Inferentializing Consequence

The proof of correctness and completeness of a logical calculus w.r.t. a given semantics can be read as telling us that the tautologies (or, more generally, the relation of consequence) specified in a model-theoretic way can be equally well specified in a proof-theoretic way, by means of the calculus (as the theorems, resp. the relation of inferability of the calculus). Thus we know that both for the classical propositional calculus and for the classical predicate calculus theorems and tautologies represent two sides of the same coin. We also know that the relation of inference as instituted by any of the common axiom systems of the classical propositional calculus coincides with the relation of consequence defined in terms of the truth tables; whereas the situation is a little bit more complicated w.r.t. the classical predicate calculus (the coincidence occurs if we restrict ourselves to closed formulas; otherwise ∀xFx is inferable from Fx without being its consequence). And of course we also know cases where a class of tautologies of a semantic system does not coincide with the class of theorems of any calculus. (The paradigmatic case is the second-order predicate calculus with standard semantics.) This may make us consider the problem of “inferentializability”. Which semantic systems are “inferentializable” in the sense that their tautologies (their relation of consequence, respectively) coincide with the class of theorems (the relation of inferability, respectively) of a calculus? One answer is ready: it is if and only if the set of tautologies is recursively enumarable. But this answer is not very informative, indeed saying that the set is recursively enumerable is only reiterating that it conicides with the class of theorems of a calculus. Moreover, paying due attention to the terms such as “calculus” and “inference” shows us that it is possible to relate them to various “levels”, whereby the problem of inferentializability becomes quite nontrivial.