With each sentential logic C, identified with a structural consequence operation in a sentential language, the class Matr * (C) of factorial matrices which validate C is associated. The paper, which is a continuation of [2], concerns the connection between the purely syntactic property imposed on C, referred to as Maehara Interpolation Property (MIP), and three diagrammatic properties of the class Matr* (C): the Amalgamation Property (AP), the (deductive) Filter Extension Property (FEP) and Injections Transferable (IT). The main theorem of (...) the paper (Theorem 2.2) is analogous to the Wroski's result for equational classes of algebras [13]. It reads that for a large class of logics the conjunction of (AP) and (FEP) is equivalent to (IT) and that the latter property is equivalent to (MIP). (shrink)

The main result of the present paper — Theorem 3 — establishes the equivalence of the interpolation and amalgamation properties for a large family of logics and their associated classes of matrices.

We examine the notion of primitive satisfaction in logical matrices. Theorem II. 1, being the matrix counterpart of Baker's well-known result for congruently distributive varieties of algebras (cf [1], Thm. 1.5), links the notions of primitive and standard satisfaction. As a corollary we give the matrix version of Jónsson's Lemma, proved earlier in [4]. Then we investigate propositional logics with disjunction. The main result, Theorem III. 2, states a necessary and sufficient condition for such logics to be finitely based.

Two connectives are of special interest in metalogical investigations — the connective of implication which is important due to its connections to the notion of inference, and the connective of equivalence. The latter connective expresses, in the material sense, the fact that two sentences have the same logical value while in the strict sense it expresses the fact that two sentences are interderivable on the basis of a given logic. The process of identification of equivalent sentences relative to theories of (...) a logic C defines a class of abstract algebras. The members of the class are called Lindenbaum-Tarski algebras of the logic C. One may abstract from the origin of these algebras and examine them by means of purely algebraic methods. (shrink)

The purpose of this paper is to present in a uniform way the commutator theory for k-deductive system of arbitrary positive dimension k. We are interested in the logical perspective of the research — an emphasis is put on an analysis of the interconnections holding between the commutator and logic. This research thus qualifies as belonging to abstract algebraic logic, an area of universal algebra that explores to a large extent the methods provided by the general theory of deductive systems. (...) In the paper the new term ‘commutator formula’ is introduced. The paper is concerned with the meanings of the above term in the models provided by the commutator theory and clarifies contexts in which these meanings occur. The work is presented in an abstracted form: main ideas are outlined but proofs are deferred to the second part of the paper. (shrink)

In the first section logics with an algebraic semantics are investigated. Section 2 is devoted to subdirect products of matrices. There, among others we give the matrix counterpart of a theorem of Jónsson from universal algebra. Some positive results concerning logics with, finite degrees of maximality are presented in Section 3.

The present paper is thought as a formal study of distributive closure systems which arise in the domain of sentential logics. Special stress is laid on the notion of a C-filter, playing the role analogous to that of a congruence in universal algebra. A sentential logic C is called filter distributive if the lattice of C-filters in every algebra similar to the language of C is distributive. Theorem IV.2 in Section IV gives a method of axiomatization of those filter distributive (...) logics for which the class Matr (C) prime of C-prime matrices (models) is axiomatizable. In Section V, the attention is focused on axiomatic strengthenings of filter distributive logics. The theorems placed there may be regarded, to some extent, as the matrix counterparts of Baker's well-known theorem from universal algebra [9, § 62, Theorem 2]. (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)

The first known statements of the deduction theorems for the first-order predicate calculus and the classical sentential logic are due to Herbrand [8] and Tarski [14], respectively. The present paper contains an analysis of closure spaces associated with those sentential logics which admit various deduction theorems. For purely algebraic reasons it is convenient to view deduction theorems in a more general form: given a sentential logic C (identified with a structural consequence operation) in a sentential language I, a quite arbitrary (...) set P of formulas of I built up with at most two distinct sentential variables p and q is called a uniform deduction theorem scheme for C if it satisfies the following condition: for every set X of formulas of I and for any formulas and , C(X{{a}}) iff P(, ) AC(X). [P(, ) denotes the set of formulas which result by the simultaneous substitution of for p and for q in all formulas in P]. The above definition encompasses many particular formulations of theorems considered in the literature to be deduction theorems. Theorem 1.3 gives necessary and sufficient conditions for a logic to have a uniform deduction theorem scheme. Then, given a sentential logic C with a uniform deduction theorem scheme, the lattices of deductive filters on the algebras A similar to the language of C are investigated. It is shown that the join-semilattice of finitely generated (= compact) deductive filters on each algebra A is dually Brouwerian. (shrink)

