In the paper we study the class of weakly algebraizable logics, characterized by the monotonicity and injectivity of the Leibniz operator on the theories of the logic. This class forms a new level in the non-linear hierarchy of protoalgebraic logics.

A logic is selfextensional if its interderivability (or mutual consequence) relation is a congruence relation on the algebra of formulas. In the paper we characterize the selfextensional logics with a conjunction as the logics that can be defined using the semilattice order induced by the interpretation of the conjunction in the algebras of their algebraic counterpart. Using the charactrization we provide simpler proofs of several results on selfextensional logics with a conjunction obtained in [13] using Gentzen systems. We also obtain (...) some results on Fregean logics with conjunction. (shrink)

A logic in a finite language is said to be finitely presentable if it is axiomatized by finitely many finite rules. It is proved that binary non-indexed products of logics that are both finitely presentable and finitely equivalential are essentially finitely presentable. This result does not extend to binary non-indexed products of arbitrary finitely presentable logics, as shown by a counterexample. Finitely presentable logics are then exploited to introduce finitely presentable Leibniz classes, and to draw a parallel between the Leibniz (...) and the Maltsev hierarchies. (shrink)

In the present paper we study systematically several consequence relations on the usual language of propositional intuitionistic logic that can be defined semantically by using Kripke frames and the same defining truth conditions for the connectives as in intuitionistic logic but without imposing some of the conditions on the Kripke frames that are required in the intuitionistic case. The logics so obtained are called subintuitionistic logics in the literature. We depart from the perspective of considering a logic just as a (...) set of theorems and also depart from the perspective taken by Restall in that we consider standard Kripke models instead of models with a base point. We study the relations between subintuitionistic logics and modal logics given by the translation considered by Došen. Moreover, we classify the logics obtained according to the hierarchy considered inAlgebraic Logic. (shrink)

A definition and some inaccurate cross-references in the paper A Survey ofAlgebraic Logic, which might confuse some readers, are clarified and corrected; a short discussion of the main one is included. We also update a dozen of bibliographic references.

A filter of a sentential logic ? is Leibniz when it is the smallest one among all the ?-filters on the same algebra having the same Leibniz congruence. This paper studies these filters and the sentential logic ?+ defined by the class of all ?-matrices whose filter is Leibniz, which is called the strong version of ?, in the context of protoalgebraic logics with theorems. Topics studied include an enhanced Correspondence Theorem, characterizations of the weak algebraizability of ?+ and of (...) the explicit definability of Leibniz filters, and several theorems of transfer of metalogical properties from ? to ?+. For finitely equivalential logics stronger results are obtained. Besides the general theory, the paper examines the examples of modal logics, quantum logics and Łukasiewicz's finitely-valued logics. One finds that in some cases the existence of a weak and a strong version of a logic corresponds to well-known situations in the literature, such as the local and the global consequences for normal modal logics; while in others these constructions give an independent interest to the study of other lesser-known logics, such as the lattice-based many-valued logics. (shrink)

The present paper introduces and studies the variety WH of weakly Heyting algebras. It corresponds to the strict implication fragment of the normal modal logic K which is also known as the subintuitionistic local consequence of the class of all Kripke models. The tools developed in the paper can be applied to the study of the subvarieties of WH; among them are the varieties determined by the strict implication fragments of normal modal logics as well as varieties that do not (...) arise in this way as the variety of Basic algebras or the variety of Heyting algebras. Apart from WH itself the paper studies the subvarieties of WH that naturally correspond to subintuitionistic logics, namely the variety of R-weakly Heyting algebras, the variety of T-weakly Heyting algebras and the varieties of Basic algebras and subresiduated lattices. (shrink)

In the present paper we study systematically several consequence relations on the usual language of propositional intuitionistic logic that can be defined semantically by using Kripke frames and the same defining truth conditions for the connectives as in intuitionistic logic but without imposing some of the conditions on the Kripke frames that are required in the intuitionistic case. The logics so obtained are called subintuitionistic logics in the literature. We depart from the perspective of considering a logic just as a (...) set of theorems and also depart from the perspective taken by Restall in that we consider standard Kripke models instead of models with a base point. We study the relations between subintuitionistic logics and modal logics given by the translation considered by Došen. Moreover, we classify the logics obtained according to the hierarchy considered inAlgebraic Logic. (shrink)

The paper provides a new semantics for positive modal logic using Kripke frames having a quasi ordering on the set of possible worlds and an accessibility relation connected to the quasi ordering by the conditions (1) that the composition of with is included in the composition of with and (2) the analogous for the inverse of and . This semantics has an advantage over the one used by Dunn in "Positive modal logic," Studia Logica (1995) and works fine for extensions (...) of the minimal system of normal positive modal logic. (shrink)

This paper is a contribution to the study of equality-free logic, that is, first-order logic without equality. We mainly devote ourselves to the study of algebraic characterizations of its relation of elementary equivalence by providing some Keisler-Shelah type ultrapower theorems and an Ehrenfeucht-Fraïssé type theorem. We also give characterizations of elementary classes in equality-free logic. As a by-product we characterize the sentences that are logically equivalent to an equality-free one.

