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.

We define and study abstract valuation semantics for logics, an algebraically well-behaved version of valuation semantics. Then, in the context of the behavioral approach to the algebraization of logics, we show, by means of meaningful bridge theorems and application examples, that abstract valuations are suited to play a role similar to the one played by logical matrices in the traditional approach to algebraization.

Tarski apresentou sua definição de operador de consequência com a intenção de expor as concepções fundamentais da consequência lógica. Um espaço de Tarski é um par ordenado determinado por um conjunto não vazio e um operador de consequência sobre este conjunto. Esta estrutura matemática caracteriza um espaço quase topológico. Este artigo mostra uma visão algébrica dos espaços de Tarski e introduz uma lógica proposicional modal que interpreta o seu operador modal nos conjuntos fechados de algum espaço de Tarski. DOI:10.5007/1808-1711.2010v14n1p47.

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)

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)

Let $\mathcal{P}_w$ be the lattice of Muchnik degrees of nonempty $\Pi^0_1$ subsets of $2^\omega$. The lattice $\mathcal{P}$ has been studied extensively in previous publications. In this note we prove that the lattice $\mathcal{P}$ is not Brouwerian.

This paper is an historical study of Tarski's methodology of deductive sciences (in which a logic S is identified with an operator Cn S, called the consequence operator, on a given set of expressions), from its appearance in 1930 to the end of the 1970s, focusing on the work done in the field by Roberto Magari, Piero Mangani and by some of their pupils between 1965 and 1974, and comparing it with the results achieved by Tarski and the Polish school (...) (Łoś, Suszko, Słupecki, Pogorzelski, Wójcicki). In the last section of the paper we will then compare these works with some recent developments in algebraic logic: this will lead to a better understanding of the results of the methodology of deductive science, but at the same time will show some intrinsic limits to such an approach to logic. Even if Magari's work on diagonizable algebras and universal algebra and Mangani's axiomatization of MV-algebras and results in model theory are rather famous, the articles on closure operators, published in the 1960s, are almost totally unknown outside Italy (mainly because of a linguistic limitation, the papers we analyse having been written and published in Italian). This paper aims to fill the gap in the literature and to enable the international community to get acquainted with this part of Italian logic. The same applies to some works published in Barcelona (in Catalan) at the end of the 1970s, analysed in the last section. (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.

It is shown that axiomatic extensions of intuitionistic propositional calculus defining univocally new connectives, including those proposed by Gabbay, are strongly complete with respect to valuations in Heyting algebras with additional operations. In all cases, the double negation of such a connective is equivalent to a formula of intuitionistic calculus. Thus, under the excluded third law it collapses to a classical formula, showing that this condition in Gabbay's definition is redundant. Moreover, such connectives can not be interpreted in all Heyting (...) algebras, unless they are already equivalent to a formula of intuitionistic calculus. These facts relativize to connectives over intermediate logics. In particular, the intermediate logic with values in the chain of length n may be "completed" conservatively by adding a single unary connective, so that the expanded system does not allow further axiomatic extensions by new connectives. (shrink)

