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 of Abstract Algebraic Logic which consists in finding interesting characterizations of classes of algebraizable logics that are strongly algebraizable. (shrink)

An extensions by new axioms and rules of an algebraizable logic in the sense of Blok and Pigozzi is not necessarily algebraizable if it involves new connective symbols, or it may be algebraizable in an essentially different way than the original logic. However, extension whose axioms and rules define implicitly the new connectives are algebraizable, via the same equivalence formulas and defining equations of the original logic, by enriched algebras of its equivalente quasivariety semantics. For certain (...) strongly algebraizable logics, all connectives defined implicitly by axiomatic extensions of the logic are explicitly definable. (shrink)

An extensions by new axioms and rules of an algebraizable logic in the sense of Blok and Pigozzi is not necessarily algebraizable if it involves new connective symbols, or it may be algebraizable in an essentially different way than the original logic. However, extension whose axioms and rules define implicitly the new connectives are algebraizable, via the same equivalence formulas and defining equations of the original logic, by enriched algebras of its equivalente quasivariety semantics. For certain (...) strongly algebraizable logics, all connectives defined implicitly by axiomatic extensions of the logic are explicitly definable. (shrink)

ABSTRACT The two main directions pursued in the present paper are the following. The first direction was started by Pigozzi in 1969. In [Mak 91] and [Mak 79] Maksimova proved that a normal modal logic has the Craig interpolation property iff the corresponding class of algebras has the superamalgamation property. In this paper we extend Maksimova's theorem to normal multi-modal logics with arbitrarily many, not necessarily unary modalities, and to not necessarily normal multi-modal logics with modalities of ranks (...) smaller than 2. To extend the characterization beyond multi-modal logics, we look at arbitrary algebraizable logics. We will introduce an algebraic property equivalent with the Craig interpolation property in algebraizable logics, and prove that the superamalgamation property implies the Craig interpolation property. The problem of extending the characterization result to non-normal non-unary modal logics also will be discussed. In the second direction pursued herein: for non-normal modal logic with one unary modality Lemmon [Lem 66] gave a possible worlds semantics. Here we give a more general possible worlds semantics for not necessarily normal multi-modal logics with arbitrarily many not necessarily unary modalities. Strongly related to the above is the theorem, proved, e.g., in Jóns son-Tarski [JT 52] and Henkin-Monk-Tarski [HMT 71], that every normal Boolean algebra with operators can be represented as a subalgebra of the complex algebra of some relational structure. We extend this result to not necessarily normal BAO's as follows. We define partial relational structures and show that every not necessarily normal BAO is embeddable into the complex algebra of a partial relational structure. This gives a possible worlds semantics for not necessarily normal multi-modal logics. (shrink)

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 project of constructing a logic of scientific inference on the basis of mathematical probability theory was first undertaken in a systematic way by the mid-nineteenth-century British logicians Augustus De Morgan, George Boole and William Stanley Jevons. This paper sketches the origins and motivation of that effort, the emergence of the inverse probability (IP) model of theory assessment, and the vicissitudes which that model suffered at the hands of its critics. Particular emphasis is given to the influence which competing interpretations (...) of probability had on the project, and to the role of the 'lottery' or 'ballot box' metaphor in the philosophical imagination of the proponents of the IP model. (shrink)

While historians of scientific method have recently called attention to the views of many of John Stuart Mill's contemporaries on the relation between probability and inductive inference, little if any note has been taken of Mill's own vigorous attack on the received "Laplacean" interpretation of probability in the first edition of the System of Logic. This paper examines the place of Mill's critique, both in the overall framework of his philosophy, and in the tradition of assessing the so-called "probability of (...) causes". It also offers an account of why, in later editions of the work, Mill appears to adopt a much more sympathetic stance toward the received view. (shrink)

The Uniformly Reflexive Structure was introduced by E. G. Wagner who showed that the theory of such structures generalized much of recursive function theory. In this paper Uniformly Reflexive Structures are constructed as factor algebras of Free nonassociative algebras. Wagner's question about the existence of a model with no computable splinter ("successor set") is answered in the affirmative by the construction of a model whose only computable sets are the finite sets and their complements. Finally, for each countable Boolean algebra (...) R of subsets of a countable set which contains the finite subsets, a model is constructed with R as its family of computable sets. (shrink)

