Showing papers on "Paraconsistent logic published in 2000"

Frontiers of Paraconsistent Logic

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Extending Classical Logic with Inductive Definitions

Abstract: The goal of this paper is to extend classical logic with a generalized notion of inductive definition supporting positive and negative induction, to investigate the properties of this logic, its relationships to other logics in the area of non-monotonic reasoning, logic programming and deductive databases, and to show its application for knowledge representation by giving a typology of definitional knowledge

Reasoning with contradictory information using quasi-classical logic

Abstract: The proof theory of quasi-classical logic (QC logic) allows the derivation of non-trivializable classical inferences from inconsistent information. A non-trivializable, or paraconsistent, logic is, by necessity, a compromise, or weakening, of classical logic. The compromises on QC logic seem to be more appropriate than other paraconsistent logics for applications in computing. In particular, the connectives behave in a classical manner. Here we motivate the need for QC logic, present a proof theory, and semantics for the logic, and compare it to other paraconsistent logics.

Dual Intuitionistic Logic Revisited

Abstract: We unify the algebraic, relational and sequent methods used by various authors to investigate “dual intuitionistic logic”. We show that restricting sequents to “singletons on the left/right” cannot capture “intuitionistic logic with dual operators”, the natural hybrid logic that arises from intuitionistic and dual-intuitionistic logic. We show that a previously reported generalised display framework does deliver the required cut-free display calculus. We also pinpoint precisely the structural rule necessary to turn this display calculus into one for classical logic.

Hybrid Probabilistic Logic Programs as Residuated Logic Programs

Abstract: In this paper we show the embedding of Hybrid Probabilistic Logic Programs into the rather general framework of Residuated Logic Programs, where the main results of (definite) logic programming are validly extrapolated, namely the extension of the immediate consequences operator of van Emden and Kowalski The importance of this result is that for the first time a framework encompassing several quite distinct logic programming semantics is described, namely Generalized Annotated Logic Programs, Fuzzy Logic Programming, Hybrid Probabilistic Logic Programs, and Possibilistic Logic Programming Moreover, the embedding provides a more general semantical structure paving the way for defining paraconsistent probabilistic reasoning logic programming semantics

The Dialogical Approach to Paraconsistency

Abstract: Being a pragmatic and not a referential approach tosemantics, the dialogical formulation ofparaconsistency allows the following semantic idea tobe expressed within a semi-formal system: In anargumentation it sometimes makes sense to distinguishbetween the contradiction of one of the argumentationpartners with himself (internal contradiction) and thecontradiction between the partners (externalcontradiction). The idea is that externalcontradiction may involve different semantic contextsin which, say A and ¬A have been asserted.The dialogical approach suggests a way of studying thedynamic process of contradictions through which thetwo contexts evolve for the sake of argumentation intoone system containing both contexts.More technically, we show a new, dialogical, way tobuild paraconsistent systems for propositional andfirst-order logic with classical and intuitionisticfeatures (i.e. paraconsistency both with and withouttertium non-datur) and present theircorresponding tableaux.

Reasoning with inconsistency in structured text

Abstract: Reasoning with inconsistency involves some compromise on classical logic. There is a range of proposals for logics (called paraconsistent logics) for reasoning with inconsistency each with pros and cons. Selecting an appropriate paraconsistent logic for an application depends upon the requirements of the application. Here we review paraconsistent logics for the potentially significant application area of technology for structured text. Structured text is a general concept that is implicit in a variety of approaches to handling information. Syntactically, an item of structured text is a number of grammatically simple phrases together with a semantic label for each phrase. Items of structured text may be nested within larger items of structured text. The semantic labels in a structured text are meant to parameterize a stereotypical situation, and so a particular item of structured text is an instance of that stereotypical situation. Much information is potentially available as structured text, including tagged text in XML, text in relational and object-oriented databases, and the output from information extraction systems in the form of instantiated templates. In this review paper, we formalize the concept of structured text, and then focus on how we can identify inconsistency in items of structured text, and reason with these inconsistencies. Then we review key approaches to paraconsistent reasoning, and discuss the application of them to reasoning with inconsistency in structured text.