Esta enciclopédia abrange, de uma forma introdutória mas desejavelmente rigorosa, uma diversidade de conceitos, temas, problemas, argumentos e teorias localizados numa área relativamente recente de estudos, os quais tem sido habitual qualificar como «estudos lógico-filosóficos». De uma forma apropriadamente genérica, e apesar de o território teórico abrangido ser extenso e de contornos por vezes difusos, podemos dizer que na área se investiga um conjunto de questões fundamentais acerca da natureza da linguagem, da mente, da cognição e do raciocínio humanos, bem (...) como questões acerca das conexões destes com a realidade não mental e extralinguística. A razão daquela qualificação é a seguinte: por um lado, a investigação em questão é qualificada como filosófica em virtude do elevado grau de generalidade e abstracção das questões examinadas (entre outras coisas); por outro, a investigação é qualificada como lógica em virtude de ser uma investigação logicamente disciplinada, no sentido de nela se fazer um uso intenso de conceitos, técnicas e métodos provenientes da disciplina de lógica. O agregado de tópicos que constitui a área de estudos lógico-filosóficos é já visível, pelo menos em parte, no Tractatus Logico-Philosophicus de Ludwig Wittgenstein, uma obra publicada em 1921. E uma boa maneira de ter uma ideia sinóptica do território disciplinar abrangido por esta enciclopédia, ou pelo menos de uma porção substancial dele, é extrair do Tractatus uma lista dos tópicos mais salientes aí discutidos; a lista incluirá certamente tópicos do seguinte género, muitos dos quais se podem encontrar ao longo desta enciclopédia: factos e estados de coisas; objectos; representação; crenças e estados mentais; pensamentos; a proposição; nomes próprios; valores de verdade e bivalência; quantificação; funções de verdade; verdade lógica; identidade; tautologia; o raciocínio matemático; a natureza da inferência; o cepticismo e o solipsismo; a indução; as constantes lógicas; a negação; a forma lógica; as leis da ciência; o número. (shrink)

Sumário: 1. El caso del método científico, Alberto Oliva; 2. Un capítulo de la prehistoria de las ciencias humanas: la defensa por Vico de la tópica, Jorge Alberto Molina; 3. La figura de lo cognoscible y los mundos, Pablo Vélez León; 4. Lebenswelt de Husserl y las neurociencias, Vanessa Fontana; 5. El uso estético del concepto de mundos posibles, Jairo Dias Carvalho; 6. Realismo normativo no naturalista y mundos morales imposibles, Alcino Eduardo Bonella; 7. En la lógica de pragmatismo, Hércules (...) de Araujo Feitosa, Luiz Henrique da Cruz Silvestrini; 8 La lógica, el tiempo y el lenguaje natural: un sistema formal para tiempos de portugués, Carlos Luciano Manholi. / Temas em filosofia contemporânea II / Jonas Rafael Becker Arenhart, Jaimir Conte, Cezar Augusto Mortari (orgs.) – Florianópolis : NEL/UFSC, 2016. 167 p. : tabs. - (Rumos da epistemologia ; v. 14); ISBN 978-85-87253-27-9 (papel); ISBN 978-85-87253-26-2 (e-book). (shrink)

A deterministic weakening \ of the Belnap–Dunn four-valued logic \ is introduced to formalize the acceptance and rejection of a proposition at a state in a linearly ordered informational frame with persistent valuations. The logic \ is formalized as a sequent calculus. The completeness and decidability of \ with respect to relational semantics are shown in terms of normal forms. From an algebraic perspective, the class of all algebras for \ is described, and found to be a subvariety of Berman’s (...) variety \. Every linearly ordered frame is logically equivalent to its dual algebra. It is proved that \ is the logic of a nine-element distributive lattice with a negation. Moreover, \ is embedded into \ by Glivenko’s double-negation translation. (shrink)

In abstract algebraic logic, the general study of propositional non-classical logics has been traditionally based on the abstraction of the Lindenbaum-Tarski process. In this process one considers the Leibniz relation of indiscernible formulae. Such approach has resulted in a classification of logics partly based on generalizations of equivalence connectives: the Leibniz hierarchy. This paper performs an analogous abstract study of non-classical logics based on the kind of generalized implication connectives they possess. It yields a new classification of logics expanding Leibniz (...) hierarchy: the hierarchy of implicational logics. In this framework the notion of implicational semilinear logic can be naturally introduced as a property of the implication, namely a logic L is an implicational semilinear logic iff it has an implication such that L is complete w.r.t. the matrices where the implication induces a linear order, a property which is typically satisfied by well-known systems of fuzzy logic. The hierarchy of implicational logics is then restricted to the semilinear case obtaining a classification of implicational semilinear logics that encompasses almost all the known examples of fuzzy logics and suggests new directions for research in the field. (shrink)

We characterize (both from a syntactic and an algebraic point of view) the normal K4-logics for which unification is filtering. We also give a sufficient semantic criterion for existence of most general unifiers, covering natural extensions of K4.2⁺ (i.e., of the modal system obtained from K4 by adding to it, as a further axiom schemata, the modal translation of the weak excluded middle principle).