This paper studies, with techniques ofAlgebraic Logic, the effects of putting a bound on the cardinality of the set of side formulas in the Deduction Theorem, viewed as a Gentzen-style rule, and of adding additional assumptions inside the formulas present in Modus Ponens, viewed as a Hilbert-style rule. As a result, a denumerable collection of new Gentzen systems and two new sentential logics have been isolated. These logics are weaker than the positive implicative logic. We have determined their algebraic models (...) and the relationships between them, and have classified them according to several standard criteria of Abstract Algebraic Logic. One of the logics is protoalgebraic but neither equivalential nor weakly algebraizable, a rare situation where very few natural examples were hitherto known. In passing we have found new, alternative presentations of positive implicative logic, both in Hilbert style and in Gentzen style, and have characterized it in terms of the restricted Deduction Theorem: it is the weakest logic satisfying Modus Ponens and the Deduction Theorem restricted to at most 2 side formulas. The algebraic part of the work has lead to the class of quasi-Hilbert algebras, a quasi-variety of implicative algebras introduced by Pla and Verdú in 1980, which is larger than the variety of Hilbert algebras. Its algebraic properties reflect those of the corresponding logics and Gentzen systems. (shrink)

This paper studies, with techniques ofAlgebraic Logic, the effects of putting a bound on the cardinality of the set of side formulas in the Deduction Theorem, viewed as a Gentzen-style rule, and of adding additional assumptions inside the formulas present in Modus Ponens, viewed as a Hilbert-style rule. As a result, a denumerable collection of new Gentzen systems and two new sentential logics have been isolated. These logics are weaker than the positive implicative logic. We have determined their algebraic models (...) and the relationships between them, and have classified them according to several standard criteria of Abstract Algebraic Logic. One of the logics is protoalgebraic but neither equivalential nor weakly algebraizable, a rare situation where very few natural examples were hitherto known. In passing we have found new, alternative presentations of positive implicative logic, both in Hilbert style and in Gentzen style, and have characterized it in terms of the restricted Deduction Theorem: it is the weakest logic satisfying Modus Ponens and the Deduction Theorem restricted to at most 2 side formulas. The algebraic part of the work has lead to the class of quasi-Hilbert algebras, a quasi-variety of implicative algebras introduced by Pla and Verdú in 1980, which is larger than the variety of Hilbert algebras. Its algebraic properties reflect those of the corresponding logics and Gentzen systems. (shrink)

The present paper is a study in abstract algebraic logic. We investigate the correspondence between the metalogical Beth property and the algebraic property of surjectivity of epimorphisms. It will be shown that this correspondence holds for the large class of equivalential logics. We apply our characterization theorem to relevance logics and many-valued logics.

There exist important deductive systems, such as the non-normal modal logics, that are not proper subjects of classical algebraic logic in the sense that their metatheory cannot be reduced to the equational metatheory of any particular class of algebras. Nevertheless, most of these systems are amenable to the methods of universal algebra when applied to the matrix models of the system. In the present paper we consider a wide class of deductive systems of this kind called protoalgebraic logics. These include (...) almost all (non-pathological) systems of prepositional logic that have occurred in the literature. The relationship between the metatheory of a protoalgebraic logic and its matrix models is studied. The following results are obtained for any finite matrix model U of a filter-distributive protoalgebraic logic : (I) The extension U of is finitely axiomatized (provided has only finitely many inference rules); (II) U has only finitely many extensions. (shrink)

In Lewin et al. 359–386) the authors proved that certain systems of annotated logics are algebraizable in the sense of Block and Rigozzi 396). Later in Lewin et al. the study of the associated quasi-varieties of annotated algebras is initiated. In this paper we continue the study of the these classes of algebras, in particular, we report some recent results about the free annotated algebras.

