Showing papers on "Paraconsistent logic published in 1995"

On logic of paradox

Abstract: G. Priest introduced nonmonotonicity into a paraconsistent logic, so-called logic of paradox LP, that yields a solution to the weakness of paraconsistent logic. The resulting logic (of minimal parades) LP/sub m/ is nonmonotonic in the sense that inconsistency is minimal. The problem of proof theory of logic LP/sub m/ left open because the base logic LP is paraconsistent so that syntactic formulations of nonmonotonic logic are not available for LP/sub m/, though LP/sub m/ is well characterized by minimal semantics. In this paper, we provide a minimal tableaus as a satisfactory proof theory for LP/sub m/. We first present a signed tableaux for LP. Then minimal tableaux for LP/sub m/ is obtained by revising signed tableaux for LP to fit LP/sub m/ in which the branches of non-minimally-inconsistent models of the tableaux are eliminated. The soundness and completeness theorems of the tableaux with respect to the semantics of LP and LP/sub m/ are proved, respectively.

Quasi-classical Logic: Non-trivializable classical reasoning from incosistent information

Abstract: Here we present a new paraconsistent logic, called quasi-classical logic (or QC logic) that allows the derivation of non-trivializable classical inferences. For this it is necessary that queries are in conjunctive normal form and the reasoning process is essentially that of clause finding. We present a proof-theoretic definition, and semantics, and show that the consequence relation observes reflexivity, monotonicity and transitivity, but fails cut and supraclassicality. Finally we discuss some of the advantages of this logic, over other paraconsistent logics, for applications in information systems.

Thinking about logic : an introduction to the philosophy of logic

Abstract: Logic deals with the inevitable - those consequences which follow inescapably from a given set of premisses. This fact has caused it to be seen as different from other more self-questioning branches of philosophy. In this book, Stephen Read sets out to rescue logic from its undeserved reputation as an inflexible, dogmatic discipline by demonstrating that its technicalities and processes are founded on assumptions which are themselves amenable to philosophical investigation. He examines the fundamental principles of consequence, logical truth and correct inference within the context of logic, and shows that the principles by which we delineate consequences are themselves not guaranteed free from error. Central to the notion of truth is the beguiling issue of paradox. Its philosophical value, Read shows, lies in exposing the invalid assumption on which the paradox is built. Thinking About Logic also discusses logical puzzles which introduce questions relating to language, the world, and their relationship. While formal logic often employs its own esoteric language, the achievement of this book is to focus on those issues which raise exciting philosophical questions, and to make them intelligible to readers with no previous knowledge of logic.

Aspects of Paraconsistent Logic

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A new axiomatic foundation of partial comparability

Abstract: The paper presents some results obtained in searching for a new axiomatic foundation for partial comparability (PC) in the frame of non-conventional preference modeling. The basic idea is to define an extended preference structure able to represent lack of information, uncertainty, ambiguity, multidimensional and conflicting preferences, using formal logic as the basic formalism. A four-valued paraconsistent logic is therefore described in the paper as a more suitable language for the purposes of the research. The concepts of partition, general binary relations properties, fundamental relational system of preferences (f.r.s.p.), maximal f.r.s.p. and well founded f.r.s.p. are then introduced and some theorems are demonstrated in order to provide the axiomatic foundation of PC. The main result obtained is a preference structure that is a maximal well founded f.r.s.p. This preference structure facilitates a more flexible, reliable and robust preference modeling. Moreover it can be viewed as a generalization of the conventional approach, so that all the results obtained until now can be used under it. Two examples are provided at the end of the paper in order to give an account of the operational potentialities of the new theory, mainly in the area of multicriteria decision aid and social choice theory. Further research directions conclude the paper.

Characterizing Belnap's Logic via De Morgan's Laws

Abstract: The aim of this paper is technically to study Belnap's four-valued sentential logic (see [2]). First, we obtain a Gentzen-style axiomatization of this logic that contains no structural rules while all they are still admissible in the Gentzen system what is proved with using some algebraic tools. Further, the mentioned logic is proved to be the least closure operator on the set of {Λ, V, ⌝}-formulas satisfying Tarski's conditions for classical conjunction and disjunction together with De Morgan's laws for negation. It is also proved that Belnap's logic is the only sentential logic satisfying the above-mentioned conditions together with Anderson-Belnap's Variable-Sharing Property. Finally, we obtain a finite Hilbert-style axiomatization of this logic. As a consequence, we obtain a finite Hilbert-style axiomatization of Priest's logic of paradox (see [12]).

