Montague semantics is a theory of natural language semantics and ofits relation with syntax. It was originally developed by the logicianRichard Montague (1930–1971) and subsequently modified andextended by linguists, philosophers, and logicians. The most importantfeatures of the theory are its use of model theoretic semantics whichis nowadays commonly used for the semantics of logical languages andits adherence to the principle of compositionality—that is, themeaning of the whole is a function of the meanings of its parts andtheir mode of syntactic combination. This entry presents the originsof Montague Semantics, summarizes important aspects of the classicaltheory, and sketches more recent developments. We conclude with asmall example, which illustrates some modern features.

• 1. Introduction
  • 1.1 Background
  • 1.2 Basic Aspects
• 2. Components of Montague Semantics
  • 2.1 Unicorns and Meaning Postulates
  • 2.2 Noun Phrases and Generalized Quantifiers
  • 2.3 Logic and Translating
  • 2.4 Intensionality and Tautologies
  • 2.5 Scope and Derivational History
• 3. Philosophical Aspects
  • 3.1 From Frege to Intensions
  • 3.2 Compositionality
  • 3.3 Syntactic Categories and Semantic Types
  • 3.4 Pragmatics
  • 3.5 Ontology
• 4. Concluding Remarks
  • 4.1 Legacy
  • 4.2 Further Reading
  • 4.3 Example
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. Introduction

1.1 Background

Montague semantics is the approach to the semantics of naturallanguage introduced by Richard Montague in the 1970s. He described theaim of his enterprise as follows:

The basic aim of semantics is to characterize the notion of a truesentence (under a given interpretation) and of entailment (Montague1970c, 373 fn).

The salient points of Montague’s approach are a model theoreticsemantics, a systematic relation between syntax and semantics, and afully explicit description of a fragment of natural language. Hisapproach constituted a revolution: after the Chomskyan revolution thatbrought mathematical methods into syntax, now such methods wereintroduced in semantics.

Montague’s approach became influential, as many authors began towork in his framework and conferences were devoted to ‘Montaguegrammar’. Later on, certain aspects of his approach were adaptedor changed, became generally accepted or were entirely abandoned.Nowadays not many authors would describe their own work as‘Montague semantics’ given the many differences that havetaken shape in semantics since Montague’s own work, but hisideas have left important traces, and changed the semantic landscapeforever. In our presentation of Montague semantics the focus will beon 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 held theview that natural language was a formal language very much in the samesense as predicate logic was a formal language. As such, inMontague’s view, the study of natural language belonged tomathematics, and not to psychology (Thomason 1974, 2). Montagueformulated his views:

There is in my opinion no important theoretical difference betweennatural languages and the artificial languages of logicians; indeed Iconsider it possible to comprehend the syntax and semantics of bothkinds of languages with a single natural and mathematically precisetheory. (Montague 1970c, 373)

Sometimes only the first part of the quote is recalled, and that mightraise the question whether he did not notice the great differences:for instance that natural languages develop without an a priori set ofrules whereas artificial languages have an explicit syntax and aredesigned for a special purpose. But the quote as a whole expressesclearly what Montague meant by ‘no important theoreticaldifference’; the ‘single natural and mathematicallyprecise theory’ which he aimed at, is presented in his paper‘Universal Grammar’ (Montague 1970c). He became mostwell-known after the appearance of Montague 1973, in which the theoryis applied to some phenomena which were discussed intensively in thephilosophical literature of those days.

According to Caponigro (forthcoming), Montague’s interest in thefield arose when preparing a seminar on the philosophy of language asa visiting professor in Amsterdam in 1966. Only a couple of yearsearlier, he had deemed the “systematic exploration of theEnglish language, indeed of what might be called the ‘logic ofordinary English’, […] either extremely laborious orimpossible” and did ‘not find it rewarding’(Montague and Kalish 1964, 10). Yet he appears to have changed hismind after perusing Quine’s (1960)Word and Object aswell as Chomsky’s (1965)Aspects of the Theory ofSyntax: the latter opened the perspective of treating the syntaxof natural language as a formal system but failed to provide anyserious analysis of linguistic meaning; the former offered asystematic connection between traditional grammar and formal logic– and much more systematically so than contemporary logic texts.In fact, Montague’s semantic work owes a lot to Quine’sdescriptive insights into the ‘logic of ordinary English’,but differs from his predecessor by making the connection betweenlanguage and logic in rigorous, mathematical terms:

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, 237)

We next describe the basic ideas of Montague semantics. Section2 presents several components of Montague semantics in more detail.Section3 includes a discussion of philosophically interesting aspects, andSection4 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, andthat natural language expressions are interpreted as elements (orsets, or functions) in this universe. Such a model should not beconceived of as a model of reality. On the one hand, the model givesmore than reality: natural language does not only speak about past,present and future of the real world, but also about situations thatmight be the case, or are imaginary, or cannot be the case at all. Onthe other hand, however, the model offers less: it merely specifiesreality as conceived by language. An example: we speak about massnouns such aswater as if every part of water is water again,as if it has no minimal parts, which physically is not correct. Formore information on natural language metaphysics, see Bach 1986b.

Montague semantics is not interested in a particular situation (e.g.the real world) but in semantical properties of language. Whenformalizing such properties, reference to a class of models has to bemade, and therefore the interpretation of a language will be definedwith respect to a set of (suitable) models. For example, in theintroduction we mentioned that the characterization of entailment wasa basic goal of semantics. That notion is defined as follows. Sentence\(A\) entails sentence \(B\) if in all models in which theinterpretation of \(A\) is true, also the interpretation of \(B\) istrue. Likewise, a tautology is true in all models, and a contradictionis true in no model.

An essential feature of Montague semantics is the systematic relationbetween syntax and semantics. This relation is described by thePrinciple of Compositionality, which reads, in a formulationthat is standard nowadays:

The meaning of a compound expression is a function of the meanings ofits parts and of the way they are syntactically combined. (Partee1984, 281)

An example: Suppose that the meaning ofwalk, orsing is (for each model in the class) defined as the set ofindividuals who share respectively the property of (being anindividual that is) walking or the property of (being an individualthat is) singing. By appealing to the principle of compositionality,if there is a rule that combines these two expressions to the verbphrasewalk and sing, there must be a corresponding rule thatdetermines the meaning of that verb phrase. In this case, theresulting meaning will be the intersection of the two sets.Consequently, in all models the meaning ofwalk and sing is asubset of the meaning ofwalk. Furthermore, we have a rulethat combines the noun phraseJohn with a verb phrase. Theresulting sentenceJohn walks and sings means that John is anelement of the set denoted by the verb phrase. Note that in any modelin which John is element of the intersection of walkers and singers,he is an element of the set of walkers. SoJohn walks andsings entailsJohn walks.

An important consequence of the principle of compositionality is thatall the parts that play a role in the syntactic composition of asentence must also have a meaning. And furthermore, each syntacticrule must be accompanied by a semantic rule which says how the meaningof the compound is obtained. Thus, the meaning of an expression isdetermined by the way in which the expression is formed, and as suchthe derivational history plays a role in determining the meaning. Forfurther discussion, see Section2.5.

The formulation of the aim of Montague semantics mentioned in theintroduction (‘to characterize truth and entailment ofsentences’) suggests that the method is restricted todeclarative sentences. But this need not be the case. In Montague 1973(241 fn) we already find suggestions for how to deal with imperativesand questions. Hamblin (1973) and Karttunen (1977) have given asemantics for questions by analyzing them as expressing sets ofpropositions, viz. those expressed by their (declarative) answers; analternative approach, taken by Groenendijk and Stokhof (1989)considers questions as partitioning logical space into mutuallyexcluding possibilities.

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 oneby one. How then to treat co-referentiality of anaphora over sentenceboundaries? The solution which was proposed first wasDiscourseRepresentation Theory (Kamp 1981). On the one hand, that was anoffspring 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 areseveral reformulations of DRT that fit into Montague’s framework(see van Eijck and Kamp 1997). A later solution was based upon achange of the logic; dynamic Montague semantics was developed and thatgave a procedure for binding free variables in logic which has aneffect on subsequent formulas (Groenendijk and Stokhof 1990, 1991).Hence the sentence boundary is not a fundamental obstacle for Montaguesemantics.

2. Components of Montague Semantics

2.1 Unicorns and Meaning Postulates

Montague’s most influential article was ‘The ProperTreatment of Quantification in Ordinary English’ (Montague1973), commonly abbreviated as ‘PTQ’. 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 sentencesJohn finds a unicorn andJohnseeks a unicorn. These are syntactically alike(subject-verb-object), but are semantically very different. From thefirst sentence follows that there exists at least one unicorn, whereasthe second sentence is ambiguous between the so calleddedicto (ornon-specific, ornotional) readingwhich does not imply the existence of unicorns, and thede re(orspecific, orobjectual) reading from whichexistence of unicorns follows.

The two sentences are examples of a traditional problem called‘quantification into intensional contexts’. Traditionally,the second sentence as a whole was seen as an intensional context, andthe novelty of Montague’s solution was that he considered theobject position ofseek as the source of the phenomenon. Heformalizedseek not as a relation between two individuals,but as a relation between an individual and a more abstract entity(see section 2.2). Under this analysis the existence of a unicorn doesnot follow. Thede re reading is obtained in a different way(see section 2.5).

It was Montague’s strategy to apply to all expressions of acategory the most general approach, and narrow this down, whenrequired, by meaning postulates (and, in some cases, logicaldecomposition). So initially,find is also considered to be arelation between an individual and such an abstract entity, but somemeaning postulate restricts the class of models in which we interpretthe fragment to only those models in which the relation forfind is the (classical) relation between individuals.

As a consequence of this strategy, Montague’s paper has manymeaning postulates. Nowadays semanticists often prefer to express thesemantic properties of individual lexical items directly in theirlexical meaning, and thenfind is directly interpreted as arelation between individuals. Meaning postulates are mainly used toexpress structural properties of the models (for instance, thestructure of the time axis), and to express relations between themeanings of words. For a discussion of the role of meaning postulates,see Zimmermann 1999.

2.2 Noun Phrases and Generalized Quantifiers

Noun phrases likea pig,every pig, andBabe, behave in many respects syntactically alike: they canoccur in the same positions, can be conjoined, etc. But a uniformsemantics seems problematic. There were proposals which said thatevery pig denotes the universally generic pig, andapig an arbitrary pig. Such proposals were famously rejected byLewis (1970), who raised, for instance, the question which would bethe color of the universal pig, all colors, or would it becolorless?

Montague proposed the denotation of a descriptive phrase to be a setof properties. For instance, the denotation ofJohn is theset consisting of properties which hold for him, and ofeveryman the set of properties which hold for every man. Thus they aresemantically uniform, and then conjunction and/or disjunction ofarbitrary quantifier phrases (including e.g.most but notall) can be dealt with in a uniform way.

This abstract approach has led to generalized quantifier theory, seeBarwise and Cooper 1981 as well as Peters and Westerståhl 2006.Among the most popular achievements of generalized quantifier theoryis a semantic characterization of so-called ‘negative polarityitems’: words likeyet andever. Theiroccurrence can be licensed by negation:The 6:05 has arrivedyet is out, whereasThe 6:05 hasn’t arrived yet isOK. But there are more contexts in which negative polarity items mayoccur, and syntacticians did not succeed in characterizing them.Ladusaw (1980) did so by using a characterization from generalizedquantifier theory. This has been widely acknowledged as 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 becausefromNo man walks it follows thatNo father walks. Anegative polarity item is acceptable only if it is interpreted in thescope of a downward entailing expression, e.g.No man everwalks. Further research showed that the analysis needed refining,and that a hierarchy of negative polarity items should be used(Ladusaw 1996, Homer 2021).

2.3 Logic and Translation

An expression may directly be associated with some element from themodel. For instance,walk with some set of individuals. Thenalso the operations on meanings have to be specified directly, andthat leads to formulations such as:

\(G_3 (\ulcorner\)is\(\urcorner)\) is that function \(f \in((2^I)^{A\times A})^{A^{ \omega}}\) such that, for all \(x \inA^{\omega}\), all \(u,t \in A\) and all \(i \in I : f(x)(t,u)(i) = 1\)if and only if \(t = u\). (Montague 1970a, 195)

Such descriptions are not easy to understand, nor convenient to workwith. Montague (1973, 228) said, ‘it is probably moreperspicuous to proceed indirectly’. For this purpose heintroduced a language, called ‘intensional logic’. Theoperation described above is then represented by \(^{\wedge}\lambdat\lambda u[t = u\)]. The \(\lambda t\) says that it is a function thattakes \(t\) as argument, likewise for \(\lambda u\). So \(\lambdat\lambda u[t = u\)] is a function which takes two arguments, andyields true if the arguments are equal, and otherwise false. Thepreceding \(^{\wedge}\) says that we consider a function from possibleworlds and moments of time to the thus defined function.

Three features of the Montague’s ‘intensional logic’attracted attention:

1. It is ahigher-order logic. Even though type logic wasalready an established logical framework in those days, linguists,philosophers and mathematicians were more familiar with first orderlogic (the logic in which there are only variables for basicentities). Since in Montague semantics the parts of expressions musthave meaning too, a higher order logic was needed (we have alreadyseen thatevery man denotes a set of properties).

2. It isintensional in that it obeys neither Leibniz’ lawof substitution of co-extensional terms nor existentialgeneralization, thereby seemingly bringing logic closer to naturallanguage. To achieve this, Montague generalized Kripke’s (1963)groundbreaking semantic techniques from first-order modal logic tohigher-order type logic. However, the same interpretive effect couldhave been achieved by using a two-sorted, extensional type logic(Gallin 1975; Zimmermann 1989; 2021), which many semanticsists prefertoday.

3. Montague used\(\lambda\)-abstraction, which at the time wasnot a standard ingredient of logic. As illustrated in section 4.1, the\(\lambda\)-operator makes it possible to express higher-orderfunctions, which may serve as the contributions that parts ofsentences make to their truth-values. The importance of\(\lambda\)-expressions was once expressed by Barbara Partee in a talkon ‘The first decade of Montague Grammar’: ‘Lambdaschanged my life’ (Partee 1996, 24). Nowadays\(\lambda\)-expressions are a standard tool in natural languagesemantics, and particularly in the type-driven approach made popularby Heim and Kratzer (1998). In section 4.1, an example will be giventhat illustrates the power of \(\lambda\)-expressions.

This motivation forindirect interpretation – by way ofcompositional translation as a tool for obtaining perspicuousrepresentations of meaning – has a number of importantconsequences:

1. Translation is a tool to obtain formulas whichrepresentmeanings. Different, but equivalent formulas are equally acceptable.In the introduction of this article it was said that Montague grammarprovided a mechanical procedure for obtaining the logical translation.As a matter of fact, the result of Montague’s translation ofEvery man runs is not identical with the traditionaltranslation, although equivalent with it, see the example in Section4.1.
2. The translation into logic is a mere notational convenience and inprinciple dispensable. So in Montague semantics, translation intological notation does not provide unique ‘logicalforms’.
3. For each syntactic construction (operation plus rule, in theterminology of Montague 1970c) there is a semantic rule that combinesthe corresponding representations of the meanings. Bach (1976) aptlycalled this interpretive strategy the ‘rule-to-rulehypothesis‘.
4. Operations depending on syntactic 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 logical notation for representing meanings has along history, going back at least to philosophers such as Dalgarno andLeibniz who developed formal languages in order to express philosophyclearly. In the 19th century, there were several proposals forartificial languages in order to make mathematical argumentation moretransparent, for instance by Frege and by Peano. Frege’s‘Begriffsschrift’ (Frege 1879) can be seen as the birth ofpredicate logic: he introduced quantifiers. His motivation came frommathematical needs; he did not use his Begriffsschrift in his paperson natural language. Russell (1905) used logic to represent themeanings of natural language. A classical example in his paper is theanalysis ofThe king of France is bald. Syntactically it hasthe form subject-predicate, but if it were constructed logically as asubject-predicate, thenthe king of France, which denotesnothing, cannot be the subject. So syntactic form and logical form maydiverge: natural language obscures the view of the real meaning. Thisbecame known as the ‘misleading form thesis’. Therefore,philosophers of language saw, in those days, the role of logic as atool to improve natural language, an aim that is alien to Montaguesemantics. In fact, using higher-order functional type theory (Church1940) as the target of his translation, Montague (1970c) developed a‘compositional’ version of Russell‘s analysis, whichdoes preserve the constituent structure of the source language(English). An interesting overview of the history of translatingnatural language into logic is given in Stokhof 2007.

