Showing papers on "Paraconsistent logic published in 2015"

Pure Inductive Logic

Abstract: Pure inductive logic is the study of rational probability treated as a branch of mathematical logic. This monograph, the first devoted to this approach, brings together the key results from the past seventy years plus the main contributions of the authors and their collaborators over the last decade to present a comprehensive account of the discipline within a single unified context. The exposition is structured around the traditional bases of rationality, such as avoiding Dutch Books, respecting symmetry and ignoring irrelevant information. The authors uncover further rationality concepts, both in the unary and in the newly emerging polyadic languages, such as conformity, spectrum exchangeability, similarity and language invariance. For logicians with a mathematical grounding, this book provides a complete self-contained course on the subject, taking the reader from the basics up to the most recent developments. It is also a useful reference for a wider audience from philosophy and computer science.

Classical Negation and Expansions of Belnap---Dunn Logic

Abstract: 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 Beziau and the logic of classical implication and a paraconsistent de Morgan negation of Zaitsev.

The logics of strict-tolerant logic

Abstract: Adding a transparent truth predicate to a language completely governed by classical logic is not possible. The trouble, as is well-known, comes from paradoxes such as the Liar and Curry. Recently, Cobreros, Egre, Ripley and van Rooij have put forward an approach based on a non-transitive notion of consequence which is suitable to deal with semantic paradoxes while having a transparent truth predicate together with classical logic. Nevertheless, there are some interesting issues concerning the set of metainferences validated by this logic. In this paper, we show that this logic, once it is adequately understood, is weaker than classical logic. Moreover, the logic is in a way similar to the paraconsistent logic LP.

Non-Classical Logics

Abstract: Classical logic is usually viewed as a masterpiece of the human mind. It serves as the basic logic of classical mathematics and almost all other sciences. However, despite its long history and venerable reputation, it is not an ideal logic. It faces serious objections which demonstrate that as a practical tool, it is inadequate. A logic is an inadequate tool if its practical use generates counterintuitive and absurd situations that are highly incompatible with common sense and natural language. Classical logic and its modal extensions that we studied in the preceding chapter are just such logics. A few examples will suffice to prove the point.

Three-Valued Paraconsistent Propositional Logics

Abstract: Three-valued matrices provide the simplest semantic framework for introducing paraconsistent logics. This paper is a comprehensive study of the main properties of propositional paraconsistent three-valued logics in general, and of the most important such logics in particular. For each logic in the latter group, we also provide a corresponding cut-free Gentzen-type system.

Generalized algebra-valued models of set theory

Abstract: We generalize the construction of lattice-valued models of set theory due to Takeuti, Titani, Kozawa and Ozawa to a wider class of algebras and show that this yields a model of a paraconsistent logic that validates all axioms of the negation-free fragment of Zermelo-Fraenkel set theory.

Paraconsistent Intelligent-Based Systems: New Trends in the Applications of Paraconsistency

Abstract: This book presents some of the latest applications of new theories based on the concept of paraconsistency and correlated topics in informatics, such as pattern recognition (bioinformatics), robotics, decision-making themes, and sample size. Each chapter is self-contained, and an introductory chapter covering the logic theoretical basis is also included. The aim of the text is twofold: to serve as an introductory text on the theories and applications of new logic, and as a textbook for undergraduate or graduate-level courses in AI. Today AI frequently has to cope with problems of vagueness, incomplete and conflicting (inconsistent) information. One of the most notable formal theories for addressing them is paraconsistent (paracomplete and non-alethic) logic.

Paraconsistency properties in degree-preserving fuzzy logics

Abstract: Paraconsistent logics are specially tailored to deal with inconsistency, while fuzzy logics primarily deal with graded truth and vagueness. Aiming to find logics that can handle inconsistency and graded truth at once, in this paper we explore the notion of paraconsistent fuzzy logic. We show that degree-preserving fuzzy logics have paraconsistency features and study them as logics of formal inconsistency. We also consider their expansions with additional negation connectives and first-order formalisms and study their paraconsistency properties. Finally, we compare our approach to other paraconsistent logics in the literature.

