Paraconsistent approaches have received little attention in the literature on vagueness (at least compared to other proposals). The reason seems to be that many philosophers have found the idea that a contradiction might be true (or that a sentence and its negation might both be true) hard to swallow. Even advocates of paraconsistency on vagueness do not look very convinced when they consider this fact; since they seem to have spent more time arguing that paraconsistent theories are at least as (...) good as their paracomplete counterparts, than giving positive reasons to believe on a particular paraconsistent proposal. But it sometimes happens that the weakness of a theory turns out to be its mayor ally, and this is what (I claim) happens in a particular paraconsistent proposal known as subvaluationism. In order to make room for truth-value gluts subvaluationism needs to endorse a notion of logical consequence that is, in some sense, weaker than standard notions of consequence. But this weakness allows the subvaluationist theory to accommodate higher-order vagueness in a way that it is not available to other theories of vagueness (such as, for example, its paracomplete counterpart, supervaluationism). (shrink)

We present a novel treatment of set theory in a four-valued paraconsistent and paracomplete logic, i.e., a logic in which propositions can be both true and false, and neither true nor false. Our approach is a significant departure from previous research in paraconsistent set theory, which has almost exclusively been motivated by a desire to avoid Russell’s paradox and fulfil naive comprehension. Instead, we prioritise setting up a system with a clear ontology of non-classical sets, which can be used to (...) reason informally about incomplete and inconsistent phenomena, and is sufficiently similar to ${\mathrm {ZFC}}$ to enable the development of interesting mathematics.We propose an axiomatic system ${\mathrm {BZFC}}$, obtained by analysing the ${\mathrm {ZFC}}$ -axioms and translating them to a four-valued setting in a careful manner, avoiding many of the obstacles encountered by other attempted formalizations. We introduce the anti-classicality axiom postulating the existence of non-classical sets, and prove a surprising results stating that the existence of a single non-classical set is sufficient to produce any other type of non-classical set.Our theory is naturally bi-interpretable with ${\mathrm {ZFC}}$, and provides a philosophically satisfying view in which non-classical sets can be seen as a natural extension of classical ones, in a similar way to the non-well-founded sets of Peter Aczel [1].Finally, we provide an interesting application concerning Tarski semantics, showing that the classical definition of the satisfaction relation yields a logic precisely reflecting the non-classicality in the meta-theory. (shrink)

It has been an open question whether or not we can define a belief revision operation that is distinct from simple belief expansion using paraconsistent logic. In this paper, we investigate the possibility of meeting the challenge of defining a belief revision operation using the resources made available by the study of dynamic epistemic logic in the presence of paraconsistent logic. We will show that it is possible to define dynamic operations of belief revision in a paraconsistent setting.

This book is the first in the field of paraconsistency to offer a comprehensive overview of the subject, including connections to other logics and applications in information processing, linguistics, reasoning and argumentation, and philosophy of science. It is recommended reading for anyone interested in the question of reasoning and argumentation in the presence of contradictions, in semantics, in the paradoxes of set theory and in the puzzling properties of negation in logic programming. Paraconsistent logic comprises a major logical theory and (...) offers the broadest possible perspective on the debate of negation in logic and philosophy. It is a powerful tool for reasoning under contradictoriness as it investigates logic systems in which contradictory information does not lead to arbitrary conclusions. Reasoning under contradictions constitutes one of most important and creative achievements in contemporary logic, with deep roots in philosophical questions involving negation and consistency This book offers an invaluable introduction to a topic of central importance in logic and philosophy. It discusses the history of paraconsistent logic; language, negation, contradiction, consistency and inconsistency; logics of formal inconsistency and the main paraconsistent propositional systems; many-valued companions, possible-translations semantics and non-deterministic semantics; paraconsistent modal logics; first-order paraconsistent logics; applications to information processing, databases and quantum computation; and applications to deontic paradoxes, connections to Eastern thought and to dialogical reasoning. (shrink)