We develop a general algebraic and proof-theoretic study of substructural logics that may lack associativity, along with other structural rules. Our study extends existing work on substructural logics over the full Lambek Calculus [34], Galatos and Ono [18], Galatos et al. [17]). We present a Gentzen-style sequent system that lacks the structural rules of contraction, weakening, exchange and associativity, and can be considered a non-associative formulation of . Moreover, we introduce an equivalent Hilbert-style system and show that the (...) logic associated with and is algebraizable, with the variety of residuated lattice-ordered groupoids with unit serving as its equivalent algebraic semantics. Overcoming technical complications arising from the lack of associativity, we introduce a generalized version of a logical matrix and apply the method of quasicompletions to obtain an algebra and a quasiembedding from the matrix to the algebra. By applying the general result to specific cases, we obtain important logical and algebraic properties, including the cut elimination of and various extensions, the strong separation of , and the finite generation of the variety of residuated lattice-ordered groupoids with unit. (shrink)

This is the second of a series of papers that investigate fragments of quasi-Nelson logic (QNL) from an algebraic logic standpoint. QNL, recently introduced as a common generalization of intuitionistic and Nelson’s constructive logic with strong negation, is the axiomatic extension of the substructural logic |$FL_{ew}$| (full Lambek calculus with exchange and weakening) by the Nelson axiom. The algebraic counterpart of QNL (quasi-Nelson algebras) is a class of commutative integral residuated lattices (a.k.a. |$FL_{ew}$|-algebras) that includes both Heyting and Nelson algebras (...) and can be characterized algebraically in several alternative ways. The present paper focuses on the algebraic counterpart (a class we dub quasi-Nelson implication algebras, QNI-algebras) of the implication–negation fragment of QNL, corresponding to the connectives that witness the algebraizability of QNL. We recall the main known results on QNI-algebras and establish a number of new ones. Among these, we show that QNI-algebras form a congruence-distributive variety (Cor. 3.15) that enjoys equationally definable principal congruences and the strong congruence extension property (Prop. 3.16); we also characterize the subdirectly irreducible QNI-algebras in terms of the underlying poset structure (Thm. 4.23). Most of these results are obtained thanks to twist representations for QNI-algebras, which generalize the known ones for Nelson and quasi-Nelson algebras; we further introduce a Hilbert-style calculus that is algebraizable and has the variety of QNI-algebras as its equivalent algebraic semantics. (shrink)

According to Frege's principle the denotation of a sentence coincides with its truth-value. The principle is investigated within the context of abstract algebraic logic, and it is shown that taken together with the deduction theorem it characterizes intuitionistic logic in a certain strong sense.A 2nd-order matrix is an algebra together with an algebraic closed set system on its universe. A deductive system is a second-order matrix over the formula algebra of some fixed but arbitrary language. A second-order matrix A is (...) Fregean if, for any subset X of A, the set of all pairs a,b such that X{a} and X{b} have the same closure is a congruence relation on A. Hence a deductive system is Fregean if interderivability is compositional. The logics intermediate between the classical and intuitionistic propositional calculi are the paradigms for Fregean logics. Normal modal logics are non-Fregean while quasi-normal modal logics are generally Fregean.The main results of the paper: Fregean deductive systems that either have the deduction theorem, or are protoalgebraic and have conjunction, are completely characterized. They are essentially the intermediate logics, possibly with additional connectives. All the full matrix models of a protoalgebraic Fregean deductive system are Fregean, and, conversely, the deductive system determined by any class of Fregean second-order matrices is Fregean. The latter result is used to construct an example of a protoalgebraic Fregean deductive system that is not strongly algebraizable. (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.

In order to develop efficient tools for automated reasoning with inconsistency (theorem provers), eventually making Logics of Formal inconsistency (_LFI_) a more appealing formalism for reasoning under uncertainty, it is important to develop the proof theory of the first-order versions of such _LFI_s. Here, we intend to make a first step in this direction. On the other hand, the logic _Ciore_ was developed to provide new logical systems in the study of inconsistent databases from the point of view of (...) _LFI_s. An interesting fact about _Ciore_ is that it has a _strong_ consistency operator, that is, a consistency operator which (forward/backward) propagates inconsistency. Also, it turns out to be an algebraizable logic (in the sense of Blok and Pigozzi) that can be characterized by means of a 3-valued logical matrix. Recently, a first-order version of _Ciore_, namely _QCiore_, was defined preserving the spirit of _Ciore_, that is, without introducing unexpected relationships between the quantifiers. Besides, some important model-theoretic results were obtained for this logic. In this paper we study some proof–theoretic aspects of both _Ciore_ and _QCiore_ respectively. In first place, we introduce a two-sided sequent system for _Ciore_. Later, we prove that this system enjoys the cut-elimination property and apply it to derive some interesting properties. Later, we extend the above-mentioned system to first-order languages and prove completeness and cut-elimination property using the well-known Shütte’s technique. (shrink)

