Intension and Extension

The extension of a predicate is its actual class of instances: the extension of "human" is the set of all actually existing humans. The extension of a sentence is its truth-value: true or false. The intension of a predicate is the property or concept that determines its extension in any possible state of the world (any "state-description"): the intension of "human" is the property of being human, which applies to the same things in any world in which they exist. The intension of a sentence is the proposition it expresses — the set of all possible worlds in which it is true. Two expressions are L-equivalent (logically equivalent) if they have the same intension; merely extensionally equivalent if they happen to have the same extension in the actual world.

The distinction resolves a longstanding puzzle about substitution in belief and modal contexts. In ordinary extensional contexts, co-referring terms can be freely substituted (if "the morning star" and "the evening star" refer to the same thing, they can be swapped without affecting truth-values). But in intensional contexts — "necessarily, the morning star is the morning star" versus "necessarily, the morning star is the evening star," or "John believes that the morning star is a planet" — substitution of co-referring terms can change truth-values. Carnap explains this by distinguishing L-equivalence (same intension) from mere extensional equivalence: in intensional contexts, what matters is not what the term refers to but what concept it expresses.

Carnap operationalises the notion of intension through the concept of state-descriptions: complete, consistent assignments of truth-values to all atomic sentences of a language, representing all logically possible ways the world might be. A sentence's intension is the set of state-descriptions in which it is true. Necessary truth becomes truth in all state-descriptions; possibility becomes truth in at least one. This semantics for modal logic anticipates Kripke's possible-worlds semantics by fifteen years — though Kripke's approach, which makes possible worlds metaphysically serious rather than merely linguistic constructs, departs substantially from Carnap's linguistic framing.

Meaning and Necessity was first published in 1947 and expanded in 1956. Quine's review of the first edition was largely critical, focusing on the notion of analyticity that Carnap's framework presupposes. The possible-worlds semantics developed by Kripke in the late 1950s and Montague grammar in the 1970s both draw on Carnap's framework while departing from his anti-realist framing of modal concepts.