The positive fragment of the local modal consequence relation defined by the class of all Kripke frames is studied in the context ofAlgebraic Logic. It is shown that this fragment is non-protoalgebraic and that its class of canonically associated algebras according to the criteria set up in [7] is the class of positive modal algebras. Moreover its full models are characterized as the models of the Gentzen calculus introduced in [3].

In this paper we consider the class of truth-functional modal many-valued logics with the complete lattice of truth-values. The conjunction and disjunction logic operators correspond to the meet and join operators of the lattices, while the negation is independently introduced as a hierarchy of antitonic operators which invert bottom and top elements. The non-constructive logic implication will be defined for a subclass of modular lattices, while the constructive implication for distributive lattices (Heyting algebras) is based on relative pseudo-complements as in (...) intuitionistic logic. We show that the complete lattices are intrinsically modal, with banal identity modal operator. We define the autoreferential set-based representation for the class of modal algebras, and show that the autoreferential Kripke-style semantics for this class of modal algebras is based on the set of possible worlds equal to the complete lattice of algebraic truth-values. The philosophical assumption is based on the consideration that each possible world represents a level of credibility, so that only propositions with the right logic value (i.e., level of credibility) can be accepted by this world, then we connect it with paraconsistent properties and LFI logics. The bottom truth value in this complete lattice corresponds to the trivial world in which each formula is satisfied, that is, to the world with explosive inconsistency. The top truth value corresponds to the world with classical logics, while all intermediate possible worlds represent the different levels of paraconsistent logics. (shrink)

We investigate bisimulations for instantial neighbourhood logic and an \-indexed collection of its fragments. For each of these logics we give a Hennessy-Milner theorem and a Van Benthem-style characterisation theorem.

In this paper we consider distributive modal logic, a setting in which we may add modalities, such as classical types of modalities as well as weak forms of negation, to the fragment of classical propositional logic given by conjunction, disjunction, true, and false. For these logics we define both algebraic semantics, in the form of distributive modal algebras, and relational semantics, in the form of ordered Kripke structures. The main contributions of this paper lie in extending the notion of Sahlqvist (...) axioms to our generalized setting and proving both a correspondence and a canonicity result for distributive modal logics axiomatized by Sahlqvist axioms. Our proof of the correspondence result relies on a reduction to the classical case, but our canonicity proof departs from the traditional style and uses the newly extended algebraic theory of canonical extensions. (shrink)

We study an application of gaggle theory to unary negative modal operators. First we treat negation as impossibility and get a minimal logic system Ki that has a perp semantics. Dunn 's kite of different negations can be dealt with in the extensions of this basic logic Ki. Next we treat negation as “unnecessity” and use a characteristic semantics for different negations in a kite which is dual to Dunn 's original one. Ku is the minimal logic that has a (...) characteristic semantics. We also show that Shramko's falsification logic FL can be incorporated into some extension of this basic logic Ku. Finally, we unite the two basic logics Ki and Ku together to get a negative modal logic K-, which is dual to the positive modal logic K+ in [7]. Shramko has suggested an extension of Dunn 's kite and also a dual version in [12]. He also suggested combining them into a “united” kite. We give a united semantics for this united kite of negations. (shrink)

This paper explores a notion of “the strong version” of a sentential logic S, initially defined in terms of the notion of a Leibniz filter, and shown to coincide with the logic determined by the matrices of S whose filter is the least S-filter in the algebra of the matrix. The paper makes a general study of this notion, which appears to unify under an abstract framework the relationships between many pairs of logics in the literature. The paradigmatic examples are (...) the local and the global consequences associated with a normal modal logic, and the logics preserving degrees of truth and preserving truth associated with certain substructural and many-valued logics. For protoalgebraic logics the results in the paper coincide with those obtained by two of the authors in 2001, so the main novelty of the approach is its suitability for all kinds of logics. The paper also studies three kinds of definability of the Leibniz filters, and their consequences for the determination of the strong version. In a second part of the paper several case studies are developed, comprising positive modal logic, Dunn–Belnap’s four-valued logic, the large family of substructural logics, and some relevance logics. (shrink)

Our concern is the completeness problem for spi-logics, that is, sets of implications between strictly positive formulas built from propositional variables, conjunction and modal diamond operators. Originated in logic, algebra and computer science, spi-logics have two natural semantics: meet-semilattices with monotone operators providing Birkhoff-style calculi and first-order relational structures (aka Kripke frames) often used as the intended structures in applications. Here we lay foundations for a completeness theory that aims to answer the question whether the two semantics define the same (...) consequence relations for a given spi-logic. (shrink)

Positive modal algebras are the$$\left\langle { \wedge, \vee,\diamondsuit,\square,0,1} \right\rangle $$-subreducts of modal algebras. We prove that the variety of positive S4-algebras is not locally finite. On the other hand, the free one-generated positive S4-algebra is shown to be finite. Moreover, we describe the bottom part of the lattice of varieties of positive S4-algebras. Building on this, we characterize structurally complete varieties of positive K4-algebras.

We consider subintuitionistic logics as an extension of positive propositional logic with a binary modality, interpreted over ordered and unordered monotone neighborhood frames, with a range of frame conditions. This change in perspective allows us to apply tools and techniques from the modal setting to subintuitionistic logics. We provide a Priestley-style duality, and transfer results from the (classical) logic of monotone neighborhood frames to obtain completeness, conservativity, and a finite model property for the basic logic, extended with a number of (...) axioms. (shrink)

We propose a new approach to positive modal logics, hereby called anodic modal logics. Our treatment is completely positive since the language has neither negation nor any falsum or minimal particle. The elimination of the minimal particle of the language requires introducing the new concept of factual sets and factual deductions which permit us to talk about deductions in the actual world. We start from a positive fragment of the standard system K, denoted by K⊃, ∧, ◊, which is a (...) bimodal system with □ and ◊ as primitive. This system is then extended to a class of fragments of the Lemmon and Scott systems (cf. (Lemmon et al., 1977)), denoted by K⊃, ∧, ◊ + Gk;l;m;n + Gm;n;k;l. It is shown that such classes of systems are characterized with respect to the usual Kripke-style semantics. The proof is by way of a Henkin-style construction, with “possible worlds” being taken to be prime theories as introduced in the modal context by J. M. Dunn in (Dunn, 1995). We also obtain a surprising limiting result showing that the incompleteness phenomenon in modal logic is independent of negation. (shrink)

Wójcicki has provided a characterization of selfextensional logics as those that can be endowed with a complete local referential semantics. His result was extended by Jansana and Palmigiano, who developed a duality between the category of reduced congruential atlases and that of reduced referential algebras over a fixed similarity type. This duality restricts to one between reduced atlas models and reduced referential algebra models of selfextensional logics. In this paper referential algebraic systems and congruential atlas systems are introduced, which abstract (...) referential algebras and congruential atlases, respectively. This enables the formulation of an analog of Wójcicki’s Theorem for logics formalized as π-institutions. Moreover, the results of Jansana and Palmigiano are generalized to obtain a duality between congruential atlas systems and referential algebraic systems over a fixed categorical algebraic signature. In future work, the duality obtained in this paper will be used to obtain one between atlas system models and referential algebraic system models of an arbitrary selfextensional π-institution. Using this latter duality, the characterization of fully selfextensional deductive systems among the selfextensional ones, that was obtained by Jansana and Palmigiano, can be extended to a similar characterization of fully selfextensional π-institutions among appropriately chosen classes of selfextensional ones. (shrink)

Matthew Spinks [35] introduces implicative BCSK-algebras, expanding implicative BCK-algebras with an additional binary operation. Subdirectly irreducible implicative BCSK-algebras can be viewed as flat posets with two operations coinciding only in the 1- and 2-element cases, each, in the latter case, giving the two-valued implication truth-function. We introduce the resulting logic (for the general case) in terms of matrix methodology in §1, showing how to reformulate the matrix semantics as a Kripke-style possible worlds semantics, thereby displaying the distinction between the two (...) implications in the more familiar language of modal logic. In §§2 and 3 we study, from this perspective, the fragments obtained by taking the two implications separately, and – after a digression (in §4) on the intuitionistic analogue of the material in §3 – consider them together in §5, closing with a discussion in §6 of issues in the theory of logical rules. Some material is treated in three appendices to prevent §§1–6 from becoming overly distended. (shrink)

We consider the equationally orderable quasivarieties and associate with them deductive systems defined using the order. The method of definition of these deductive systems encompasses the definition of logics preserving degrees of truth we find in the research areas of substructural logics and mathematical fuzzy logic. We prove several general results, for example that the deductive systems so defined are finitary and that the ones associated with equationally orderable varieties are congruential.