Paraconsistent Weak Kleene logic is the 3-valued logic with two designated values defined through the weak Kleene tables. This paper is a first attempt to investigate PWK within the perspective and methods of abstract algebraic logic. We give a Hilbert-style system for PWK and prove a normal form theorem. We examine some algebraic structures for PWK, called involutive bisemilattices, showing that they are distributive as bisemilattices and that they form a variety, \, generated by the 3-element algebra WK; we also (...) prove that every involutive bisemilattice is representable as the Płonka sum over a direct system of Boolean algebras. We then study PWK from the viewpoint of AAL. We show that \ is not the equivalent algebraic semantics of any algebraisable logic and that PWK is neither protoalgebraic nor selfextensional, not assertional, but it is truth-equational. We fully characterise the deductive filters of PWK on members of \ and the reduced matrix models of PWK. Finally, we investigate PWK with the methods of second-order AAL—we describe the class \ of PWK-algebras, algebra reducts of basic full generalised matrix models of PWK, showing that they coincide with the quasivariety generated by WK—which differs from \—and explicitly providing a quasiequational basis for it. (shrink)

ABSTRACT The concept of paraconsistent logical consequence is usually negatively defined as a validity semantics in which not every sentences is deducible or in which inferential explosion does not occur. Paraconsistency has been negatively characterized in this way because paraconsistent logics have been designed specifically to avoid the trivialization of deductive inference entailed by the classical paradoxes of material implication for applications in a system that tolerates syntactical contradictions. The effect of the negative characterization of paraconsistency has been to encourage (...) an unsystematic development of distinct versions of paraconsistent logic. It has also contributed to a restricted overview of the full range of paraconsistent formalisms. After reviewing a standard negative characterization of paraconsistency and commenting on its limitations, I propose a positive characterization that makes possible a constructivistic description of the complete spectrum of distinct families of paraconsistent logics, and the description of a new type of maximally relevant paraconsistency, which I argue compares favorably with previously identified categories of paraconsistent logics. (shrink)

In this paper we propose to take seriously the claim that at least some kinds of paraconsistent negations are subcontrariety forming operators. We shall argue that from an intuitive point of view, by considering paraconsistent negations as formalizing that particular kind of opposition, one needs not worry with issues about the meaning of true contradictions and the like, given that “true contradictions” are not involved in these paraconsistent logics. Our strategy will consist in showing that, on the one hand, the (...) natural translation for subcontrariety in formal languages is not a contradiction in natural language, and on the other, translating alleged cases of contradiction in natural language to paraconsistent formal systems works only provided we transform them into a subcontrariety. Transforming contradictions into subcontrariety shall provide for an intuitive interpretation for paraconsistent negation, which we also discuss here. By putting all those pieces together, we hope a clearer sense of paraconsistency can be made, one which may liberate us from the need to tame contradictions. (shrink)

Scientific inquiry is typically focused on particular questions about particular objects and properties. This leads to a multiplicity of models which, even when they draw on a single, consistent body of concepts and principles, often employ different methods and assumptions to model different systems. Pluralists have remarked on how scientists draw on different assumptions to model different systems, different aspects of systems and systems under different conditions and defended the value of distinct, incompatible models within science at any given time. (...) Paraconsistentists have proposed logical strategies to avoid trivialization when inconsistencies arise by a variety of means. Here we examine how chunk and permeate, a simple approach to paraconsistent reasoning which avoids heterodox logic by confining commitments to separate contexts in which reasoning with them is taken to be reliable while allowing ‘permeation’ of some conclusions into other contexts, can help to systematize pluralistic reasoning across the boundaries of plural contexts, using regional climate models as an example. The result is a kind of unity for science—but a unity achieved by the constrained exchange of specified information between different contexts, rather than the closure of all commitments under some paraconsistent consequence relation. (shrink)

In some logics, anything whatsoever follows from a contradiction; call these logics explosive. Paraconsistent logics are logics that are not explosive. Paraconsistent logics have a long and fruitful history, and no doubt a long and fruitful future. To give some sense of the situation, I’ll spend Section 1 exploring exactly what it takes for a logic to be paraconsistent. It will emerge that there is considerable open texture to the idea. In Section 2, I’ll give some examples of techniques for (...) developing paraconsistent logics. In Section 3, I’ll discuss what seem to me to be some promising applications of certain paraconsistent logics. In fact, however, I don’t think there’s all that much to the concept ‘paraconsistent’ itself; the collection of paraconsistent logics is far too heterogenous to be very productively dealt with under a single label. Perhaps that will emerge as we go. (shrink)

The paper highlights the import of the paraconsistent movement, list some motivations for its origin, and distinguishes some stands with respect to para-consistency. It then discusses some sources of inconsistency that are specific for worldviews, and the import of the paraconsistent turn for the worldviews enterprise.

