A formal system for the non-theorems of the propositional calculus

Introduction The completeness of the classical propositional calculus allows us to give a deductive system consisting of finitely many axiom schemas and finitely many rules of inference, that permit us to pass from a formula or a pair of formulae to a syntactically related formula, in such a manner that the formulae obtained inductively from the axioms by repeated application of the rules are exactly the tautologies. In this paper we give an analogous deductive system (more concretely, a Hubert type system) such that the formulae deduced are exactly those that are not tautologies, the non-theorems of the propositional calculus. Obviously, this has to be the most non-standard of the non-classical logics. It is important to note that there are many other algorithms to generate recursively the nontheorems, since the propositional calculus is decidable. Usually they are based in the methodical search for a counterexample, but they lack the inductive character of a Hubert type system, where every formula involved in a deduction is itself deductible. In our system, unlike semantic tableaux or refutation trees, every formula introduced in a deduction is a nontautology, and it is introduced only if it is a non-tautological axiom, or it follows by one of the non-tautological rules of inference from nontautologies introduced earlier in the deduction.