A paraconsistent relational data model

Abstract: We present a generalisation of the relational data model based on a 4-valued paraconsistent logic. Our data model is capable of manipulating incomplete as well as inconsistent information. For this model, we define algebraic operators that are generalisations of the usual operators, such as union, selection, join, on ordinary relations. Our data model can underlie any database management system that deals with incomplete or inconsistent information. As another application of our model and its algebra, we present a bottom-up method for constructing the weak well-founded model of general deductive databases. This method can be very simply extended to construct the well-founded model.

Maximal weakly-intuitionistic logics

Abstract: This article introduces the three-valuedweakly-intuitionistic logicI 1 as a counterpart of theparaconsistent calculusP 1 studied in [11].I 1 is shown to be complete with respect to certainthree-valued matrices. We also show that in the sense that any proper extension ofI 1 collapses to classical logic. The second part shows thatI 1 is algebraizable in the sense of Block and Pigozzi (cf. [2]) in a way very similar to the algebraization ofP 1 given in [8]. In the last part of the paper we suggest the definition of certain hierarchies of finite-valued propositional paraconsistent and weakly-intuitionistic calculi, and comment on their intrinsic interest.

Description logic in practice: a CLASSIC application

Abstract: Descr ipt ion logic-based configurat ion applications have been used w i t h i n A T & T since 1990 to process over two and a half b i l l ion dollars wor th of orders. Whi le this fami ly of applications[4] has widely acknowledged importance, it is di f f icul t to use for pedagogical purposes since the typical product configured is a highly interconnected, complicated technical piece of equipment like the DACS IV -2000 . 1 We have developed a smaller-scale configurat ion appl icat ion that has analogous reasoning processes but a more approachable domain—that of bui ld ing home theater systems. Th is appl icat ion provides a p la t form for expla in ing how Descr ipt ion Logic-based Systems (DLSs) work, in our case the C L A S S I C knowledge representation system[ l ] , and how they can support industr ia l applications like conf igurat ion.

A natural deduction approach to dynamic logic

Abstract: Natural Deduction style presentations of program logics are useful in view of the implementation of such logics in interactive proof development environments, based on type theory, such as LEGO, Coq, etc. In fact, ND-style systems are the kind of systems which can take best advantage of the possibility of reasoning “under assumptions” offered by proof assistants generated by Logical Frameworks. In this paper we introduce and discuss sound and complete proof systems in Natural Deduction style for representing various “truth” consequence relations of Dynamic Logic. We discuss the design decisions which lead to adequate encodings of these logics in Coq. We derive in Dynamic Logic a set of rules representing a ND-style system for Hoare Logic.

On Priest's logic of paradox

Abstract: The present paper concerns a technical study of PRIEST'S logic of paradox [Pri 79], We prove that this logic has no proper paraconsistent strengthening. It is also proved that the mentioned logic is the largest paraconsistent one satisfaying TARSKI'S conditions for the classical conjunction and disjunction together with DE MORGAN'S laws for negation. Finally, we obtain for the logic of paradox an algebraic completeness result related to Kleene lattices.

Games and Trees in Infinitary Logic: A Survey

Abstract: We describe the work and underlying ideas of the Helsinki Logic Group in infinitary logic. The central idea is to use trees and Ehrenfeucht-Fraisse games to measure differences between uncountable models. These differences can be expressed by sentences of the so-called infinitely deep languages. This study has ramified to purely set-theoretical problems related to properties of trees, descriptive set theory in ω1ω1, a detailed study of transfinite Ehrenfeucht-Fraisse games, new constructions of uncountable models, non-well-founded induction, infinitely deep languages, non-structure theorems, and stability theory. The aim of this paper is to give an overview of the underlying ideas of this reasearch together with a survey of the main results.

Algebraic study of Sette's maximal paraconsistent logic

Abstract: The aim of this paper is to study the paraconsistent deductive systemP1 within the context of Algebraic Logic. It is well known due to Lewin, Mikenberg and Schwarse thatP1 is algebraizable in the sense of Blok and Pigozzi, the quasivariety generated by Sette's three-element algebraS being the unique quasivariety semantics forP1. In the present paper we prove that the mentioned quasivariety is not a variety by showing that the variety generated byS is not equivalent to any algebraizable deductive system. We also show thatP1 has no algebraic semantics in the sense of Czelakowski. Among other results, we study the variety generated by the algebraS. This enables us to prove in a purely algebraic way that the only proper non-trivial axiomatic extension ofP1 is the classical deductive systemPC. Throughout the paper we also study those abstract logics which are in a way similar toP1, and are called hereabstract Sette logics. We obtain for them results similar to those obtained for distributive abstract logics by Font, Verdu and the author.

