Search

We consider the Gödel bi-modal logic determined by fuzzy Kripke models where both the propositions and the accessibility relation are infinitely valued over the standard Gödel algebra [0,1], and prove strong completeness of the Fischer Servi intuitionistic modal logic IK plus the prelinearity axiom with respect to this semantics. We axiomatize also the bi-modal analogues of classical T, S4 and S5, obtained by restricting to models over frames satisfying the [0,1]-valued versions of the structural properties which characterize these logics. As an application of the completeness theorems we obtain a representation theorem for bi-modal Gödel algebras. [ABSTRACT FROM AUTHOR]

The article presents a new semantic framework for modal propositional language. The basic structures of the semantics are Boolean algebras with operators. However, the semantics is not algebraic but rather relational; in it, Boolean algebras with operators play a similar role as Kripke models in standard relational semantics, and the semantics is based on a relation between the elements of Boolean algebras enriched with operators and formulas from modal language. Some basic connections between the new semantic framework and standard algebraic and relational semantics are studied. [ABSTRACT FROM AUTHOR]

Recently, algebraic Routley–Meyer-style semantics was introduced for basic substructural logics. This paper extends it to fuzzy logics. First, we recall the basic substructural core fuzzy logic MIAL (Mianorm logic) and its axiomatic extensions, together with their algebraic semantics. Next, we introduce two kinds of ternary relational semantics, called here linear Urquhart-style and Fine-style Routley–Meyer semantics, for them as algebraic Routley–Meyer-style semantics. [ABSTRACT FROM AUTHOR]

We take up a suggestion by Odintsov (2009, Studia Logica, 91, 407–428) and define intuitionistic variants of certain logics arising from the trilattice SIXTEEN3 introduced in Shramko and Wansing (2005, Journal of Philosophical Logic, 34, 121–153 and 2006, Journal of Logic, Language and Information, 15, 403–424). In a first step, a logic I16 is presented as a Gentzen-type sequent calculus for an intuitionistic version of Odintsov’s Hilbert-style axiom system LT (Kamide and Wansing, 2009, Review of Symbolic Logic, 2, 374–395; Odintsov, 2009, Studia Logica, 91, 407–428). The cut-elimination theorem for I16 is proved using an embedding of I16 into Gentzen’s LJ. The completeness theorem with respect to a Kripke-style semantics is also proved for I16. The framework of I16 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 I16, and (ii) sequent calculi for Belnap and Dunn’s four-valued logic (Anderson et al., 1992, Entailment: The Logic of Relevance and Necessity; Belnap, 1977, A useful four-valued logic, In Modern uses of Multiple Valued Logic, pp. 5–37; Dunn, 1976, Philosophical Studies, 29, 149–168) and for Nelson’s constructive four-valued logic (Almukdad and Nelson, 1984, Journal of Symbolic Logic, 49, 231–233) are included as natural subsystems of I16. In a second step, a logic IT16 is introduced as a tableau calculus. The tableau system IT16 is an intuitionistic counterpart of Odintsov’s axiom system for truth entailment ⊨t in SIXTEEN3 and of the sequent calculus for ⊨t presented in Wansing (2010, Journal of Philosophical Logic, to appear). 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. [ABSTRACT FROM PUBLISHER]