A unified and modular falsification-aware single-succedent Gentzen-style framework is introduced for classical, paradefinite, paraconsistent, and paracomplete logics. This framework is composed of two special inference rules, referred to as the rules of explosion and excluded middle, which correspond to the principle of explosion and the law of excluded middle, respectively. Similar to the cut rule in Gentzen’s LK for classical logic, these rules are admissible in cut-free LK. A falsification-aware single-succedent Gentzen-style sequent calculus fsCL for classical logic is formalized based (...) on the proposed framework. The calculus fsCL is obtained from the existing falsification-aware single-succedent Gentzen-style sequent calculus GN4 for Nelson’s paradefinite (or paraconsistent) four-valued logic N4 by adding the rules of explosion and excluded middle. A falsification-aware single-succedent Gentzen-style sequent calculus GN3 for Nelson’s paracomplete three-valued logic N3 is also obtained from GN4 by adding the rule of explosion. The cut-elimination theorems for fsCL, GN3, and some of their neighbors as well as the Glivenko theorem for fsCL are proved. (shrink)

Over the past 50 years, Nelson algebras have been extensively studied by distinguished scholars as the algebraic counterpart of Nelson's constructive logic with strong negation. Despite these studies, a comprehensive survey of the topic is currently lacking, and the theory of Nelson algebras remains largely unknown to most logicians. This paper aims to fill this gap by focussing on the essential developments in the field over the past two decades. Additionally, we explore generalisations of Nelson algebras, such as N4-lattices which (...) correspond to the paraconsistent version of Nelson's logic, as well as their applications to other areas of interest to logicians, such as duality and rough set theory. A general representation theorem states that each Nelson algebra is isomorphic to a subalgebra of a rough set-based Nelson algebra induced by a quasiorder. Furthermore, a formula is a theorem of Nelson logic if and only if it is valid in every finite Nelson algebra induced by a quasiorder. (shrink)

Il s’agit de comprendre dans cet article l’opposition formulée par Gilles-Gaston Granger entre deux types de négation : la négation « radicale », d’un côté, et les négations « appliquées » de l’autre. Nous examinerons les propriétés de cette opposition, ainsi que les enseignements à en tirer sur la philosophie de la logique de Granger. Puis nous proposerons une théorie constructive des valeurs logiques considérées comme des objets structurés, consolidant à la fois l’unité de la théorie logique de Granger et (...) son pluralisme philosophique. (shrink)

In this paper, we extend the expressive power of the logics K3, LP and FDE with anormality operator, which is able to express whether a for-mula is assigned a classical truth value or not. We then establish classical recapture theorems for the resulting logics. Finally, we compare the approach via normality operator with the classical collapse approach devisedby Jc Beall.

The present article shows that there are consistent and decidable manyvalued systems of propositional logic which satisfy two or all the three criteria for non-trivial inconsistent theories by da Costa . The weaker one of these paraconsistent system is also able to avoid a series of paradoxes which come up when classical logic is applied to empirical sciences. These paraconsistent systems are based on a 6-valued system of propositional logic for avoiding difficulties in several domains of empirical science ).

We investigate the notion of classical negation from a non-classical perspective. In particular, one aim is to determine what classical negation amounts to in a paracomplete and paraconsistent four-valued setting. We first give a general semantic characterization of classical negation and then consider an axiomatic expansion BD+ of four-valued Belnap–Dunn logic by classical negation. We show the expansion complete and maximal. Finally, we compare BD+ to some related systems found in the literature, specifically a four-valued modal logic of Béziau and (...) the logic of classical implication and a paraconsistent de Morgan negation of Zaitsev. (shrink)

Dentro del conjunto de las lógicas no clásicas, las lógicas paraconsistentes han suscitado de manera particular el interés de diversos filósofos. Además de las definiciones tradicionales, en los últimos años, se han propuesto nuevas maneras de caracterizar a la paraconsistencia. Lo que tienen en común todas estas definiciones es que alguna forma de la regla o de la metarregla de explosión debe ser rechazada. En este artículo, presentaré dichas definiciones y evaluaré el rol que juegan la negación y la transitividad (...) en ellas. Finalmente, propondré una nueva caracterización de la paraconsistencia, la que llamaré paraconsistencia pura y mostraré que la regla de monotonía juega un rol crucial en todas las presentaciones de dicho concepto. (shrink)

This paper develops the first deontic logic in the context of paraconsistent logics.

