Showing papers on "Paraconsistent logic published in 2007"

Paraconsistent Logics and Paraconsistency

Abstract: Publisher Summary This chapter discusses paraconsistent logics (PL) and paraconsistency. PL are the logics of inconsistent but nontrivial theories. A deductive theory is paraconsistent if its underlying logic is paraconsistent. A theory is inconsistent if there is a formula (a grammatically well-formed expression of its language) such that the formula and its negation are both theorems of the theory; otherwise, the theory is called consistent. A theory is trivial if all formulas of its language are theorems. In a trivial theory “everything” (expressed in its language) can be proved. If the underlying logic of a theory is classical logic, or even any of the standard logical systems such as intuitionistic logic, inconsistency entails triviality, and conversely. This chapter discusses da Costa's C-logics. This chapter elaborates on paraconsistent set theories, and shows, in particular, how they accommodate inconsistent objects, such as the Russell set. Ja´skowski's discussive logic is examined, and it is showed how it can be used in the formulation of the concept of partial truth. The chapter also examines annotated logic, and some of its applications.

Paraconsistency and dialetheism

Abstract: This chapter discusses paraconsistent logic, in which contradictions do not entail everything. However, the roots of paraconsistency lie deep in the history of logic, its modern developments date to just before the middle of the 20th century. Since then, the paraconsistent logic have been proposed and constructed for many and for different reasons. The most philosophically challenging of these reasons is dialetheism, the view that some contradictions are true. The chapter discusses the history of paraconsistency and the history of dialetheism. The chapter also discusses the modern developments of paraconsistency and dialetheism, those since about 1950. Some important issues that bear on paraconsistency, or on which paraconsistency bears the foundations of mathematics, the notion of negation, and rationality are discussed in the chapter.

The many valued and nonmonotonic turn in logic

Abstract: Preface List of Contributors Chapter 1. Many-valued Logic (Grzegorz Malinowski) Chapter 2. Paraconsistent Logic: Preservationist Variations (Bryson Brown) Chapter 3. Paraconsistent Logic: Dialethic Variations (Graham Priest) Chapter 4. Quantum Logic (M. Dalla Chiara, Roberto Giuntini and Miklos Redei) Chapter 5. Logic of Vagueness (Dominic Hyde) Chapter 6. Fuzzy Logic (Didier Dubois, Henri Prade and Lluis Godo) Chapter 7. Non-monotonic Logic (Karl Schlechta) Chapter 8. Default Logic (Grigoris Antoniou and Kewen Wang) Chapter 9. Non-monotonic Reasoning and Belief Change (Alexander Bochman) Chapter 10. Free Logic (Carl Posy) Index

Measuring Inconsistency for Description Logics Based on Paraconsistent Semantics.

Abstract: In this paper, we present an approach for measuring inconsistency in a knowledge base. We first define the degree of inconsistency using a four-valued semantics for the description logic $\mathcal{ALC}$. Then an ordering over knowledge bases is given by considering their inconsistency degrees. Our measure of inconsistency can provide important information for inconsistency handling.

Logica Universalis: Towards a General Theory of Logic

Abstract: Universal Logic is not a new logic, but a general theory of logics, considered as mathematical structures. The name was introduced about ten years ago, but the subject is as old as the beginning of modern logic: Alfred Tarski and other Polish logicians such as Adolf Lindenbaum developed a general theory of logics at the end of the 1920s based on consequence operations and logical matrices. The subject was revived after the flowering of thousands of new logics during the last thirty years: there was a need for a systematic theory of logics to put some order in this chaotic multiplicity. This book contains recent works on universal logic by first-class researchers from all around the world. The book is full of new and challenging ideas that will guide the future of this exciting subject. It will be of interest for people who want to better understand what logic is. Tools and concepts are provided here for those who want to study classes of already existing logics or want to design and build new ones.

An Intelligent Action Control System Based on Extended Vector Annotated Logic Program and its Hardware Implementation

Abstract: Vazious intelligent control systems for autonomous machines have been developed, and such intelligent control systems often require real time processing. However, it is difficult for software control to realize fast real time processing. Therefore, intelligent control systems that can be easily hardware implemented are expected. We propose a theoretical framework for intelligent autonomous action control systems based on a paraconsistent logic program called an EVALPSN. The paraconsistent logic program EVALPSN can deal with defeasible deontic reasoning and conflicts, and is appropriate for hazdware implementation. In order to show the feature of the intelligent action control, we take a virtual beetle robot action control system as an example. Moreover, we introduce the electronic circuit implementation for the robot action control system with experimental results.

