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'''Montague grammar''' is an approach to [[Philosophy:Natural language|natural language]] [[Philosophy:Semantics|semantics]], named after American logician [[Biography:Richard Montague|Richard Montague]]. The Montague grammar is based on [[Mathematical logic|mathematical logic]], especially [[Philosophy:Higher-order logic|higher-order]] predicate logic and [[Lambda calculus|lambda calculus]], and makes use of the notions of [[Philosophy:Intensional logic|intensional logic]], via [[Kripke semantics|Kripke models]]. Montague pioneered this approach in the 1960s and early 1970s.

==Overview==
Montague's thesis was that [[Philosophy:Natural language|natural language]]s (like [[Social:English language|English]]) and [[Formal language|formal language]]s (like [[Programming language|programming language]]s) can be treated in the same way:
<blockquote>
There is in my opinion no important theoretical difference between natural languages and the artificial languages of logicians; indeed, I consider it possible to comprehend the syntax and semantics of both kinds of language within a single natural and mathematically precise theory. On this point I differ from a number of philosophers, but agree, I believe, with [[Biography:Noam Chomsky|Chomsky]] and his associates. ("Universal Grammar" 1970)
</blockquote>

Montague published what soon became known as Montague grammar<ref>The linguist Barbara Partee credibly claims to have invented the term in 1971 “for the system spelled out in Montague's“ UG, EFL and “especially in PTQ”. See her essay [http://people.umass.edu/partee/docs/BHP_Essay_Feb05.pdf "Reflections of a Formal Semanticist as of Feb 2005"], p. 14, footnote 36.</ref> in three papers:
* 1970: "Universal grammar" (= UG)<ref>"Universal grammar". ''Theoria'' 36 (1970), 373–398. (reprinted in Thomason, 1974)</ref>
* 1970: "English as a Formal Language" (= EFL)<ref>"English as a Formal Language". In: Bruno Visentini (ed.): ''Linguaggi nella società e nella tecnica''. Mailand 1970, 189–223. (reprinted in Thomason, 1974)</ref>
* 1973: "The Proper Treatment of Quantification in Ordinary English" (= PTQ)<ref>[http://semantics.uchicago.edu/kennedy/classes/s08/semantics2/montague73.pdf "The Proper Treatment of Quantification in Ordinary English]". In: [[Biography:Jaakko Hintikka|Jaakko Hintikka]], Julius Moravcsik, [[Biography:Patrick Suppes|Patrick Suppes]] (eds.): ''Approaches to Natural Language''. Dordrecht 1973, 221–242. (reprinted in Thomason, 1974)</ref>

==Illustration==

Montague grammar can represent the meanings of quite complex sentences
compactly. Below is a grammar presented in Eijck and Unger's textbook.<ref>J. van Eijck and C. Unger. Computational Semantics with Functional Programming. Cambridge University Press, 2010.</ref>

The types of the syntactic categories in the grammar are as follows, with ''t''
denoting a term (a reference to an entity) and ''f'' denoting a formula.

{| class=wikitable
|-
! category
! symbol
! type
|-
|  Sentence
| S
|<math>f</math>
|-
|  [[Social:Verb phrase|Verb phrase]]
| VP
|<math>t \rightarrow f</math>
|-
|  [[Social:Noun phrase|Noun phrase]]
| NP
|<math>(t \rightarrow f)\rightarrow f</math>
|-
| Common noun
| CN
|<math>t \rightarrow f</math>
|-
| [[Social:Determiner|Determiner]]
| DET
| <math>(t\rightarrow f)\rightarrow((t\rightarrow f)\rightarrow f)</math>
|-
|  Transitive verb
| TV
|<math>t \rightarrow (t \rightarrow f)</math>
|-
|}

The meaning of a sentence obtained by the rule <math>S : \mathit{NP}\ \mathit{VP}</math> is obtained by
applying the function for NP to the function for VP.

The types of VP and NP might appear unintuitive because of the question as to the meaning of a noun phrase that is not simply a term. This is because meanings of many noun phrases, such as "the man who whistles", are not just terms in predicate logic, but also include a predicate for the activity, like "whistles", which cannot be represented in the term (consisting of constant and function symbols but not of predicates). So we need some term, for example ''x'', and a formula ''whistles(x)'' to refer to the man who whistles. The meaning of verb phrases VP can be expressed with that term, for example stating that a particular ''x'' satisfies sleeps(x) <math>\wedge</math> snores(x) (expressed as a function from ''x'' to that formula). Now the function associated with NP takes that kind of function and combines it with the formulas needed to express the meaning of the noun phrase. This particular way of stating NP and VP is not the only possible one.