In this paper, we use a ‘normality operator’ in order to generate logics of formal inconsistency and logics of formal undeterminedness from any subclassical many-valued logic that enjoys a truth-functional semantics. Normality operators express, in any many-valued logic, that a given formula has a classical truth value. In the first part of the paper we provide some setup and focus on many-valued logics that satisfy some of the three properties, namely subclassicality and two properties that we call fixed-point negation property (...) and conservativeness. In the second part of the paper, we introduce normality operators and explore their formal behaviour. In the third and final part of the paper, we establish a number of classical recapture results for systems of formal inconsistency and formal undeterminedness that satisfy some or all the properties above. These are the main formal results of the paper. Also, we illustrate concrete cases of recapture by discussing the logics $\mathsf{K}^{\circledast }_{3}$, $\mathsf{LP}^{\circledast }$, $\mathsf{K}^{w\circledast }_{3}$, $\mathsf{PWK}^{\circledast }$ and $\mathsf{E_{fde}}^{\circledast }$, that are in turn extensions of $\mathsf{{K}_{3}}$, $\mathsf{LP}$, $\mathsf{K}^{w}_{3}$, $\mathsf{PWK}$ and $\mathsf{E_{fde}}$, respectively. (shrink)

Within the framework of Meinongianism, nothingness turns out to have contradictory features—it seems to be an object and not. In this paper, we explore two different kinds of Meinongian accounts of nothingness. The first one is the consistent account, which rejects the contradiction of nothingness, while the second one is the inconsistent account, which accepts the contradiction of nothingness. First of all, after showing that the consistent account of nothingness defended by Jacquette fails, we express some concerns on the general (...) possibility of consistently characterizing nothingness. Secondly, starting from Priest’s inconsistent characterization of nothingness :146–158, 2014b), we will introduce our own inconsistent account. The key idea of our account is to take nothingness as the complement of the totality. Finally, we will make formal sense of it by constructing an inconsistent mereological system, which is the development of the paraconsistent mereology proposed by Weber and Cotnoir. (shrink)

This paper proposes a new dialetheic logic, a Dialetheic Logic with Exclusive Assumptions and Conclusions ), including classical logic as a particular case. In \, exclusivity is expressed via the speech acts of assuming and concluding. In the paper we adopt the semantics of the logic of paradox extended with a generalized notion of model and we modify its proof theory by refining the notions of assumption and conclusion. The paper starts with an explanation of the adopted philosophical perspective, then (...) we propose our \ logic. Finally, we show how \ supports the dialetheic solution of the liar paradox. (shrink)

There are two foundational, but not fully developed, ideas in paraconsistency, namely, the duality between paraconsistent and intuitionistic paradigms, and the introduction of logical operators that express metalogical notions in the object language. The aim of this paper is to show how these two ideas can be adequately accomplished by the logics of formal inconsistency and by the logics of formal undeterminedness. LFIs recover the validity of the principle of explosion in a paraconsistent scenario, while LFUs recover the validity of (...) the principle of excluded middle in a paracomplete scenario. We introduce definitions of duality between inference rules and connectives that allow comparing rules and connectives that belong to different logics. Two formal systems are studied, the logics mbC and mbD, that display the duality between paraconsistency and paracompleteness as a duality between inference rules added to a common core—in the case studied here, this common core is classical positive propositional logic. The logics mbC and mbD are equipped with recovery operators that restore classical logic for, respectively, consistent and determined propositions. These two logics are then combined obtaining a pair of LFI and undeterminedness, namely, mbCD and mbCDE. The logic mbCDE exhibits some nice duality properties. Besides, it is simultaneously paraconsistent and paracomplete, and able to recover the principles of excluded middle and explosion one at a time. The last sections offer an algebraic account for such logics by adapting the swap structures semantics framework of the LFIs the LFUs. This semantics highlights some subtle aspects of these logics, and allows us to prove decidability by means of finite nondeterministic matrices. (shrink)

