In this paper we exhibit a non-finitely based, finitely generated quasi-variety of De Morgan algebras and determine the bottom of the lattices of sub-quasi-varieties of Kleene and De Morgan algebras.

The notion of an algebraic semantics of a deductive system was proposed in [3], and a preliminary study was begun. The focus of [3] was the definition and investigation of algebraizable deductive systems, i.e., the deductive systems that possess an equivalent algebraic semantics. The present paper explores the more general property of possessing an algebraic semantics. While a deductive system can have at most one equivalent algebraic semantics, it may have numerous different algebraic semantics. All of these give rise to (...) an algebraic completeness theorem for the deductive system, but their algebraic properties, unlike those of equivalent algebraic semantics, need not reflect the metalogical properties of the deductive system. Many deductive systems that don't have an equivalent algebraic semantics do possess an algebraic semantics; examples of these phenomena are provided. It is shown that all extensions of a deductive system that possesses an algebraic semantics themselves possess an algebraic semantics. Necessary conditions for the existence of an algebraic semantics are given, and an example of a protoalgebraic deductive system that does not have an algebraic semantics is provided. The mono-unary deductive systems possessing an algebraic semantics are characterized. Finally, weak conditions on a deductive system are formulated that guarantee the existence of an algebraic semantics. These conditions are used to show that various classes of non-algebraizable deductive systems of modal logic, relevance logic and linear logic do possess an algebraic semantics. (shrink)

We look at extensions (i.e., stronger logics in the same language) of the Belnap–Dunn four-valued logic. We prove the existence of a countable chain of logics that extend the Belnap–Dunn and do not coincide with any of the known extensions (Kleene’s logics, Priest’s logic of paradox). We characterise the reduced algebraic models of these new logics and prove a completeness result for the first and last element of the chain stating that both logics are determined by a single finite logical (...) matrix. We show that the last logic of the chain is not finitely axiomatisable. (shrink)

The aim of this paper is technically to study Belnap's four-valued sentential logic . First, we obtain a Gentzen-style axiomatization of this logic that contains no structural rules while all they are still admissible in the Gentzen system what is proved with using some algebraic tools. Further, the mentioned logic is proved to be the least closure operator on the set of {Λ, V, ⌝}-formulas satisfying Tarski's conditions for classical conjunction and disjunction together with De Morgan's laws for negation. It (...) is also proved that Belnap's logic is the only sentential logic satisfying the above-mentioned conditions together with Anderson-Belnap's Variable-Sharing Property. Finally, we obtain a finite Hilbert-style axiomatization of this logic. As a consequence, we obtain a finite Hilbert-style axiomatization of Priest's logic of paradox. (shrink)

The present paper concerns a technical study of PRIEST'S logic of paradox [Pri 79], We prove that this logic has no proper paraconsistent strengthening. It is also proved that the mentioned logic is the largest paraconsistent one satisfaying TARSKI'S conditions for the classical conjunction and disjunction together with DE MORGAN'S laws for negation. Finally, we obtain for the logic of paradox an algebraic completeness result related to Kleene lattices.

A curious feature of Belnap’s “useful four-valued logic”, also known as first-degree entailment (FDE), is that the overdetermined value B (both true and false) is treated as a designated value. Although there are good theoretical reasons for this, it seems prima facie more plausible to have only one of the four values designated, namely T (exactly true). This paper follows this route and investigates the resulting logic, which we call Exactly True Logic.

A formal theory of truth, alternative to tarski's 'orthodox' theory, based on truth-value gaps, is presented. the theory is proposed as a fairly plausible model for natural language and as one which allows rigorous definitions to be given for various intuitive concepts, such as those of 'grounded' and 'paradoxical' sentences.

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 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)

In this paper we exhibit a non-finitely based, finitely generated quasi-variety of De Morgan algebras and determine the bottom of the lattices of sub-quasi-varieties of Kleene and De Morgan algebras.

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)

This paper contains some contributions to the study of Belnap's four-valued logic from an algebraic point of view. We introduce a finite Hilbert-style axiomatization of this logic, along with its well-known semantical presentation, and a Gentzen calculus that slightly differs from the usual one in that it is closer to Anderson and Belnap's formalization of their “logic of first-degree entailments”. We prove several Completeness Theorems and reduce every formula to an equivalent normal form. The Hilbert-style presentation allows us to characterize (...) the Leibniz congruence of the matrix models of the logic, and to find that the class of algebraic reducts of its reduced matrices is strictly smaller than the variety of De Morgan lattices. This means that the links between the logic and this class of algebras cannot be fully explained in terms of matrices, as in more classical logics. It is through the use of abstract logics as models that we are able to confirm that De Morgan lattices are indeed the algebraic counterpart of Belnap's logic, in the sense of Font and Jansana's recent theory of full models for sentential logics. Among other characterizations, we prove that its full models are those abstract logics that are finitary, do not have theorems, and satisfy the metalogical properties of Conjunction, Disjunction, Double Negation, and Weak Contraposition. As a consequence, we find that the Gentzen calculus presented at the beginning is strongly adequate for Belnap's logic and is algebraizable in the sense of Rebagliato and Verdú, having the variety of De Morgan lattices as its equivalent algebraic semantics. (shrink)