Key is the meaning of an expression is obtained as a function of its components, either by function application (indicated by boldface parentheses enclosing function and argument) or by constructing a new function from the functions associated with the component. This compositionality makes it possible to assign meanings reliably to arbitrarily complex sentence structures, with auxiliary clauses and many other complications.

The meanings of other categories of expressions are either similarly [[Function application|function application]]s, or [[Higher-order function|higher-order function]]s. The following are the rules of the grammar, with
the first column indicating a non-terminal symbol, the second column one possible
way of producing that non-terminal from other non-terminals and terminals,
and the third column indicating the corresponding meaning.

{| class=wikitable
|-
!
!
! meaning
|-
| S
| NP VP
|<math>\boldsymbol{(}\mathit{NP}\ \mathit{VP}\boldsymbol{)}</math>
|-
| NP	
|  name
| <math>\lambda P.\boldsymbol{(}P\  name\boldsymbol{)}</math>
|-
| NP	
|  DET CN
|<math>\boldsymbol{(}\mathit{DET}\ \mathit{CN}\boldsymbol{)}</math>
|-
| NP	
|  DET RCN
|<math>\boldsymbol{(}\mathit{DET}\ \mathit{RCN}\boldsymbol{)}</math>
|-
| DET	
|  "some"
| <math>\lambda P.\lambda Q. \exists x(\boldsymbol{(}P\ x\boldsymbol{)}\wedge\boldsymbol{(}Q\ x\boldsymbol{)})</math>
|-
| DET	
|  "a"
| <math>\lambda P.\lambda Q. \exists x(\boldsymbol{(}P\ x\boldsymbol{)}\wedge\boldsymbol{(}Q\ x\boldsymbol{)})</math>
|-
| DET	
|  "every"
| <math>\lambda P.\lambda Q.\forall x(\boldsymbol{(}P\ x\boldsymbol{)}\rightarrow\boldsymbol{(}Q\ x\boldsymbol{)})</math>
|-
| DET	
|  "no"
| <math>\lambda P.\lambda Q.\forall x(\boldsymbol{(}P\ x\boldsymbol{)}\rightarrow\neg\boldsymbol{(}Q\ x\boldsymbol{)})</math>
|-
| VP	
|  intransverb
| <math>\lambda x.intransverb(x)</math>
|-
| VP
| TV NP
|<math>\lambda x.\boldsymbol{(}\mathit{NP}\ \lambda y.\boldsymbol{(}\mathit{TV}\ y\ x\boldsymbol{)}\boldsymbol{)}</math>
|-
| TV	
|  transverb
| <math>\lambda y.\lambda x.transverb(x,y)</math>
|-
| RCN	
|  CN "that" VP
|<math>\lambda x.(\boldsymbol{(}\mathit{CN}\ x\boldsymbol{)}\wedge\boldsymbol{(}\mathit{VP}\ x\boldsymbol{)})</math>
|-
| RCN
| CN "that" NP TV
|<math>\lambda x.(\boldsymbol{(}\mathit{CN}\ x\boldsymbol{)}\wedge\boldsymbol{(}\mathit{NP}\ \lambda y.\boldsymbol{(}\mathit{TV}\ y\ x\boldsymbol{)}\boldsymbol{)})</math>
|-
| CN	
|  predicate
| <math>\lambda x.predicate(x)</math>
|-
|}

Here are example expressions and their associated meaning, according to the above grammar, showing that the meaning of a given sentence is formed from its constituent
expressions, either by forming a new higher-order function, or by applying
a higher-order function for one expression to the meaning of another.

