Montague grammar is an approach tonatural languagesemantics, named after AmericanlogicianRichard Montague. The Montague grammar is based onmathematical logic, especiallyhigher-orderpredicate logic andlambda calculus, and makes use of the notions ofintensional logic, viaKripke models. Montague pioneered this approach in the 1960s and early 1970s.

Montague's thesis was thatnatural languages (likeEnglish) andformal languages (likeprogramming languages) can be treated in the same way:

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, withChomsky and his associates. ("Universal Grammar" 1970)

Montague published what soon became known as Montague grammar^[1] in three papers:

• 1970: "Universal grammar" (= UG)^[2]
• 1970: "English as a Formal Language" (= EFL)^[3]
• 1973: "The Proper Treatment of Quantification in Ordinary English" (= PTQ)^[4]

Montague grammar can represent the meanings of quite complex sentencescompactly. Below is a grammar presented in Eijck and Unger's textbook.^[5]

The types of the syntactic categories in the grammar are as follows, withtdenoting a term (a reference to an entity) andf denoting a formula.

category	symbol	type
Sentence	S	${\displaystyle f}$
Verb phrase	VP	${\displaystyle t\to f}$
Noun phrase	NP	${\displaystyle (t\to f)\to f}$
Common noun	CN	${\displaystyle t\to f}$
Determiner	DET	${\displaystyle (t\to f)\to ((t\to f)\to f)}$
Transitive verb	TV	${\displaystyle t\to (t\to f)}$

The meaning of a sentence obtained by the rule${\displaystyle S:\mathit{N}\mathit{P}\mathit{V}\mathit{P}}$ is obtained byapplying 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 examplex, and a formulawhistles(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 particularx satisfies sleeps(x)${\displaystyle \wedge }$ snores(x) (expressed as a function fromx 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 similarlyfunction applications, orhigher-order functions. The following are the rules of the grammar, withthe first column indicating anon-terminal symbol, the second column one possibleway of producing that non-terminal from other non-terminals and terminals,and the third column indicating the corresponding meaning.

		meaning
S	NP VP	${\displaystyle \mathbf{(}\mathit{N}\mathit{P}\mathit{V}\mathit{P}\mathbf{)}}$
NP	name	${\displaystyle \lambda P.\mathbf{(}Pname\mathbf{)}}$
NP	DET CN	${\displaystyle \mathbf{(}\mathit{D}\mathit{E}\mathit{T}\mathit{C}\mathit{N}\mathbf{)}}$
NP	DET RCN	${\displaystyle \mathbf{(}\mathit{D}\mathit{E}\mathit{T}\mathit{R}\mathit{C}\mathit{N}\mathbf{)}}$
DET	"some"	${\displaystyle \lambda P.\lambda Q.\mathrm{\exists }x(\mathbf{(}Px\mathbf{)}\wedge \mathbf{(}Qx\mathbf{)})}$
DET	"a"	${\displaystyle \lambda P.\lambda Q.\mathrm{\exists }x(\mathbf{(}Px\mathbf{)}\wedge \mathbf{(}Qx\mathbf{)})}$
DET	"every"	${\displaystyle \lambda P.\lambda Q.\mathrm{\forall }x(\mathbf{(}Px\mathbf{)}\to \mathbf{(}Qx\mathbf{)})}$
DET	"no"	${\displaystyle \lambda P.\lambda Q.\mathrm{\forall }x(\mathbf{(}Px\mathbf{)}\to \mathrm{\neg }\mathbf{(}Qx\mathbf{)})}$
VP	intransverb	${\displaystyle \lambda x.intransverb(x)}$
VP	TV NP	${\displaystyle \lambda x.\mathbf{(}\mathit{N}\mathit{P}\lambda y.\mathbf{(}\mathit{T}\mathit{V}yx\mathbf{)}\mathbf{)}}$
TV	transverb	${\displaystyle \lambda y.\lambda x.transverb(x,y)}$
RCN	CN "that" VP	${\displaystyle \lambda x.(\mathbf{(}\mathit{C}\mathit{N}x\mathbf{)}\wedge \mathbf{(}\mathit{V}\mathit{P}x\mathbf{)})}$
RCN	CN "that" NP TV	${\displaystyle \lambda x.(\mathbf{(}\mathit{C}\mathit{N}x\mathbf{)}\wedge \mathbf{(}\mathit{N}\mathit{P}\lambda y.\mathbf{(}\mathit{T}\mathit{V}yx\mathbf{)}\mathbf{)})}$
CN	predicate	${\displaystyle \lambda x.predicate(x)}$