Paraconsistent reasoning as an analytic tool

Abstract: The study of logic usually focuses on either the proof theoretic or the model theoretic properties of logic. Yet the pragmatics of logic is often ignored. In this paper we would like to demonstrate that a logic can be practical in the sense that it can assist us in evaluating and measuring the amount of information in an inconsistent set of data. The underlying notion of information is inspired by Shannon’s communication theory. It denes the amount of information of a message in terms of the probability of the message being true. The logic presented here is the paraconsistent logic QC. As such QC logic can be seen as an analytical tool for evaluating data.

Applications of paraconsistency in data and knowledge bases

Abstract: The study of paraconsistent logic as a branch of mathematics and logic has been pioneered by Newton da Costa. With the growing advent of distributed and often inconsistent databases over the last ten years, there has been growing interest in paraconsistency amongst researchers in databases and knowledge bases. In this paper, we provide a brief survey of work in paraconsistent databases and knowledge bases affected by Newton da Costa's important and lasting contributions to the field.

Topological separation principles and logical theories

Abstract: This paper is dedicated to Newton da Costa, who,among his many achievements, was the first toaim at dualising intuitionism in order to produce paraconsistent logics,the C-systems. This paper similarly dualises intuitionism to aparaconsistent logic, but the dual is a different logic, namely closed setlogic. We study the interaction between the properties of topologicalspaces, particularly separation properties, and logical theories on thosespaces. The paper begins with a brief survey of what is known about therelation between topology and modal logic, intuitionist logic and paraconsistentlogic in respect of the incompleteness and inconsistency of theories.Necessary and sufficient conditions which relate the T1-property to theproperties of logical theories, are obtained. The result is then extendedto Hausdorff and Normal spaces. In the final section these methods areused to vary the modelling conditions for identity.

Paraconsistent logics and translations

Abstract: In 1999, da Silva, D'Ottaviano and Sette proposed a general definition for the term translation between logics and presented an initial segment of its theory. Logics are characterized, in the most general sense, as sets with consequence relations and translations between logics as consequence-relation preserving maps. In a previous paper the authors introduced the concept of conservative translation between logics and studied some general properties of the co-complete category constituted by logics and conservative translations between them. In this paper we present some conservative translations involving classical logic, Lukasiewicz three-valued system L 3, the intuitionistic system I 1 and several paraconsistent logics, as for instance Sette's system P 1, the D'Ottaviano and da Costa system J 3 and da Costa's systems C n, 1≤ n≤ω.

How Philosophical is Informal Logic

Abstract: Consider the proposition, "Informal logic is a subdiscipline of philosophy". The best chance of showing this to be true is showing that informal logic is part of logic, which in turn is a part of philosophy. Part 1 is given over to the task of sorting out these connections. If successful, informal logic can indeed be seen as part of philosophy; but there is no question of an exclusive relationship. Part 2 is a critical appraisal of the suggestion that informal logic is applied epistemology. Part 3 examines the claim that informal logic has failed to penetrate into mainstream philosophy, and suggestions for amelioration are considered.

Subprevarieties Versus Extensions. Application to the Logic of Paradox

Abstract: Abstract In the present paper we prove that the poset of all extensions of the logic defined by a class of matrices whose sets of distinguished values are equationally definable by their algebra reducts is the retract, under a Galois connection, of the poset of all subprevarieties of the prevariety generated by the class of the algebra reducts of the matrices involved. We apply this general result to the problem of finding and studying all extensions of the logic of paradox (viz., the implication-free fragment of any non-classical normal extension of the relevance-mingle logic). In order to solve this problem, we first study the structure of prevarieties of Kleene lattices. Then, we show that the poset of extensions of the logic of paradox forms a four-element chain, all the extensions being finitely many-valued and finitely-axiomatizable logics. There are just two proper consistent extensions of the logic of paradox. The first is the classical logic that is relatively axiomatized by the Modus ponens rule for the material implication. The second extension, being intermediate between the logic of paradox and the classical logic, is the one relatively axiomatized by the Ex Contradictione Quodlibet rule.