In this paper we consider the structure of the class FGModS of full generalized models of a deductive system S from a universal-algebraic point of view, and the structure of the set of all the full generalized models of S on a fixed algebra A from the lattice-theoretical point of view; this set is represented by the lattice FACSs A of all algebraic closed-set systems C on A such that (A, C) ε FGModS. We relate some properties of these structures (...) with tipically logical properties of the sentential logic S. The main algebraic properties we consider are the closure of FGModS under substructures and under reduced products, and the property that for any A the lattice FACSs A is a complete sublattice of the lattice of all algebraic closed-set systems over A. The logical properties are the existence of a fully adequate Gentzen system for S, the Local Deduction Theorem and the Deduction Theorem for S. Some of the results are established for arbitrary deductive systems, while some are found to hold only for deductive systems in more restricted classes like the protoalgebraic or the weakly algebraizable ones. The paper ends with a section on examples and counterexamples. (shrink)

The best known algebraizable logics with a conjunction and an implication have the property that the conjunction defines a meet semi-lattice in the algebras of their algebraic counterpart. This property makes it possible to associate with them a semi-lattice based deductive system as a companion. Moreover, the order of the semi-lattice is also definable using the implication. This makes that the connection between the properties of the logic and the properties of its semi-lattice based companion is strong. We introduce a (...) class of algebraizable deductive systems that includes those systems, and study some of their properties and of their semi-lattice based companions. We also study conditions which, when satisfied by a deductive system in the class, imply that it is strongly algebraizable. This brings some information on the open area of research ofAlgebraic Logic which consists in finding interesting characterizations of classes of algebraizable logics that are strongly algebraizable. (shrink)

Given a structure for a first-order language L, two objects of its domain can be indiscernible relative to the properties expressible in L, without using the equality symbol, and without actually being the same. It is this relation that interests us in this paper. It is called Leibniz equality. In the paper we study systematically the problem of its definibility mainly for classes of structures that are the models of some equality-free universal Horn class in an infinitary language Lκκ, where (...) κ is an infinite regular cardinal. (shrink)

Leibniz filters play a prominent role in the theory of protoalgebraic logics. In [3] the problem of the definability of Leibniz filters is considered. Here we study the definability of Leibniz filters with parameters. The main result of the paper says that a protoalgebraic logic S has its strong version weakly algebraizable iff it has its Leibniz filters explicitly definable with parameters.

We generalize Priestley duality for distributive lattices to a duality for distributive meet-semilattices. On the one hand, our generalized Priestley spaces are easier to work with than Celani’s DS-spaces, and are similar to Hansoul’s Priestley structures. On the other hand, our generalized Priestley morphisms are similar to Celani’s meet-relations and are more general than Hansoul’s morphisms. As a result, our duality extends Hansoul’s duality and is an improvement of Celani’s duality.

This paper presents a unified framework that explains and extends the already successful applications of the Leibniz operator, the Suszko operator, and the Tarski operator in recent developments in abstract algebraic logic. To this end, we refine Czelakowski’s notion of an S-compatibility operator, and introduce the notion of coherent family of S-compatibility operators, for a sentential logic S. The notion of coherence is a restricted property of commutativity with inverse images by surjective homomorphisms, which is satisfied by both the Leibniz (...) and the Suszko operators. We generalize several constructions and results already existing for the mentioned operators; in particular, the well-known classes of algebras associated with a logic through each of them, and the notions of full generalized model of a logic and a special kind of S-filters. We obtain a General Correspondence Theorem, extending the well-known one from the theory of protoalgebraic logics to arbitrary logics and to more general operators, and strengthening its formulation. We apply the general results to the Leibniz and the Suszko operators, and obtain several characterizations of the main classes of logics in the Leibniz hierarchy by the form of their full generalized models, by old and new properties of the Leibniz operator, and by the behaviour of the Suszko operator. Some of these characterizations complete or extend known ones, for some classes in the hierarchy, thus offering an integrated approach to the Leibniz hierarchy that uncovers some new, nice symmetries. (shrink)

In the paper we study the class of weakly algebraizable logics, characterized by the monotonicity and injectivity of the Leibniz operator on the theories of the logic. This class forms a new level in the non-linear hierarchy of protoalgebraic logics.

