When Rules Define Logical Operators: Rules as Second-Order Definitions

Abstract Logical inferentialists hold that the meaning of logical operators is given by their rules of inference. Arthur Prior cast doubt on this by introducing rules for his tonk operator that would allow for the derivation of any sentence whatsoever from any sentence whatsoever. The obvious inferentialist reply was to require some constraints on the defining rules, such as conservativeness (Belnap) or harmony (Dummett). In this paper, I will defend and investigate a different constraint: rules define a classical logical operator just in case they translate into an explicit definition in pure classical second-order logic. The right-hand side of this criterion will be found (i) to be philosophically principled in taking the idea of rules as definitions perfectly seriously, (ii) to explain how the semantic meaning of the operators can be determined from their rules, (iii) to be local in a similar sense as harmony, (iv) to validate the intuitionistic natural deduction rules and the intuitionistic/classical sequent calculus rules as defining the classical propositional operators while ruling out Prior’s rules for tonk, (v) to make clear why already the intuitionistic natural deduction rules define the classical meaning of logical operators so long as metavariables are interpreted as expressing classical propositions, (vi) to validate the classical natural deduction rules as analytic, (vii) to entail consistency, and, in the case of propositional operators (not quantifiers), (viii) to be decidable and (ix) to determine precisely those operators to be definable by rules that correspond to truth-functions.