Towards a philosophical understanding of the logics of formal inconsistency

Abstract: In this paper we present a philosophical motivation for the logics of formal inconsistency, a family of paraconsistent logics whose distinctive feature is that of having resources for expressing the notion of consistency within the object language in such a way that consistency may be logically independent of non-contradiction. We defend the view according to which logics of formal inconsistency may be interpreted as theories of logical consequence of an epistemological character. We also argue that in order to philosophically justify paraconsistency there is no need to endorse dialetheism, the thesis that there are true contradictions. Furthermore, we show that mbC, a logic of formal inconsistency based on classical logic, may be enhanced in order to express the basic ideas of an intuitive interpretation of contradictions as conflicting evidence.

Strong Three-Valued Paraconsistent Logics

Abstract: After describing the two formulations of the principle of non contradiction in modern logic \(T \vdash \lnot (p \wedge \lnot p)\) (NC) and \(T, p, \lnot p \vdash q\) (EC) and explaining that three-valued matrices can be used to easily prove their independence, we investigate the possibilities to construct strong paraconsistent negations, i.e., for which neither (NC) nor (EC) holds, using three-valued logical matrices.

Introduction to Annotated Logics: Foundations for Paracomplete and Paraconsistent Reasoning

Abstract: This book is written as an introduction to annotated logics. It provides logical foundations for annotated logics, discusses some interesting applications of these logics and also includes the authors' contributions to annotated logics. The central idea of the book is to show how annotated logic can be applied as a tool to solve problems of technology and of applied science. The book will be of interest to pure and applied logicians, philosophers and computer scientists as a monograph on a kind of paraconsistent logic. But, the layman will also take profit from its reading.

Paraconsistent Logic

Abstract: 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.

Infinity and a Critical View of Logic

Abstract: The paper explores the view that in mathematics, in particular where the infinite is involved, the application of classical logic to statements involving the infinite cannot be taken for granted. L. E. J. Brouwer’s well-known rejection of classical logic is sketched, and the views of David Hilbert and especially Hermann Weyl, both of whom used classical logic in their mathematical practice, are explored. We inquire whether arguments for a critical view can be found that are independent of constructivist premises and consider the entanglement of logic and mathematics. This offers a convincing case regarding second-order logic, but for first-order logic, it is not so clear. Still, we ask whether we understand the application of logic to the higher infinite better than we understand the higher infinite itself.

On the Philosophy and Mathematics of the Logics of Formal Inconsistency

Abstract: The aim of this text is to present the philosophical motivations for the Logics of Formal Inconsistency (LFIs), along with some relevant technical results. The text is divided into two main parts (besides a short introduction). In Sect. 3.2, we present and discuss philosophical issues related to paraconsistency in general, and especially to logics of formal inconsistency. We argue that there are two basic and philosophically legitimate approaches to paraconsistency that depend on whether the contradictions are understood ontologically or epistemologically. LFIs are suitable to both options, but we emphasize the epistemological interpretation of contradictions. The main argument depends on the duality between paraconsistency and paracompleteness. In a few words, the idea is as follows: just as excluded middle may be rejected by intuitionistic logic due to epistemological reasons, explosion may also be rejected by paraconsistent logics due to epistemological reasons. In Sect. 3.3, some formal systems and a few basic technical results about them are presented.

Liberating Paraconsistency from Contradiction

Abstract: 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.

Inconsistent boundaries

Abstract: Mereotopology is a theory of connected parts. The existence of boundaries, as parts of everyday objects, is basic to any such theory; but in classical mereotopology, there is a problem: if boundaries exist, then either distinct entities cannot be in contact, or else space is not topologically connected (Varzi in Nous 31:26–58, 1997). In this paper we urge that this problem can be met with a paraconsistent mereotopology, and sketch the details of one such approach. The resulting theory focuses attention on the role of empty parts, in delivering a balanced and bounded metaphysics of naive space.

