) presents this sentence and others, and says: ‘In order tomaintain the Compositionality Principle, theorists have resorted to anumber of devices which are all more or less unmotivated (except tomaintain the Principle): Montagovian “quantifying-in” rules, traces, gaps,[…].’ Pelletier's objection can be appreciated if one assumes that meaningassignment is directly linked with constituent structure. But, as explainedin Section 1.2, this is not the case. The derivation specifies which rulesare combined in which order, and this derivation constitutes the input tothe meaning assignment function. The constituent structure is determinedby the output of the syntactic rules, and different derivation processesmay generate one and the same constituent structure. In this way,semantic ambiguities are accounted for. One should not call something‘constituent structure’ if it is not intended as such, and next refute itbecause it does not have the desired properties.

The distinction between a derivation tree and a constituent tree is made invarious theories of grammar. In Tree Adjoining Grammars (TAG's) thedifferent scope readings of the sentence about loving a woman differ inthe order in which the noun-phrases are substituted in the basic tree. Aclassical example in Chomskyan grammar is The shooting of the hunterswas bloody, which is ambiguous between the hunters shooting, or thehunters being shot at. The two readings come from two different sources:one in which the hunters is the subject of the sentence, and one in whichit is the object.

3. Philosophical Aspects

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3.1 From Frege to Intensions

Frege (1892) introduced the distinction between ‘sense’ and ‘reference’. Ithas been said that Montague followed this distinction, and that ‘intension’coincides with ‘sense’. But that is not correct. Let us first consider Frege'sargumentation. It concerns The Greeks did not know that the morning staris the evening star. During classical antiquity, it had not yet beendiscovered that both the morning star and the evening star are the planetVenus. We would, however, not like to analyze the sentence as statingthat the Greeks did not know that Venus is the same as Venus, i.e. thatthey did not recognize an obvious truth. Frege's theory is that in ordinarycontexts the expression the morning star denotes its referent (a celestialobject), but in indirect contexts it denotes something different that iscalled ‘its sense’. This notion includes not only the referent, but also theway in which one refers to an object. Since referring to a celestial objectby the morning star differs from referring to it by the evening star, thesentence The morning star is the evening star does not express an analytictruth.

Frege's approach was abandoned because it was not really satisfactory.The phrase the morning star is not what one would call a lexicalambiguity: there is no sentence that has different readings due to thatphrase. Nevertheless, Frege associated with that expression twodenotations. The situation gets even worse: Carnap (1947) noted thatunder Frege's approach we would also need the ‘sense of a sense’ etc.Consequently, Frege's approach requires an infinite hierarchy of semanticdenotations (and that for an expression which never gives rise to theambiguity of a sentence). Carnap proposed another formalization of thesame idea, but in which with one expression only one denotation isassociated. His solution, however, had problems, caused by the fact thathe identified possible worlds with models. For these reasons Montague

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(1970c, 233) introduced his ‘intensional logic’, in which the problems donot arise. The difference with Frege (one denotation for an expression,instead of infinitely many) was possible due to two novelties (seeMontague 1970a, 217–218): ‘descriptive phrases do not denoteindividuals’, and ‘the denotation of a sentence is not a truth value, but aproposition’. He explains: ‘Frege's argument that sentences cannot denotepropositions of course depends on the assumption that descriptive phrasesdenote individuals.’

For an elaborated discussion, see Janssen (2001) and Janssen (2011); forinformation on the history of intensional logic, see Montague (1970b,145).

3.2 Compositionality

For Montague the principle of compositionality was not a subject ofdeliberation or discussion, because for him, as a mathematical logician, itwas the only way to proceed. He describes his method in side remarkswith phrases like ‘following Tarski’, or ‘following Frege’, without evercalling it a principle. Later authors identified the Principle ofCompositionality as the cornerstone of Montague's work. The reason wasthat discussions arose, and an investigation of the foundations ofMontague grammar was asked for.

It has been claimed that Montague himself did not work compositionallyin the case of pronouns. This is, however, not the case. In order to explainthe compositional nature of his treatment of pronouns, both Janssen(1997) and Dowty (2007) explain how variables are interpreted in logic;we follow their explanations. Consider the following clauses from thetraditional Tarskian interpretation of predicate logic.

1. ⟦ϕ ∧ ψ⟧g = 1 if and only if ⟦ϕ⟧g = 1 and ⟦ψ⟧g = 12. ⟦∀xϕ⟧g = 1 if and only if for all h ∼xg holds ⟦ϕ⟧h = 1

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The first clause says: ϕ ∧ ψ is true when using assignment g if and onlyif ϕ and ψ are true when the assignment g is used. In the second clauseassignments h are introduced (by ∼x g) which are equal to g except maybefor the value they assign to variable x. Montague uses the same format,with the difference that besides g he also has i, the time of reference and j,the possible world, as superscripts.