We show that intuitionistic logic is deductively equivalent to Connexive Heyting Logic ($$\textrm{CHL}$$ CHL ), hereby introduced as an example of a strongly connexive logic with an intuitive semantics. We use the reverse algebraisation paradigm: $$\textrm{CHL}$$ CHL is presented as the assertional logic of a point regular variety (whose structure theory is examined in detail) that turns out to be term equivalent to the variety of Heyting algebras. We provide Hilbert-style and Gentzen-style proof systems for $$\textrm{CHL}$$ CHL ; moreover, we (...) suggest a possible computational interpretation of its connexive conditional, and we revisit Kapsner’s idea of superconnexivity. (shrink)

Czelakowski introduced the Suszko operator as a basis for the development of a hierarchy of non-protoalgebraic logics, paralleling the well-known abstract algebraic hierarchy of protoalgebraic logics based on the Leibniz operator of Blok and Pigozzi. The scope of the theory of the Leibniz operator was recently extended to cover the case of, the so-called, protoalgebraic π-institutions. In the present work, following the lead of Czelakowski, an attempt is made at lifting parts of the theory of the Suszko operator to the (...) π-institution framework. (shrink)

Finitely algebraizable deductive systems were introduced by Blok and Pigozzi to capture the essential properties of those deductive systems that are very tightly connected to quasivarieties of universal algebras. They include the equivalential logics of Czelakowski. Based on Blok and Pigozzi’s work, Herrmann defined algebraizable deductive systems. These are the equivalential deductive systems that are also truth-equational, in the sense that the truth predicate of the class of their reduced matrix models is explicitly definable by some set of unary equations. (...) Raftery undertook the task of characterizing the property of truth-equationality for arbitrary deductive systems. In this paper, following Raftery, we extend the notion of truth-equationality for logics formalized as $\pi$-institutions and abstract several of the results that hold for deductive systems in this more general categorical context. (shrink)

Two classes of π are studied whose properties are similar to those of the protoalgebraic deductive systems of Blok and Pigozzi. The first is the class of N-protoalgebraic π-institutions and the second is the wider class of N-prealgebraic π-institutions. Several characterizations are provided. For instance, N-prealgebraic π-institutions are exactly those π-institutions that satisfy monotonicity of the N-Leibniz operator on theory systems and N-protoalgebraic π-institutions those that satisfy monotonicity of the N-Leibniz operator on theory families. Analogs of the correspondence property of (...) Blok and Pigozzi for π-institutions are also introduced and their connections with preand protoalgebraicity are explored. Finally, relations of these two classes with the (, N)-algebraic systems, introduced previously by the author as an analog of the -algebras of Font and Jansana, and with an analog of the Suszko operator of Czelakowski for π-institutions are also investigated. (shrink)

Recently, Caleiro, Gon¸calves and Martins introduced the notion of behaviorally algebraizable logic. The main idea behind their work is to replace, in the traditional theory of algebraizability of Blok and Pigozzi, unsorted equational logic with multi-sorted behavioral logic. The new notion accommodates logics over many-sorted languages and with non-truth-functional connectives. Moreover, it treats logics that are not algebraizable in the traditional sense while, at the same time, shedding new light to the equivalent algebraic semantics of logics that are algebraizable according (...) to the original theory. In this paper, the notion of an abstract multi-sorted π-institution is introduced so as to transfer elements of the theory of behavioral algebraizability to the categorical setting. Institutions formalize a wider variety of logics than deductive systems, including logics involving multiple signatures and quantifiers. The framework developed has the same relation to behavioral algebraizability as the classical categorical abstract algebraic logic framework has to the original theory of algebraizability of Blok and Pigozzi. (shrink)

