Truth tables for modal logics T and S4, by using three-valued non-deterministic level semantics

Abstract

Novel three-valued non-deterministic level semantics for modal logics $\textbf {T}$ and $\textbf {S4}$ are presented. A criterion for partial level valuations is given, making it possible to create truth tables. Additionally, semantics and truth tables for $\textbf {0}$ (defined as $\textbf {PC}$ plus rule of necessitation) and $\textbf {0T}$ with only two values are based on Ivlev’s work. We need Kearns’ notion of level valuations: a generalization of Dugundji’s theorem shows that there is no non-deterministic semantics for modal logics up to $\textbf {S5}$, containing the rule of necessitation.

Tableau systems for some Ivlev-like (quantified) modal logics

“…Concerning the full (normal) version of the propositional modal systems discussed here, it should be noticed that, recently, Grätz has modified the four-valued Nmatrix semantics with level valuations for S4 introduced by Kearns in [Kearns, 1981], obtaining so a threevalued Nmatrix semantics with level valuations in which the criteria for choosing the level valuations is effective (see [Grätz, 2021b]). This constitutes a novel decision procedure for modal systems T and S4, overcoming a criticism to the original method of Kearns we made in [Coniglio et al, 2017, Section 4], precisely concerning its status as a decision procedure.…”

“…Closely related results were obtained independently in [Omori and Skurt, 2016] and [Omori and Skurt, 2020]). New results in this subject were presented in [Grätz, 2021b] and [Pawlowski and La Rosa, 2021] (see Section 4.3).…”

“…It should be noted that most part of modal logic manuals -for instance[Hughes and Cresswell, 1996],[Fitting and Mendelsohn, 1998] and[Garson, 2006] -only consider basic predicate signatures, that is, do not consider function symbols among the symbols of their language (an exception is[Carnielli and Pizzi, 2008, p. 241]).…”

Effective Semantics for the Modal Logics K and KT via Non-deterministic Matrices

“…One of these approaches, initiated by Kearns [10], is based on an infinite sequence of sets of valuations in a non-deterministic many-valued semantics. Since then, several non-deterministic many-valued semantics, without possible worlds, were developed for modal logics (see, e.g., [4,8,12,14]). The current paper is a part of that body of work.…”

Restricted swap structures for da Costa's $C_n$ and their category

“…Kearns' RNmatrices approach to modal logics was afterwards considered in [10,11,31,32]. In [20], a decision procedure for several normal modal logics was obtained by Grätz by refining the original RNmatrices originally proposed by Kearns. RNmatrices were also considered by Pawlowski and Urbaniak in the context of logics of informal provability ( [33,34]).…”