The Correspondence between Propositional Modal Logic with Axiom □_Φ↔◇_Φ and the Propositional Logic

摘要

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 □_Φ↔◇_Φ 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 (-) only occurs before a pseudo-atomic formula of form □~kp. 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 □~kp_i(k,i ≥ 0) is corresponding to a propositional variable q_(ki), 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~+.