In the formulation of the clauses there is nothing which can be pointed atas ‘the meaning’, in fact it is a definition of truth with g and h asparameters. So how is it possible that this (and Montague's work) arecompositional?

The answer requires a shift in perspective. The meaning of a formula ϕ,shortly M(ϕ), is the set of assignments for which the formula is true.Then the first clause says that M(ϕ ∧ ψ) = M(ϕ) ∩ M(ψ), so a simple set-theoretic combination on the two meanings is performed. And M(∀xϕ) ={h ∼xg∣g ∈ M(ϕ)}, which can be described as: extend the set M(ϕ) withall x-variants. Likewise, in Montague semantics the meaning of anexpression is a function which has as domain the triples <moment of time,possible world, assignment to variables>.

Is it possible to achieve compositionality for natural language? Obviouscandidates for counterexamples are idioms, because their meanings seemnot to be built from their constituting words. However, Westerståhl(2002) presents a collection of methods, varying from compound basicexpressions, to deviant meanings for constituting parts. Janssen (1997)refutes several other counterexamples that are put forward in theliterature.

How strong is compositionality? Mathematical results show that anylanguage can be given a compositional semantics, either by using anunorthodox syntax (Janssen (1986) and Janssen (1997)) or by using an

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unorthodox semantics Zadrozny (1994). However their proofs are nothelpful in practice. Hodges (2001) showed how a given compositionalsemantics for a fragment can be extended to a larger language.

Among formal semanticists one can find the following attitudes towardscompositionality (about the same list is given by Partee (1996)):

1. It is a methodological principle. Any proposal should obey it.Janssen (1997) is an advocate of this position.

2. It is a good method. Compositionality by itself has no special status,but it is an example how a systematic relation between syntax andsemantics could be given. Other methods are acceptable as well,such as using representations essentially (as in DRT Kamp (1981)),or allowing a few systematic exceptions to compositionality.

3. It is a kind of goal, but fulfilling it can be put aside. A description ofthe meaning of phrases exhibiting a certain phenomenon is given,and a remark is added that it is assumed that the results can beachieved in a compositional way.

4. It is an empirical principle. It is a claim about the organization of thegrammar, and formal semantics is an enterprise that investigates howfar this can be maintained. Maybe there are situations where the pricewould be too high, or where it would not be possible at all (see e.g.Dowty (2007)).

An extensive discussion of compositionality is given in Janssen (1997)and, in this encyclopedia, in the contribution by Szabó (2007).

3.3 Syntactic Categories and Semantic Types

According to Montague the purpose of syntax is to produce the input forthe semantics:

I fail to see any interest in syntax except as a preliminary to

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Although the syntax was in his eyes subordinate, he was fully explicit inhis rules. He used all kinds of ad hoc solutions, but did not usesophisticated tools, such as syntactic features. Moreover his rules produceflat strings.

In Montague (1970a) the relation between syntactic categories andsemantic types is given by a list, but in Montague (1973) a systematicrelation is defined, which amounts to the same relation as one would havein categorial grammar. However, his syntax is not a categorial syntaxbecause the rules are not always category driven and because some of therules are not concatenation rules.

For each of these two aspects, there have been proposals put forward. Onedirection was to stay closer to the ideals of categorial grammar, with onlytype driven rules, sometimes allowing for a restricted extension of thepower of concatenation rules. Typical examples are Morrill (1994) andCarpenter (1998). The other approach was to incorporate in Montaguegrammar as much as possible the insights from syntactic theories,especially originating from the tradition of Chomsky. A first step wasmade by Partee (1973), who let the grammar produce structures (labelledbracketings). A syntactically sophisticated grammar (with Chomskyanmovement rules) was used in the Rosetta translation project Rosetta(1994).

In order to cover linguistic generalizations, it is proposed to relax therelation between an expression and its semantic type (and therefore withits meaning). Montague had to introduce the and in John wants to catchand eat a fish by a different rule than the and in John walks and Marysings, because syntactically the second one is a conjunction of sentencesand the first is between verb phrases. However, the two meanings of andare closely related and usually one does not make a distinction in the

semantics. (Montague 1970c, 223)

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syntax. As a general solution it was proposed to use rules (or generalprinciples) that ‘lift’ the type of an expression to another category,corresponding with semantic rules that change the meaning. For instance,the meaning of and as a connective between verb phrases is obtained bylifting the sentence connective ∧ to λPλQλx[P(x) ∧ Q(x)]. Classicalpapers about the approach with lifting rules are Partee and Rooth (1983),Partee (1987), Hendriks (2001) and a monograph in which the wholecomplex of conjoined phrases is considered, is Winter (2001).