A logic is called 'paraconsistent' if it rejects the rule called 'ex contradictione quodlibet', according to which any conclusion follows from inconsistent premises. While logicians have proposed many technically developed paraconsistent logical systems and contemporary philosophers like Graham Priest have advanced the view that some contradictions can be true, and advocated a paraconsistent logic to deal with them, until recent times these systems have been little understood by philosophers. This book presents a comprehensive overview on paraconsistent logical systems to change (...) this situation. The book includes almost every major author currently working in the field. The papers are on the cutting edge of the literature some of which discuss current debates and others present important new ideas. The editors have avoided papers about technical details of paraconsistent logic, but instead concentrated upon works that discuss more 'big picture' ideas. Different treatments of paradoxes takes centre stage in many of the papers, but also there are several papers on how to interpret paraconistent logic and some on how it can be applied to philosophy of mathematics, the philosophy of language, and metaphysics. (shrink)

The purpose of this paper is mainly to give a model of paraconsistent logic satisfying the "Frege comprehension scheme" in which we can develop standard set theory (and even much more as we shall see). This is the continuation of the work of Hinnion and Libert.

William Parry conceived in the early thirties a theory of entail-
ment, the theory of analytic implication, intended to give a formal expression to the idea that the content of the conclusion of a valid argument must be included in the content of its premises. This paper introduces a system of analytic, paraconsistent and quasi-classical propositional logic that does not validate the paradoxes of Parry’s analytic implication. The interpretation of the expressions of this logic will be given in terms of a (...) four-valued semantics,and its proof theory will be provided by a system of signed semantic tableaux that incorporates the techniques developed to improve the efficiency of the tableaux method for many-valued logics.
1. Introduction. (shrink)

In this book we present a collection of papers on the topic of applying paraconsistent logic to solve inconsistency related problems in science, mathematics and computer science. The goal is to develop, compare, and evaluate different ways of applying paraconsistent logic. After more than 60 years of mainly theoretical developments in many independent systems of paraconsistent logic, we believe the time has come to compare and apply the developed systems in order to increase our philosophical understanding of reasoning when faced (...) with inconsistencies. This book wants to be a first step toward an application based, constructive debate to tackle the question which systems are best applied for which kind of problems and which philosophical conclusions can be drawn from such applications. (shrink)

In this paper, we recover the idea cast by Graham Priest to our ears in 2000: That it was possible to experience Ontological Paraconsistency in life. He had, back then, as a translation of his thinking, a painting by Escher: The stairs could be going up or down, and one could not tell where they were going by simply examining the painting. The most obvious argument as to why that was not an instance of Ontological Paraconsistency found in reality was (...) that the perspective from which you would have to stare at the painting to see something different would be different too, so that it was impossible that we were getting up and down at the same time, that is, from the same perspective. That would happen with anything we picked in this world. We recently found something that does not entirely belong to this world, however, something that could finally satisfy the requirements of Priest, and therefore prove to us that there is a place for Ontological Paraconsistency. We observe that the paraconsistent robot, Emmy (Abe et al., 2006), is an application of the Non-ontological Paraconsistency, which we always believed to be passive of existence, but we here talk about another type of paraconsistency, which would be intrinsic to the being. The purpose of this paper is then providing a definite answer to the questions: Is there any real life instance of Ontological Paraconsistency? Is Ontological Paraconsistency a useful concept in terms of logical theories? (shrink)

We introduce a family of matrices that define logics in which paraconsistency and/or paracompleteness occurs only at the level of literals, that is, formulas that are propositional letters or their iterated negations. We give a sound and complete axiomatization for the logic defined by the class of all these matrices, we give conditions for the maximality of these logics and we study in detail several relevant examples.