How is it possible that beginning from the negation of rational thoughts one comes to produce knowledge? This problem, besides its intrinsic interest, acquires a great relevance when the representation of a knowledge is settled, for example, on data and automatic reasoning. Many treatment ways have been tried, as in the case of the non-monotonic logics; logics that intend to formalize an idea of reasoning by default, etc. These attempts are incomplete and are subject to failure. A possible solution would (...) be to formulate a logic of the irrational, which offers a model for reasoning permitting to support contradictions as well as to produce knowledge from such situations. An intuition underlying the foundation of such a logic consists of the da Costa's paraconsistent logics presenting however, a different deduction theory and a whole distinct semantics, called here "the semantics of possible translations". The present proposing, following our argumentation, intends to enlight all this question, by a whole satisfactory logical point of view, being practically applicable and philosophically acceptable.Como é possível que a partir da negação do racional se possa obter conhecimento adicional? Esse problema, além de seu interesse intrínseco, adquire uma relevância adicional quando o encontramos na representação do conhecimento em bases de dados e raciocínio automático, por exemplo. Nesse caso, diversas tentativas de tratamento têm sido propostas, como as lógicas não-monotônicas, as lógicas que tentam formalizar a ideia do raciocínio por falha . Tais tentativas de solução, porém, são falhas e incompletas; proponho que uma solução possível seria formular uma lógica do irracional, que oferecesse um modelo para o raciocínio permitindo não só suportar contradições, como conseguir obter conhecimento, a partir de tais situações. A intuição subjacente à formulação de tal lógica são as lógicas paraconsistentes de da Costa, mas com uma teoria da dedução diferente e uma semântica completamente distinta . Tal proposta, como pretendo argumentar, fornece um enfoque para a questão que é ao mesmo tempo completamente satisfatório, aplicável do ponto de vista prático e aceitável do ponto de vista filosófico. (shrink)

This paper investigates the question of characterizing first-order LFIs (logics of formal inconsistency) by means of two-valued semantics. LFIs are powerful paraconsistent logics that encode classical logic and permit a finer distinction between contradictions and inconsistencies, with a deep involvement in philosophical and foundational questions. Although focused on just one particular case, namely, the quantified logic QmbC, the method proposed here is completely general for this kind of logics, and can be easily extended to a large family of quantified paraconsistent (...) logics, supplying a sound and complete semantical interpretation for such logics. However, certain subtleties involving term substitution and replacement, that are hidden in classical structures, have to be taken into account when one ventures into the realm of nonclassical reasoning. This paper shows how such difficulties can be overcome, and offers detailed proofs showing that a smooth treatment of semantical characterization can be given to all such logics. Although the paper is well-endowed in technical details and results, it has a significant philosophical aside: it shows how slight extensions of classical methods can be used to construct the basic model theory of logics that are weaker than traditional logic due to the absence of certain rules present in classical logic. Several such logics, however, as in the case of the LFIs treated here, are notorious for their wealth of models precisely because they do not make indiscriminate use of certain rules; these models thus require new methods. In the case of this paper, by just appealing to a refined version of the Principle of Explosion, or Pseudo-Scotus, some new constructions and crafty solutions to certain non-obvious subtleties are proposed. The result is that a richer extension of model theory can be inaugurated, with interest not only for paraconsistency, but hopefully to other enlargements of traditional logic. (shrink)

This is the ASL report on the 7th Latin American Symposium on Mathematical Logic held in Campinas, SP, Brazil, from July 29- August 02, 1985.

This paper contributes to the study of paracompleteness and paraconsistency. We present two logics that address the following questions in novel ways. How can the paracomplete theorist characterize the formulas that defy excluded middle while maintaining that not all formulas are of this kind? How can the paraconsistent theorist characterize the formulas that obey explosion while still maintaining that there are some formulas not of this kind?

The purpose of this paper is to present a paraconsistent formal system and a corresponding intended interpretation according to which true contradictions are not tolerated. Contradictions are, instead, epistemically understood as conflicting evidence, where evidence for a proposition A is understood as reasons for believing that A is true. The paper defines a paraconsistent and paracomplete natural deduction system, called the Basic Logic of Evidence, and extends it to the Logic of Evidence and Truth. The latter is a logic of (...) formal inconsistency and undeterminedness that is able to express not only preservation of evidence but also preservation of truth. LETj is anti-dialetheist in the sense that, according to the intuitive interpretation proposed here, its consequence relation is trivial in the presence of any true contradiction. Adequate semantics and a decision method are presented for both BLE and LETj, as well as some technical results that fit the intended interpretation. (shrink)

It is well-known that da Costa's C-systems of paraconsistent logic do not admit a Blok-Pigozzi algebraization. Still, an algebraic flavored semantics for them has been proposed in the literature, namely using the class of so-called da Costa algebras. However, the precise connection between these semantic structures and the C-systems was never established at the light of the theory of algebraizable logics. In this paper we propose to study the C-systems from an algebraic point of view, and to fill in this (...) gap by using the tools and techniques of the newly developed behavioral approach to abstract algebraic logic. As a by-product of the approach, we also rediscover the bivaluation semantics of the logics. (shrink)