Nowadays the syntactic side usually plays no important role inpublications on Montague semantics. One rather focuses on asemantically interesting phenomenon, suggesting rules which are onlyexplicit concerning the semantic side. Whether and how the phenomenonfits together with the treatment of other phenomena is not considered.Montague's method to present fragments with a fully explicit syntax hasbeen abandoned.

3.4 Pragmatics

The meaning of sentences is sometimes determined by factors from thecontext of use; e.g. whether I am happy is true, depends on who thespeaker is. Other examples are here and this. Montague writes about thesefactors in his paper ‘Pragmatics’ Montague (1968) and in Montague(1970b), and he indicates how this could be done by introducingadditional parameters (besides the time and the possible world). Hispapers focus on the formal apparatus, and he works it out only for thepronoun I.

Several authors followed Montague's approach, and extended, whenneeded, the list of parameters. A classical example is given by theapproach of Kaplan (1989) for dealing with demonstratives andindexicals. He uses ‘context’ as a parameter, which consists at least inagent, moment of time, location, and possible world. The content of a

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sentence, with respect to a context, is a proposition, and the linguisticmeaning, or character, of an expression is a function from contexts tocontents. This difference between content and meaning is exploited todevelop his (influential) theory of demonstratives (she, her, that) andindexicals (I, today).

Cresswell (1973, 111) has another opinion. He argues that the approachwith parameters requires that a finite list of contextual features is given inan advance. He considers that to be impossible and provides analternative. His proposal is not followed by other authors.

Presuppositions and implicatures are often considered as belonging topragmatics. The aim of a recursive approach to presupposition wasalways in the air, for the practical reason that it seems the only way todeal with presuppositions for infinitely many sentences. An example of acompositional treatment is Peters (1979). But the phenomena arecomplex, and later treatments are not always completely compositional,several correcting factors have to be taken into consideration, see Beaver(1997).

Finally, there is pragmatics in the sense of using a language in practicalsituations. Declarative sentences can be used to ask questions, and to giveorders, and sometimes sentences are not used literally, but metaphorically.On this aspect of pragmatics not much has been written, but Cresswell(1973) explains that formal semantics has all the ingredients to cope withit.

3.5 Ontology

Montague's ‘intensional logic’ is a higher order logic. This aspectprovoked a very critical attack by Hintikka:

It seems to me that this is the strategy employed by Montague

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He also says:

And he proposes his own game theoretical semantics.

Some comments (the first two originate from Groenendijk & Stokhof,personal communication).

1. If first-order analysis is so natural and psychologically realistic, itwould be extremely interesting to have an explanation why it tookmore than two thousand years since Aristotle before the notion ‘firstorder’ was introduced by Frege.

2. It is difficult to see why the first-order notation matters. If there areontological commitments, then the notions used in the interpretationof the logic, in the metatheory, are crucial, and not the notation itself.It is, for instance difficult to understand why a winning strategy for agame is more natural than a function from properties to truth values.

3. If it is a point of axiomatizability, it would be interesting to have anaxiomatization of game theoretical semantics. As concernsintensional logic, one might use generalized models; with respect tothese models there is an axiomatization Gallin (1975).

Grammarians, who are in fact strongly committed tocompositionality. […]. There is a price to be paid however. Thehigher order entities evoked in this “type theoretical ascent” aremuch less realistic philosophically and psycholinguistically thanour original individuals. Hence the ascent is bound to detract fromthe psycholinguistic and methodological realism of one's theory.(Hintikka 1983, 20)

Moreover, the first order formulations have other advantages overhigher order ones. In first-order languages we can achieve anaxiomatization of logical truths and of valid inferences. (Hintikka1983, 285)

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Hintikka's criticism has not found supporters. Ironically, Hintikka'salternative (game theoretical semantics), is encapsulated in the traditionalTarskian approach (see Hodges (1997) or Caicedo et al. (2009)); theydefine the meaning of a formula as a collection of sets of assignments.

In Montague's approach possible worlds are basic objects without internalor external structure. Phenomena having to do with belief, requireexternal structure, such as an accessibility relation for belief-alternatives.Counterfactuals require a distance notion to characterize worlds whichdiffer minimally from each other. Structures on possible worlds are usedfrequently.