An algebraization of multi-signature first-order logic without terms is presented. Rather than following the traditional method of choosing a type of algebras and constructing an appropriate variety, as is done in the case of cylindric and polyadic algebras, a new categorical algebraization method is used: The substitutions of formulas of one signature for relation symbols in another are treated in the object language. This enables the automatic generation via an adjunction of an algebraic theory. The algebras of this theory are (...) then used to algebraize first-order logic. (shrink)

Equivalent deductive systems were introduced in [4] with the goal of treating 1‐deductive systems and algebraic 2‐deductive systems in a uniform way. Results of [3], appropriately translated and strengthened, show that two deductive systems over the same language type are equivalent if and only if their lattices of theories are isomorphic via an isomorphism that commutes with substitutions. Deductive equivalence of π‐institutions [14, 15] generalizes the notion of equivalence of deductive systems. In [15, Theorem 10.26] this criterion for the equivalence (...) of deductive systems was generalized to a criterion for the deductive equivalence of term π‐institutions, forming a subclass of all π‐institutions that contains those π‐institutions directly corresponding to deductive systems. This criterion is generalized here to cover the case of arbitrary π‐institutions. (shrink)

Equivalent deductive systems were introduced in [4] with the goal of treating 1-deductive systems and algebraic 2-deductive systems in a uniform way. Results of [3], appropriately translated and strengthened, show that two deductive systems over the same language type are equivalent if and only if their lattices of theories are isomorphic via an isomorphism that commutes with substitutions. Deductive equivalence of π-institutions [14, 15] generalizes the notion of equivalence of deductive systems. In [15, Theorem 10.26] this criterion for the equivalence (...) of deductive systems was generalized to a criterion for the deductive equivalence of term π-institutions, forming a subclass of all π-institutions that contains those π-institutions directly corresponding to deductive systems. This criterion is generalized here to cover the case of arbitrary π-institutions. (shrink)

This paper develops a new framework for combining propositional logics, called "juxtaposition". Several general metalogical theorems are proved concerning the combination of logics by juxtaposition. In particular, it is shown that under reasonable conditions, juxtaposition preserves strong soundness. Under reasonable conditions, the juxtaposition of two consequence relations is a conservative extension of each of them. A general strong completeness result is proved. The paper then examines the philosophically important case of the combination of classical and intuitionist logics. Particular attention is (...) paid to the phenomenon of collapse. It is shown that there are logics with two stocks of classical or intuitionist connectives that do not collapse. Finally, the paper briefy investigates the question of which rules, when added to these logics, lead to collapse. (shrink)

This paper provides a semantic analysis of admissible rules and associated completeness conditions for arbitrary deductive systems, using the framework of abstract algebraic logic. Algebraizability is not assumed, so the meaning and significance of the principal notions vary with the level of the Leibniz hierarchy at which they are presented. As a case study of the resulting theory, the nonalgebraizable fragments of relevance logic are considered.

The aim of this paper is to study the paraconsistent deductive systemP 1 within the context of Algebraic Logic. It is well known due to Lewin, Mikenberg and Schwarse thatP 1 is algebraizable in the sense of Blok and Pigozzi, the quasivariety generated by Sette's three-element algebraS being the unique quasivariety semantics forP 1. In the present paper we prove that the mentioned quasivariety is not a variety by showing that the variety generated byS is not equivalent to any algebraizable (...) deductive system. We also show thatP 1 has no algebraic semantics in the sense of Czelakowski. Among other results, we study the variety generated by the algebraS. This enables us to prove in a purely algebraic way that the only proper non-trivial axiomatic extension ofP 1 is the classical deductive systemPC. Throughout the paper we also study those abstract logics which are in a way similar toP 1, and are called hereabstract Sette logics. We obtain for them results similar to those obtained for distributive abstract logics by Font, Verdú and the author. (shrink)

This paper discusses the general problem of translation functions between logics, given in axiomatic form, and in particular, the problem of determining when two such logics are "synonymous" or "translationally equivalent." We discuss a proposed formal definition of translational equivalence, show why it is reasonable, and also discuss its relation to earlier definitions in the literature. We also give a simple criterion for showing that two modal logics are not translationally equivalent, and apply this to well-known examples. Some philosophical morals (...) are drawn concerning the possibility of having two logical systems that are "empirically distinct" but are both translationally equivalent to a common logic. (shrink)

We show that a finitely generated protoalgebraic strict universal Horn class that is filter-distributive is finitely based. Equivalently, every protoalgebraic and filter-distributive multidimensional deductive system determined by a finite set of finite matrices can be presented by finitely many axioms and rules.