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 constituentexpressions, either by forming a new higher-order function, or by applyinga higher-order function for one expression to the meaning of another.

expression	meaning
a	${\displaystyle \lambda P.\lambda Q.\mathrm{\exists }x(\mathbf{(}Px\mathbf{)}\wedge \mathbf{(}Qx\mathbf{)})}$
man	${\displaystyle \lambda x.\mathit{M}\mathit{A}\mathit{N}(x)}$
a man	${\displaystyle \lambda Q.\mathrm{\exists }x(\mathit{M}\mathit{A}\mathit{N}(x)\wedge \mathbf{(}Qx\mathbf{)})}$
sleeps	${\displaystyle \lambda x.SLEEPS(x)}$
a man sleeps	${\displaystyle \mathrm{\exists }x(\mathit{M}\mathit{A}\mathit{N}(x)\wedge \mathit{S}\mathit{L}\mathit{E}\mathit{E}\mathit{P}\mathit{S}(x))}$
man that dreams	${\displaystyle \lambda x.(\mathit{M}\mathit{A}\mathit{N}(x)\wedge \mathit{D}\mathit{R}\mathit{E}\mathit{A}\mathit{M}\mathit{S}(x))}$
a man that dreams	${\displaystyle \lambda Q.\mathrm{\exists }x(\mathit{M}\mathit{A}\mathit{N}(x)\wedge \mathit{D}\mathit{R}\mathit{E}\mathit{A}\mathit{M}\mathit{S}(x)\wedge \mathbf{(}Qx\mathbf{)})}$
a man that dreams sleeps	${\displaystyle \mathrm{\exists }x(\mathit{M}\mathit{A}\mathit{N}(x)\wedge \mathit{D}\mathit{R}\mathit{E}\mathit{A}\mathit{M}\mathit{S}(x)\wedge \mathit{S}\mathit{L}\mathit{E}\mathit{E}\mathit{P}\mathit{S}(x))}$

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

sentence	translation to logic
Jill sees Jack	${\displaystyle sees(Jill,Jack)}$
every woman sees a man	${\displaystyle \mathrm{\forall }x(woman(x)\to (\mathrm{\exists }y(man(y)\wedge sees(x,y))))}$
every woman sees a man that sleeps	${\displaystyle \mathrm{\forall }x(woman(x)\to (\mathrm{\exists }y(man(y)\wedge sleeps(y)\wedge sees(x,y))))}$
a woman that eats sees a man that sleeps	${\displaystyle \mathrm{\exists }x(woman(x)\wedge eats(x)\wedge \mathrm{\exists }y(man(y)\wedge sleeps(y)\wedge sees(x,y)))}$

InDavid Foster Wallace's novelInfinite Jest, the protagonist Hal Incandenza has written an essay entitledMontague Grammar and the Semantics of Physical Modality. Montague grammar is also referenced explicitly and implicitly several times throughout the book.

• Categorial grammar – Family of formalisms in natural language syntax
• Continuation-passing style – Programming style in which control is passed explicitly
• Kripke semantics – Formal semantics for non-classical logic systems
• Situation semantics – Concept in situation theory
• Temperature paradox – Logical paradox

• 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.J.F.A.K. van Benthem andA. G. B. ter MeulenElsevier/MIT Press, 1997, pp. 5–92.ISBN 0-262-22053-9
• Reinhard Muskens Type-logical Semantics to appear in the Routledge Encyclopedia of Philosophy Online (contains an annotated bibliography).

• A Free Montague Parser in anon-deterministic extension ofCommon Lisp.
• Montague Grammar in historical context. / The theory and the substance of Montague grammar. Central principles. / Further developments and controversies. by Barbara H. Partee.

• Grammar
• Semantics
• Formal languages
• Lambda calculus

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