Intuitionistic Trilattice Logics

Abstract

We will take up a suggestion by Odintsov (Studia Logica 93:407–428, 2009) and define intuitionistic variants of certain logics arising from the trilattice $SIXTEEN_3.$ In a first step, a logic $\hbox{I}_{16}$ is presented as a Gentzen-type sequent calculus for an intuitionistic version of Odintsov’s Hilbert-style axiom system $L_{T}$ from Chap. 5. The cut-elimination theorem for $\hbox{I}_{16}$ is proved using an embedding of $\hbox{I}_{16}$ into Gentzen’s sequent system LJ for intuitionistic logic. The completeness theorem with respect to a Kripke-style semantics is also proved for $\hbox{I}_{16}.$ The framework of $\hbox{I}_{16}$ is regarded as plausible and natural for the following reasons: (i) the properties of constructible falsity and paraconsistency with respect to some negation connectives hold for $\hbox{I}_{16},$ and (ii) sequent calculi for Belnap and Dunn’s four-valued logic of first-degree entailment and for Nelson’s constructive paraconsistent logic N4 are included as natural subsystems of $\hbox{I}_{16}.$ In a second step, a logic $\hbox{IT}_{16}$ is introduced as a tableau calculus. The tableau system $\hbox{IT}_{16}$ is an intuitionistic counterpart of Odintsov’s axiom system for truth entailment $\models_t$ in $SIXTEEN_3$ and of the sequent calculus for $\models_t$ presented in Chap. 6. The tableau calculus is also shown to be sound and complete with respect to a Kripke-style semantics. A tableau calculus for falsity entailment can be obtained by suitably modifying the notion of provability.