This paper contains a joint study of two sentential logics that combine a many-valued character, namely tetravalence, with a modal character; one of them is normal and the other one quasinormal. The method is to study their algebraic counterparts and their abstract models with the tools of Abstract Algebraic Logic, and particularly with those of Brown and Suszko's theory of abstract logics as recently developed by Font and Jansana in their "A General Algebraic Semantics for Sentential Logics". (...) The logics studied here arise from the algebraic and lattice-theoretical properties we review of Tetravalent Modal Algebras, a class of algebras studied mainly by Loureiro, and also by Figallo, Landini and Ziliani, at the suggestion of the late Antonio Monteiro. (shrink)

A category theoretic generalization of the theory of algebraizable deductive systems of Blok and Pigozzi is developed. The theory of institutions of Goguen and Burstall is used to provide the underlying framework which replaces and generalizes the universal algebraic framework based on the notion of a deductive system. The notion of a term -institution is introduced first. Then the notions of quasi-equivalence, strong quasi-equivalence and deductive equivalence are defined for -institutions. Necessary and sufficient conditions are given for the quasi-equivalence (...) and the deductive equivalence of two term -institutions, based on the relationship between their categories of theories. The results carry over without any complications to institutions, via their associated -institutions. The -institution associated with a deductive system and the institution of equational logic are examined in some detail and serve to illustrate the general theory. (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 general methodology of "algebraizing" logics is used here for combining different logics. The combination of logics is represented as taking the colimit of the constituent logics in the category of algebraizable logics. The cocompleteness of this category as well as its isomorphism to the corresponding category of certain first-order theories are proved.

Protoalgebraic logics are characterized by the monotonicity of the Leibniz operator on their theory lattices and are at the lower end of the Leibniz hierarchy of abstract algebraic logic. They have been shown to be the most primitive among those logics with a strong enough algebraic character to be amenable to algebraic study techniques. Protoalgebraic π-institutions were introduced recently as an analog of protoalgebraic sentential logics with the goal of extending the Leibniz hierarchy from the sentential framework (...) to the π-institution framework. Many properties of protoalgebraic logics, studied in the sentential logic framework by Blok and Pigozzi, Czelakowski, and Font and Jansana, among others, have already been adapted in previous work by the author to the categorical level. This work aims at further advancing that study by exploring in this new level some more properties of protoalgebraic sentential logics. (shrink)

The general methodology of "algebraizing" logics is used here for combining different logics. The combination of logics is represented as taking the colimit of the constituent logics in the category of algebraizable logics. The cocompleteness of this category as well as its isomorphism to the corresponding category of certain first-order theories are proved.

This paper develops an order-theoretic generalization of Blok and Pigozziʼs notion of an algebraizable logic. Unavoidably, the ordered model class of a logic, when it exists, is not unique. For uniqueness, the definition must be relativized, either syntactically or semantically. In sentential systems, for instance, the order algebraization process may be required to respect a given but arbitrary polarity on the signature. With every deductive filter of an algebra of the pertinent type, the polarity associates a reflexive and transitive (...) relation called a Leibniz order, analogous to the Leibniz congruence of abstract algebraic logic . Some core results of AAL are extended here to sentential systems with a polarity. In particular, such a system is order algebraizable if the Leibniz order operator has the following four independent properties: it is injective, it is isotonic, it commutes with the inverse image operator of any algebraic homomorphism, and it produces anti-symmetric orders when applied to filters that define reduced matrix models. Conversely, if a sentential system is order algebraizable in some way, then the order algebraization process naturally induces a polarity for which the Leibniz order operator has properties –. (shrink)

A category theoretic generalization of the theory of algebraizable deductive systems of Blok and Pigozzi is developed. The theory of institutions of Goguen and Burstall is used to provide the underlying framework which replaces and generalizes the universal algebraic framework based on the notion of a deductive system. The notion of a term π-institution is introduced first. Then the notions of quasi-equivalence, strong quasi-equivalence and deductive equivalence are defined for π-institutions. Necessary and sufficient conditions are given for the quasi-equivalence (...) and the deductive equivalence of two term π-institutions, based on the relationship between their categories of theories. The results carry over without any complications to institutions, via their associated π-institutions. The π-institution associated with a deductive system and the institution of equational logic are examined in some detail and serve to illustrate the general theory. (shrink)