2.4 Intensionality and Tautologies

Montague defined the denotation of a sentence as a function frompossible worlds and moments of time to truth values. Such a functionis called an ‘intension’. As he said (Montague 1970a,220), this made it possible to deal with the semantics of commonphenomena such as modifiers, e.g. inNecessarily the father ofCain is Adam. Its denotation cannot be obtained from the truthvalue ofThe father of Cain is Adam: one needs to know thetruth value for other possible worlds and moments of time. Theintensional approach also made it possible to deal with severalclassical puzzles. Two examples from Montague 1973 are:Thetemperature is rising, which should not be analyzed as statingthat some number is rising; andJohn wishes to catch a fish andeat it, which should not be analyzed as implying that John has aparticular fish in mind.

Intensional semantics has been criticized for the fact that alltautologies get the same meaning (are synonymous). Indeed, a tautologyasJohn is ill or he is not ill gets as intension thefunction that constantly yieldstrue, and the sameholds 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 distinctions between tautologies, and‘equivalence’ should be sensitive for the thus refinednotion of meaning. The oldest proposals to account for this problemgoes back to Carnap (1947, §14) and was later taken up by Lewis(1970, sec. 5): propositions are structured by including in theirmeanings also the meanings of their parts. Then indeedGreen grassis green andWhite snow is white have differentmeanings. However, lexical synonyms still pose a problem. Sincewoodchuck andgroundhog are names for the samespecies,John believes that Phil is a groundhog is, underthis view, equivalent withJohn believes that Phil is awoodchuck. One could consider belief contexts a separate problem,but most authors see it as part of the problem of equivalence of alltautologies.

Later several proposals for dealing with this have been developed;surveys can be found in Bäuerle and Cresswell (2003), Fox andLappin (2005), and Egré (2021). The latter authors explain thatthere are two strategies: the first is to introduce impossible worldsin whichwoodchuck andgroundhog are not equivalent,and the second is to introduce an entailment relation with theproperty that identity does not follow from reciprocal entailment. Foxand Lappin follow the second strategy.

2.5 Scope and Derivational History

A well known example of scope ambiguity isEvery man loves awoman. Is there only one woman involved (e.g. Mother Mary), ordoes every man love a different woman? The sentence has no lexicallyambiguous words, and there are no syntactic arguments to assign themmore than one constituent structure. How to account for theambiguity?

In Montague 1973, the scope ambiguity is dealt with by providing forthe sentence two different derivations. On the reading thatevery has wide scope, the sentence is produced fromeveryman andloves a woman. On the reading that only onewoman is involved, the sentence is obtained fromEvery man loveshim\(_1\). Thehim\(_1\) is an artifact, a placeholder,or, one might say, a syntactic variable. A special kind of rule,called a ‘quantifying-in rule’, will replace thishim\(_1\) by a noun phrase or a pronoun (in case there aremore occurrences of this placeholder). The placeholder correspondswith a logical variable that becomes bound by the semantic counterpartof the quantifying-in rule. For the sentence under discussion, theeffect of the application of the quantifying-in rule toawoman andEvery man loves him\(_1\) is that the desiredsentence is produced and that the quantifier corresponding withawoman gets wide scope. When we would depict its derivation as atree, this tree would be larger than the constituent structure of thesentence due to the introduction and later removal ofhim\(_1\).

This quantifying-in rule is used by Montague for other phenomena aswell. An example is co-referentiality:Mary loves the man whom shekissed is obtained fromHe\(_1\)loves the man whomhe\(_1\)kissed. And thede re reading ofJohn seeks a unicorn is obtained froma unicorn andJohn seeks him\(_1\).

Many researchers did not like this analysis in which powerfulsyntactic rules and artificial symbols (him\(_1)\) are used.Below we consider two strategies to remedy.

The first strategy was to deny the ambiguity. Some linguists haveargued that the scope order is the same as the surface order; this isknown as ‘Jackendoff’s principle’ (Jackendoff 1972).But there are sentences where this does not work. Others said that itis sufficient only to obtain the weakest reading (every widescope), and that the stronger reading is inferred when additionalinformation is available. But there are sentences for which thedifferent scope readings are logically independent, as inEverywoman loves one man.

The second strategy was to capture the ambiguity in another way thanby the quantifying-in rules. Historically the first method was to putthe interpretations 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 asa grammar in which the direct correspondence between syntax andsemantics has been relaxed. The method is called ‘CooperStore’, after the author who proposed this (Cooper 1983). Alater proposal is DRT \((=\) Discourse Representation Theory), whererepresentations are used to account for such ambiguities (van Eijckand Kamp 1997).

A recent method is by means of ‘lifting rules’ (seesection 3.3): the meaning of a noun-phrase is ‘lifted’ toa more abstract level, and different levels yield different scopereadings (see Hendriks 2001 and Jacobson 2014).

Even if the role of derivational history can be avoided for scope andco-referentiality, other phenomena remain for which derivationalhistories have a role. An example isJohn wondered when Alice saidshe would leave. This is ambiguous between John asking for thetime of leaving, or for the time of saying. So the sentence isambiguous, even though there are no arguments for assigning to it morethan one constituent structure. Pelletier (1993) presents thissentence and others, and says: ‘In order to maintain theCompositionality Principle, theorists have resorted to a number ofdevices which are all more or less unmotivated (except to maintain thePrinciple): Montagovian “quantifying-in” rules, traces,gaps, […].’ Pelletier’s objection can beappreciated if one assumes that meaning assignment is directly linkedwith constituent structure. But, as explained in Section1.2, this is not the case. The derivation specifies which rules arecombined in which order, and this derivation constitutes the input tothe meaning assignment function. The constituent structure isdetermined by the output of the syntactic rules, and differentderivation processes may generate one and the same constituentstructure. In this way, semantic ambiguities are accounted for. Oneshould not call something ‘constituent structure’ if it isnot intended as such, and next refute it because it does not have thedesired properties.

The distinction between a derivation tree and a constituent tree ismade in several theories of grammar. In Tree Adjoining Grammars (TAGs)the different scope readings of the sentence about loving a womandiffer in the order in which the noun-phrases are substituted in thebasic tree. A classical example in Chomskyan grammar isTheshooting of the hunters was bloody, which is ambiguous betweenthe hunters shooting, or the hunters being shot at. The two readingscome from two different sources: one in whichthe hunters isthe subject of the sentence, and one in which it is the object.

3. Philosophical Aspects

3.1 From Frege to Intensions

Throughout most of his semantic work, Montague avowedly adopted aversion of Frege’s (1892) distinction between‘sense’ and ‘denotation’. Frege’soriginal line of thought concerns sentences likeThe Greeks didnot know that the morning star is the evening star, which doesnot seem to express that the Greeks were confused about theself-identity of Venus. Frege’s analysis accounts for thisobservation by having descriptive names likethe morning stardenote their referents in ordinary contexts, but something differentin embedded clauses (or, more generally, in ‘indirectcontexts’): their ‘sense’ – a semantic valuethat captures the way in which an object is referred to. Sincereferring to a celestial object bythe morning star differsfrom referring to it bythe evening star, the embedded clausedoes not denote an analytic truth but a contingent proposition, whosetruth may well have escaped the Greeks.