The basic constructive logic for a weak sense of consistency

Abstract: In this paper, consistency is understood as the absence of the negation of a theorem, and not, in general, as the absence of any contradiction. We define the basic constructive logic BKc1 adequate to this sense of consistency in the ternary relational semantics without a set of designated points. Then we show how to define a series of logics extending BKc1 within the spectrum delimited by contractionless minimal intuitionistic logic. All logics defined in the paper are paraconsistent logics.

Many-valued non-deterministic semantics for first-order logics of formal (in)consistency

Abstract: A paraconsistent logic is a logic which allows non-trivial inconsistent theories. One of the oldest and best known approaches to the problem of designing useful paraconsistent logics is da Costa's approach, which seeks to allow the use of classical logic whenever it is safe to do so, but behaves completely differently when contradictions are involved. da Costa's approach has led to the family of Logics of Formal (In)consistency (LFIs). In this paper we provide non-deterministic semantics for a very large family of first-order LFIs (which includes da Costa's original system C1*, as well as thousands of other logics). We show that our semantics is effective and modular, and we use this effectiveness to derive some important properties of logics in this family.

Dynamic Description Logic: Embracing Actions into Description Logic.

Abstract: We present a dynamic description logic D-ALCO@ for representing knowledge about dynamic application domains. D-ALCO@ is a combination of a typical action theory and the description logic ALCO@, in such a way that actions are treated as citizens of the logic. Actions of D-ALCO@ are explicitly specified with the help of formulas, and are then used in the construction of concepts and formulas. Based on a regression operator introduced to deal with actions, we provide a tableau-based decision algorithm for this logic.

Electric Power Systems Contingencies Analysis by Paraconsistent Logic Application

Abstract: In this paper we present paraconsistent logic application at electric power system contingency analysis with risks identification. Para-consistent Logic is a non-classic logic whose foundations allow for the treatment of contradictions without invalidating the conclusions through algorithms named PANs - para-consistent analysis nodes. In the equations of the PANs are calculated signals representing restrictions, risks and configurations of electric power systems networks. The "pre- Fault state" represented by the resulting evidence degree from PAN is evaluated together with the "post-fault" information in a contingency evaluation. In that way the occurrence type and their parameters are classified by PANet with the purpose of offering an optimized re-establishment sequence to the power system. This method is being validated through off-line tests applied at the Eletropaulo-Eletricidade de Sao Paulo, an electric power distribution utility company of Sao Paulo state, Brazil, which has a distribution network and a model substation of medium range.

Large-scale organizational computing requires unstratified reflection and strong paraconsistency

Abstract: Organizational Computing is a computational model for using the principles, practices, and methods of human organizations. Organizations of Restricted Generality (ORGs) have been proposed as a foundation for Organizational Computing. ORGs are the natural extension of Web Services, which are rapidly becoming the overwhelming standard for distributed computing and application interoperability in Organizational Computing. The thesis of this paper is that large-scale Organizational Computing requires reflection and strong paraconsistency for organizational practices, policies, and norms. Strong paraconsistency is required because the practices, policies, and norms of large-scale Organizational Computing are pervasively inconsistent. By the standard rules of logic, anything and everything can be inferred from an inconsistency, e.g., "The moon is made of green cheese." The purpose of strongly paraconsistent logic is to develop principles of reasoning so that irrelevances cannot be inferred from the fact of inconsistency while preserving all natural inferences that do not explode in the face of inconsistency. Reflection is required in order that the practices, policies, and norms can mutually refer to each other and make inferences. Reflection and strong paraconsistency are important properties of Direct Logic [Hewitt 2007] for large software systems. Godel first formalized and proved that it is not possible to decide all mathematical questions by inference in his 1st incompleteness theorem. However, the incompleteness theorem (as generalized by Rosser) relies on the assumption of consistency! This paper proves a generalization of the Godel/Rosser incompleteness theorem: theories of Direct Logic are incomplete. However, there is a further consequence. Although the semi-classical mathematical fragment of Direct Logic is evidently consistent, since the Godelian paradoxical proposition is selfprovable, every theory in Direct Logic has an inconsistency!

Aristotle's Square Revisited to Frame Discovery Science

Abstract: The paper attempts to give a formal framework to capture the entire process of scientific discovery including hypothesis formation, reasoning, identifying contradictions, peer reviewing, reformulating and so on. Data mining can be seen as one step in this complex process of interactive learning of an empirical theory This paper uses the terminology from paraconsistent logic and paracomplete logic that extends Aristotle square in a hypercube of oppositions which defines or substantiates any step of the discovery process. The central formal notions are validated on a mathematical scientific discovery game, and an industrial application in the field of Drug Discovery illustrates how the presented framework combines different learning processes to predict pharmaco-kinetic properties (ADME-T) and adverse side effects of therapeutic drug molecules.