Paraconsistent logic makes it possible to study inconsistent theories in a coherent way. From its modern start in the mid-20th century, paraconsistency was intended for use in mathematics, providing a rigorous framework for describing abstract objects and structures where some contradictions are allowed, without collapse into incoherence. Over the past decades, this initiative has evolved into an area of non-classical mathematics known as inconsistent or paraconsistent mathematics. This Element provides a selective introductory survey of this research program, distinguishing between `moderate' (...) and `radical' approaches. The emphasis is on philosophical issues and future challenges. (shrink)

In the context of non-classical logics, many philosophers have been particularly interested in the paraconsistent logics. In addition to traditional definitions, in recent years, new ways of characterizing the notion of paraconsistency have been proposed. In all of these definitions the rule or the meta-rule of explosion is abandoned. In this article, I present those definitions and evaluate the role that the negation and the transitivity play in each of them. Finally, I propose a new definition of paraconsistency which I (...) call total paraconsistency and show that the rule of weakening plays a crucial role in all of the characterizations of such a concept. (shrink)

We define in precise terms the basic properties that an ‘ideal propositional paraconsistent logic’ is expected to have, and investigate the relations between them. This leads to a precise characterization of ideal propositional paraconsistent logics. We show that every three-valued paraconsistent logic which is contained in classical logic, and has a proper implication connective, is ideal. Then we show that for every n > 2 there exists an extensive family of ideal n -valued logics, each one of which is not (...) equivalent to any k -valued logic with k < n. (shrink)

Paraconsistency is the study of logical systems with a non-explosive negation such that a pair of contradictory formulas (with respect to such negation) does not necessarily imply triviality, discordant to what would be expected by contemporary logical orthodoxy. From a purely logical point of view, the significance of paraconsistency relies on the meticulous distinction between the general notions of contradictoriness and triviality of a theory—respectively, the fact that a given theory proves a proposition and its negation, and the fact that (...) a given theory proves any proposition (in the language of its underlying logic). Aside from this simple rationale, the formal techniques and approaches that meet the latter definitional requirement are manifold. Furthermore, it is not solely the logical-mathematical properties of such systems that are open to debate. Rather, there are several foundational and philosophical questions worth studying, including the very question about the nature of the contradictions allowed by paraconsistentists. This entry aims to advance a brief account of some distinct approaches to paraconsistency, providing a panorama on the development of paraconsistent logic. (shrink)

This paper discusses how to define logics as deductive limits of sequences of other logics. The case of da Costa's hierarchy of increasingly weaker paraconsistent calculi, known as $ \mathcal {C}$n, 1 $ \leq$ n $ \leq$ $ \omega$, is carefully studied. The calculus $ \mathcal {C}$$\scriptstyle \omega$, in particular, constitutes no more than a lower deductive bound to this hierarchy and differs considerably from its companions. A long standing problem in the literature (open for more than 35 years) is (...) to define the deductive limit to this hierarchy, that is, its greatest lower deductive bound. The calculus $ \mathcal {C}$min, stronger than $ \mathcal {C}$$\scriptstyle \omega$, is first presented as a step toward this limit. As an alternative to the bivaluation semantics of $ \mathcal {C}$min presented thereupon, possible-translations semantics are then introduced and suggested as the standard technique both to give this calculus a more reasonable semantics and to derive some interesting properties about it. Possible-translations semantics are then used to provide both a semantics and a decision procedure for $ \mathcal {C}$Lim, the real deductive limit of da Costa's hierarchy. Possible-translations semantics also make it possible to characterize a precise sense of duality: as an example, $ \mathcal {D}$min is proposed as the dual to $ \mathcal {C}$min. (shrink)

This paper concerns the paraconsistent logic LPQ⊃,F and an application of it in the area of relational database theory. The notions of a relational database, a query applicable to a relational database, and a consistent answer to a query with respect to a possibly inconsistent relational database are considered from the perspective of this logic. This perspective enables among other things the definition of a consistent answer to a query with respect to a possibly inconsistent database without resort to database (...) repairs. In an earlier paper, LPQ⊃,F is presented with a sequent-style natural deduction proof system. In this paper, a sequent calculus proof system is presented instead because such proof systems are generally considered more suitable as the basis of proof search procedures than natural deduction proof systems and proof search procedures can serve as the core of algorithms for computing consistent answers to queries. (shrink)