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)

Within the technical frame supplied by the algebraic variety of diagonalizable algebras, defined by R. Magari in [2], we prove the following: Let T be any first-order theory with a predicate Pr satisfying the canonical derivability conditions, including Löb's property. Then any formula in T built up from the propositional variables $q,p_{1},...,p_{n}$ , using logical connectives and the predicate Pr, has the same "fixed-points" relative to q (that is, formulas $\psi (p_{1},...,p_{n})$ for which for all $p_{1},...,p_{n}\vdash _{T}\phi (\psi (p_{1},...,p_{n}),p_{1},...,p_{n})\leftrightarrow \psi (...) (p_{1},...,p_{n})$ ) of a formula $\phi ^{\ast}$ of the same kind, obtained from φ in an effective way. Moreover, such $\phi ^{\ast}$ is provably equivalent to the formula obtained from φ substituting with $\phi ^{\ast}$ itself all the occurrences of q which are under Pr. In the particular case where q is always under Pr in φ, $\phi ^{\ast}$ is the unique (up to provable equivalence) "fixed-point" of φ. Since this result is proved only assuming Pr to be canonical, it can be deduced that Löb's property is, in a sense, equivalent to Gödel's diagonalization lemma. All the results are proved more generally in the intuitionistic case. (shrink)

We prove that no logic (i.e. consequence operation) determined by any class of orthomodular lattices admits the deduction theorem (Theorem 2.7). We extend those results to some broader class of logics determined by ortholattices (Corollary 2.6).

Orthologic is defined by weakening the axioms and rules of inference of the classical propositional calculus. The resulting Lindenbaum-Tarski quotient algebra is an orthoimplication algebra which generalizes the author's implication algebra. The associated order structure is a semi-orthomodular lattice. The theory of orthomodular lattices is obtained by adjoining a falsity symbol to the underlying orthologic or a least element to the orthoimplication algebra.

In this paper, we introduce the notion of dual Post’s negation and an infinite class of Dual Post’s finitely-valued logics which differ from Post’s ones with respect to the definitions of negation and the sets of designated truth values. We present adequate natural deduction systems for all Post’s k-valued ) logics as well as for all Dual Post’s k-valued logics.

A contraction-free and cut-free sequent calculus \ for semi-De Morgan algebras, and a structural-rule-free and single-succedent sequent calculus \ for De Morgan algebras are developed. The cut rule is admissible in both sequent calculi. Both calculi enjoy the decidability and Craig interpolation. The sequent calculi are applied to prove some embedding theorems: \ is embedded into \ via Gödel–Gentzen translation. \ is embedded into a sequent calculus for classical propositional logic. \ is embedded into the sequent calculus \ for intuitionistic (...) propositional logic. (shrink)

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

In this paper we characterize the MV-algebras containing as subalgebras Post algebras of finitely many orders. For this we study cyclic elements in MV-algebras which are the generators of the fundamental chain of the Post algebras.

It was proved by Odintsov and Pearce that the logic is a deductive base for paraconsistent answer set semantics of logic programs with two kinds of negation. Here we describe the lattice of logics extending, characterise these logics via classes of -models, and prove that none of the proper extensions of is a deductive base for PAS.

The logic TK was introduced as a propositional logic extending the classical propositional calculus with a new unary operator which interprets some conceptions of Tarski’s consequence operator. TK-algebras were introduced as models to TK . Thus, by using algebraic tools, the adequacy (soundness and completeness) of TK relatively to the TK-algebras was proved. This work presents a neighbourhood semantics for TK , which turns out to be deductively equivalent to the non-normal modal logic EMT4 . DOI:10.5007/1808-1711.2011v15n2p287.