Philosophy of logic

Abstract: General Preface (Dov Gabbay, Paul Thagard and John Woods) Preface List of Contributors Introduction: Philosophy of Logic Today (Dale Jacquette) What is Logic? (Jaakko Hintikka and Gabriel Sandu) The Scope and Limits of Logic (Wilfrid Hodges) Logic in Philosophy (Johan van Benthem) Informal Logic and the Concept of Argument (David Hitchcock) On the Relation of Informal to Symbolic Logic (Dale Jacquette) Vagueness and the Logic of Ordinary Language (Roy A. Sorensen) Logic and Semantic Analysis (Ernest Lepore and Matthew Stone) Justificatory Irrelevance of Formal Semantics (Charles F. Kielkopf) A Brief History of Truth (Stewart Candlish and Nic Damnjanovic) Truth and Paradox: A Philosophical Sketch (J.C. Beall) Hilbert's Program Then and Now (Richard Zach) Logicism and its Contemporary Legacy (Herbert Hochberg) Classical Logic's Coming of Age (John W. Dawson, Jr.) Infinity (Peter Fletcher) Lowenheim-Skolem Theorems (Heinz-Dieter Ebbinghaus) The Mathematics of Skolem's Paradox (Timothy Bays) Objectual and Substitutional Interpretations of the Quantifiers (Michael Hand) Many-Valued Logics (Siegfried Gottwald) Relevance Logics (Katalin Bimbo) Paraconsistent Lgoics and Paraconsistency (Newton C.A. da Costa, Otavio Bueno and Decio Krause) Extensional vs Intensional Logic (Jaroslav Peregrin) Logically Possible Worlds and Counterpart Semantics for Modal Logic (Marcus Kracht and Oliver Kutz) Modal Realism and its Roots in Mathematical Realism (Charles S. Chihara) Free Logics (John Nolt) Fictions and their Logic (John Woods) Counterfactuals, Causation, and Preemption (John Collins) Logic, Mathematics, and the Natural Sciences (Neil Tennant) Default Reasoning (Nicholas Rescher) Index

A paraconsistent extension of Sylvan¿s logic

Abstract: We deal with Sylvan’s logic CCω. It is proved that this logic is a conservative extension of positive intuitionistic logic. Moreover, a paraconsistent extension of Sylvan’s logic is constructed, which is also a conservative extension of positive intuitionistic logic and has the property of being decidable. The constructed logic, in which negation is defined via a total accessibility relation, is a natural intuitionistic analog of the modal system S5. For this logic, an axiomatization is given and the completeness theorem is proved.

Extended full computation-tree logics for paraconsistent model checking

Abstract: It is known that the full computation-tree logic CTL * is an important base logic for model checking. The bisimulation theorem for CTL* is known to be useful for abstraction in model checking. In this paper, the bisimulation theorems for two paraconsistent four-valued extensions 4CTL* and 4LCTL* of CTL* are shown, and a translation from 4CTL* into CTL* is presented. By using 4CTL* and 4LCTL*, inconsistency-tolerant and spatiotemporal reasoning can be expressed as a model checking framework.

Paraconsistent Resolution for Four-valued Description Logics.

Abstract: In this paper, we propose an approach to translating any ALC ontology (possible inconsistent) into a logically consistent set of disjunctive datalog rules. We achieve this in two steps: First we give a simple way to make anyALC based ontology 4-valued satisfiable, and then we study a sound and complete paraconsistent ordered-resolution decision procedure for our 4-valuedALC. Our approach can be viewed as a paraconsistent version of KAON2 algorithm.

A logical expression of reasoning

Abstract: A non-monotonic logic, the Logic of Plausible Reasoning (LPR), capable of coping with the demands of what we call complex reasoning, is introduced. It is argued that creative complex reasoning is the way of reasoning required in many instances of scientific thought, professional practice and common life decision taking. For managing the simultaneous consideration of multiple scenarios inherent in these activities, two new modalities, weak and strong plausibility, are introduced as part of the Logic of Plausible Deduction (LPD), a deductive logic specially designed to serve as the monotonic support for LPR. Axiomatics and semantics for LPD, together with a completeness proof, are provided. Once LPD has been given, LPR may be defined via a concept of extension over LPD. Although the construction of LPR extensions is first presented in standard style, for the sake of comparison with existing non-monotonic formalisms, alternative more elegant and intuitive ways for constructing non-monotonic LPR extensions are also given and proofs of their equivalence are presented.