Natural deduction for paraconsistent logic

Abstract: In this paper, by using the method of natural deduction, via the method of subordinate proofs, we develop a hierarchy of natural deduction logical systems NDC n containing just deduction rules (or deduction schemata) withno axiom schema. We prove that these systems NDC n , 1≤n≤ω, are logically equivalent to the systems of Da Costa's hierarchy of paraconsistent logics C n , 1≤n≤ω. Some of the deduction rules used to introduce these systems are new and do not correspond to Da Costa's axioms rewritten, permitting the definition of a new paraconsistent semantics, such that soundness and completeness of the systems NDC n , 1≤n≤ω, may be directly obtained. Other natural deduction systems logically equivalent to Da Costa's systems C n , 1≤n≤ω, are also introduced.

Sociative Logics and Their Applications: Essays by the Late Richard Sylvan

Abstract: This title was first published in 2003. Richard Sylvan died in 1996, he had made contributions to many areas of philosophy, such as, relevant and paraconsistent logic, Meinongianism and metaphysics and environmental ethics. One of his “trademarks�? was the taking up of unpopular views and defending them. To Richard Sylvan ideas were important, wether they were his or not. This is a book of ideas, based on a collection of work found after his death, a chance for readers to see his vision of his projects. This collected works represents material drafted between 1982 and 1996, and the theme is that a small band of logics, namely pararelevant logics, offer solutions to many problems, puzzles and paradoxes in the philosophy of science.

On the Role of Implication in Formal Logic

Abstract: Evidence is given that implication (and its special case, negation) carry the logical strength of a system of formal logic. This is done by proving normalization and cut elimination for a system based on combinatory logic or λ-calculus with logical constants for and, or, all, and exists, but with none for either implication or negation. The proof is strictly finitary, showing that this system is very weak. The results can be extended to a “classical” version of the system. They can also be extended to a system with a restricted set of rules for implication: the result is a system of intuitionistic higher-order BCK logic with unrestricted comprehension and without restriction on the rules for disjunction elimination and existential elimination. The result does not extend to the classical version of the BCK logic. 1991 AMS (MOS) Classification: 03B40, 03F05, 03B20

Paraconsistent ideas in quantum logic

Abstract: Etude des approches non-strictes de la theorie et de la logique quantiques qui revendiquent une liberalisation de la contrepartie mathematique de la notion intuitive de proposition experimentale. A partir de la definition des structures algebriques des quantites physiques, l'A. compare la logique paraconsistante de Brouwer-Zadeh a la logique quantique de Lukasiewicz.

A paraconsistent system of autonomous agents for Brazilian bank check treatment

Abstract: This work is part of the Multicheck Project that defines the architecture of autonomous agents for the automatic treatment of handwritten Brazilian bank checks. The competence of these agents is implemented in two layers. The first corresponds to pattern recognition algorithms directly applied to image segments. The second one corresponds to reasoning mechanisms applied to the information from the first layer, either to validate or to interpret it. The interpretation process also involves information obtained from other agents. This information can present inconsistencies. This problem is treated properly and naturally through the concepts and operators of paraconsistent logic. This paper focuses on the second layer, on task distribution problems and on communication between agents. The first layer information was obtained through a simulated database.

Vasil'Év and Imaginary Logic

Abstract: This paper is about the ‘Imaginary Logic’ developed by the Russian logician Nicholas Vasil'ev between about 1910 and 1913, a logic that is often claimed to be a forerunner of different sorts of modern nonclassical logics. The paper describes the content of that logic (not by trying to interpret it in modern logic, as some commentators have done, but by describing it in its own terms). It then looks at the philosophical underpinnings of the logic. Finally, in the light of the preceding, it discusses Vasil‘ev's place in the history of logic.

On the relationship between fuzzy logic and four-valued relevance logic

Abstract: In fuzzy propositional logic, to a proposition a partial truth in [0,1] is assigned It is well known that under certain circumstances, fuzzy logic collapses to classical logic In this paper, we will show that under dual conditions, fuzzy logic collapses to four-valued (relevance) logic, where propositions have truth-value true, false, unknown, or contradiction As a consequence, fuzzy entailment may be considered as ``in between'' four-valued (relevance) entailment and classical entailment

Paraconsistent Deontic Logic with Enforceable Rights

Abstract: En: Frontiers of Paraconsistent Logic ed. por D. Batens, Ch. Mortensen, G. Priest & J.-P. van Bendegem Baldford (England): Research Studies Press Ltd. (RSP) [Logic and Computation Series], 2000. ISBN 086302532, pp. 29-47

