Two-Dimensional Semantics

Two-dimensional approaches to semantics, broadly understood, recognize two “dimensions” of the meaning or content of linguistic items. On these approaches, expressions and their utterances are associated with two different sorts of semantic values, which play different explanatory roles. Typically, one semantic value is associated with reference and ordinary truth-conditions, while the other is associated with the way that reference and truth-conditions depend on the external world. The second sort of semantic value is often held to play a distinctive role in analyzing matters of cognitive significance and/or context-dependence. In this broad sense, even Frege’s theory of sense and reference might qualify as a sort of two-dimensional approach. More commonly, two-dimensional approaches are understood more narrowly to be a species of possible-worlds semantics, on which each dimension is understood in terms of possible worlds and related modal notions. In possible-world semantics, linguistic expressions and/or their utterances are first associated with an extension. The extension of a sentence is its truth-value: for example, the extension of ‘Plato was a philosopher’ is true. The extension of a singular term is its referent: for example, the extension of ‘Don Bradman’ is Bradman. The extension of a general term is the class of individuals that fall under the term: for example, the extension of ‘cat’ is the class of cats. Other expressions work similarly. One can then associate expressions with an intension, which is a function from possible worlds to extensions. The intension of a sentence is a function that is true at a possible world if and only if the sentence is true there: the intension of ‘Plato was a philosopher’ is true at all worlds where Plato was a philosopher. The intension of a singular term maps a possible world to the referent of a term in that possible world: