Semantic Pollution and Syntactic purity

Logical inferentialism claims that the meaning of the logical constants should be given, not model-theoretically, but by the rules of inference of a suitable calculus. It has been claimed that certain proof-theoretical systems, most particularly, labelled deductive systems for modal logic, are unsuitable, on the grounds that they are semantically polluted and suffer from an untoward intrusion of semantics into syntax. The charge is shown to be mistaken. It is argued on inferentialist grounds that labelled deductive systems are as syntactically pure as any formal system in which the rules define the meanings of the logical constants. §1. Semantic pollution. Poggiolesi & Restall (2012) discuss how to frame a proof theory for modal logic which accords with Gentzen’s idea (1969) that the operational rules should be separable (that is, each rule should mention only one connective, so that there are separate rules for each logical constant), and should divide into those rules which define the meaning of the connective by introducing a formula exhibiting the connective as its main connective, and those which eliminate it. The standard rules, due to Curry (1950), Fitch (1952) and Prawitz (1965), do not do this, for two reasons: first, they do not easily generalize (if at all) from rules for the systems S4 and S5 to the full range of (even) normal modal logics; secondly, although the rules have the form of introduction and elimination rules, and indeed normalize,1 it is clear that, e.g., the introduction-rule for ‘♦’ (possibility) does not capture its full meaning in the way inferentialism expects: