The formalization of semantics

By ‘logical semantics’ is here meant the study of meaning with the aid of mathematical logic. The term is commonly used by logicians in a narrower sense than this: to refer to the investigation of the meaning, or interpretation, of expressions in specially constructed logical systems. (The term ‘expression’ will be employed throughout this chapter in the sense in which it is customarily employed by logicians: cf. 1.5). Logical semantics in this narrower and more technical sense may be referred to, following Carnap (1942, 1956), as pure* semantics. It is a highly specialized branch of modern logic, which we shall be concerned with only in so far as it furnishes us with concepts and symbolic notation useful for the analysis of language. The present chapter is not therefore intended as an introduction to pure semantics; and it should not be treated as such by the reader. We will not discuss such questions as consistency and completeness; and no reference will be made to axiomatization or methods of proof.

Constructed logical systems are frequently referred to as languages. But we will not adopt this usage. We will refer to them, instead, as calculi*, keeping the term ‘language’ for natural languages. This will enable us to oppose linguistic semantics* (a branch of linguistics) to pure semantics* (a branch of logic or mathematics). Linguistic semantics, like other branches of linguistics, will have a theoretical and a descriptive section.