. A paraconsistent logic is a logical system that attempts to deal with contradictions in a discriminating way. In an earlier paper [Notre Dame J. Form. Log. 49, 401–424], we developed the systems of weakening of intuitionistic negation logic, called and, in the spirit of da Costa's approach by preserving, differently from da Costa, the fundamental properties of negation: antitonicity, inversion and additivity for distributive lattices. Taking into account these results, we make some observations on the modified systems of and, (...) and their paraconsistent properties. (shrink)

In this paper, the notion of degree of inconsistency is introduced as a tool to evaluate the sensitivity of the Full Bayesian Significance Test (FBST) value of evidence with respect to changes in the prior or reference density. For that, both the definition of the FBST, a possibilistic approach to hypothesis testing based on Bayesian probability procedures, and the use of bilattice structures, as introduced by Ginsberg and Fitting, in paraconsistent logics, are reviewed. The computational and theoretical advantages of using (...) the proposed degree of inconsistency based sensitivity evaluation as an alternative to traditional statistical power analysis is also discussed. (shrink)

Paraconsistent quantum logics are weak forms of quantum logic, where the noncontradiction and the excluded-middle laws are violated. These logics find interesting applications in the operational approach to quantum mechanics. In this paper, we present an axiomatization, a Kripke-style, and an algebraic semantical characterization for two forms of paraconsistent quantum logic. Further developments are contained in Giuntini and Greuling's paper in this issue.

In this paper two systems of AGM-like Paraconsistent Belief Revision are overviewed, both defined over Logics of Formal Inconsistency (LFIs) due to the possibility of defining a formal consistency operator within these logics. The AGM° system is strongly based on this operator and internalize the notion of formal consistency in the explicit constructions and postulates. Alternatively, the AGMp system uses the AGM-compliance of LFIs and thus assumes a wider notion of paraconsistency - not necessarily related to the notion of formal (...) consistency. (shrink)

Digraphs provide an alternative syntax for propositional logic, with digraph kernels corresponding to classical models. Semikernels generalize kernels and we identify a subset of well-behaved semikernels that provides nontrivial models for inconsistent theories, specializing to the classical semantics for the consistent ones. Direct (instead of refutational) reasoning with classical resolution is sound and complete for this semantics, when augmented with a specific weakening which, in particular, excludes Ex Falso. Dropping all forms of weakening yields reasoning which also avoids typical fallacies (...) of relevance. (shrink)

A classical paraconsistent logic, which is regarded as a modified extension of first-degree entailment logic, is introduced as a Gentzen-type sequent calculus. This logic can simulate the classical negation in classical logic by paraconsistent double negation in CP. Theorems for syntactically and semantically embedding CP into a Gentzen-type sequent calculus LK for classical logic and vice versa are proved. The cut-elimination and completeness theorems for CP are also shown using these embedding theorems. Similar results are also obtained for an intuitionistic (...) paraconsistent logic, and several versions of Glivenko and Gödel-Gentzen translation theorems are proved for CP and IP. (shrink)

Some strategies to turn any logic into a paraconsistent system are examined. In the environment of universal logic, we show how to paraconsistentize logics at the abstract level using a transformation in the class of all abstract logics called paraconsistentization by consistent sets. Moreover, by means of the notions of paradeduction and paraconsequence we go on applying the process of changing a logic converting it into a paraconsistent system. We also examine how this transformation can be performed using multideductive abstract (...) logics. To conclude, the conceptual notion paraconsistent orbit of a logic is proposed. (shrink)

This chapter contains sections titled: What is Paraconsistency? Motives for Paraconsistency The Sources of Trivialization A Natural Taxonomy for Paraconsistent Logics Paraconsistent Logics Current Issues.

