A Resolution Calculus for Modal Logics

In this paper, we present a resolution calculus for the first-order modal logic S4. The formulas are given not necessary in a clausal form. This method can be used for automatizable proof procedure of a quantified modal logic. We will consider formulas for which the following conditions hold: 1. the formulas F contain only logical connectives ¬,&,∨, and no logical or modal symbol in F lies in the scope of a negation, 2. the formulas are closed, i.e., we consider the formulas without free variables, 3. the formulas are transformed into Skolem normal form (see [1],[2]), 4. the formulas are of the formG1 ∨G2 ∨ ...∨Gs, where Gi is a literal or a formula beginning with ,3. The order of formulas is not fixed in a disjunction or in a conjunction. In what follows, P, P1, P2 denote the atomic formulas. Formulas are denoted by F,G,K,H and M . Moreover, H and M can be the empty formulas as well. The symbol⊥ denotes an empty formula.