Proof-Nets, Hybrid Logics and Minimalist Representations

Abstract: In this paper, we aim at giving a logical account of the representationalist view of minimalist grammars by referring to the notion of Proof-Net in Linear Logic. We propose, at the same time, a hybrid logic which mixes one logic (Lambek calculus) for building up elementary proofs and another one for combining the proofs so obtained. Because the first logic is non-commutative and the second one is commutative, this brings us a way to combine commutativity and non-commutativity in the same framework. The dynamic of cut-elimination in proof-nets is used to formalise the move-operation. Otherwise, we advocate a proof-net formalism which allows us to consider formulae as nodes to which it is possible to assign weights which determine the final phonological interpretation.

Goal-Directed Proof Search in Multiple-Conclusions Intuitionistic Logic

Abstract: A key property in the definition of logic programming languages is the completeness of goal-directed proofs. This concept originated in the study of logic programming languages for intuitionistic logic in the (single-conclusioned) sequent calculus LJ, but has subsequently been adapted to multiple-conclusioned systems such as those for linear logic. Given these developments, it seems interesting to investigate the notion of goal-directed proofs for a multiple-conclusioned sequent calculus for intuitionistic logic, in that this is a logic for which there are both single-conclusioned and multiple-conclusioned systems (although the latter are less well known). In this paper we show that the language obtained for the multiple-conclusioned system differs from that for the single-conclusioned case, show how hereditary Harrop formulae can be recovered, and investigate contraction-free fragments of the logic.

Fibring (Para)consistent Logics

Abstract: The problem of fibring paraconsistent logics is addressed. Such logics raise new problems in the semantics of fibring since previous work assumed verum-functional models. The solution is found in a general notion of interpretation system presentation that “specifies” the intended valuations in some appropriate meta language. Fibring appears as a universal construction in the category of interpretation system presentations, generalizing the results for systems with verum-functional semantics. As an illustration, the fibring of paraconsistent system C1 and modal system K, while sharing propositional symbols, conjunction, disjunction and implication, is obtained. The fibring of the whole hierarchy {Cn}n∈N leads to the limit paraconsistent logic Clim. Fibring is shown to be a promising technique for generating new paraconsistent logics.

Absolute Continuity of States on Concrete Logics

Abstract: By concrete logic we mean a quantum logic which is set-representable, and byVitali—Hahn—Saks logic (VHS-logic) we mean a concrete logic for which theVitali—Hahn—Saks theorem holds true. In this note we investigate the size of theclass of VHS-logics, showing among others that each concrete logic can beconcretely enlarged to a VHS-logic as well as to a non-VHS-logic.

Inconsistency and preservation

Abstract: One of the main goals of paraconsistent logics is to develop a theory of reasoning that can tolerate inconsistencies In this paper we present a novel way to analyze and compare several paraconsistent reasoning mechanisms in terms of their preservational properties The main idea is that although an inconsistent set of data cannot all be true, such a set may nevertheless carry useful properties that are worthy of preservation One of these properties provides a theoretically interesting way to meaisure the relative incoherence of a data set; another one provides a way to measure the quantity of empirical information in an inconsistent set

Continuous logic – I. Basic concepts

Abstract: A general description of a continuous (‐valued) logic is given, basic operations of the logic are defined, and some problems and particulars of their solutions are discussed. First, we define algebra of continuous logic and enumerate its basic unary, binary and ternary functions. All laws of continuous logic are compared with laws of discrete binary logic. We discuss how to enumerate all functions of continuous logic with specified number of variables and how to represent the functions in a standard form. Procedures of minimization of continuous logical functions and their decomposition into the functions with less clarity are exploited. The procedures are compared with their counterparts from binary logic. We also tackle problems of analysis and synthesis of continuous logical functions, and show that the problem of synthesis may not have a solution. Basics of differential and integral calculus are applied to continuous valued logic. We demonstrate that any continuous logical function has the points where a derivative does not exist. At the end of the paper we briefly discuss an incompleteness problem of continuous logic, application of continuous logic in mathematics, engineering and economy, give examples, draw a perspective of further development and supply extensive bibliography of Russian works in the field.