We introduce and study a new approach to the theory of abstract algebraic logic (AAL) that explores the use of many-sorted behavioral logic in the role traditionally played by unsorted equational logic. Our aim is to extend the range of applicability of AAL toward providing a meaningful algebraic counterpart also to logics with a many-sorted language, and possibly including non-truth-functional connectives. The proposed behavioral approach covers logics which are not algebraizable according to the standard approach, while also bringing a new (...) algebraic perspective to logics which are algebraizable using the standard tools of AAL. Furthermore, we pave the way toward a robust behavioral theory of AAL, namely by providing a behavioral version of the Leibniz operator which allows us to generalize the traditional Leibniz hierarchy, as well as several well-known characterization results. A number of meaningful examples will be used to illustrate the novelties and advantages of the approach. (shrink)

In positive logic the negation of a propositionA is defined byA X whereX is some fixed proposition. A number of standard properties of negation, includingreductio ad absurdum, can then be proved, but not the law of noncontradiction so that this forms a paraconsistent logic. Various stronger paraconsistent logics are then generated by putting in particular propositions forX. These propositions range from true through contingent to false.

According to the trivialist, everything is true. But why would anyone believe that? It turns out that trivialism emerges naturally from a certain inconsistency view of language, and it has significant benefits that need to be acknowledged. But trivialism also encounters some troubles along the way. After discussing them, I sketch a couple of alternatives that can preserve the benefits of trivialism without the corresponding costs.

Second-order logic has a number of attractive features, in particular the strong expressive resources it offers, and the possibility of articulating categorical mathematical theories (such as arithmetic and analysis). But it also has its costs. Five major charges have been launched against second-order logic: (1) It is not axiomatizable; as opposed to first-order logic, it is inherently incomplete. (2) It also has several semantics, and there is no criterion to choose between them (Putnam, J Symbol Logic 45:464–482, 1980 ). Therefore, (...) it is not clear how this logic should be interpreted. (3) Second-order logic also has strong ontological commitments: (a) it is ontologically committed to classes (Resnik, J Phil 85:75–87, 1988 ), and (b) according to Quine (Philosophy of logic, Prentice-Hall: Englewood Cliffs, 1970 ), it is nothing more than “set theory in sheep’s clothing”. (4) It is also not better than its first-order counterpart, in the following sense: if first-order logic does not characterize adequately mathematical systems, given the existence of non - isomorphic first-order interpretations, second-order logic does not characterize them either, given the existence of different interpretations of second-order theories (Melia, Analysis 55:127–134, 1995 ). (5) Finally, as opposed to what is claimed by defenders of second-order logic [such as Shapiro (J Symbol Logic 50:714–742, 1985 )], this logic does not solve the problem of referential access to mathematical objects (Azzouni, Metaphysical myths, mathematical practice: the logic and epistemology of the exact sciences, Cambridge University Press, Cambridge, 1994 ). In this paper, I argue that the second-order theorist can solve each of these difficulties. As a result, second-order logic provides the benefits of a rich framework without the associated costs. (shrink)

A proof method for automation of reasoning in a paraconsistent logic, the calculus C1* of da Costa, is presented. The method is analytical, using a specially designed tableau system. Actually two tableau systems were created. A first one, with a small number of rules in order to be mathematically convenient, is used to prove the soundness and the completeness of the method. The other one, which is equivalent to the former, is a system of derived rules designed to enhance computational (...) efficiency. A prototype based on this second system was effectively implemented. (shrink)

We argue that many-valued logics can be useful in analysing informational conflicts by using society semantics. This work concentrates on four-valued Łukasiewicz logic. SSs were proposed by Carnielli and Lima-Marques to deal with conflicts of information involving rational agents that make judgements about propositions according to a given logic within a society, where a society is understood as a collection $\mathcal{A}$ of agents. The interesting point of such semantics is that a new logic can be obtained by combining the logic (...) of the agents under some appropriate rules. Carnielli and Lima-Marques defined SSs for the three-valued logics $I^{1}$ and $P^{1}$. In this kind of semantics, all the agents reason according to classical logic and the molecular formulas behave in the same way as in CL. Marcos provided SSs with classical agents for the three-valued Łukasiewicz logic Ł$_{3}$, but in this case, the molecular formulas do not behave classically. We prove here that one can characterize Ł$_{4}^{\prime}$, a conservative extension of Ł$_{4}$ obtained by adding a connective $\blacktriangledown$, by means of a closed society where the agents reason according to Ł$_{3}$. We shall emphasize the importance of recovery operators in the construction of this class of societies. Moreover, we shall relate this semantics to Suszko’s view on the ‘two-valuedness’ of logic. (shrink)