From intuitionistic logic to dynamic operational quantum logic

Abstract: Research within the operational approach to the logical foundations of physics has recently pointed out a new perspective in which quantum logic can be viewed as an intuitionistic logic with an additional operator to capture its essential, i.e., non-distributive, properties. In this paper we will offer an introduction to this approach. We will focus further on why quantum logic has an inherent dynamic nature which is captured in the meaning of “orthomodularity” and on how it motivates physically the introduction of dynamic implication operators, each for which a deduction theorem holds with respect to a dynamic conjunction. As such we can offer a positive answer to the many who pondered about whether quantum logic should really be called a logic. Doubts to answer the question positively were in first instance due to the former lack of an implication connective which satisfies the deduction theorem within quantum logic.

The lattice of extensions of the minimal logic

Abstract: In this article, we survey the results on the lattice of extensions of the minimal logic Lj, a paraconsistent analog of the intuitionistic logic Li. Unlike the well-studied classes of explosive logics, the class of extensions of the minimal logic has an interesting global structure. This class decomposes into the disjoint union of the class Int of intermediate logics, the class Neg of negative logics with a degenerate negation, and the class Par of properly paraconsistent extensions of the minimal logic. The classes Int and Neg are well studied, whereas the study of Par can be reduced to some extent to the classes Int and Neg.

Fictions and their Logic

Abstract: Publisher Summary This chapter discusses fictions and their logic The logic of fiction came into its own in the 1970s with the publication of a number of seminal works Fiction has been a stand-alone research program for logicians only since that time Logicians whose interests are primarily mathematical might see logic of fiction as any mature and coherent system of mathematical logic, in which referring terms and quantifiers for objects of fiction, and truth and closure conditions for fictional sentences are dealt with as part of the logic's overall handling of reference, truth, and inference Three levels of logical treatment of the fiction can be identified as: (1) mathematical, (2) conceptual, and (3) conceptually purpose-built formalization A mathematical logic of the fictional is a first-level logic to the extent that it is an adaptation of an existing logic without principled regard for the conceptual adequacy of the adaptations A philosophical logic of the fictional is a conceptual analysis of the concept of the fictional At third level, logic of fiction is a mathematical structure whose primary target properties involve all those features of a conceptually adequate account of the fictional, as well as the traditional targets of entailment and logical truth

Is Dialetheism an Idealism? The Russellian Fallacy and the Dialetheist’s Dilemma

Abstract: In his famous work on vagueness, Russell named 'fallacy of verbalism' the fallacy that consists in mistaking the properties of words for the properties of things. In this paper, I examine two (clusters of) mainstream paraconsistent logical theories - the non-adjunctive and relevant approaches -, and show that, if they are given a strongly paraconsistent or dialetheic reading, the charge of committing the Russellian Fallacy can be raised against them in a sophisticated way, by appealing to the intuitive reading of their underlying semantics. The meaning of 'intuitive reading' is clarified by exploiting a well-established distinction between pure and applied semantics. If the proposed arguments go through, the dialetheist or strong paraconsistentist faces the following Dilemma: either she must withdraw her claim to have exhibited true contradictions in a metaphysically robust sense - therefore, inconsistent objects and/or states of affairs that make those contradictions true; or she has to give up realism on truth, and embrace some form of anti-realistic (idealistic, or broadly constructivist) metaphysics. Sticking to the second horn of the Dilemma, though, appears to be promising: it could lead to a collapse of the very distinction, commonly held in the literature, between a weak and a strong form of paraconsistency - and this could be a welcome result for a dialetheist.

Varieties of Finitism

Abstract: We consider here several versions of finitism or conceptions that try to work around postulating sets of infinite size. Restricting oneself to the so-called potential infinite seems to rest either on temporal readings of infinity (or infinite series) or on anti-realistic background assumptions. Both these motivations may be considered problematic. The approaches centering on the indefinitely large and the use of schemata would provide a work-around to circumvent usage of actual infinities if we had a clear understanding of how schemata work and where to draw the conceptual line between the indefinitely large and the infinite. Neither of this seems to be clear enough. Versions of strict finitism in contrast provide a clear picture of a (realistic) finite number theory. One can recapture standard arithmetic without being committed to actual infinities. The major problem of them is their usage of a paraconsistent logic with an accompanying theory of inconsistent objects. If we are, however, already using a paraconsistent approach for other reasons (in semantics, epistemology or set theory), we get finitism for free. This strengthens the case for paraconsistency.