Montague Grammar, developed by the logician and philosopher Richard Montague in the late 1960s and early 1970s, demonstrated that natural languages can be treated with the same formal precision as the artificial languages of logic. Montague's key insight was that there is no important theoretical difference between natural languages and formal languages: both can be given rigorous model-theoretic interpretations through a compositional translation into intensional logic. This work laid the foundation for the entire field of formal semantics in linguistics.
Architecture of the Framework
⟦John walks⟧ = walk′(j)
⟦every man walks⟧ = ∀x[man′(x) → walk′(x)]
Types: e (entities), t (truth values), s (possible worlds)
⟨s, ⟨e, t⟩⟩ = intension of a property
Montague's system works through a homomorphic mapping from syntactic analysis trees to expressions of intensional logic (IL). Each syntactic rule is paired with a semantic translation rule, ensuring that every well-formed expression of English receives a unique translation into IL. The IL expressions are then interpreted model-theoretically, with a model providing a set of possible worlds, a domain of entities, and interpretation functions. This three-stage pipeline—syntax, translation, interpretation—ensures full compositionality.
Intensional Logic and Possible Worlds
A distinctive feature of Montague Grammar is its use of intensional logic to handle phenomena that resist extensional treatment. The intension of an expression is a function from possible worlds (and times) to its extension. For instance, the intension of "the president" is a function from world-time pairs to individuals, capturing the fact that different individuals may hold the office in different circumstances. This apparatus handles propositional attitudes, modality, and intensional transitive verbs like "seek" and "believe."
Montague's most influential paper, "The Proper Treatment of Quantification in Ordinary English" (PTQ, 1973), provided a fragment of English with a complete formal grammar and model-theoretic semantics. PTQ demonstrated how quantified noun phrases like "every man" and "a unicorn" could be given uniform semantic treatments using generalized quantifiers and lambda abstraction. The paper's treatment of quantifier scope ambiguity via quantifying-in rules became a touchstone for subsequent work on scope phenomena in formal semantics.
Legacy in Computational Linguistics
Montague Grammar profoundly influenced computational semantics. The idea that syntactic derivations can be paired with semantic composition rules is the basis for virtually all compositional semantic parsing systems. Combinatory Categorial Grammar (CCG), used in modern semantic parsers, inherits Montague's principle of surface compositionality. Lambda calculus, the compositional engine of Montague's system, remains the standard tool for building meanings in computational semantics.
Modern extensions of Montague's approach include Glue Semantics (which replaces the syntax-semantics homomorphism with linear logic), Abstract Categorial Grammars, and continuation-based semantics. These frameworks address limitations of the original system, such as its handling of anaphora and discourse, while preserving its core insight: natural language meaning can be specified with full mathematical precision.