I propose a comprehensive account of negation as a modal operator, vindicating a moderate logical pluralism. Negation is taken as a quantifier on worlds, restricted by an accessibility relation encoding the basic concept of compatibility. This latter captures the core meaning of the operator. While some candidate negations are then ruled out as violating plausible constraints on compatibility, different specifications of the notion of world support different logical conducts for negations. The approach unifies in a philosophically motivated picture the following (...) results: nothing can be called a negation properly if it does not satisfy Contraposition and Double Negation Introduction; the pair consisting of two split or Galois negations encodes a distinction without a difference; some paraconsistent negations also fail to count as real negations, but others may; intuitionistic negation qualifies as real negation, and classical Boolean negation does as well, to the extent that constructivist and paraconsistent doubts on it do not turn on the basic concept of compatibility but rather on the interpretation of worlds. (shrink)

1. Real-world applications, such as monitoring urban wastewater networks, commonly process large volumes of multi-source, heterogeneous data to support reasoning, query answering and decision-makin...

There are a considerable number of logics that do not seem to share the same inferential principles. Intuitionistic logics do not include the law of the exclude.

The logics of formal inconsistency are logics tolerant to some amount of contradiction, but in which some versions of explosion still hold. The main result of this paper is a reconstruction of two such logics in the dialogical framework. By doing so, we achieve two things. On the one hand, we provide a formal approach to argumentative situations where some contradictions may occur while keeping the idea that there may still be situations in which some propositions are ‘safe’ in the (...) sense of immunity to the contradictions. On the other hand, we open a new line of study on these logics, in the context of the game-theoretical approach to semantics born in the 1960s, with various interesting perspectives, some of which are discussed at the end of this article. (shrink)

In some recent articles, Cobreros, Egré, Ripley, & van Rooij have defended the idea that abandoning transitivity may lead to a solution to the trouble caused by semantic paradoxes. For that purpose, they develop the Strict-Tolerant approach, which leads them to entertain a nontransitive theory of truth, where the structural rule of Cut is not generally valid. However, that Cut fails in general in the target theory of truth does not mean that there are not certain safe instances of Cut (...) involving semantic notions. In this article we intend to meet the challenge of answering how to regain all the safe instances of Cut, in the language of the theory, making essential use of a unary recovery operator. To fulfill this goal, we will work within the so-called Goodship Project, which suggests that in order to have nontrivial naïve theories it is sufficient to formulate the corresponding self-referential sentences with suitable biconditionals. Nevertheless, a secondary aim of this article is to propose a novel way to carry this project out, showing that the biconditionals in question can be totally classical. In the context of this article, these biconditionals will be essentially used in expressing the self-referential sentences and, thus, as a collateral result of our work we will prove that none of the recoveries expected of the target theory can be nontrivially achieved if self-reference is expressed through identities. (shrink)

In this paper, we present a non-trivial and expressively complete paraconsistent naïve theory of truth, as a step in the route towards semantic closure. We achieve this goal by expressing self-reference with a weak procedure, that uses equivalences between expressions of the language, as opposed to a strong procedure, that uses identities. Finally, we make some remarks regarding the sense in which the theory of truth discussed has a property closely related to functional completeness, and we present a sound and (...) complete three-sided sequent calculus for this expressively rich theory. (shrink)

In this work, we investigate the relationship between paraconsistent semantics and some well-known topological spaces such as connected and continuous spaces. We also discuss homotopies as truth preserving operations in paraconsistent topological models.