In the context of non-classical logics, many philosophers have been particularly interested in the paraconsistent logics. In addition to traditional definitions, in recent years, new ways of characterizing the notion of paraconsistency have been proposed. In all of these definitions the rule or the meta-rule of explosion is abandoned. In this article, I present those definitions and evaluate the role that the negation and the transitivity play in each of them. Finally, I propose a new definition of paraconsistency which I (...) call total paraconsistency and show that the rule of weakening plays a crucial role in all of the characterizations of such a concept. (shrink)

In this article, our aim is to take a step towards a full understanding of the notion of paraconsistency in the context of metainferential logics. Following the work initiated by Barrio et al. [2018], we will consider a metainferential logic to be paraconsistent whenever the metainferential version of Explosion is invalid. However, our contribution consists in modifying the definition of meta-Explosion by extending the standard framework and introducing a negation for inferences and metainferences. From this new perspective, Tarskian paraconsistent logics (...) such as LP will not turn out to be metainferentially paraconsistent, in contrast to, for instance, non-transitive logics like ST. Finally, we will end up by defining a logic which is metainferentially paraconsistent at every level, and discussing whether this logic is uniform through translations. (shrink)

This article is a preliminary presentation of conjunctive paraconsistency, the claim that there might be non-explosive true contradictions, but contradictory propositions cannot be considered separately true. In case of true ‘p and not p’, the conjuncts must be held untrue, Simplification fails. The conjunctive approach is dual to non-adjunctive conceptions of inconsistency, informed by the idea that there might be cases in which a proposition is true and its negation is true too, but the conjunction is untrue, Adjunction fails. While (...) non-adjunctivism is a well-known option, the other view is not so much studied nowadays, but it was not unknown in the tradition, and there are some positive suggestions, in recent literature, that the position is plausible and deserves to be developed. The article compares conjunctivism, non-adjunctivism and dialetheism, then focuses on some possible justifications, costs and benefits of the conjunctive view. (shrink)

The study of the theory of operators over modal pseudocomplemented De Morgan algebras was begun in papers [20] and [21]. In this paper, we introduce and study the class of modal pseudocomplemented De Morgan algebras enriched by a k-periodic automorphism -algebras). We denote by \ the automorphism where k is a positive integer. For \, the class coincides with the one studied in [20] where the automorphism works as a new unary operator which can be considered as a negation. In (...) the first place, we develop an algebraic study of the class of \-algebras; as consequence, we prove the class \-algebras is a semisimple variety and determine the generating algebras. After doing the algebraic study and using these properties, we built two families of sentential logics that we denote with \ and \ for every k. \ is a 1-assertional logic and \ is the degree-preserving logic both associated with the class of \-algebras. Working over these logics, we prove that \ is paraconsistent with respect to the de Morgan negation \, which is protoalgebraic and finitely equivalential but not algebraizable. In contrast, we prove that \ is algebraizable, sharing the same theorems with \, but not paraconsistent with respect to \. Furthermore, we show that \ and \ are paracomplete logics with respect to \ and \ and paraconsistent logics with respecto to \, for every k. (shrink)

The constructions of sequential calculi are based on the idea of application to the deduction process not only single logical constants but complexes of them as well. Surely, making use of this idea is not obligatory in paraconsistent logic. Nevertheless, using it in this field gives us a convenient tool for seeking proofs in formulation of many paraconsistent logics. Each of sequential propositional logics discussed in this paper is obtained as a result of a transformation of a starting calculus GCL (...) – one of sequential formalizations of classical propositional logic. Let α be a standard propositional language with a vocabulary {p1, p2, p3, . . . , −,∨, &, ⊃,¬,), we shall denote any formulas of α and by the letters Γ,∆, Σ, Θ finite sets of α-formulas. (shrink)

The terms “model” and “model-building” have been used to characterize the field of formal philosophy, to evaluate philosophy’s and philosophical logic’s progress and to define philosophical logic itself. A model is an idealization, in the sense of being a deliberate simplification of something relatively complex in which several important aspects are left aside, but also in the sense of being a view too perfect or excellent, not found in reality, of this thing. Paraconsistent logic is a branch of philosophical logic. (...) It is however not clear how paraconsistent logic can be seen as model-building. What exactly is modeled? In this paper I adopt the perspective of looking at a particular instance of paraconsistent logic—paranormal modal logic—which might be seen as a model of a specific kind of agent: inductive agents. After ntroducing what I call the highlevel and low-level models of inductive agents, I analyze the extent to which the above-mentioned idealizing features of model-building appear in paranormal modal logic and how they affect its philosophical significance. (shrink)

This paper begins an analysis of the real line using an inconsistency-tolerant (paraconsistent) logic. We show that basic field and compactness properties hold, by way of novel proofs that make no use of consistency-reliant inferences; some techniques from constructive analysis are used instead. While no inconsistencies are found in the algebraic operations on the real number field, prospects for other non-trivializing contradictions are left open.

