# Montague grammar

Montague grammar is an approach to natural language semantics, named after American logician Richard Montague. The Montague grammar is based on mathematical logic, especially higher-order predicate logic and lambda calculus, and makes use of the notions of intensional logic, via Kripke models. Montague pioneered this approach in the 1960s and early 1970s.

## Overview

Montague's thesis was that natural languages (like English) and formal languages (like programming 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, with Chomsky 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]

In a 2004 paper,[5] Chris Barker linked Montague's treatment of quantification to the notion of continuation in programming language semantics.

## Illustration

Montague grammar can represent the meanings of quite complex sentences compactly. Below is a grammar presented in Eijck and Unger's textbook [6].

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.

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

The meaning of a sentence obtained by the rule ${\displaystyle S:{\mathit {NP}}\ {\mathit {VP}}}$  is obtained by applying the function for NP to the function for VP.

The Types of VP and NP might appear a bit unintuitive: why is the meaning of a noun phrase not simply a term? This is because meanings of many noun phrases, such as "the man who whistles", is not just a term in predicate logic, but also includes 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) ${\displaystyle \wedge }$  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. Note that this particular way of typing NP and VP is not the only possible one.

The important thing here is that the meaning of an expression is obtained as a function of its components, either by function application 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 applications, or higher-order functions. 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.

meaning
S NP VP ${\displaystyle ({\mathit {NP}}\ {\mathit {VP}})}$
NP name ${\displaystyle \lambda P.(P\ name)}$
NP DET CN ${\displaystyle ({\mathit {DET}}\ {\mathit {CN}})}$
NP DET RCN ${\displaystyle ({\mathit {DET}}\ {\mathit {RCN}})}$
DET "some" ${\displaystyle \lambda P.\lambda Q.\exists x((P\ x)\wedge (Q\ x))}$
DET "a" ${\displaystyle \lambda P.\lambda Q.\exists x((P\ x)\wedge (Q\ x))}$
DET "every" ${\displaystyle \lambda P.\lambda Q.\forall x((P\ x)\rightarrow (Q\ x))}$
DET "no" ${\displaystyle \lambda P.\lambda Q.\forall x((P\ x)\rightarrow (\neg (Q\ x)))}$
VP intransverb ${\displaystyle \lambda x.intransverb(x)}$
VP TV NP ${\displaystyle \lambda x.({\mathit {NP}}\ (\lambda y.({\mathit {TV}}\ y\ x)))}$
TV transverb ${\displaystyle \lambda x.\lambda y.transverb(x,y)}$
RCN CN "that" VP ${\displaystyle \lambda x.(({\mathit {CN}}\ x)\wedge ({\mathit {VP}}\ x))}$
RCN CN "that" NP TV ${\displaystyle \lambda x.(({\mathit {CN}}\ x)\wedge ({\mathit {NP}}\ (\lambda y.({\mathit {TV}}\ y\ x))))}$
CN predicate ${\displaystyle \lambda x.predicate(x)}$

Here are example expressions and their associated meaning according to the above grammar. You can look at how 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.

expression meaning
a ${\displaystyle \lambda P.\lambda Q.\exists x((P\ x)\wedge (Q\ x))}$
man ${\displaystyle \lambda x.{\mathit {MAN}}(x)}$
a man ${\displaystyle \lambda Q.\exists x({\mathit {MAN}}(x)\wedge (Q\ x))}$
sleeps ${\displaystyle \lambda x.SLEEPS(x)}$
a man sleeps ${\displaystyle \exists x({\mathit {MAN}}(x)\wedge {\mathit {SLEEPS}}(x))}$
man that dreams ${\displaystyle \lambda x.({\mathit {MAN}}(x)\wedge {\mathit {DREAMS}}(x))}$
a man that dreams ${\displaystyle \lambda Q.\exists x({\mathit {MAN}}(x)\wedge {\mathit {DREAMS}}(x)\wedge (Q\ x))}$
a man that dreams sleeps ${\displaystyle \exists x({\mathit {MAN}}(x)\wedge {\mathit {DREAMS}}(x)\wedge {\mathit {SLEEPS}}(x))}$

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 ${\displaystyle \forall x(woman(x)\rightarrow (\exists y(man(y)\wedge sees(x,y))))}$
every woman sees a man that sleeps ${\displaystyle \forall x(woman(x)\rightarrow (\exists y(man(y)\wedge sleeps(y)\wedge sees(x,y))))}$
a woman that eats sees a man that sleeps ${\displaystyle \exists x(woman(x)\wedge eats(x)\wedge \exists y(man(y)\wedge sleeps(y)\wedge sees(x,y)))}$

## 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.

## References

1. ^ 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 "Reflections of a Formal Semanticist as of Feb 2005", p. 14, footnote 36.
2. ^ "Universal grammar". Theoria 36 (1970), 373–398. (reprinted in Thomason, 1974)
3. ^ "English as a Formal Language". In: Bruno Visentini (ed.): Linguaggi nella società e nella tecnica. Mailand 1970, 189–223. (reprinted in Thomason, 1974)
4. ^ "The Proper Treatment of Quantification in Ordinary English". In: Jaakko Hintikka, Julius Moravcsik, Patrick Suppes (eds.): Approaches to Natural Language. Dordrecht 1973, 221–242. (reprinted in Thomason, 1974)
5. ^ See Continuations in Natural Language, Chris Barker, extended abstract for Fourth ACM-SIGPLAN Continuation Workshop ’04 Venice, Italy
6. ^ J. van Eijck and C. Unger. Computational Semantics with Functional Programming. Cambridge University Press, 2010.