Sometimes an internal structure for possible worlds is proposed. Apossible world determines a set of propositions (those propositions whichare true with respect to that world), and in Fox and Lappin (2005) thereverse order is followed. They have propositions as primitive notions,and define possible worlds on the basis of them. Also Cresswell (1973)provides a method for obtaining possible worlds with internal structure:he describes how to build possible worlds from basic facts. None of theseproposals for internal structure have been applied by other authors thanthe proposers.

The philosophical status of certain entities is not so clear, such as pains,tasks, obligations and events. These are needed when evaluating sentenceslike e.g. Jones had a pain similar to the one he had yesterday. In ‘On thenature of certain philosophical entities’, Montague (1960) describes howthese notions can be described using his intentional logic; they areproperties of moments of time in a possible world. Of these notions, onlyevents occur in papers by other authors, albeit not in the way Montaguesuggested. They are seen as basic, but provided with an algebraicstructure allowing for e.g. subevents, (Link 1998, ch. 10–12 and Bach1986a).

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The set E may include whatever one would like to consider as basicentities: numbers, possible objects, and possible individuals. Whether anindividual is considered to be really living or existing at a certain momentof time or in a certain possible world is not given directly by the model;one has to introduce a predicate expressing this. Normally the set E hasno internal structure, but for mass nouns (which have the characteristicproperty that any part of water is water), a structure is needed, seePelletier and Schubert (2003). Also plurals might need a structure on theset E, e.g. when sum-individuals are used, see Link (1983), Link (1998,ch. 1–4), and Bach (1986a). Also when properties (loving John) areconsidered as entities for which predicates may hold (Mary likes lovingJohn) structure is needed: property theory gives the tools to incorporatethem, see Turner (1983).

3.6 Psychology

When Montague grammar emerged, the leading theory about syntax wasChomskyan grammar. That approach claimed that it revealed processesthat went on in the brain, and that linguistics was a branch of biology. Inthose days it was experimentally shown that the passive transformationwas a real process in the brain. Chomskyan grammar still is a leadingtheory, and although most of the theory has changed considerably (thereis no passive transformation anymore), it still considers itself to berevealing psychologically real processes. Montague had no psychologicalclaim for his theory; on the contrary, he considered linguistics as a branchof mathematics and not of psychology (Thomason 1974, 2). But thescientific context caused that the issue could not be avoided.

Partee (1977) explained to a meeting of psychologists that the theorycannot be applied directly to psychology because of the huge numbers ofentities in the models (infinite numbers of functions from functions tofunctions). Partee (1979) argues that there is a deep gap between the

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mathematical view and the psychological view, especially concerningpropositional attitude verbs and the behavior of proper names in suchcontexts, and she says that this gap has, somehow, to be bridged.

An argument often put forward in defense of compositionality concernsits psychological motivation. The principle explains how a person canunderstand sentences he has never heard before. This motivation forcompositionality is attacked by Schiffer (1987). On the one hand heargues that compositionality is not needed in an explanation of thatpower, and on the other hand that a compositional approach does notwork. His argumentation is illustrated by Tanya believes that Gustav is adog. Schiffer considers several compositional theories and argues thatnone of these theories offers a plausible account for the proposition that issupposed to be the content of Tanya's belief. So there is nothing fromwhich the meaning of the sentence can be formed compositionally. Hencecompositionality cannot hold. Partee (1988) discusses Schiffer'sarguments against compositionality, and explains that Schiffer does notmake a sufficient distinction between semantic facts and psychologicalfacts. There is a fundamental difference between semantic factsconcerning belief contexts (as implication and synonymy), and questionsthat come closer to psychological processes (how can a person sincerelyutter such a sentence?). What Schiffer showed was that problems arise ifone attempts to connect semantic theories with the relation betweenhuman beings and their language. Partee points out the analogy betweenthese problems with belief and those with the semantics of proper names(how can one correctly use proper names without being acquainted withthe referent). The latter is discussed and explained by Kripke (1972).Partee proposes to solve the problems of belief along the same lines. Herpaper is followed by a reaction of Schiffer (1988). However, he does notreact to this suggestion, nor to the main point: that a semantic theory is tobe distinguished from a psychological theory.

