Truth-conditional semantics is a theory of the meaning of natural language sentences. It takes the language–world relation as the basic concern of semantics rather than the language–mind relation: language is about states of affairs in the world. The semantic competence of a speaker–hearer is said to consist in his/her knowledge, for any sentence of his/her language, of how the world would have to be for that sentence to be true. The context in which the truth-theoretic approach to semantics arose was that of the nineteenth century investigations into the interpretation of mathematical logic and other formal systems. The work of Gottlob Frege was foundational. One of his ideas was that these formal systems could be given an interpretation, that is, could be assigned meaning, by providing each sentence of the system with truth-conditions. He is also credited with the principle of compositionality which, as we have seen, is central to the project of formulating a truth theory for a language with an infinite number of sentences. The logician Alfred Tarski is celebrated for his definition of the concept of truth for formal logical languages. Incorporating possible worlds into the semantics is a way of reflecting our knowledge that language can express ways in which the world might be, as well as the way it is. Montague, Lewis and others have used possible worlds in giving the truth-conditional semantics of sentences which contain modals, counterfactual conditionals and propositional attitudes. Very broadly speaking, twentieth century philosophers working on the semantics of natural language have taken one of two approaches: (a) the truth-conditional, or (b) the intention-convention. While the language–world connection is fundamental on the first approach, the language–mind connection is central on the second, in which linguistic meaning is accounted for in terms of psychological entities such as intentions and beliefs. Grice’s theory of meaning (to be distinguished from his pragmatic ‘theory of conversation’) is the primary exemplar of this second approach. Within early generative linguistics, a third approach to natural language semantics developed. Generative linguistics takes as its subject matter a speaker–hearer’s linguistic competence: phonological, syntactic, semantic. This knowledge is characterised as systems of mental representations and rules for transforming one sort of representation into another. Although Lewis (1970) and other truth-conditionalists have pointed out the fundamental failing of these ‘translational’ accounts, there are theories who would say that there is a place for both accounts, such as the relevance-theoretic approach to natural language semantics. It follows from the richly articulated program of J. A. Fodor (1975, 1987, 1990) in the philosophy of mind and language. The dominant framework for modern pragmatics has been provided by Grice (1975, 1989 [1967]), with his crucial distinction between saying and implicating. As a first approximation, then, it may seem reasonable to suppose that the domains of truth-conditional semantics and pragmatics are strictly complementary, as Gazdar (1979) maintained. This is assessed by considering two fundamental facts about natural languages: (a) their indexicality; (b) the existence of meaningful linguistic elements that do not seem to contribute to truth-conditional content. These present challenges to the truth-conditional approach that did not arise in its application to formal systems.