What is the difference between intension and extension?

The extension of an expression is its denotation in a given world (e.g. the actual set of dogs), while its intension is the function that yields an extension for each possible world (roughly, the concept), which is what is needed to interpret modal and opaque contexts.

Model-Theoretic and Montague Semantics

Montague semantics interprets natural language model-theoretically, using typed intensional logic to assign denotations to expressions in a fully compositional way.

Definition

Model-theoretic semantics interprets expressions relative to a formal model; Montague semantics is the specific program of doing so for natural language using typed intensional logic and a homomorphic syntax-semantics mapping.

Scope

This topic covers the formal apparatus of model-theoretic semantics for natural language: models with domains of individuals, possible worlds, and times; semantic types and the typed lambda calculus; and the interpretation of expressions as functions over these domains. It centres on Montague's 'The Proper Treatment of Quantification in Ordinary English' (PTQ) and the broader program of treating English with the rigour of formal logic, including intension and extension and the analysis of noun phrases as generalized quantifiers.

Core questions

How are expressions of a natural language assigned denotations in a formal model?
What roles do semantic types and the lambda calculus play in compositional interpretation?
How are intension and extension distinguished, and why are possible worlds needed?
How does Montague's treatment handle quantifier phrases and scope?

Key concepts

possible worlds
intension vs. extension
semantic type
typed lambda calculus
generalized quantifier
homomorphic syntax-semantics mapping
intensional logic

Key theories

The Proper Treatment of Quantification (PTQ): Montague's fragment of English in which syntactic rules are paired one-to-one with semantic interpretation rules, noun phrases denote generalized quantifiers, and intensional logic handles opacity and modality.
Type theory and intensional logic: Expressions are assigned types built from basic types (entities, truth values) and possibly worlds/times; intensions are functions from possible worlds to extensions, capturing the meaning of opaque and modal contexts.

History

Building on Frege, Carnap's intension/extension distinction, and Tarskian model theory, Montague published a series of papers around 1970, most famously PTQ, arguing that English can be treated as a formal language with a model-theoretic semantics. The textbook by Dowty, Wall, and Peters made the technical apparatus accessible to linguists, and the framework became the foundation of modern formal semantics.

Debates

Possible-worlds intensions and the problem of granularity: Treating meanings as functions from possible worlds to extensions wrongly equates all necessarily equivalent expressions, prompting proposals for more fine-grained, structured, or hyperintensional notions of meaning.

Key figures

Richard Montague
Barbara Partee
David Dowty
Stanley Peters

Seminal works

montague1974
dowtywallpeters1981

Frequently asked questions

What is the difference between intension and extension?: The extension of an expression is its denotation in a given world (e.g. the actual set of dogs), while its intension is the function that yields an extension for each possible world (roughly, the concept), which is what is needed to interpret modal and opaque contexts.