This paper contains some contributions to the study of Belnap's four-valued logic from an algebraic point of view. We introduce a finite Hilbert-style axiomatization of this logic, along with its well-known semantical presentation, and a Gentzen calculus that slightly differs from the usual one in that it is closer to Anderson and Belnap's formalization of their “logic of first-degree entailments”. We prove several Completeness Theorems and reduce every formula to an equivalent normal form. The Hilbert-style presentation allows us to characterize (...) the Leibniz congruence of the matrix models of the logic, and to find that the class of algebraic reducts of its reduced matrices is strictly smaller than the variety of De Morgan lattices. This means that the links between the logic and this class of algebras cannot be fully explained in terms of matrices, as in more classical logics. It is through the use of abstract logics as models that we are able to confirm that De Morgan lattices are indeed the algebraic counterpart of Belnap's logic, in the sense of Font and Jansana's recent theory of full models for sentential logics. Among other characterizations, we prove that its full models are those abstract logics that are finitary, do not have theorems, and satisfy the metalogical properties of Conjunction, Disjunction, Double Negation, and Weak Contraposition. As a consequence, we find that the Gentzen calculus presented at the beginning is strongly adequate for Belnap's logic and is algebraizable in the sense of Rebagliato and Verdú, having the variety of De Morgan lattices as its equivalent algebraic semantics. (shrink)

This paper explores allowing truth value assignments to be undetermined or "partial" and overdetermined or "inconsistent", thus returning to an investigation of the four-valued semantics that I initiated in the sixties. I examine some natural consequence relations and show how they are related to existing logics, including ukasiewicz's three-valued logic, Kleene's three-valued logic, Anderson and Belnap's relevant entailments, Priest's "Logic of Paradox", and the first-degree fragment of the Dunn-McCall system "R-mingle". None of these systems have nested implications, and I investigate (...) twelve natural extensions containing nested implications, all of which can be viewed as coming from natural variations on Kripke's semantics for intuitionistic logic. Many of these logics exist antecedently in the literature, in particular Nelson 's "constructible falsity". (shrink)

This paper is a contribution to the study of the rôle of disjunction inAlgebraic Logic. Several kinds of (generalized) disjunctions, usually defined using a suitable variant of the proof by cases property, were introduced and extensively studied in the literature mainly in the context of finitary logics. The goals of this paper are to extend these results to all logics, to systematize the multitude of notions of disjunction (both those already considered in the literature and those introduced in this paper), (...) and to show several interesting applications allowed by the presence of a suitable disjunction in a given logic. (shrink)

The notion of an algebraic semantics of a deductive system was proposed in [3], and a preliminary study was begun. The focus of [3] was the definition and investigation of algebraizable deductive systems, i.e., the deductive systems that possess an equivalent algebraic semantics. The present paper explores the more general property of possessing an algebraic semantics. While a deductive system can have at most one equivalent algebraic semantics, it may have numerous different algebraic semantics. All of these give rise to (...) an algebraic completeness theorem for the deductive system, but their algebraic properties, unlike those of equivalent algebraic semantics, need not reflect the metalogical properties of the deductive system. Many deductive systems that don't have an equivalent algebraic semantics do possess an algebraic semantics; examples of these phenomena are provided. It is shown that all extensions of a deductive system that possesses an algebraic semantics themselves possess an algebraic semantics. Necessary conditions for the existence of an algebraic semantics are given, and an example of a protoalgebraic deductive system that does not have an algebraic semantics is provided. The mono-unary deductive systems possessing an algebraic semantics are characterized. Finally, weak conditions on a deductive system are formulated that guarantee the existence of an algebraic semantics. These conditions are used to show that various classes of non-algebraizable deductive systems of modal logic, relevance logic and linear logic do possess an algebraic semantics. (shrink)

We shed light on an old problem by showing that the logic LP cannot define a binary connective $\odot$ obeying detachment in the sense that every valuation satisfying $\varphi$ and $(\varphi\odot\psi)$ also satisfies $\psi$ , except trivially. We derive this as a corollary of a more general result concerning variable sharing.

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 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)

Frege's Principle asserts that the denotation of a propositional sentence coincides with its truth value. In the context of algebraizable logics the principle can be interpreted as the compositionality of interderivability relation $\Fr{S}$, defined formally by $\Fr{S}T=\{\langle \phi, \psi\rangle\in\Fml^2\mid T,\phi \dashv\vdash_{\mathcal S}T,\psi \}$, for given deductive system $\mathcal S$ and any $\mathcal S$-theory $T$. Of special interest are the deductive systems for which the property of being Fregean is inherited by all full 2nd-order models, so called, \it{fully Fregean} deductive systems. (...) The main result of this paper is a characterization of fully Fregean deductive systems over countable languages using properties of the strong Frege operator on the formula algebra. The example of a Fregean, but not fully Fregean deductive system $\mathcal S$ is provided. $\mathcal S$ also turns out to be selfextensional, but not fully selfextensional, and, in addition, the three principal algebraic semantics for $\mathcal S$ are different, i.e., $\Alg^\ast\mathcal S\subsetneq\Alg\mathcal S\subsetneq\Var$. (shrink)