In this paper, we introduce and study a corresponding logic to equality-algebras and obtain some basic properties of this logic. We prove the soundness and completeness of this logic based on equality-algebras and local deduction theorem. We show that this logic is regularly algebraizable with respect to the variety of equality∆-algebras but it is not Fregean. Then we introduce the concept of equality∆-algebras and investigate some related properties. Also, we study ∆-deductive systems of equality∆-algebras. In particular, we prove that (...) every prelinear equality ∆-algebra is a subdirect product of linearly ordered equality∆-algebras. Finally, we construct prelinear equality ∆ logic and prove the soundness and strong completeness of this logic respect to prelinear equality∆-algebras. (shrink)

Given a π -institution I , a hierarchy of π -institutions I is constructed, for n ≥ 1. We call I the n-th order counterpart of I . The second-order counterpart of a deductive π -institution is a Gentzen π -institution, i.e. a π -institution associated with a structural Gentzen system in a canonical way. So, by analogy, the second order counterpart I of I is also called the “Gentzenization” of I . In the main result of the paper, it (...) is shown that I is strongly Gentzen , i.e. it is deductively equivalent to its Gentzenization via a special deductive equivalence, if and only if it has the deduction-detachment property. (shrink)

Various connexive FDE-based modal logics are studied. Some of these logics contain a conditional that is both connexive and strict, thereby highlighting that strictness and connexivity of a conditional do not exclude each other. In particular, the connexive modal logics cBK-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{-}$$\end{document}, cKN4, scBK-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{-}$$\end{document}, scKN4, cMBL, and scMBL are introduced semantically by means of classes of Kripke models. The logics (...) cBK-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{-}$$\end{document} and cKN4 are connexive variants of the FDE-based modal logics BK-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{-}$$\end{document} and KN4 with a weak and a strong implication, respectively. The system cMBL is a connexive variant of the modal bilattice logic MBL. The latter is a modal extension of Arieli and Avron’s logic of logical bilattices and is characterized by a class of Kripke models with a four-valued accessibility relation. In the systems scBK-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{-}$$\end{document}, scKN4, and scMBL, the conditional is both connexive and strict. Sound and complete tableau calculi for all these logics are presented and used to show that the entailment relations of the systems under consideration are decidable for finite premise set. Moreover, the logics cBK-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {cBK}^-$$\end{document} and cMBL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {cMBL}$$\end{document} are shown to be algebraizable. The algebraizability of cMBL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {cMBL}$$\end{document} is derived from proving cMBL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {cMBL}$$\end{document} to be definitionally equivalent to MBL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {MBL}$$\end{document}. All connexive modal logics studied in this paper are decidable, paraconsistent, and inconsistent but non-trivial logics. (shrink)

There is a well-established correspondence between interpolation and amalgamation for algebraizable logics that satisfy certain additional assumptions. In this paper, we introduce the Robinson property of a logic and show that a conditionally algebraizable logic without any additional assumptions has the Robinson property if and only if the corresponding class of Lindenbaum–Tarski algebras has the amalgamation property. Moreover, we give the logical characterization of the strong amalgamation property, solving an open problem of Andréka–Németi–Sain. It is also shown (...) that given the mentioned extra assumptions the Robinson property implies the interpolation property. As conditionally algebraizable logics cover algebraizable logics as well as various quantifier logics such as classical first order logic, our results yield a generalization of some of the results concerning interpolation and amalgamation. (shrink)

The notion of an algebraizable logic in the sense of Blok and Pigozzi [3] is generalized to that of a possibly infinitely algebraizable, for short, p.i.-algebraizable logic by admitting infinite sets of equivalence formulas and defining equations. An example of the new class is given. Many ideas of this paper have been present in [3] and [4]. By a consequent matrix semantics approach the theory of algebraizable and p.i.-algebraizable logics is developed in a different (...) way. It is related to the theory of equivalential logics in the sense of Prucnal and Wroski [18], and it is extended to nonfinitary logics. The main result states that a logic is algebraizable (p.i.-algebraizable) iff it is finitely equivalential (equivalential) and the truth predicate in the reduced matrix models is equationally definable. (shrink)