Existence Assumptions and Logical Principles: Choice Operators in Intuitionistic Logic

Abstract: Hilbert’s choice operators τ and e, when added to intuitionistic logic, strengthen it. In the presence of certain extensionality axioms they produce classical logic, while in the presence of weaker decidability conditions for terms they produce various superintuitionistic intermediate logics. In this thesis, I argue that there are important philosophical lessons to be learned from these results. To make the case, I begin with a historical discussion situating the development of Hilbert’s operators in relation to his evolving program in the foundations of mathematics and in relation to philosophical motivations leading to the development of intuitionistic logic. This sets the stage for a brief description of the relevant part of Dummett’s program to recast debates in metaphysics, and in particular disputes about realism and anti-realism, as closely intertwined with issues in philosophical logic, with the acceptance of classical logic for a domain reflecting a commitment to realism for that domain. Then I review extant results about what is provable and what is not when one adds epsilon to intuitionistic logic, largely due to Bell and DeVidi, and I give several new proofs of intermediate logics from intuitionistic logic+e without identity. With all this in hand, I turn to a discussion of the philosophical significance of choice operators. Among the conclusions I defend are that these results provide a finer-grained basis for Dummett’s contention that commitment to classically valid but intuitionistically invalid principles reflect metaphysical commitments by showing those principles to be derivable from certain existence assumptions; that Dummett’s framework is improved by these results as they show that questions of realism and anti-realism are not an “all or nothing” matter, but that there are plausibly metaphysical stances between the poles of anti-realism (corresponding to acceptance just of intutionistic logic) and realism (corresponding to acceptance of classical logic), because different sorts of ontological assumptions yield intermediate rather than classical logic; and that these intermediate positions between classical and intuitionistic logic link up in interesting ways with our intuitions about issues of objectivity and reality, and do so usefully by linking to questions around intriguing everyday concepts such as “is smart,” which I suggest involve a number of distinct dimensions which might themselves be objective, but because of their multivalent structure are themselves intermediate between being objective and not. Finally, I discuss the implications of these results for ongoing debates about the status of arbitrary and ideal objects in the foundations of logic, showing among other things that much of the discussion is flawed because it does not recognize the degree to which the claims being made depend on the presumption that one is working with a very strong (i.e., classical) logic.

Chrysippus’ Indemonstrables and Mental Logic

Abstract: Stoic logic assumes fi ve inference schemata attributed to Chrysippus of Soli. Those schemata are the well-known indemonstrables. A problem related to them can be that, according to standard propositional calculus, only one of them, modus ponens, is clearly indemonstrable. Nevertheless, I try to show in this paper that the mental logic theory enables to understand why the Stoics considered such schemata to be basic kinds of arguments. Following that theory, four of them can be linked to ‘Core Schemata’ of mental logic and the only one that is more controversial is modus tollens. However, as I also comment, some assumptions of Stoic philosophy, which can be interpreted from the mental logic theory, can explain why this last argument was included into the set of the indemonstrables as well.

Tutorial on Inconsistency-Adaptive Logics

Abstract: This paper contains a concise introduction to a few central features of inconsistency-adaptive logics. The focus is on the aim of the program, on logics that may be useful with respect to applications, and on insights that are central for judging the importance of the research goals and the adequacy of results. Given the nature of adaptive logics, the paper may be read as a peculiar introduction to defeasible reasoning.

An Approach to Paraconsistent Multivalued Logic: Evaluation by Complex Truth Values

Abstract: The main purpose of the paper is to connect some kind of dialetheism to the use of complex truth values, with new definitions of basic truth-functional connectives that allow for p, \(\backslash \)not p to both be true. ‘True’ is interpreted as \(\left| p \right| = 1\), ‘False’ as \(\left| p \right| = 0\); other values are dispensed with. New definitions of basic truth-functional connectives then allow for “p and not p” to be true. A propositional logic is discussed with the set of connectives including negation, conjunction, disjunction, implication, concordance, discordance, complementary, and equivalence. The authors introduce truth values of propositions, which belong to a subset E, of an uncountable semi-ring F and valuations of propositions, which can be obtained from truth values with the help of a function \(V:E \rightarrow \left[ {0,1} \right] \) satisfying simple properties. Finally, a paraconsistent Boolean logic is introduced.

