Errata and Addenda to 'Finite non-deterministic semantics for some modal systems'

Trying to overcome Dugundji’s result on uncharacterizability of modal logics by finite logical matrices, J. Kearns (in 1981) and J. Ivlev (in 1988) propose, independently, a characterization of some modal systems by means of four-valued multivalued truth-functions (by restricting the valuations using level valuations, in Kearns’ approach), as an alternative to Kripke semantics. This constitutes an antecedent of the non-deterministic matrices introduced in 2001 by A. Avron and I. Lev. In this paper we propose a reconstruction of Kearns and Ivlev’s results (which did not have the dissemination or impact they deserved) in an uniform way, obtaining an extension to another modal systems. The first part of the paper is devoted to four-valued Nmatrices, including Kearns and Ivlev’s. Besides proving with full details Kearns’ results for T, S4 and S5, we also obtain a characterization of the system B by four-valued Nmatrices with level valuations. Concerning Ivlev’s results, two new modal systems are introduced and characterized by Nmatrices. In the second part of this paper, six-valued Nmatrices are introduced which characterize a variant of the eight systems studied in the first part, by replacing axiom (T ) by axiom (D). As a by-product, novel decision procedures for T, S4, S5, D, KDB, KD4 and KD45 are obtained, which open interesting possibilities in the study of the complexity of modal logics and, in particular, of Intuitionistic propositional logic (IPC), taking into account the Gödel-McKinsey-Tarski translation between IPC and S4. ∗[email protected] †[email protected] ‡[email protected]