Frege’s approach is known to run into a number of problems. Oneof them concerns the iteration of indirect contexts, as inGottlobsuspected that the Greeks did not know that the morning star is theevening star. Though he did not explicitly address the issue,Frege is usually understood as resorting to an infinite hierarchy ofever more indirect senses to be associated with each otherwisenon-ambiguous expression (Dummett 1981, 267; Carnap 1947, §30;Kripke 2008, 183; see however Parsons 1981 for a more cautiousinterpretation). The purported Fregean line of analysis has beencriticized for multiplying ambiguity beyond necessity (Janssen 2012)as well as raising serious learnability issues (Davidson 1968, 11).Though Montague did acknowledge a hierarchy of senses, he did notemploy it for the analysis of iterated indirect contexts. Instead, heidentified Frege’s (1892) senses withintensions alongthe lines of Carnap (1947) – set theoretic functions on alogical space of possible worlds (or world-time-pairs) whose valuesare the denotations of expressions – theirextensions.In particular, the way in which a description refers to its referentis captured by its dependence on contingent facts. As a case in point,the famous Fregean descriptions differ in intension as long as thereis a possible world in which the brightest star at dawn is not thesame object as the brightest star at night.

The replacement of senses by intensions paves the way to analternative approach to iterated intensionality: generalizingKripke’s (1963) semantics of modality, Montague (1970b, 73)accounted for clausal embedding in terms of propositional operatorswhose extension, like that of their argument, depends on a given pointin logical space. As it turns out, this so-called ‘neighborhoodsemantics’ of clausal embedding does without reference to asense hierarchy even in iterated indirect environments(ibid., 76), which is why Montague used it as the basis forhis general compositional analysis of natural language. Montague(ibid., 75f.) still presented his approach as being in linewith Frege’s, thereby emphasizing the commonalities in theoverall architecture of semantic theory, which he identified as‘Frege’s functionality principle’:

the extension of a formula is a function of the extensions (ordinaryextensions) of those of its parts not standing within indirectcontexts (that is […] not standing within the scope of anoperator), together with the intensions (what Frege also calledindirect extensions) of those parts that do stand withinindirect contexts. (Montague 1970b, 74f.)

Moreover, Montague (1970c, 390) called one of the key constructions ofhis general theory of reference ‘Fregean interpretation’;and in his type-logical hierarchy, intensions are marked by the letter‘\(s\)’, which is short for ‘sense’(ibid., 379). This notation has become quite common inlinguistic semantics, although the ‘\(s\)’ is frequentlytaken to stand for possible \(s\)ituations!

Only at one point in his semantic work did Montague abandon hisFregean stance: in his essay ‘English as a formallanguage’ (1970a), he employed a one-level architecture of‘Russellian’ denotations and expressed his doubts aboutthe cogency of Frege’s motivation for non-propositional senses(ibid., sec. 9, remark xi), thereby foreshadowingKaplan’s (1975) comparison between the frameworks of Frege 1892and Russell 1905. Yet in his ‘Universal Grammar’, Montaguecommented:

I should like, however, to withdraw my emphasis […] on thepossibility of doing without a distinction between sense anddenotation. While such a distinction can be avoided in special cases,it remains necessary for the general theory, and probably provides theclearest approach even to the special cases in question. (Montague1970c, 374, fn.)

Even though Montague tended to play down the difference, the switchfrom senses to intensions is known to have dramatic consequences onthe fine-grainedness of semantic analysis. In particular, as mentionedin section 2.4, any two logically equivalent sentences come out ashaving the same intension; yet their senses will diverge if theirtruth value is not determined in the same way. Montague indicated howthis unwelcome consequence may be avoided in terms of mismatchesbetween worlds and contexts, creating what he called‘“unactualizable” points of reference’(ibid., 382), but he did not provide a detailed analysis tosubstantiate his sketchy remarks.

3.2 Compositionality

For Montague the principle of compositionality did not seem to be asubject of deliberation or discussion, but the only way to proceed. Ineffect he made compositionality the core part of his ‘theory ofmeaning’ (Montague 1970c, 378), which was later summed up in theslogan: ‘Syntax is an algebra, semantics is an algebra, andmeaning is a homomorphism between them’ (Janssen 1983, 25). Yetalthough Montague used the term ‘Frege’s functionalityprinciple’ for the way in which extension and intension arecompositionally intertwined, he did not have a special term forcompositionality in general. Later authors, who identified thePrinciple of Compositionality as a cornerstone of Montague’swork, also used the term ‘Frege’s Principle’(originating with Cresswell 1973, 75); Thomason 1980 is an earlysource for the term ‘compositional’.

It has been claimed that Montague’s analysis of pronouns is notcompositional. This is, however, not the case. In order to explain thecompositional nature of his treatment of pronouns, both Janssen (1997)and Dowty (2007) explain how variables are interpreted in logic; wefollow their explanations. Consider the following clauses from thetraditional Tarskian interpretation of predicate logic.

1. \(\llbracket \varphi \wedge \psi \rrbracket^g = \mathbf{1}\) ifand only if \(\llbracket \varphi \rrbracket^g = \mathbf{1}\) and\(\llbracket \psi \rrbracket^g = \mathbf{1}\)
2. \(\llbracket \exists \ x\varphi \rrbracket^g = \mathbf{1}\) if andonly if for some \(h, h \sim_x g\) and \(\llbracket \varphi\rrbracket^h = \mathbf{1}\)

The first clause says: \(\varphi \wedge \psi\) is true when usingassignment \(g\) if and only if \(\varphi\) and \(\psi\) are true whenthe assignment \(g\) is used. In the second clause assignments \(h\)are introduced (by \(\sim_x g)\) which are equal to \(g\) except maybefor the value they assign to variable \(x\). Montague uses the sameformat, with the difference that besides \(g\) he also has \(i\), thetime of reference and \(j\), the possible world, as superscripts.

In the formulation of the clauses there is nothing which can bepointed at as ‘the meaning’, in fact it is a definition oftruth with \(g\) and \(h\) as parameters. So how is it possible thatthis (and Montague’s work) are compositional?

The answer requires a shift in perspective. The meaning of a formula\(\varphi\), shortly \(M(\varphi)\), is the set of assignments forwhich the formula is true. Then the first clause says that

so a simple set-theoretic combination on the two meanings isperformed. And

i.e., \(\{g \mid \text{for some }h \in M(\varphi), g \sim_x h \}\),which can be described as: extend the set \(M(\varphi)\) with all\(x\)-variants. (The reference to ‘\(x\)’ may be felt asproblematic, but Montague even eliminated this trace ofnon-compositionality by assigning appropriate meanings to variables;see Zimmermann and Sternefeld 2013, ch. 10, for pertinent details.) Ingeneral, in Montague semantics the meaning of an expression is afunction which has as domain the triples \(\langle\)moment of time,possible world, assignment to variables\(\rangle\).

Is it possible to achieve compositionality for natural language?Obvious candidates for counterexamples are idioms, because theirmeanings seem not to be built from their constituting words. However,Westerståhl (2002) presents a collection of methods, varyingfrom compound basic expressions, to deviant meanings for constitutingparts. Janssen (1997) refutes several other counterexamples that areput forward in the literature.

How strong is compositionality? Mathematical results show that anylanguage can be given a compositional semantics, either by using anunorthodox syntax (Janssen 1997) or by using an unorthodox semantics(Zadrozny 1994). However their proofs are not helpful in practice.Hodges (2001) showed under which circumstances a given compositionalsemantics for a fragment can be extended to a larger language.

There is no general agreement among formal semanticists about the roleand status of compositionality; at least the following four positionshave been held (nearly the same list is given in Partee 1996):

1. Compositionality is a basic methodological principle; any proposalshould obey it. Janssen (1997) and Jacobson (2014) are adherents ofthis position.
2. Compositionality is a good method, but other methods can be usedas well. For instance formal meaning representation can be used in anessential way. An example is DRT (discourse representation theory,Kamp 1981).
3. Compositionality is an ideal, but a proposal need not satisfyit.
4. It is an empirical issue whether compositionality can be achieved.See Dowty 2007 for a discussion.

An extensive discussion of compositionality is given in Janssen 1997,and in the entry oncompositionality.

3.3 Syntactic Categories and Semantic Types

According to Montague, the purpose of syntax is to provide the inputto semantics:

I fail to see any interest in syntax except as a preliminary tosemantics. (Montague 1970c, 223)

Although syntax was in his eyes subordinate, he was fully explicit inhis rules in which he used some ad hoc syntactic tools.