We show that a finitely generated protoalgebraic strict universal Horn class that is filter-distributive is finitely based. Equivalently, every protoalgebraic and filter-distributive multidimensional deductive system determined by a finite set of finite matrices can be presented by finitely many axioms and rules.

We provide a new proof of the following Pałasińska's theorem: Every finitely generated protoalgebraic relation distributive equality free quasivariety is finitely axiomatizable. The main tool we use are ${\mathcal{Q}}$ Q -relation formulas for a protoalgebraic equality free quasivariety ${\mathcal{Q}}$ Q . They are the counterparts of the congruence formulas used for describing the generation of congruences in algebras. Having this tool in hand, we prove a finite axiomatization theorem for ${\mathcal{Q}}$ Q when it has definable principal ${\mathcal{Q}}$ Q -subrelations. This (...) is a property obtained by carrying over the definability of principal subcongruences, invented by Baker and Wang for varieties, and which holds for finitely generated protoalgebraic relation distributive equality free quasivarieties. (shrink)

This paper is a study of similarities and differences between strong and weak quantum consequence operations determined by a given class of ortholattices. We prove that the only strong orthologics which admits the deduction theorem (the only strong orthologics with algebraic semantics, the only equivalential strong orthologics, respectively) is the classical logic.

In this paper we investigate the relation between the axiomatization of a given logical consequence operation and axiom systems defining the class of algebras related to that consequence operation. We show examples which prove that, in general there are no natural relation between both ways of axiomatization.

In Lewin et al. 359–386) the authors proved that certain systems of annotated logics are algebraizable in the sense of Block and Rigozzi 396). Later in Lewin et al. the study of the associated quasi-varieties of annotated algebras is initiated. In this paper we continue the study of the these classes of algebras, in particular, we report some recent results about the free annotated algebras.

Varieties like groups, rings, or Boolean algebras have the property that, in any of their members, the lattice of congruences is isomorphic to a lattice of more manageable objects, for example normal subgroups of groups, two-sided ideals of rings, filters (or ideals) of Boolean algebras.algebraic logic can explain these phenomena at a rather satisfactory level of generality: in every member A of a τ-regular variety ������ the lattice of congruences of A is isomorphic to the lattice of deductive filters on (...) A of the τ-assertional logic of ������. Moreover, if ������ has a constant 1 in its type and is 1-subtractive, the deductive filters on A ∈ ������ of the 1-assertional logic of ������ coincide with the ������-ideals of A in the sense of Gumm and Ursini, for which we have a manageable concept of ideal generation. However, there are isomorphism theorems, for example, in the theories of residuated lattices, pseudointerior algebras and quasi-MV algebras that cannot be subsumed by these general results. The aim of the present paper is to appropriately generalise the concepts of subtractivity and τ-regularity in such a way as to shed some light on the deep reason behind such theorems. The tools and concepts we develop hereby provide a common umbrella for the algebraic investigation of several families of logics, including substructural logics, modal logics, quantum logics, and logics of constructive mathematics. (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.