Paraconsistency and Sette’s calculus P1

Abstract: In 1973, Sette presented a calculus, called P1, which is recognized as one of the most remarkable paraconsistent systems. The aim of this paper is to propose a new axiomatization of P1. The axiom schemata are chosen to show that P1 behaves in a paraconsistent way only at the atomic level, i.e. the rule: α , ~ α / β holds in P1 only if α is not a propositional variable.

The relativity and universality of logic

Abstract: After recalling the distinction between logic as reasoning and logic as theory of reasoning, we first examine the question of relativity of logic arguing that the theory of reasoning as any other science is relative. In a second part we discuss the emergence of universal logic as a general theory of logical systems, making comparison with universal algebra and the project of mathesis universalis. In a third part we critically present three lines of research connected to universal logic: logical pluralism, non-classical logics and cognitive science.

Logic as metaphysics

Abstract: How should logic be understood? I outline and defend the view that I call “realism” about logic. This is a thesis with two elements: first, that certain logical constants are features or constituents of the world; and, second, that logical constants concepts and words have the function of referring to these features or constituents of the world.

Logics and Falsifications: A New Perspective on Constructivist Semantics

Abstract: Introduction.- Part 1. Background.- Introduction to Part One.- Constructivism.- Intuitionism.- Gaps, Gluts and Paraconsistency.- Part 2. Falsifications.- Introduction to Part Two.- From Proofs to Verifications, and on to Falsifications.- Falsificationism.-Part 3. Logics.- Introduction to Part Three.- Stage Five: Pure Falsificationism and Dual Intuitionistic Logic.- Stage Two: Expanded Verificationism and the Logic N3.- Stage Four.-Stage Three: Hybrid Strategies.- Summary.- Appendix.

The Relation Between Logic, Set Theory and Topos Theory as it Is Used by Alain Badiou

Abstract: Topos theory plays, in Alain Badiou’s philosophical model, the role of inner logic of mathematics, given its power to explore possible mathematical universes; whereas set theory, because of its axiomatics, plays the role of ontology. However, in category theory, which is a vaster theory, topos theory embodies a particular axiomatic choice, the fundamental consequence of which consists in imposing an internal intuitionist logic, that is a non-contradictory logic which gets rid of the principle of excluded middle. Category theory shows that the dual axiomatic choice exists, namely the one imposing, this time, a logic of the excluded middle which accepts true contradictions without deducing from them everything, and this is called a paraconsistent logic. Therefore, after recalling the basics of category and topos theory necessary to demonstrate the categorical duality of paracompleteness (i.e. intuitionism) and paraconsistency, we will be able to introduce into Badiou’s thought category theory seen as a logic of the possible ontologies, a logic which demonstrates the strong symmetry of the axioms of excluded middle and of non-contradiction.

Trivial Dialetheism and the Logic of Paradox

Abstract: In this paper we explain that the paraconsistent logic LP (Logic of Paradox) promoted by Graham Priest can only be supported by trivial dialetheists, i.e., those who believe that all sentences are dialetheias.

None of the Above: The Catuṣkoṭi in Indian Buddhist Logic

Abstract: The catuṣkoṭi (Greek: tetralemma; English: four corners) is a venerable principle of Indian logic, which has been central to important aspects of reasoning in the Buddhist tradition. What, exactly, it is, and how it is applied, are, however, moot—though one thing that does seem clear is that it has been applied in different ways at different times and by different people. Of course, Indian logicians did not incorporate the various interpretations of the principle in anything like a theory of validity in the modern Western sense; but the tools of modern non-classical logic show exactly how to do this. The tools are those of the paraconsistent logic of First Degree Entailment and some of its modifications.