Some classical modal logics with a necessity/impossibility operator.

In this paper we examine modal logics in which the modal operator 2 can be read as necessity, or impossibility, or both. Consider classical modal logics, for example; that is, the logics closed under the following rule of inference: A ↔ B / 2A ↔ 2B. Here 2 usually represents necessity. But it also can be read as possibility, impossibility, contingency, non-necessity, and even negation. On the other hand, in the rule A → B / 2B → 2A the 2 operator can no longer be read as necessity, or possibility — but it makes sense to read it as impossibility or negation: if A implies B and B is impossible, so is A. In this paper we will deal with only the necessity/impossibility readings of 2. We consider several modal formulas and, using neighborhood semantics, identify, on frames, conditions corresponding to them. We consider several logics obtained by adding one or more of these formulas as axioms, and prove determination theorems for them. Besides the preliminary results presented in this paper, we conclude indicating some topics for further research.

Authors

• Cezar A. Mortari

Keywords

• classical modal logics
• neighborhood semantics
• impossibility operator