Bivalent modalities and multiple-valued modalities: the case of trivalued modal logics

The connection between many-valued logics and modal logics is old. Łukasiewicz›s pioneering investigations in 1920 about future contingents through multivalued logics attempted to characterize modalities through truth values. Although modal and multivalued logics constitute autonomous fields of investigation, their interactions are very prolific. In this article, we investigate three-valued modal logics based on the three-valued logics LP, K₃, TS and ST. More specifically, we discuss two possible ways to interpret modal sentences: (i) modal sentences ◻A and ◇A can receive only true or false; and (ii) modal sentences can also receive intermediate values. We argue that both characterizations have virtues and problems. In case (i), we argue that they validate modal principles that contradict their inferences at the propositional level. In case (ii), we argue that its proximity to classical logic imposes certain difficulties in some applications of these logics.

Keywords:
modal logics; many-valued logics; substructural logics.