On Non-Deterministic Four-Valued Semantics for Some Normal and Very Weak T-Normal Logics

Abstract In this paper, for some modal logics, we analyse eight sets of non-deterministic four-valued truth tables with two pairs of truths and falsehoods—necessary and contingent. Three of them were given by Kearns for S5 , S4 , KT . Two of them were given by Coniglio, Fariñas del Cerro, and Peron for S4 , KTB . To these five sets, we add three new ones for S5 and KT . We show that these eight sets are the only ones of this kind that satisfy certain additional natural conditions. From these eight sets of tables, one can create sixteen non-deterministic matrices, depending on whether we distinguish only the necessary truth or both truths. From these matrices, one can create various semantics by adding to them an appropriate set of valuations. Kearns singled out a countably infinite decreasing sequence of sets of valuations of a given level. For S5 , S4 , KT , Kearns applied valuations with infinite level, which belong to all these sets (for these valuations, the choice of the set of distinguished values is not important). Omori and Skurt prove that Kearns’ tables for S5 and S4 can be used with 1st-level valuations. We add the set of PL-valuations (which assign necessary truth to all substitution instances of classical tautologies), which generally contains the set of 1st-level valuations. We showed that only for Kearns’ tables for S5 , the sets of PL- and 1st-level valuations are equal. We prove that the remaining seven tables with two distinguished values and PL-valuations are adequate for S0.5 and six very weak logics containing it. We show that three of these seven tables with 1st-level valuations are adequate for S5 and S4 . The remaining four sets of tables with the infinite level valuations are adequate for S5 , KTB , KT .