In the last few decades many formal systems of fuzzy logics have been developed. Since the main differences between fuzzy and classical logics lie at the propositional level, the fuzzy predicate logics have developed more slowly (compared to the propositional ones). In this text we aim to promote interest in fuzzy predicate logics by contributing to the model theory of fuzzy predicate logics. First, we generalize the completeness theorem, then we use it to get results on conservative extensions of theories (...) and on witnessed models. (shrink)

Modal counterparts of intermediate predicate logics will be studied by means of algebraic devise. Our main tool will be a construction of algebraic semantics for modal logics from algebraic frames for predicate logics. Uncountably many examples of modal counterparts of intermediate predicate logics will be given.

The object of this paper is to make a study of four systems of modal logic (S4, S5, and their intuitionistic analogues IM4 and IM5) with the techniques of the theory of abstract logics set up by Suszko, Bloom, Brown, Verdú and others. The abstract concepts corresponding to such systems are defined as generalizations of the logics naturally associated to their algebraic models (topological Boolean or Heyting algebras, general or semisimple). By considering new suitably defined connectives and by distinguishing between (...) having the rule of necessitation only for theorems or as a full inference rule (which amounts to dealing with all filters or with open filters of the algebras) we are able to reduce the study of a modal (abstract) logic L to that of two nonmodal logics L - and L + associated with L. We find that L is "of IM4 type" if and only if L - and L + are both intuitionistic and have the same theorems, and logics of type S4, IM5 or S5 are obtained from those of type IM4 simply by making classical L -, L + or both. We compare this situation with that found in recent approaches to intuitionistic modal logic using birelational models or using higher-level sequent-systems. The treatment of modal systems with abstract logics is rather new, and in our way to it we find several general constructions and results which can also be applied to other modal systems weaker than those we study in detail. (shrink)

The aim of this paper is to give, using the Kripke semantics for intuitionism, a representation of finitely generated free Heyting algebras. By means of the representation we determine in a constructive way some set of "special elements" of such algebras. Furthermore, we show that many algebraic properties which are satisfied by the free algebra on one generator are not satisfied by free algebras on more than one generator.

We continue work of our earlier paper :219–241, 2009) where abstract logics and particularly intuitionistic abstract logics are studied.logics can be topologized in a direct and natural way. This facilitates a topological study of classes of concrete logics whenever they are given in abstract form. Moreover, such a direct topological approach avoids the often complex algebraic and lattice-theoretic machinery usually applied to represent logics. Motivated by that point of view, we define in this paper the category of intuitionistic abstract logics (...) with stable logic maps as morphisms, and the category of implicative spectral spaces with spectral maps as morphisms. We show the equivalence of these categories and conclude that the larger categories of distributive abstract logics and distributive sober spaces are equivalent, too. (shrink)

RésuméCet article discute diverses confusions, actueles ou potentielles, liées á la bivalence et au tiers exclu. Il s'agit, en particulier, 1) d'examiner divers cas illustrant les rapports entre la bivalence et le tiers exclu ; 2) de discuter la thése selon laquelle le tiers exclu et le schéma-T de Tarskipour la vérité entraînent la bivalence; 3) de proposer quelques remarques sur les rapports entre la bivalence, le tiers exclu et la preuve par l'absurde; 4) de scruter un argument répandu selon (...) lequel le tiers exclu vaut pour les termes «précis» et 5) de commenter l'histoire de la terminologie. L'objectif ici est d'attirer l'attention des philosophes sur une distinction importante qui ne reçoit pas souvent le traitement minutieux qu'elle mérite. (shrink)

We consider the logics determined by the set of all natural implicative expansions of Kleene’s strong 3-valued matrix and select the class of all logics functionally equivalent to Łukasiewicz’s 3-valued logic Ł3. The concept of a “natural implicative matrix” is based upon the notion of a “natural conditional” defined in Tomova.

Motivated by the definition of semi-Nelson algebras, a propositional calculus called semi-intuitionistic logic with strong negation is introduced and proved to be complete with respect to that class of algebras. An axiomatic extension is proved to have as algebraic semantics the class of Nelson algebras.