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For each n ∈ N , we introduce the algebraic semantics for the minimum many-valued modal logic over the MV-chain with n elements. We prove that this quasivariety of algebras is generated by the complex algebras, obtaining as a consequence the completeness result of the logic with respect to the Kripke semantics. As a first step towards the study of this algebraic semantics, we study some congruences of the complex algebras. [ABSTRACT FROM AUTHOR]

The present paper establishes systematic connections among the first-order correspondents of Sahlqvist modal reduction principles in various relational semantic settings, including crisp and many-valued Kripke frames, and crisp and many-valued polarity-based frames (aka enriched formal contexts). Building on unified correspondence theory, we aim at introducing a theoretical environment which makes it possible to: (a) compare and inter-relate the various frame correspondents (in different relational settings) of any given Sahlqvist modal reduction principle; (b) recognize when first-order sentences in the frame-correspondence languages of different types of relational structures encode the same "modal content"; (c) meaningfully transfer and represent well known relational properties such as reflexivity, transitivity, symmetry, seriality, confluence, density, across different semantic contexts. These results can be understood as a first step in a research program aimed at making correspondence theory not just (methodologically) unified, but also (effectively) parametric. [ABSTRACT FROM AUTHOR]

This paper continues the investigation of the logic of competing theories (be they scientific, social, political etc.) initiated in [4]. We introduce a many-valued, multi-type modal language which we endow with relational semantics based on enriched reflexive graphs, inspired by Ploščica's representation of general lattices. We axiomatize the resulting many-valued, non-distributive modal logic of these structures and prove a completeness theorem. We illustrate the application of this logic through a case study in which we model competition among interacting political promises and social demands within an arena of political parties social groups. [ABSTRACT FROM AUTHOR]

We shall be concerned with the modal logic BK—which is based on the Belnap–Dunn four-valued matrix, and can be viewed as being obtained from the least normal modal logic K by adding 'strong negation'. Though all four values 'truth', 'falsity', 'neither' and 'both' are employed in its Kripke semantics, only the first two are expressible as terms. We show that expanding the original language of BK to include constants for 'neither' or/and 'both' leads to quite unexpected results. To be more precise, adding one of these constants has the effect of eliminating the respective value at the level of BK-extensions. In particular, if one adds both of these, then the corresponding lattice of extensions turns out to be isomorphic to that of ordinary normal modal logics. [ABSTRACT FROM AUTHOR]

We present and defend the Australian Plan semantics for negation. This is a comprehensive account, suitable for a variety of different logics. It is based on two ideas. The first is that negation is an exclusion-expressing device: we utter negations to express incompatibilities. The second is that, because incompatibility is modal, negation is a modal operator as well. It can, then, be modelled as a quantifier over points in frames, restricted by accessibility relations representing compatibilities and incompatibilities between such points. We defuse a number of objections to this Plan, raised by supporters of the American Plan for negation, in which negation is handled via a many-valued semantics. We show that the Australian Plan has substantial advantages over the American Plan. [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]

The majority of works on modal many-valued logics consider Kripke-style possible worlds frames as the principal semantics despite their well-known axiomatizability issues when considering non-Boolean accessibility relations. The present work explores a more general semantical picture, namely a many-valued version of the classical neighborhood semantics. We present it in two levels of generality. First, we work with modal languages containing only the two usual unary modalities, define neighborhood frames over algebras of the logic FL ew with operators, and show their relation with the usual Kripke semantics (this is actually the highest level of generality where one can give a straightforward definition of the Kripke-style semantics). Second, we define generalized neighborhood frames for arbitrary modal languages over a given class of algebras for an arbitrary protoalgebraic logic and, assuming certain additional conditions, axiomatize the logic of all such frames (which generalizes the completeness theorem of the classical modal logic E with respect to classical neighborhood frames). [ABSTRACT FROM AUTHOR]

A many-valued modal logic is introduced that combines the usual Kripke frame semantics of the modal logic K with connectives interpreted locally at worlds by lattice and group operations over the real numbers. A labelled tableau system is provided and a coNEXPTIME upper bound obtained for checking validity in the logic. Focussing on the modal-multiplicative fragment, the labelled tableau system is then used to establish completeness for a sequent calculus that admits cut-elimination and an axiom system that extends the multiplicative fragment of Abelian logic. [ABSTRACT FROM AUTHOR]

We study two notions of definability for classes of relational structures based on modal extensions of Ł ukasiewicz finitely-valued logics. The main results of the paper are the equivalent of the G oldblatt-T homason theorem for these notions of definability. [ABSTRACT FROM AUTHOR]

This paper deals with one kind of Kripke-style semantics, which we shall call algebraic Kripke-style semantics, for relevance logics. We first recall the logic R of relevant implication and some closely related systems, their corresponding algebraic structures, and algebraic completeness results. We provide simpler algebraic completeness proofs. We then introduce various types of algebraic Kripke-style semantics for these systems and connect them with algebraic semantics. [ABSTRACT FROM AUTHOR]