Extension, Intension, Character, and Beyond

This article explains some of the technical terms commonly used in semantic theory. One such term is extension. The notion of an extension is a generalization of the notion of reference. Proper names refer to objects, and a proper name's extension is the object to which it refers. For instance, the extension of 'Barack Obama' is Barack Obama, and the extension of 'Los Angeles' is Los Angeles. The extension of a predicate is the set of objects to which it correctly applies. For instance, the extension of the common noun 'dog' is the set of all dogs, the extension of the verb 'run' is the set of all things that run, and the extension of the adjective 'happy' is the set of all happy things. Binary predicates correctly apply to pairs of things, and so the extension of a binary predicate is a set of ordered pairs. For example, the extension of the binary predicate 'kiss' is the set of pairs of objects such that x kisses y. The extension of a ternary predicate (such as 'give') is a set of ordered triples. Generally, the extension of an n-place predicate (where n is greater than 1) is a set of n-tuples. Complex expressions also have extensions. For instance, the extension of 'kisses Barack Obama' is the set of objects to which this phrase correctly applies, namely the set of objects that kiss Barack Obama. The extension of this complex expression is completely determined by the extensions of the simple expressions in it, for an individual z is a member of the extension of 'kiss Barack Obama' if and only if there is a pair of objects such that (i) is a member of the extension of 'kiss', (ii) y is identical with the extension of 'Barack Obama' (that is, y is identical with Barack Obama), and (iii) z is identical with x. Declarative sentences are