The idea that the phenomenon of vagueness might be modelled by a paraconsistent logic has been little discussed in contemporary work on vagueness, just as the idea that paraconsistent logics might be fruitfully applied to the phenomenon of vagueness has been little discussed in contemporary work on paraconsistency. This is prima facie surprising given that the earliest formalisations of paraconsistent logics presented in Jaśkowski(1948) and Halldén(1949) were presented as logics of vagueness. One possible explanation for this is that, despite initial (...) advocacy by pioneers of paraconsistency, the prospects for a paraconsistent account of vagueness are so poor as to warrant little further consideration. In this paper we look at the reasons that might be offered in defence of this negative claim. As we shall show, they are far from compelling. Paraconsistent accounts of vagueness deserve further attention. (shrink)

Classical propositional logic can be characterized, indirectly, by means of a complementary formal system whose theorems are exactly those formulas that are not classical tautologies, i.e., contradictions and truth-functional contingencies. Since a formula is contingent if and only if its negation is also contingent, the system in question is paraconsistent. Hence classical propositional logic itself admits of a paraconsistent characterization, albeit “in the negative”. More generally, any decidable logic with a syntactically incomplete proof theory allows for a paraconsistent characterization of (...) its set of theorems. This, we note, has important bearing on the very nature of paraconsistency as standardly characterized. (shrink)

In a famous and controversial paper, B. H. Slater has argued against the possibility of paraconsistent logics. Our reply is centred on the distinction between two aspects of the meaning of a logical constant *c* in a given logic: its operational meaning, given by the operational rules for *c* in a cut-free sequent calculus for the logic at issue, and its global meaning, specified by the sequents containing *c* which can be proved in the same calculus. Subsequently, we use the (...) same strategy to counter Quine's meaning variance argument against deviant logics. In a nutshell, we claim that genuine rivalry between (similar) logics *L* and *L'* is possible whenever each constant in *L* has the same operational meaning as its counterpart in *L'* although differences in global meaning arise in at least one case. (shrink)

The prepositional calculiC n , 1 n introduced by N.C.A. da Costa constitute special kinds of paraconsistent logics. A question which remained open for some time concerned whether it was possible to obtain a Lindenbaum''s algebra forC n . C. Mortensen settled the problem, proving that no equivalence relation forC n . determines a non-trivial quotient algebra.The concept of da Costa algebra, which reflects most of the logical properties ofC n , as well as the concept of paraconsistent closure system, (...) are introduced in this paper. (shrink)

Paraconsistent logics may be viewed as one of the last elementsin a series of rapid developments in science in the 19th and early 20th c.,triggered by the appearance of non-Euclidean geometries. The philosophyof conventionalism, which gave a metatheoretical framework to the basicchanges involved, may also help in evaluating the truth import of logic and in determining its relation to ontology.

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

“Paraconsistent” means “beyond the consistent” [3, 15]. Paraconsistent logics tolerate inconsistencies in a way that traditional logics do not. In a paraconsistent logic, the inference of explosion A, ∼AB is rejected. This may be for any of a number of reasons [16]. For proponents of relevance [1, 2] the argument has gone awry when we infer an irrelevant B from the inconsistent premises. Those who argue that inconsistent theories may have some logical content but do not commit us to everything, (...) have reason to think that these theories are closed under a relation of paraconsistent logical consequence [12, 18]. Another reason to adopt a paraconsistent logic is more extreme. You may take the world to be inconsistent [14], and a true theory incorporating this inconsistency must be governed by a paraconsistent logic. (shrink)

This chapter contains sections titled: Introduction Paraconsistent Logic and Latin America Thinking about Logic The Nature of Paraconsistent Logic A History of Paraconsistent Logic Philosophical Aspects of Paraconsistent Logic References Further Reading.

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)