In Montague 1970a and 1970c, the relation between syntactic categoriesand semantic types is given only by a list. Montague (1973) defines asystematic relation which amounts to the same relation as one wouldhave in categorial grammar. However, Montague’s syntax is not acategorial syntax because the rules are not always category driven andbecause some of the rules are not concatenation rules.

For each of these two aspects, proposals have been put forward tochange the situation. One direction was to stay closer to the idealsof categorial grammar, with only type-driven rules, sometimes allowingfor a restricted extension of the power of concatenation rules (see,for example, Morrill 1994, Carpenter 1998). The other approach was toincorporate in Montague grammar as much as possible the insights fromsyntactic theories, especially originating from the tradition ofChomsky. A first step was made by Partee (1973), who let the grammarproduce structures (labelled bracketings); a syntacticallysophisticated grammar (with Chomskyan movement rules) was used in theRosetta translation project (Rosetta 1994). The influential textbookby Heim and Kratzer (1998) combined the two approaches by applyingtype-driven interpretation to the syntactic level of (Chomskyan)Logical Forms.

In his syntactic accounts, Montague tended to treat‘logical‘ words like determiners (the, a,every) and conjunctions (and,or, not)syncategorematically, i.e., not by means of lexical entries, but asthe effect of specific syntactic rules; the reason for this decisionis unknown, but it may be speculated that it was part of acharacterization of grammatical meaning in terms of logicality,presumably along the lines of Tarski’s 1986 invariancecriterion. As a consequence, a different rule is needed forJohnwalks and sings than forJohn walks and Mary sings:syntactically the first one is a conjunction of verb phrases and thesecond one of sentences. However, the two meanings ofand areclosely related and a generalization is missed. As a general solutionit was proposed to use rules (or alternatively general principles)that change the category of an expression – a change thatcorresponds with a semantic rule that ‘lifts’ the meaning.For instance, the meaning ofand as a connective between verbphrases is obtained by lifting the meaning of the sentence connective\(\wedge\) to \(\lambda P\lambda Q\lambda x[P(x) \wedge Q(x)].\) Theline of analysis has been extensively studied in Partee and Rooth1983, Partee 1987, Hendriks 2001, and Winter 2001.

Montague’s method of defining fragments with a fully explicitsyntax has become far less popular than it was in the heyday ofMontague Grammar in the 1980s. Nowadays semanticists prefer to focuson specific phenomena, suggesting rules which are only explicitconcerning the semantic side. This tendency has been criticized byPartee in Janssen 1997 and Jacobson 2014, where a fragment is actuallyprovided.

3.4 Pragmatics

The truth conditions of sentences sometimes vary with the context ofuse. Thus, whetherI am happy is true, depends on who thespeaker is; other examples include the referents ofhere andthis. Montague (1968; 1970b) addressed these factors,indicating that they could be treated by introducing additionalparameters besides the time and the possible world. Despite occasionalcritcism (Cresswell 1973, 111; Lewis 1980, 86f.), the treatment ofcontextual dependence by way of a fixed finite list of parameters hasbecome quite standard in formal semantics.

Montague initially treated contextual parameters on a par with timesand worlds, but in ‘Universal Grammar’ (Montague 1970c) heindicated that a distinction should be made between those thatdetermine the content (which, following Frege 1892, is what is denotedin indirect contexts) from those that constitute it:

Thus meanings are functions of two arguments– a possible worldand a context of use. The second argument is introduced in order topermit a treatment […] of such indexical locutions asdemonstratives, first- and second-person singular pronouns, and freevariables (which are treated […] as a kind of demonstrative).Senses on the other hand […] are functions of only oneargument, regarded as a possible world. The intuitive distinction isthis: meanings are those entities that serve as interpretations ofexpressions (and hence, if the interpretation of a compound is alwaysto be a function of the interpretations of its components, cannot beidentified with functions of possible worlds alone), while senses arethose intensional entities that are sometimes denoted by expressions.(Montague 1970c, 379)

While these remarks are still a far cry from double-indexingapproaches to context dependence (Kamp 1971), they do exhibit thebasic idea underlying the shiftability criterion for distinguishingcontext and index (Lewis 1980). In particular, Montague’smeanings share a core feature with Kaplan’s (1989) characters:both map paramteterized contexts to propositions, understood as(characteristic functions of) sets of possible worlds.

Montague (1970c, 68) followed Bar-Hillel 1954 in treating contextdependence as part of pragmatics. It was only after his death, thathis framework was connected to other aspects of pragmatics. Inparticular, in early work on Montague grammar, various proposals weremade to give compositional characterizations of presuppositions and(conventional) implicatures (Peters 1979; Karttunen and Peters 1979),but later treatments were not always completely compositional, takingseveral contextual factors into account (Beaver 1997). In a similarvein, early work in the tradition was rather optimistic about directlyapplying Montague semantics to non-declarative uses of (declarative)sentences (Cresswell 1973), but later accounts had to invoke a lotmore than linguistic meaning, including models of interlocutors’perspectives (Gunlogson 2003).

3.5 Ontology

Montague’s semantic analyses were given in terms of atype-logical hierarchy whose basic ingredients were truth values,possible individuals, and possible worlds. While the exact nature ofindividuals and worlds depends on the (arbitrary) choice of aparticular model (or ‘Fregean interpretation’), the truthvalues 1 (true) and 0 (false) transcend the class of all models, thusemphasizing their status as logical objects. A lot of work in currentlinguistic semantics still applies Montague’s type-logicalhierarchy, which is however often enriched byevents (or,more generally:eventualities) that serve as the referents ofverbs and verb phrases (Bach 1986a; Parsons 1990).

In early work on intensional analysis (Carnap 1947, Kaplan 1964),possible worlds had been identified with models of a suitableextensional language. For reasons indicated in section 3.1, Montague(1969, 164) broke with this tradition, appealing to Kripke’saccount of modality based on possible worlds as unstructured basicobjects. In his essay ‘On the nature of certain philosophicalentities’ (Montague 1969), he argued that this seemingly minortechnical innovation opens a new perspective in philosophicalanalysis, by reducing certain ‘dubious’ entities topredicate intensions orproperties – functions mappingpossible worlds to sets of objects. The idea was that, once theconceptual and techical problems of the semantics of intensionallanguages had been overcome, they may replace extensional predicatelogic as a basis of philosophical argument:

Philosophy is always capable of enlarging itself; that is, bymetamathematical or model-theoretic means – means availablewithin set theory – one can “justify” a language ortheory that transcends set theory, and then proceed to transact a newbranch of philosophy within the new language. It is now time to takesuch a step and to lay the foundations of intensional languages.(Montague 1969,165f.)

Montague illustrated his claim by detailed analyses of (talk about)pains, tasks, obligations, and events in terms of second-orderintensional logic, which contained the core elements of his (slightly)later compositional interpretation of English.

Although it has since become common in linguistic semantics to analysecontent in terms of possible worlds, they are not always taken to betotally devoid of structure. As a case in point, Kratzer (2002) hasargued that the verbknow relates subjects to facts and thusits interpretation requires appeal to the mereology of worlds: factsare concrete parts worlds. Moreover, as in Kripke’s originalapproach, semantic theory frequently imposes some external structureon logical space. Thus, accessibility relations and distance measuresbetween worlds are invoked to account for, respectively, propositionalattitudes (along the lines of Hinitkka 1969) and counterfactualconditionals (following Lewis 1973). In a similar vein, the universeof individuals (or ‘entities’, in Montague’sparlance) nowadays gives way to a richer domain of structured objects,including substances and their parts, which may serve as extensions ofmass nouns such aswater (Pelletier & Schubert 2003), aswell as groups and their members, which are denoted by plural nounphrases (Link 1983). Also when properties (loving John) areconsidered as entities for which predicates may hold (Mary likesloving John), additional structure is needed: property theorygives the tools to incorporate them (Turner 1983).