A Leibniz class is a class of logics closed under the formation of term-equivalent logics, compatible expansions, and non-indexed products of sets of logics. We study the complete lattice of all Leibniz classes, called the Leibniz hierarchy. In particular, it is proved that the classes of truth-equational and assertional logics are meet-prime in the Leibniz hierarchy, while the classes of protoalgebraic and equivalential logics are meet-reducible. However, the last two classes are shown to be determined by Leibniz conditions consisting of (...) meet-prime logics only. (shrink)

The paper is a brief survey of the most important semantic constructions founded on the concept of possible world. It is impossible to capture in one short paper the whole variety of the problems connected with manifold applications of possible worlds. Hence, after a brief explanation of some philosophical matters I take a look at possible worlds from rather technical standpoint of logic and focus on the applications in formal semantics. In particular, I would like to focus on the fruitful (...) marriage of possible world semantics and algebra and its evolution leading to very general construction of Wójcicki called referential semantics and some of its refinements. The presentation is informal and sketchy; the main purpose is to put in one place a short, and readable I hope, description of the most important constructions and to point out the main sources of these solutions. (shrink)

A purported axiomatization, by P. Gärdenfors, of intuitionistic propositional logic is shown to be incomplete, and that the mistaken claim to completeness is seen to result from carelessness in the choice of primitive logical vocabulary. This leads to a consideration of various ways of conceiving the distinction between primitive and defined vocabularies, along with the bearing of these differences on such matters as are discussed in connection with Gärdenfors.

The totality of extensional 1-ary connectives distinguishable in a logical framework allowing sequents with multiple or empty (alongside singleton) succedents form a lattice under a natural partial ordering relating one connective to another if all the inferential properties of the former are possessed by the latter. Here we give a complete description of that lattice; its Hasse diagram appears as Figure 1 in §2. Simple syntactic descriptions of the lattice elements are provided in §3; §§4 and 5 give some additional (...) remarks on matrix methods and on alternative terminology. Background: The size of this lattice was underestimated in [3]; some missing cases were noted in [4] in the course of correcting an example from [3] purporting to show the non-distributivity of the lattice. All the 'missing cases' (as well as those originally noted) are covered here. (The present discussion is self-contained.). (shrink)

We study a range of issues connected with the idea of replacing one formula by another in a fixed context. The replacement core of a consequence relation ⊢ is the relation holding between a set of formulas {A1,..., Am,...} and a formula B when for every context C, we have C,..., C,... ⊢ C. Section 1 looks at some differences between which inferences are lost on passing to the replacement cores of the classical and intuitionistic consequence relations. For example, we (...) find that while the inference from A and B to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$A \land B$\end{document}, sanctioned by both these initial consequence relations, is retained on passage to the replacement core in the classical case, it is lost in the intuitionistic case. Further discussion of these two logics occupies Sections 3 and 4. Section 2 looks at the m = 1 case, describing A as replaceable by B according to ⊢ when B is a consequence of A by the replacement core of ⊢, and inquiring as to which choices of ⊢ render this induced replaceability relation symmetric. Section 5 investigates further conceptual refinements— such as a contrast between horizontal and vertical replaceability—suggested by some work of R. B. Angell and R. Harrop in the 1950s and 1960s. Appendix 1 examines a related aspect of term-for-term replacement in connection with identity in predicate logic. Appendix 2 is a repository for proofs which would otherwise clutter up Section 3. (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)

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)

In her well-known book, Rasiowa states without proof that in implicative algebras there is a one-to-one correspondence between kernels of epimorphisms and the so-called special implicative filters, and that in the logic whose algebraic counterpart is the class of implicative algebras the deductive filters coincide with the special implicative filters. We show that neither claim is true, and how to repair the situation by redefining some of the notions involved. We answer other questions concerning special implicative filters, taking the theory (...) of algebraizable logics of Blok and Pigozzi as a framework to approach the question in a systematic way. (shrink)

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)

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.

We continue in this article the abstract algebraic treatment of quantum sentential logics Wil. The Notions borrowed from the field of Model Theory and Abstract Algebraic Logic - AAL (i.e., consequence relation, variety, logical matrix, deductive filter, reduced product, ultraproduct, ultrapower, Frege relation, Leibniz congruence, Suszko congruence, Leibniz operator) are applied to quantum logics. We also proved several equivalences between state property systems (Jauch-Piron-Aerts line of investigations) and AAL treatment of quantum logics (corollary 18 and 19). We show that there (...) exist the uniquely defined correspondence between state property system and consequence relation defined on quantum logics. We also signalize that a metalogical property - Lindenbaum property does not hold for the set of quantum logics. (shrink)