The Correspondence between Propositional Modal Logic with Axiom $\Box\varphi \leftrightarrow \Diamond \varphi $ and the Propositional Logic

Abstract : The propositional modal logic is obtained by adding the necessity operator □ to the propositional logic. Each formula in the propositional logic is equivalent to a formula in the disjunctive normal form. In order to obtain the correspondence between the propositional modal logic and the propositional logic, we add the axiom $\Box\varphi \leftrightarrow\Diamond\varphi $ to K and get a new system K + . Each formula in such a logic is equivalent to a formula in the disjunctive normal form, where □k(k ≥ 0) only occurs before an atomic formula p, and $\lnot$ only occurs before a pseudo-atomic formula of form □k p. Maximally consistent sets of K +  have a property holding in the propositional logic: a set of pseudo-atom-complete formulas uniquely determines a maximally consistent set. When a pseudo-atomic formula □k pi (k,i ≥ 0) is corresponding to a propositional variable qki, each formula in K +  then can be corresponding to a formula in the propositional logic P + . We can also get the correspondence of models between K +  and P + . Then we get correspondences of theorems and valid formulas between them. So, the soundness theorem and the completeness theorem of K +  follow directly from those of P + .