Occasional doubts have been raised as to the adequacy ofMontague’s higher-order intensional logic as a tool for thesemantic interpretation of natural language:

It seems to me that this is the strategy employed by MontagueGrammarians, who are in fact strongly committed to compositionality.[…]. There is a price to be paid however. The higher orderentities evoked in this “type theoretical ascent” are muchless realistic philosophically and psycholinguistically than ouroriginal individuals. Hence the ascent is bound to detract from thepsycholinguistic and methodological realism of one’s theory.(Hintikka 1983, 20)

This objection does not appreciate the role played by higher-orderabstraction in compositional semantics, which is not to form sentencesabout higher-order functions. Rather, \(\lambda\)-abstraction is usedas a heuristic tool to describe compositional contributions ofexpressions to larger syntactic environments (cf. Zimmermann 2021,sec. 2.1). Thus, e.g., the extension of a determiner is defined as itscontribution to the truth value of a sentence in which it occurs (insubject position), which can be described in terms of the extensionsof the nouns and verb phrases it combines with – and theseextensions are themselves sets (by a similar reasoning). The abstracthigher-order objects are thus merely convenient ways of describing thekinematics of compositionality and do not serve as the objects thatthe sentences of the language so described are about, or that itsterms refer to. As a case in point, it can be shown that even thoughthe (indirect) interpretation of the English fragment of Montague 1973makes use of \(\lambda\)-abstraction over second-order variables, itsexpressive power is much weaker than higher-order type logic and doesnot even have the resources to formulate certain meaning postulates towhich its lexical items abide (Zimmermann 1983). In fact,Hintikka’s alternative (game-theoretical) semantics fares nobetter once it is formulated in a compositional way (see Hodges 1997or Caicedoet al. 2009).

4. Concluding Remarks

4.1 Legacy

Montague revolutionized the field of semantic theory. He introducedmethods and tools from mathematical logic, and set standards forexplicitness in semantics. Now all semanticists know that logic hasmore to offer than first-order logic only.

4.2 Further Reading

A recent introduction is Jacobson 2014. It is a gentle introduction tothe field, especially for linguists and philosophers. It presentsseveral successes obtained by the approach. Older introductions areDowtyet al. 1981 and Gamut 1991, which are more technicaland prepare for Montague’s original papers. An overview of thehistory of the field is given by Partee and Hendriks (1997) as well asPartee (2011); Caponigro (forthcoming) provides an extensivebiographical background on Montague. Collections of important papersare Portner and Partee (eds.) 2002 and Partee 2004; furtherinformation is provided in the volume edited by McNally andSzabó (2022), which includes a short account of PTQ (Zimmermann2022). The ‘Handbook of compositionality’(Werning etal. 2011) discusses many aspects of the approach. The most importantjournals in the field areLinguistics and Philosophy, theJournal ofSemantics,Natural Language Semantics, andSemanticsand Pragmatics.

4.3 Example

A small example is presented below, it consists of the two sentencesJohn is singing, andEvery man is singing. Theexample is not presented in Montague’s original way, butmodernized: there is a lifting rule, the determiner is a basicexpression, and intensional aspects are not considered.

The grammar has four basic expressions:

1. John is an expression of the category Proper Name. Itsdenotation is an individual represented in logic byJohn.
2. The Intransitive Verbsing denotes a set (the set ofsingers), and is represented by the predicate symbolsing.
3. The Common Nounman, which denotes a set, represented byman.
4. The Determinerevery. Its denotation is \(\lambdaP\lambda Q\forall x[P(x) \rightarrow Q(x)\)]; an explanation of thisformula 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 islifted to a more abstract, a ‘higher’ meaning: therepresentation of the denotation ofJohn as Noun Phrase is\(\lambda P[P(\textbf{John})\)]. 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 ornot, i.e. whether \(P(\textbf{John})\) is true. The \(\lambda P\)abstracts from the possible interpretations of \(P\): the expression\(\lambda P[P(\textbf{John})\)] denotes a function that takes as inputproperties and yieldstrue if the property holds forJohn, andfalse otherwise. So the denotation ofJohn is the characteristic function of the set of propertieshe has.
2. A rule that takes as input a Noun Phrase and an Intransitive Verb,and yields as output a Sentence: fromJohn andsingit producesJohn is singing. The corresponding semantic rulerequires the denotation of the Noun Phrase to be applied to thedenotation of the Intransitive Verb. This is represented as \(\lambdaP[P(\textbf{John})](\textbf{sing})\). When applied to the argumentsing, the function represented by \(\lambdaP[P(\textbf{John})\)] yieldstrue if the predicatesing holds for John, so precisely in casesing(John) istrue.So \(\lambda P[P(\textbf{John})](\textbf{sing})\) andsing(John) are equivalent. Thelatter formula can be obtained by removing the \(\lambda P\) andsubstitutingsing for \(P\). This is called‘lambda-conversion’.
3. A rule that takes as inputs a Determiner and a Common Noun, andyields a Noun Phrase: fromevery andman it producesevery man. Semantically the denotation of the Determiner hasto be applied to the denotation of the Common Noun, hence \(\lambdaP\lambda Q\forall x[P(x) \rightarrow Q(x)](\textbf{man})\). By lambdaconversion (just explained) this is simplified to \(\lambda Q\forallx[\textbf{man}(x) \rightarrow Q(x)\)]. This result denotes a functionthat, when applied to property \(A\), yieldstruejust in case all man have property \(A\).

The example given with the last rule helps us to understand theformula forevery : that denotes a relation betweenproperties \(A\) and \(B\) which holds in case every \(A\) hasproperty \(B\).

The next step is now easy. Apply the rule for combining a Noun Phraseand an Intransitive Verb to the last result, producingEvery manis singing. The output of the semantic rule is \(\lambda Q\forallx[\textbf{man}(x) \rightarrow Q(x)](\textbf{sing})\). By lambdaconversion we obtain \(\forall x[\textbf{man}(x) \rightarrow\textbf{sing}(x)],\) which is the traditional logical representationofEvery man is singing.

Note the role of lambda-operators:

1. John andevery man are interpreted in a similarway: sets of properties. These sets can be represented due tolambda-operators.
2. Every man andsing are syntactically on the samelevel, but semanticallysing has a subordinated role: itoccurs embedded in the formula. This switch of level is possible dueto lambda-operators.