{| class=wikitable
|-
! expression
! meaning
|-
| a
| <math>\lambda P.\lambda Q.\exists x(\boldsymbol{(}P\ x\boldsymbol{)}\wedge\boldsymbol{}\boldsymbol{(}Q\ x\boldsymbol{)})</math>
|-
| man
|<math>\lambda x.\mathit{MAN}(x)</math>
|-
| a man
|<math>\lambda Q.\exists x(\mathit{MAN}(x)\wedge\boldsymbol{(}Q\ x\boldsymbol{)})</math>
|-
| sleeps
| <math>\lambda x.SLEEPS(x)</math>
|-
| a man sleeps
|<math>\exists x(\mathit{MAN}(x)\wedge \mathit{SLEEPS}(x))</math>
|-
| man that dreams
|<math>\lambda x.(\mathit{MAN}(x)\wedge \mathit{DREAMS}(x))</math>
|-
| a man that dreams
|<math>\lambda Q.\exists x(\mathit{MAN}(x)\wedge \mathit{DREAMS}(x)\wedge \boldsymbol{(}Q\ x\boldsymbol{)})</math>
|-
| a man that dreams sleeps
|<math>\exists x(\mathit{MAN}(x)\wedge \mathit{DREAMS}(x)\wedge \mathit{SLEEPS}(x))</math>
|-
|}

The following are other examples of sentences translated into the predicate logic by the grammar.

{| class=wikitable
|-
! sentence
! translation to logic
|-
| Jill sees Jack
| <math>sees(Jill,Jack)</math>
|-
| every woman sees a man
| <math>\forall x(woman(x) \rightarrow (\exists y(man(y) \wedge  sees(x,y))))</math>
|-
| every woman sees a man that sleeps
| <math>\forall x(woman(x) \rightarrow (\exists y(man(y) \wedge  sleeps(y) \wedge  sees(x,y))))</math>
|-
| a woman that eats sees a man that sleeps
| <math>\exists x(woman(x)\wedge  eats(x)\wedge \exists y(man(y) \wedge  sleeps(y) \wedge  sees(x,y)))</math>
|-
|}

==In popular culture==
In David Foster Wallace's novel ''Infinite Jest'', the protagonist Hal Incandenza has written an essay entitled ''Montague Grammar and the Semantics of Physical Modality''. Montague grammar is also referenced explicitly and implicitly several times throughout the book.

==See also==
* {{annotated link|Categorial grammar}}
* {{annotated link|Continuation-passing style}}
* {{annotated link|Kripke semantics}}
* {{annotated link|Temperature paradox}}

==References==
<references/>

==Further reading==
* Richmond Thomason (ed.): ''Formal Philosophy. Selected Papers by Richard Montague.'' New Haven, 1974, {{ISBN|0-300-02412-6}}
* Paul Portner, Barbara H. Partee (eds.): ''Formal Semantics: The Essential Readings'', Blackwell, 2002. {{ISBN|0-631-21542-5}}
* D. R. Dowty, R.E. Wall and S. Peters: ''Introduction to Montague Semantics.'' Kluwer Academic Publishers, 1981, {{ISBN|90-277-1142-9}}
* Emmon Bach: ''Informal Lectures on Formal Semantics.'' SUNY Press, 1989, {{ISBN|0-88706-771-9}}
* B. H. Partee, A.G.B. ter Meulen and R.E. Wall: ''Mathematical Methods in Linguistics.'' Kluwer Academic Publishers, 1990, {{ISBN|90-277-2245-5}}
* B. H. Partee with Herman Hendriks: ''Montague Grammar.'' In: ''Handbook of Logic and Language'', eds. [[Biography:Johan van Benthem (logician)|J.F.A.K. van Benthem]] and A. G. B. ter Meulen [[Company:Elsevier|Elsevier]]/MIT Press, 1997, pp.&nbsp;5–92. {{ISBN|0-262-22053-9}}
* [http://freevariable.nl/pubs/rep.pdf Reinhard Muskens Type-logical Semantics] to appear in the Routledge Encyclopedia of Philosophy Online (contains an annotated bibliography).

==External links==
* [https://www.cs.cmu.edu/afs/cs/project/ai-repository/ai/areas/nlp/parsing/montague/0.html A Free Montague Parser] in a non-deterministic extension of [[Common Lisp]].
* [http://people.umass.edu/partee/docs/MontagueGrammarElsevier.PDF Montague Grammar in historical context. / The theory and the substance of Montague grammar. Central principles. / Further developments and controversies.] by Barbara H. Partee.


{{DEFAULTSORT:Montague Grammar}}
[[Category:Formal languages]]
[[Category:Lambda calculus]]

{{Sourceattribution|Montague grammar}}