In [14] we used the term finitely algebraizable for algebraizable logics in the sense of Blok and Pigozzi [2] and we introduced possibly infinitely algebraizable, for short, p.i.-algebraizable logics. In the present paper, we characterize the hierarchy of protoalgebraic, equivalential, finitely equivalential, p.i.-algebraizable, and finitely algebraizable logics by properties of the Leibniz operator. A Beth-style definability result yields that finitely equivalential and finitely algebraizable as well as equivalential and p.i.-algebraizable (...) logics can be distinguished by injectivity of the Leibniz operator. Thus, from a characterization of equivalential logics we obtain a new short proof of the main result of [2] that a finitary logic is finitely algebraizable iff the Leibniz operator is injective and preserves unions of directed systems. It is generalized to nonfinitary logics. We characterize equivalential and, by adding injectivity, p.i.-algebraizable logics. (shrink)

This paper, which in its subject matter goes back to works on strongly nite logics , is concerned with the following problems: Let Cn1; Cn2 be two strongly nite logics over the same propositional language. Is the supremum of Cn1 and Cn2 also a strongly nite operation? Is any nite matrix axiomatizable by a nite set of standard rules? The rst question can be found in [9] . The second conjec- ture was formulated by Wolfgang (...) Rautenberg, but investigations into this problem had been carried out earlier in works of many logicians . Moreover, Stephen Bloom [1] posed a conjecture stronger than that: the consequence de- termined by a nite matrix is nitely based, i.e. it is the consequence generated by a nite set of standard rules. This hypothesis was, however, disproved by Andrzej Wronski [10] . In the present paper it is shown that neither nor holds true. The negative answer to can be viewed as a generalization of the result given by Andrzej Wronski [10]. (shrink)

Part I of this paper is developed in the tradition of Stone-type dualities, where we present a new topological representation for general lattices (influenced by and abstracting over both Goldblatt's [17] and Urquhart's [46]), identifying them as the lattices of stable compact-opens of their dual Stone spaces (stability refering to a closure operator on subsets). The representation is functorial and is extended to a full duality.In part II, we consider lattice-ordered algebras (lattices with additional operators), extending the Jónsson and Tarski (...) representation results [30] for Boolean algebras with Operators. Our work can be seen as developing, and indeed completing, Dunn's project of gaggle theory [13, 14]. We consider general lattices (rather than Boolean algebras), with a broad class of operators, which we dubb normal, and which includes the Jónsson-Tarski additive operators. Representation of l-algebras is extended to full duality. (shrink)

In a classical paper [15] V. Glivenko showed that a proposition is classically demonstrable if and only if its double negation is intuitionistically demonstrable. This result has an algebraic formulation: the double negation is a homomorphism from each Heyting algebra onto the Boolean algebra of its regular elements. Versions of both the logical and algebraic formulations of Glivenko’s theorem, adapted to other systems of logics and to algebras not necessarily related to logic can be found in the literature (see (...) [2, 9, 8, 14] and [13, 7, 14]). The aim of this paper is to offer a general frame for studying both logical and algebraic generalizations of Glivenko’s theorem. We give abstract formulations for quasivarieties of algebras and for equivalential and algebraizable deductive systems and both formulations are compared when the quasivariety and the deductive system are related. We also analyse Glivenko’s theorem for compatible expansions of both cases. (shrink)

Even though it is not very often admitted, partial functions do play a significant role in many practical applications of deduction systems. Kleene has already given a semantic account of partial functions using three-valued logic decades ago, but there has not been a satisfactory mechanization. Recent years have seen a thorough investigation of the framework of many-valued truth-functional logics. However, strong Kleene logic, where quantification is restricted and therefore not truthfunctional, does not fit the framework directly. We solve this (...) problem by applying recent methods from sorted logics. This paper presents a resolution calculus that combines the proper treatment of partial functions with the efficiency of sorted calculi. (shrink)

Multialgebras (or hyperalgebras) have been very much studied in the literature. In the realm of Logic, they were considered by Avron and his collaborators under the name of non-deterministic matrices (or Nmatrices) as a useful semantics tool for characterizing some logics (in particular, several logics of formal inconsistency or LFIs) which cannot be characterized by a single finite matrix. In particular, these LFIs are not algebraizable by any method, including Blok and Pigozzi general theory. Carnielli and Coniglio (...) introduced a semantics of swap structures for LFIs, which are Nmatrices defined over triples in a Boolean algebra, generalizing Avron's semantics. In this paper we develop the first steps towards the possibility of defining an algebraic theory of swap structures for LFIs, by adapting concepts of universal algebra to multialgebras in a suitable way. (shrink)