Bibliography

• Angelelli, I. (ed.), 1967,Gottlob Frege. KleineSchriften, Hildesheim: Olms.
• Bach, E., 1976, “An extension of classical transformationalgrammar”, in J. M. Sadock, D. J. Dwyer, S. C. Song, and E. W.Bach (eds.),Problems in Linguistic Metatheory (Proceedingsof the 1976 conference), pp. 183–224. East Lansing, Michigan:Michigan State University.
• –––, 1986a, “The algebra of events”,Linguistics and Philosophy, 9: 5–16; reprinted inPortner and Partee (eds.) 2002, pp. 324–33.
• –––, 1986b, “Natural languagemetaphysics”, in R. Barcan Marcus, G. J. W. Dorn, and P.Weingartner (eds.),Logic, Methodology, and Philosophy of ScienceVII, pp. 573–595, Amsterdam: North Holland.
• Bar-Hillel, Y., 1954, “Indexical Expressions”,Mind, 63: 359–376.
• Barwise, J. and Cooper, R., 1981, “Generalized quantifiersand natural language”,Linguistics and Philosophy, 4:159–219. Reprinted in Portner and Partee (eds.) 2002, pp.75–126.
• Bäuerle, R. and Cresswell, M. J., 2003, “Propositionalattitudes”, in Gabbay and Guenthner (eds.) 2003, pp.121–141.
• Bäuerle, R., Schwarze, C., and von Stechow, A. (eds.), 1983,Meaning, Use and Interpretation of Language, Berlin: deGruyter.
• Beaver, D. I., 1997, “Presupposition”, in J. vanBenthem and A. ter Meulen (eds.) 1997, pp. 939–1008.
• van Benthem, J. and ter Meulen, A. (eds.), 1997,Handbook ofLogic and Language, Amsterdam: Elsevier.
• van Benthem, J. and ter Meulen, A. (eds.), 2011,Handbook ofLogic and Language. Second edition, Amsterdam: Elsevier MITPress.
• Caicedo, X., Dechesne, F., and Janssen, T. M. V., 2009,“Equivalence and quantifier rules for logics with imperfectinformation”,Logic Journal of the IGPL, 17:91–129.
• Caponigro, I., forthcoming:Richard Montague: The Simplicityof Language, the Complexity of Life, New York: Oxford UniversityPress.
• Carnap, R., 1947,Meaning and Necessity: A Study in Semanticsand Modal Logic, Chicago: University of Chicago Press.
• Carpenter, B., 1998,Type-Logical Semantics, Cambridge,MA: MIT Press.
• Chomsky, N., 1965,Aspects of the Theory of Syntax,Cambridge, MA: MIT Press.
• Church, A., 1940, “A Formulation of the Simple Theory ofTypes”,Journal of Symbolic Logic, 5: 56–68.
• Cooper, R., 1983,Quantification and Syntactic Theory(Synthese Language Library, 21), Dordrecht: Reidel.
• Cresswell, M. J., 1973,Logics and Languages, London:Methuen.
• Davidson, D., 1968, “On saying that”,Synthese, 19: 130–146.
• Davidson, D. and Harman, G. (eds.), 1972,Semantics of NaturalLanguage (Synthese Library, 40), Dordrecht: Reidel.
• Dowty, D., 2007, “Compositionality as an empiricalproblem”, in C. Barker and P. Jacobson (eds.),DirectCompositionality (Oxford Studies in Theoretical Linguistics, 14),Oxford: Oxford University Press, pp. 23–101.
• Dowty, D., Wall, R., and Peters, S., 1981,Introduction toMontague Semantics (Synthese Language Library, 11), Dordrecht:Reidel.
• Dummett, M., 1981,Frege – Philosophy of Language,second edition, Cambridge, MA: Harvard University Press.
• Egré, P., 2021, “Logical Omniscience”, in D.Gutzmann, L. Matthewson, C. Meier, H. Rullmann, and T. E. Zimmermann(eds.),The Wiley Blackwell Companion to Semantics (Volume3), Hoboken: Wiley-Blackwell, Hoboken: Wiley-Blackwell, pp.1725–1749.
• van Eijck, J. and Kamp, H., 1997, “Representing discourse incontext”, in J. van Benthem and A. ter Meulen (eds.) 1997, pp.179–237; revised reprint in J. van Benthem and A. ter Meulen(eds.) 2011, pp. 181–252.
• Fox, C. and Lappin, S., 2005,Foundations of IntensionalSemantics, Malden, MA: Blackwell.
• Frege, G., 1879,Begriffsschrift, eine der arithmetischennachgebildete Formelsprache des reinen Denkens, Halle:Nebert.
• –––, 1892, “Über Sinn undBedeutung”,Zeitschrift für Philosophie undphilosophische Kritik, 100: 25–50; reprinted in Angelelli1967, pp. 143–162; translated as ‘On Sense andReference’ in Geach and Black 1952, pp. 56–58.
• Gabbay, D. M. and Guenthner, F. (eds.), 2003,Handbook ofPhilosophical Logic, second edition, Dordrecht: Kluwer.
• Gallin, D., 1975,Intensional and higher-order ModalLogic, Amsterdam: North-Holland.
• Geach, P. T. and Black, M. (eds.), 1952,Translations from thePhilosophical Writings of Gottlob Frege, Oxford: Blackwell.
• Gamut, L. T. F., 1991,Logic, Language and Meaning. Volume 2:Intensional Logic and Logical Grammar, Chicago and London: TheUniversity of Chicago Press.
• Groenendijk, J. and Stokhof, M., 1989, “Type-shifting rulesand the semantics of questions”, inProperties, Types andMeanings (Volume 2: Semantic Issues), G. Chierchia, B. H. Partee,and R. Turner (eds.), Dordrecht: Reidel, pp. 21–68; reprinted inPortner and Partee (eds.) 2002, pp. 421–457.
• –––, 1990, “Dynamic MontagueGrammar”, inProceedings of the Second Symposion on Logicand Language, L. Kalman & L Polos (eds.), Budapest: EotvosLorand University Press, pp. 3–48.
• –––, 1991, “Dynamic predicatelogic”,Linguistics and Philosophy, 14:39–100.
• Gunlogson, C., 2003,True to Form, New York:Routledge.
• Hamblin, C. L., 1973, “Questions in Montague English”,Foundations of Language, 10: 41–53.
• Heim, I. and Kratzer, A., 1998,Semantics in GenerativeGrammar, Oxford: Blackwell.
• Hendriks, H., 2001, “Compositionality and model-theoreticinterpretation”,Journal of Logic, Language andInformation, 10: 29–48.
• Hintikka, J., 1969, “Semantics for PropositionalAttitudes”, inPhilosophical Logic, J. W. Davisetal. (eds.), Dordrecht: Reidel, pp. 21–45.
• –––, 1983,The Game of Language. Studies inGame-Theoretical Semantics and Its Applications (SyntheseLanguage Library, 22), Dordrecht: Reidel.
• Hodges, W., 1997, “Compositional semantics for a language ofimperfect information”,Logic Journal of the IGPL,5(4): 539–563.
• –––, 2001, “Formal features ofcompositionality”,Journal for Logic Language andComputation, 10: 7–28.
• Homer, V., 2021, “Negative Polarity”, in D. Gutzmann,L. Matthewson, C. Meier, H. Rullmann, and T. E. Zimmermann (eds.),The Wiley Blackwell Companion to Semantics (Volume4), Hoboken: Wiley-Blackwell, pp. 2025–2063.
• Jackendoff, R. S., 1972,Semantic Interpretation in GenerativeGrammar, Cambridge, MA: MIT Press.
• Jacobson, P., 2014,Compositional Semantics. An introductionto the syntax/semantics interface, Oxford: Oxford UniversityPress.
• Janssen, T. M. V., 1983,Foundations and Applications ofMontague Grammar, Ph.D. Dissertation, University ofAmsterdam.
• –––, 1997, “Compositionality (with anappendix by B. Partee)”, in van Benthem and ter Meulen (eds.)1997, pp. 417–473.
• –––, 2001, “Frege, contextuality andcompositionality”,Journal of Logic, Language andInformation, 10: 115–136.
• –––, 2012, “Compositionality: its historiccontext”, inThe Oxford Handbook of Compositionality,M. Werning et al. (eds.), Oxford: Oxford University Press,pp. 19–46.
• Kamp, H., 1971, “Formal Properties of‘Now’”,Theoria, 37: 227–273.
• –––, 1981, “A theory of truth and semanticrepresentation”, in J. Groenendijk, T. Janssen, and M. Stokhof(eds.),Formal Methods in the Study of Language, Centre forMathematics and Computer Science, Amsterdam, pp. 1–14; reprintedin Portner and Partee (eds.) 2002, pp. 189–222.
• Kaplan, D., 1964,Foundations of Intensional Logic, UCLAdissertation.
• –––, 1975, “How to Russell a Frege-Church”,Journal of Philosophy, 72: 716–729.
• –––, 1989, “Demonstratives”, in J.Almog, J. Perry, and H. Wettstein (eds.),Themes from Kaplan,Oxford: Oxford University Press, pp. 481–563.
• Karttunen, L., 1977, “Syntax and semantics ofquestions”,Linguistics and Philosophy, 1: 3–44.Reprinted in Portner and Partee (eds.) 2002, pp. 382–420.
