Combining Swap Structures: The Case of Paradefinite Ivlev-Like Modal Logics Based on $$FDE$$

Abstract

The aim of this paper is to combine several Ivlev-like modal systems characterized by 4-valued non-deterministic matrices (Nmatrices) with \(\mathcal {IDM}4\), a 4-valued expansion of Belnap–Dunn’s logic \(FDE\) with an implication introduced by Pynko in 1999. In order to do this, we introduce a new methodology for combining logics which are characterized by means of swap structures, based on what we call superposition of snapshots. In particular, the combination of \(\mathcal {IDM}4\) with \(Tm\), the 4-valued Ivlev’s version of KT, will be analyzed with more details. From the semantical perspective, the idea is to combine the 4-valued swap structures (Nmatrices) for \(Tm\) (and several of its extensions) with the 4-valued twist structure (logical matrix) for \(\mathcal {IDM}4\). This superposition produces a universe of 6 snapshots, with 3 of them being designated. The multioperators over the new universe are defined by combining the specifications of the given swap and twist structures. This gives rise to 6 different paradefinite Ivlev-like modal logics, each one of them characterized by a 6-valued Nmatrix, and conservatively extending the original modal logic and \(\mathcal {IDM}4\). This important feature allows to consider the proposed construction as a genuine technique for combining logics. In addition, it is possible to define in the combined logics a classicality operator in the sense of logics of evidence and truth (LETs). A sound and complete Hilbert-style axiomatization is also presented for the 6 combined systems, as well as a Prolog program which implements the swap structures semantics for the 6 systems, producing a decision procedure for satisfiability, refutability and validity of formulas in these logics.