Theories in Classical Propositional Logic and the Converse of Substitution

Abstract

We study theories based on the classical propositional logic. As follows from the Sushko lemma, for any classical propositional theory T and any substitution ε (where formulas stand in place of propositional variables), the set ε⁻¹(T) is also a classical propositional theory. In this paper, we strengthen this assertion, namely, we prove that for any consistent finitely axiomatizable classical propositional theory T there exists a substitution e such that T is the inverse image of the set of all tautologies under ε. We propose an algorithm for constructing such a substitution for a given axiom of the theory.