Montague Grammar

The most famous Theory of Linguistic Logicism

In this article, I already explained logicism and its relation to commonsense reasoning. While logicism is about an axiomatic theory of the given topic and deduction rules given by logic, here I’m going to give an overview of the most famous theory of linguistic logicism.

When talking about formal semantics and linguistic logicism, the point of reference nowadays still remains Montague grammar (Partee and Hendriks,
1997). Richard Montague as an advocate for philosophical logicism is probably best known for his contributions to the formal semantics of natural language. Following the methodological approach of logicism as described in the last article, he adapted a logical formalism adequate to encapsulate a variety of linguistic phenomena. This filled a gap between the linguistic theories of the time and mathematics.

The idea was to translate a fragment of the English language to predicate logic by representing the single constituents of a sentence with lambda calculus. Lambda calculus is all about terms that look like this: λx.M. The basic is that this term describes a function definition where x is a bound variable in it (it isn’t the same as M(x) since we are not applying the function yet).

Now, assuming two types e (semantic entities) and t (truth values), we associate a type to each linguistic expression. For example, we write for an intransitive verb phrase VP: e → t where we map an entity to a truth value depending on whether this entity is performing the denoted action of the verb. In lambda calculus, this is interpreted as the corresponding function λx.VP (x).

In this example, the verb phrase thus defines a subset of all semantic entities. This forms either new higher-order functions or by applying the meaning of an expression to another, we obtain the meaning of a sentence.

For a more concrete example, let’s consider the sentence “every woman sees a man” where the corresponding part of the expression is interpreted to its meaning:

• woman, man: λx.WOMAN(x), λx.MAN(x)
• a: λP.λQ.∃x(P(x) ∧ Q(x))
  Here we are making use of such a higher-order function. Her it expects that it exists ∃ an entity with property P doing Q.
• sees: λx.λy(SEES(x, y))
  In this one, the function has to bound variables as some entity x sees another entity y.
• every: λP.λQ.∀x(P(x) → Q(x))
  Again, we have here two functions such that every entity that has property P also has property Q.

In the last expression, we can already see that P has to be the property of being a woman:
every woman: λQ.∀x(WOMAN(x) → Q(x)).

On the other hand, the term a expects the existence of a man in P that is being seen:
sees a man: λy.∃x(MAN(x) ∧ SEES(x,y)).

Finally, we can just replace Q with the second expression such that y is now also bound: every woman sees a man:
∀x(W OM AN (x) → ∃y(M AN (y) ∧ SEES(x, y))

Sources

Barbara H. Partee and Herman L.W. Hendriks. 1997. Handbook of Logic and Language. North-Holland.