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It is introduced an immune dynamic n-valued Łukasiewicz logic I D Ł n on the base of n-valued Łukasiewicz logic Ł n and corresponding to it immune dynamic M V n -algebra ( I D L n -algebra), 1 < n < ω , which are algebraic counterparts of the logic, that in turn represent two-sorted algebras (M , R , ◊) that combine the varieties of M V n -algebras M = (M , ⊕ , ⊙ , ∼ , 0 , 1) and regular algebras R = (R , ∪ , ; , ∗) into a single finitely axiomatized variety resembling R-module with "scalar" multiplication ◊ . Kripke semantics is developed for immune dynamic Łukasiewicz logic I D Ł n with application in immune system. [ABSTRACT FROM AUTHOR]

Frame semantics, given by Kripke or neighborhood frames, do not give completeness theorems for all modal logics extending, respectively, K and E. Such shortcoming can be overcome by means of general frames, i.e., frames equipped with a collection of admissible sets of worlds (which is the range of possible valuations over such frame). We export this approach from the classical paradigm to modal many-valued logics by defining general A-frames over a given residuated lattice A (i.e., the usual frames with a collection of admissible A-valued sets). We describe in detail the relation between general Kripke and neighborhood A-frames and prove that, if the logic of A is finitary, all extensions of the corresponding logic E of A are complete w.r.t. general neighborhood frames. Our work provides a new approach to the current research trend of generalizing relational semantics for non-classical modal logics to circumvent axiomatization problems. [ABSTRACT FROM AUTHOR]

Kripke frames (and models) provide a suitable semantics for sub-classical logics; for example, intuitionistic logic (of Brouwer and Heyting) axiomatizes the reflexive and transitive Kripke frames (with persistent satisfaction relations), and the basic logic (of Visser) axiomatizes transitive Kripke frames (with persistent satisfaction relations). Here, we investigate whether Kripke frames/models could provide a semantics for fuzzy logics. For each axiom of the basic fuzzy logic, necessary and sufficient conditions are sought for Kripke frames/models which satisfy them. It turns out that the only fuzzy logics (logics containing the basic fuzzy logic) which are sound and complete with respect to a class of Kripke frames/models are the extensions of the Gödel logic (or the super-intuitionistic logic of Dummett); indeed this logic is sound and strongly complete with respect to reflexive, transitive and connected (linear) Kripke frames (with persistent satisfaction relations). This provides a semantic characterization for the Gödel logic among (propositional) fuzzy logics. [ABSTRACT FROM AUTHOR]

Minimal logic, i.e., intuitionistic logic without the ex falso principle, is investigated in its original form with a negation symbol instead of a symbol denoting the contradiction. A Kripke semantics is developed for minimal logic and its sublogics with a still weaker negation by introducing a function on the upward closed sets of the models. The basic logic is a logic in which the negation has no properties but the one of being a unary operator. A number of extensions is studied of which the most important ones are contraposition logic and negative ex falso, a weak form of the ex falso principle. Completeness is proved, and the created semantics is further studied. The negative translation of classical logic into intuitionistic logic is made part of a chain of translations by introducing translations from minimal logic into contraposition logic and intuitionistic logic into minimal logic, the latter having been discovered in the correspondence between Johansson and Heyting. Finally, as a bridge to the work of Franco Montagna a start is made of a study of linear models of these logics. [ABSTRACT FROM AUTHOR]

We show that logics based on linear Kripke frames-with or without constant domains-that have a scattered end piece are not recursively enumerable. This is done by reduction to validity in all finite classical models. [ABSTRACT FROM AUTHOR]