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An extensive discussion of the relation between Montague semantics andpsychology is given in the last chapter in Dowty (1979). He starts hischapter with a description of the situation. ‘Contemporary linguists, unlikemany philosophers of language, almost invariably profess to be concernedwith the “psychological reality” of the theoretical concepts they postulatein semantics analysis’ Dowty (1979). He works out this point and thendescribes his own position. ‘To get the point right away, let me confessthat I believe that the model theoretic intension of a word has in principlenothing whatsoever to do with what goes on in a person's head when heuses a word.’ Nevertheless, he tries to show that the notion of intension isa fundamental and indispensable concept from the point of view of‘psychological semantics’. There are three reasons. The first is thatsemantics provides a theory that explains entailment (and synonymity,validity contradiction etc.’), all notions that must somehow be part of atheory of language understanding. Secondly, the theory of truth andreference must be a bottom line in any general account of ‘meaning’ innatural language. And thirdly, when certain ways of compositionallyderiving the meanings from their parts can be shown to be necessary in atheory of truth and reference, then it may be concluded that the samecompositional analysis is necessary in a theory of languageunderstanding.

These examples illustrate the general opinion that psychological realitycan only very indirectly be associated with what is going on in Montaguesemantics; only a few articles discuss the connection.

4. Concluding Remarks

4.1 Legacy

Montague revolutionized the field of semantic theory. He introducedmethods and tools from mathematical logic, and set standards for

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explicitness in semantics. Now all semanticists know that logic has moreto offer than first order logic only. Finally, recall that Barbara Partee said:‘lambdas really changed my life’; in fact lambdas changed the lives of allsemanticists.

4.2 Example

A small example is presented below, it consists of the two sentences Johnis singing, and Every man is singing. The example is not presented inMontague's original way, but modernized: there is a lifting rule, thedeterminer is a basic expression, and intensional aspects are notconsidered.

The grammar has four basic expressions:1. John is an expression of the category Proper Name. Its denotation is anindividual represented in logic by John.2. The Intransitive Verb sing denotes a set (the set of singers), and isrepresented by the predicate symbol sing. 3. The Common Noun man, which denotes a set, represented by man. 4. The Determiner every. Its denotation is λPλQ∀x[P(x) → Q(x)]; anexplanation of this formula will be given below.

The grammar has three rules.

1. A rule which takes as input a Proper Name, and produces a NounPhrase. The input word is not changed: it is lifted to a ‘higher’grammatical category. Semantically its meaning is lifted to a moreabstract, a ‘higher’ meaning: the representation of the denotation of Johnas Noun Phrase is λP[P(John)]. An explanation of the formula is asfollows. P is a variable over properties: if we have chosen aninterpretation for P, we may say whether P holds for John or not, i.e.whether P(John) is true. The λP abstracts from the possibleinterpretations of P: the expression λP[P(John)] denotes a function that

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takes as input properties and yields true if the property holds for John,and false otherwise. So the denotation of John is the characteristicfunction of the set of properties he has.

2. A rule that takes as input a Noun Phrase and an Intransitive Verb, andyields as output a Sentence: from John and sing it produces John issinging. The corresponding semantic rule requires the denotation of theNoun Phrase to be applied to the denotation of the Intransitive Verb. Thisis represented as λP[P(John)](sing). When applied to the argument sing,the function represented by λP[P(John)] yields true if the predicate singholds for John, so precisely in case sing(John) is true. SoλP[P(John)](sing) and sing(John) are equivalent. The latter formula canbe obtained by removing the λP and substituting sing for P. This is called‘lambda-conversion’.

3. A rule that takes as inputs a Determiner and a Common Noun, andyields a Noun Phrase: from every and man it produces every man.Semantically the denotation of the Determiner has to be applied to thedenotation of the Common Noun, hence λPλQ∀x[P(x) → Q(x)](man).By lambda conversion (just explained) this is simplified to λQ∀x[man(x)→ Q(x)]. This result denotes a function that, when applied to property A,yields true just in case all man have property A.

The example given with the last rule helps us to understand the formulafor every : that denotes a relation between properties A and B which holdsin case every A has property B.

The next step is now easy. Apply the rule for combining a Noun Phraseand an Intransitive Verb to the last result, producing Every man issinging. The output of the semantic rule is λQ∀x[man(x) → Q(x)](sing).By lambda conversion we obtain ∀x[man(x) →sing(x)], which is thetraditional logical representation of Every man is singing.

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Note the role of lambda-operators: 1. John and every man are interpreted in a similar way: sets of properties.These sets can be represented due to lambda-operators.2. Every man and sing are syntactically on the same level, butsemantically sing has a subordinated role: it occurs embedded in theformula. This switch of level is possible due to lambda-operators.

4.3 Further Reading

The standard introductory book for Montague semantics is Dowty et al.(1981). An overview of the development of Montague semantics, withspecial attention for linguistic aspects, is given by Partee and Hendriks(1997). Collections of the most important papers in the field are Partee(1976), Portner and Partee (2002) and Partee (2004). Articles about newdevelopments are usually published in Linguistics and Philosophy andNatural language semantics.

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