Paraconsistent logics are the formal systems in which absurdities do not trivialise the logic. In this paper, we give Hintikka-style game theoretical semantics for a variety of paraconsistent and non-classical logics. For this purpose, we consider Priest’s Logic of Paradox, Dunn’s First-Degree Entailment, Routleys’ Relevant Logics, McCall’s Connexive Logic and Belnap’s four-valued logic. We also present a game theoretical characterisation of a translation between Logic of Paradox/Kleene’s K3 and S5. We underline how non-classical logics require different verification games and prove (...) the correctness theorems of their respective game theoretical semantics. This allows us to observe that paraconsistent logics break the classical bidirectional connection between winning strategies and truth values. (shrink)

A logic \ is called self-extensional if it allows to replace occurrences of a formula by occurrences of an \-equivalent one in the context of claims about logical consequence and logical validity. It is known that no three-valued paraconsistent logic which has an implication can be self-extensional. In this paper we show that in contrast, the famous Dunn–Belnap four-valued logic has exactly one self-extensional four-valued extension which has an implication. We also investigate the main properties of this logic, determine the (...) expressive power of its language, and provide a cut-free Gentzen-type proof system for it. (shrink)

A proof is presented showing that there is no paraconsistent logics with a standard implication which have a three-valued characteristic matrix, and in which the replacement principle holds.

A logic \ is called self-extensional if it allows to replace occurrences of a formula by occurrences of an \-equivalent one in the context of claims about logical consequence and logical validity. It is known that no three-valued paraconsistent logic which has an implication can be self-extensional. In this paper we show that in contrast, there is exactly one self-extensional three-valued paraconsistent logic in the language of \ for which \ is a disjunction, and \ is a conjunction. We also (...) investigate the main properties of this logic, determine the expressive power of its language, and provide a cut-free Gentzen-type proof system for it. (shrink)

Paraconsistent logics are logics that, in contrast to classical and intuitionistic logic, do not trivialize inconsistent theories. In this paper we take a paraconsistent view on two famous modal logics: B and S5. We use for this a well-known general method for turning modal logics to paraconsistent logics by defining a new negation as $\neg \varphi =_{Def} \sim \Box \varphi$. We show that while that makes both B and S5 members of the well-studied family of paraconsistent C-systems, they differ from (...) most other C-systems in having the important replacement property. We further show that B is a very robust C-system in the sense that almost any axiom which has been considered in the context of C-systems is either already a theorem of B or its addition to B leads to a logic that is no longer paraconsistent. There is exactly one notable exception, and the result of adding this exception to B leads to the other logic studied here, S5. (shrink)

In the paper we explore the idea of describing Pawlak’s rough sets using three-valued logic, whereby the value t corresponds to the positive region of a set, the value f — to the negative region, and the undefined value u — to the border of the set. Due to the properties of the above regions in rough set theory, the semantics of the logic is described using a non-deterministic matrix (Nmatrix). With the strong semantics, where only the value t is (...) treated as designated, the above logic is a “common denominator” for Kleene and Łukasiewicz 3-valued logics, which represent its two different “determinizations”. In turn, the weak semantics—where both t and u are treated as designated—represents such a “common denominator” for two major 3-valued paraconsistent logics. We give sound and complete, cut-free sequent calculi for both versions of the logic generated by the rough set Nmatrix. Then we derive from these calculi sequent calculi with the same properties for the various “determinizations” of those two versions of the logic (including Łukasiewicz 3-valued logic). Finally, we show how to embed the four above-mentioned determinizations in extensions of the basic rough set logics obtained by adding to those logics a special two-valued “definedness” or “crispness” operator. (shrink)

According to Suszko's Thesis,any multi-valued semantics for a logical system can be replaced by an equivalent bivalent one. Moreover: bivalent semantics for families of logics can frequently be developed in a modular way. On the other hand bivalent semantics usually lacks the crucial property of analycity, a property which is guaranteed for the semantics of multi-valued matrices. We show that one can get both modularity and analycity by using the semantic framework of multi-valued non-deterministic matrices. We further show that for (...) using this framework in a constructive way it is best to view "truth-values" as information carriers, or "information-values". (shrink)

We investigate two large families of logics, differing from each other by the treatment of negation. The logics in one of them are obtained from the positive fragment of classical logic (with or without a propositional constant ff for “the false”) by adding various standard Gentzen-type rules for negation. The logics in the other family are similarly obtained from LJ+, the positive fragment of intuitionistic logic (again, with or without ff). For all the systems, we provide simple semantics which is (...) based on non-deterministic four-valued or three-valued structures, and prove soundness and completeness for all of them. We show that the role of each rule is to reduce the degree of non-determinism in the corresponding systems. We also show that all the systems considered are decidable, and our semantics can be used for the corresponding decision procedures. Most of the extensions of LJ+ (with or without ff) are shown to be conservative over the underlying logic, and it is determined which of them are not. (shrink)