Logics that do not have a deduction-detachment theorem (briefly, a DDT) may still possess a contextual DDT —a syntactic notion introduced here for arbitrary deductive systems, along with a local variant. Substructural logics without sentential constants are natural witnesses to these phenomena. In the presence of a contextual DDT, we can still upgrade many weak completeness results to strong ones, e.g., the finite model property implies the strong finite model property. It turns out that a finitary system has (...) a contextual DDT iff it is protoalgebraic and gives rise to a dually Brouwerian semilattice of compact deductive filters in every finitely generated algebra of the corresponding type. Any such system is filter distributive, although it may lack the filter extension property. More generally, filter distributivity and modularity are characterized for all finitary systems with a local contextual DDT, and several examples are discussed. For algebraizable logics, the well-known correspondence between the DDT and the equational definability of principal congruences is adapted to the contextual case. (shrink)

The aim of the paper is to analyze the expressive power of the square operator of Łukasiewicz logic: ∗x=x⊙x⁠, where ⊙ is the strong Łukasiewicz conjunction. In particular, we aim at understanding and characterizing those cases in which the square operator is enough to construct a finite MV-chain from a finite totally ordered set endowed with an involutive negation. The first of our main results shows that, indeed, the whole structure of MV-chain can be reconstructed from the involution and the (...) Łukasiewicz square operator if and only if the obtained structure has only trivial subalgebras and, equivalently, if and only if the cardinality of the starting chain is of the form n+1 where n belongs to a class of prime numbers that we fully characterize. Secondly, we axiomatize the algebraizable matrix logic whose semantics is given by the variety generated by a finite totally ordered set endowed with an involutive negation and Łukasiewicz square operator. Finally, we propose an alternative way to account for Łukasiewicz square operator on involutive Gödel chains. In this setting, we show that such an operator can be captured by a rather intuitive set of equations. (shrink)

Hybrid logics are extensions of standard modal logics, which significantly increase the expressive power of the latter. Since most of hybrid logics are known to be decidable, decision procedures for them is a widely investigated field of research. So far, several tableau calculi for hybrid logics have been presented in the literature. In this paper we introduce a sound, complete and terminating tableau calculus T H(@,E,D, ♦ −) for hybrid logics with the satisfaction operators, the (...) universal modality, the difference modality and the inverse modality as well as the corresponding sequent calculus S H(@,E,D, ♦ −) . They not only uniformly cover relatively wide range of various hybrid logics but they are also conceptually simple and enable effective search for a minimal model for a satisfiable formula. The main novelty is the exploitation of the unrestricted blocking mechanism introduced as an explicit, sound tableau rule. (shrink)

Annotated logics were introduced by V.S. Subrahmanian as logical foundations for computer programming. One of the difficulties of these systems from the logical point of view is that they are not structural, i.e., their consequence relations are not closed under substitutions. In this paper we give systems of annotated logics that are equivalent to those of Subrahmanian in the sense that everything provable in one type of system has a translation that is provable in the other. Moreover these (...) new systems are structural. We prove that these systems are weakly congruential, namely, they have an infinite system of congruence 1-formulas. Moreover, we prove that an annotated logic is algebraizable (i.e., it has a finite system of congruence formulas,) if and only if the lattice of annotation constants is finite. (shrink)

A modified filtrations argument is used to prove that the relevant logic S of [2] is decidable.

In this paper we define and study a generalized notion of a logical system that covers on an equal formal basis sentential, equational and sequential systems. We develop a general theory of equivalence between generalized logics that provides, first, a conception of algebraizable logic , second, a formal concept of equivalence between sequential systems and, third, a notion of equivalence between sentential and sequential systems. We also use our theory of equivalence for developing a general algebraic approach to (...) conjunctive non-pseudo-axiomatic self-extensional sentential logics. Finally, we consider within the framework of the mentioned approach various sequential formulations for some well-known sentential logics. (shrink)

In this paper we consider the implicational fragment of Abelian logic \ . We show that although the Abelian groups provide an semantics for the set of theorems of \ they do not for the associated consequence relation. We then show that the consequence relation is not algebraizable in the sense of Blok and Pigozzi . In the second part of the paper, we investigate an extension of \ in the same language and having the same set of theorems (...) and show that this new consequence relation is algebraizable with the Abelian groups as its equivalent algebraic semantics. Finally, we show that although \ is not algebraizable, it is order-algebraizable in the sense of Raftery. (shrink)