Meaning, the Context Principle, and the Sequent Calculus

Natural deduction inference rules seem to reflect the way we actually reason. Hence, many if not most inferentialist theories maintain that meaning is conferred on linguistic expressions by natural deduction rules, rather than the more abstract alternative of sequent rules. In the present paper, I argue, to the contrary, that an inferentialist theory of meaning must take a somewhat metainferential form, whereby the meanings of linguistic expressions—in particular, the logical constants—are conferred by sequent rules, conceived of as licensing inferences between inferences. To that end, I propose that sequent rules are to be understood as inferential semantic clauses, i.e., as playing a role analogous to that of the semantic clauses in truth conditional theories of meaning. I establish this proposal on the basis of an argument to the effect that inferentialists must adopt an extended version of Frege's context principle, according to which only within a complete sentence potentially serving either as a premise or as a conclusion of an inference do words have meaning. The resultant inferentialist theory is not disconnected from the way we actually reason. Moreover, on its basis one can make a meaning-theoretic case for rather unintuitive features of many sequent calculi, such as multiple conclusions and various ways of going substructural.