Logical constants

A fundamental problem in the philosophy of logic is to characterize the concepts of ‘logical consequence’ and ‘logical truth’ in such a way as to explain what is semantically, metaphysically or epistemologically distinctive about them. One traditionally says that a sentence p is a logical consequence of a set S of sentences in a language L if and only if (1) the truth of the sentences of S in L guarantees the truth of p and (2) this guarantee is due to the ‘logical form’ of the sentences of S and the sentence p. A sentence is said to be logically true if its truth is guaranteed by its logical form (for example, ‘2 is even or 2 is not even’). There are three problems presented by this picture: to explicate the notion of logical form or structure; to explain how the logical forms of sentences give rise to the fact that the truth of certain sentences guarantees the truth of others; and to explain what such a guarantee consists in.

The logical form of a sentence may be exhibited by replacing nonlogical expressions with a schematic letter. Two sentences have the same logical form when they can be mapped onto the same schema using this procedure (‘2 is even or 2 is not even’ and ‘3 is prime or 3 is not prime’ have the same logical form: ‘p or not-p’). If a sentence is logically true then each sentence sharing its logical form is true. Any characterization of logical consequence, then, presupposes a conception of logical form, which in turn assumes a prior demarcation of the logical constants. Such a demarcation yields an answer to the first problem above; the goal is to generate the demarcation in such a way as to enable a solution of the remaining two.

Approaches to the characterization of logical constants and logical consequence are affected by developments in mathematical logic. One way of viewing logical constanthood is as a semantic property; a property that an expression possesses by virtue of the sort of contribution it makes to determining the truth conditions of sentences containing it. Another way is proof-theoretical: appealing to aspects of cognitive or operational role as the defining characteristics of logical expressions. Broadly, proof-theoretic accounts go naturally with the conception of logic as a theory of formal deductive inference; model-theoretic accounts complement a conception of logic as an instrument for the characterization of structure.