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].

Willem Blok was one of the founders of the field Abstract Algebraic Logic. The paper describes his research in this field.

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)

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)

A pair of deductive systems (S,S’) is Leibniz-linked when S’ is an extension of S and on every algebra there is a map sending each filter of S to a filter of S’ with the same Leibniz congruence. We study this generalization to arbitrary deductive systems of the notion of the strong version of a protoalgebraic deductive system, studied in earlier papers, and of some results recently found for particular non-protoalgebraic deductive systems. The necessary examples and counterexamples found in the (...) literature are described. (shrink)

We introduce a new and general notion of canonical extension for algebras in the algebraic counterpart of any finitary and congruential logic . This definition is logic-based rather than purely order-theoretic and is in general different from the definition of canonical extensions for monotone poset expansions, but the two definitions agree whenever the algebras in are based on lattices. As a case study on logics purely based on implication, we prove that the varieties of Hilbert and Tarski algebras are canonical (...) in this new sense. (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.

In this paper we study some logics related to the logic of place introduced by von Wright and studied by Segerberg. For every we study the logic of the class of frames whose accessibility relation R satisfies the following condition: if then there is such that . For a fixed the logic is the one axiomatized by K , which we call Kn.4B, where . We prove that these logics are canonical and hence complete, and that they have the finite (...) model property, being thus decidable. We also characterize their classes of frames. In the way of studying them we also study the logics , called Kn.4, and , called Kn.B. A translation between these logics and S5 is also presented, and the relation among them all is established. (shrink)

A notion of interpretation between arbitrary logics is introduced, and the poset $\mathsf {Log}$ of all logics ordered under interpretability is studied. It is shown that in $\mathsf {Log}$ infima of arbitrarily large sets exist, but binary suprema in general do not. On the other hand, the existence of suprema of sets of equivalential logics is established. The relations between $\mathsf {Log}$ and the lattice of interpretability types of varieties are investigated.

Hilbert algebras provide the equivalent algebraic semantics in the sense of Blok and Pigozzi to the implication fragment of intuitionistic logic. They are closely related to implicative semilattices. Porta proved that every Hilbert algebra has a free implicative semilattice extension. In this paper we introduce the notion of an optimal deductive filter of a Hilbert algebra and use it to provide a different proof of the existence of the free implicative semilattice extension of a Hilbert algebra as well as a (...) simplified characterization of it. The optimal deductive filters turn out to be the traces in the Hilbert algebra of the prime filters of the distributive lattice free extension of the free implicative semilattice extension of the Hilbert algebra. To define the concept of optimal deductive filter we need to introduce the concept of a strong Frink ideal for Hilbert algebras which generalizes the concept of a Frink ideal for posets. (shrink)

A pair of deductive systems is Leibniz-linked when S’ is an extension of S and on every algebra there is a map sending each filter of S to a filter of S’ with the same Leibniz congruence. We study this generalization to arbitrary deductive systems of the notion of the strong version of a protoalgebraic deductive system, studied in earlier papers, and of some results recently found for particular non-protoalgebraic deductive systems. The necessary examples and counterexamples found in the literature (...) are described. (shrink)

In this paper we develop a general framework to deal with abstract logics associated with a given modal logic. In particular we study the abstract logics associated with the weak and strong deductive systems of the normal modal logicK and its intuitionistic version. We also study the abstract logics that satisfy the conditionC +(X)=C( in I n X) and find the modal deductive systems whose abstract logics, in addition to being classical or intuitionistic, satisfy that condition. Finally we study the (...) deductive systems whose abstract logics satisfy, in addition to the already mentioned properties, the property that the operatorC + is classical relative to some new defined operations. (shrink)

In this paper we introduce and study a variety of algebras that properly includes integral distributive commutative residuated lattices and weak Heyting algebras. Our main goal is to give a characterization of the principal congruences in this variety. We apply this description in order to study compatible functions.

In this paper we introduce and study a variety of algebras that properly includes integral distributive commutative residuated lattices and weak Heyting algebras. Our main goal is to give a characterization of the principal congruences in this variety. We apply this description in order to study compatible functions.