• Karttunen, L. and Peters, S., 1979, “ConventionalImplicature”, inPresupposition (Syntax and Semantics,11), New York: Academic Press, pp. 1–56.
• Kratzer, A., 2002, “Facts: Particulars or InformationUnits?”,Linguistics and Philosophy, 25:655–670.
• Kripke, S., 1963, “Semantical Considerations on ModalLogic”,Acta Philosophica Fennica, 16:83–94.
• –––, 2008 “Frege’s Theory of Senseand Reference: Some Exegetical Notes”,Theoria, 74:181–218.
• Ladusaw, W. A., 1980, “On the notionAffective inthe analysis of negative-polarity items”,Journal ofLinguistic Research, 1: 1–16; reprinted in Portner andPartee (eds.) 2002, pp. 457–470.
• –––, 1996, “Negation and polarityitems”, in Lappin (ed.) 1996, chapter 12, pp.321–341.
• Lappin, S. (ed.), 1996,The Handbook of Contemporary SemanticTheory (Blackwell Handbooks in Linguistics, 3), Oxford:Blackwell.
• Lewis, D., 1970, “General semantics”,Synthese, 22: 18–67.
• –––, 1973,Counterfactuals, Oxford:Blackwell.
• –––, 1980, “Index, Context, andContent”, in S. Kanger and S. Öhman (eds.),Philosophyand Grammar, Dordrecht: Reidel, pp. 79–100.
• Link, G., 1983, “The logical analysis of plurals and massterms: a lattice-theoretic approach”, in Bäuerleetal. 1983, 302–323; reprinted in Portner and Partee (eds.)2002, pp. 127–146.
• McNally, L. and Szabó, Z. (eds.), 2022,AReader’s Guide to Classic Papers in Formal Semantics(Studies in Linguistics and Philosophiy, 100), Cham: Springer.
• Montague, R., 1968, “Pragmatics”, in R. Klibansky(ed.),Contemporary Philosophy. A Survey, Florence: La NuovaItalia Editrice, pp. 102–122; reprinted in Thomason (ed.) 1974,pp. 95–118.
• –––, 1969, “On the nature of certainphilosophical entities”,The Monist, 353:159–194; reprinted in Thomason (ed.) 1974, pp.148–187.
• –––, 1970a, “English as a formallanguage”, in B. Visentini, et al. (eds.),Linguaggi nellaSocietà e nella Tecnica, Milan: Edizioni di Communita, pp.189–224; reprinted in Thomason (ed.) 1974, pp.188–221.
• –––, 1970b, “Pragmatics and intensionallogic”,Synthese, 22: 68–94; reprinted inThomason (ed.) 1974, pp. 119–147.
• –––, 1970c, “Universal grammar”,Theoria, 36: 373–398; reprinted in Thomason (ed.) 1974,pp. 7–27.
• –––, 1973, “The proper treatment ofquantification in ordinary English”, in K. J. J. Hintikka, J. M.E. Moravcsik, and P. Suppes (eds.),Approaches to NaturalLanguage (Synthese Library, 49), Dordrecht: Reidel,221–242; reprinted in Portner and Partee (eds.) 2002, pp.17–35.
• Montague, R. and Kalish, D., 1964,Logic: Techniques of FormalReasoning, New York: Harcourt, Brace, and Jovanovich.
• Morrill, G. V., 1994,Type Logical Grammar. The Logic ofSigns, Dordrecht: Kluwer.
• Parsons, T., 1981, “Frege’s Hierarchies of IndirectSenses and the Paradox of Analysis”,Midwest Studies inPhilosophy, 6: 37–57.
• –––, 1990,Events in the semantics ofEnglish. A study in subatomic semantics, Cambridge, MA: MITPress.
• Partee, B. H., 1973, “Some transformational extensions ofMontague grammar”,Journal of Philosophical Logic, 2:509–534; reprinted in Partee 1976, pp. 51–76.
• –––, 1984, “Compositionality”, in F.Landman and F. Veltman (eds.),Varieties of Formal Semantics:Proceedings of the 4th Amsterdam Colloquium (Groningen-AmsterdamStudies in Semantics, No. 3), Dordrecht: Foris, pp. 281–311;reprinted in Partee 2004, pp. 153–181.
• –––, 1987, “Noun phrase interpretation andtype-shifting principles”, in J. Groenendijk, D. de Jongh, andM. Stokhof (eds.),Studies in Discourse Representation Theory andthe Theory of Generalized Quantifiers (Groningen-AmsterdamStudies in Semantics, 8), Dordrecht: Foris, pp. 115–143;reprinted in Partee 2004, pp. 203–224.
• –––, 1996, “The development of formalsemantics in linguistic theory”, in Lappin (ed.) 1996, pp.11–38.
• –––, 2004,Compositionality in FormalSemantics. Selected Papers by B. H. Partee (Explorations inSemantics 1), Malden, MA: Blackwell.
• –––, 2011, “Formal Semantics: Origins,Issues, Early Impact”,Baltic International Yearbook ofCognition, Logic and Communication, 6: 1–52.
• Partee, B. H. with Hendriks, H., 1997, “Montaguegrammar”, in van Benthem and ter Meulen (eds.) 1997, pp.7–91; reprinted in van Benthem and ter Meulen (eds.) 2011, pp.3–94
• Partee, B. and Rooth, M., 1983, “Generalized conjunction andtype ambiguity”, in Bäuerleet al. (eds.) 1983,pp. 361–383; reprinted in Portner and Partee (eds.) 2002, pp.334–356.
• Pelletier, F. J., 1993, “Remarks involving internal andexternal semantics”, in J. MacNamara and G. Reges (eds.),The Logical Foundations of Cognition, Oxford: OxfordUniversity Press, 283–295.
• Pelletier, F. J. and Schubert, L. K., 2003, “Massexpressions”, in Gabbay and Guenthner (eds.) 2003,249–335.
• Peters, S., 1979, “A truth-conditional formulation ofKarttunen’s account of presupposition”,Synthese,40: 301–316.
• Peters, S. and Westerståhl, D., 2006,Quantifiers inLanguage and Logic, Oxford: Oxford University Press.
• Portner, P. and Partee, B. (eds.), 2002,Formal Semantics: theEssential Readings, Oxford: Blackwell.
• Quine, W. V. O., 1960,Word and Object, Cambridge, MA:MIT Press.
• Rosetta, M. T., 1994,Compositional Translation (KluwerInternational Series in Engineering and Computer Science, 230),Dordrecht: Kluwer.
• Russell, B., 1905, “On denoting”,Mind, 14:175–226; reprinted in R. C. Marsh (ed.),Logic andKnowledge. Essays 1901–1950, New York:Macmillan, pp. 33–56.
• Stokhof, M., 2007, “Hand or hammer? On formal and naturallanguages in semantics”,Journal of Indian Philosophy,35: 597–626.
• Tarski, A., 1986, “What are logical notions?”, editedby J. Corcoran,History and Philosophy of Logic, 7:143–154.
• Thomason, R. H. (ed.), 1974,Formal Philosophy. SelectedPapers of Richard Montague, New Haven: Yale UniversityPress.
• –––, 1980, “A Model Theory forPropositional Attitudes”,Linguistics and Philosophy,4: 47–70.
• Turner, R., 1983, “Montague semantics, nominalizations andScott’s domains”,Linguistics and Philosophy, 6:259–288.
• Werning, M., Hinzen, W. and Machery, E. (eds.), 2012,TheOxford Handbook of Compositionality, Oxford: Oxford UniversityPress
• Westerståhl, D., 2002, “On the compositionality ofidioms. An abstract approach”, in D. Barker-Plummer, D. I.Beaver, J. van Benthem, and P. S. di Luzio (eds.),Words, Proofsand Diagrams (CSLI Lecture Notes 141), Stanford: CSLIPublications, pp. 241–271.
• Winter, Y., 2001,Flexibility Principles in Boolean Semantics:Coordination, Plurality and Scope in Natural Language (CurrentStudies in Linguistics, 37), Cambridge, MA: MIT Press.
• Zadrozny, W., 1994, “From compositional to systematicsemantics”,Linguistics and Philosophy, 17:329–342.
• Zimmermann, T. E., 1983, “A Note on TransparencyPostulates”,Journal of Semantics, 4: 67–77.
• –––, 1989, “Intensional logic andtwo-sorted type theory”,Journal of Symbolic Logic, 54:65–77.
• –––, 1999, “Meaning postulates and themodel-theoretic approach to natural language semantics”,Linguistics and Philosophy, 22: 529–561.
• –––, 2021, “Representingintensionality”, in D. Gutzmann, L. Matthewson, C. Meier, H.Rullmann, and T. E. Zimmermann (eds.),The Wiley BlackwellCompanion to Semantics (Volume 4), Hoboken:Wiley-Blackwell, pp. 2607–2643.
• –––, 2022, “On Montague’s ‘Theproper treatment of quantification in ordinary English’”,in L. McNally & Z. Szabó (eds.), 2022: 331–366.
• Zimmermann, T. E. and Sternefeld, W., 2013,Introduction toSemantics, Berlin: De Gruyter.

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