A Model Theory for Paraconsistent Logic Programming

Abstract: We provide a nine-valued logic to characterize the models of logic programs under a paraconsistent well-founded semantics with explicit negation WFSXp We define a truth-functional logic, \(\mathcal{N}\mathcal{I}\mathcal{N}\mathcal{E}\), based on the bilattice construction of Ginsberg and Fitting The models identified by WFSXp are models of logic \(\mathcal{N}\mathcal{I}\mathcal{N}\mathcal{E}\) We conclude with a discussion on the conditions to obtain an isomorphism between the two definitions, and thereby characterizing WFSXp model-theoretically

Inconsistent Systems of Linear Equations

Abstract: The existence of the inconsistent case of a system of linear equations (or for that matter any system of constraints, not necessarily linear) has been known for a long time, but there has been no attempt to analyse its structure. There would seem to be good reason to do so, if only because the state of affairs might arise in a real life control system (see sections 3 and 4). Using the methods developed so far, it is possible to say something about the structure of solutions to such cases; though it must be confessed that in the end the situation remains less than satisfactory.

Discursive logic towards a logic of rational discourse

Abstract: Both logic and philosophy of science investigate formal aspects of scientific discourse, i.e. properties of (non-monotonic) consequence operations for discursive logic. In the present paper we handle two of them: paraconsistency and enthymematycity.

Default Consequence Relations as a Logical Framework for Logic Programs

Abstract: We consider the use of default consequence relations suggested in [1,2] as a ‘logical basis’ for normal logic programs. We give a representation of major semantics for logic programs in this framework and study the question what kind of default reasoning is appropriate for them. It is shown, in particular, that default consequence relations based on three-valued inference are adequate for these semantics.

Contributions to a Theory of Existential Termination for Definite Logic Programs.

Abstract: We suggest a new formalization of the existential termination problem of logic programs under the PROLOG leftmost selection rule and depth-rst computation rule. First of all, we give a characterization of the problem in terms of occurrence sets, by proving that a hprogram; goali existentially terminates if and only if there exists a nite correct occurrence set. Then we show that in order to study existential termination, we do not need to specify the occurrences of the atoms, since existential termination turns out to be decidable, when instances of atoms are used more than once (up to renaming). We then reduce the veriication of existential termination to the search of a suitable semi occurrence set for the pair hprogram; goali, by providing an algorithm for proving that the proposed semi occurrence set is a correct occurrence set. Finally we propose a simple method (based on abstract interpretation techniques) for generating such semi occurrence sets.

Paraconsistent circumscription: first-order case

Abstract: In this paper we describe paraconsistent circumscription by application of predicate circumscription in a paraconsistent logic. In addition to circumscribe the predicates, we also circumscribe the inconsistency. Paraconsistent circumscription can be well characterized by the minimal semantics that is both nonmonotonic and paraconsistent. It brings us advantages at least in two respects: nonmonotonic logic would be nontrivial while there was a contradiction, and paraconsistent logic would be equivalent to classical logic while there was no direct effect of a contradiction.

On axiomatizing free logic - and inclusive logic in the bargain*

Abstract: Free Logic owes its name to its being free of two presuppositions of Standard Logic, one to the effect that something exists, a patent truth but surely a factual rather than a logical one, and the other to the effect that every singular term designates something, a patent falsehood1 In Standard Logic with Identity, it is this familiar law: $$ \left( {\exists {\rm X}} \right)\left( {{\rm X} - {\rm T}} \right)$$ (A)

What Is a Skeptical Proof

Abstract: We investigate the task of skeptically reasoning in extension-based, nonmonotonic logics by concentrating on general argumentation theories. The restricted applicability of Dung's notion of skeptical provability in his well-known argumentation framework is illustrated, and a new approach based on the notion of a claim associated with each argument is proposed. We provide a formal definition of a skeptical proof in our framework. As a concrete formalism, default logic in case of normal default theories is embedded in the general framework. We prove a formal correspondence between the two notions of skeptical provability, which enables us to adopt the general concept of a skeptical proof into default logic.

Reflections on “difficult” embeddings

Abstract: The main purpose of this note is to present “difficult” embeddings of minimal and full intuitionistic logic into classical linear logic, and to prove their soundness and faithfulness. Moreover, it is also pointed out that Girard's translation of intuitionistic logic into classical linear logic is provably equivalent to one of the translations considered in this paper.