Showing papers on "Paraconsistent logic published in 2002"

Dialogues as a Foundation for Intuitionistic Logic

Abstract: The principal content of this article is a (new) foundation for intuitionistic logic, based on an analysis of argumentative processes as codified in the concepts of a dialogue and a strategy for dialogues. This work is presented in Section 3. A general historical introduction is given in Section2. Since already there the reader will need to know exactly what a dialogue and a strategy shall be, these basic concepts are defined in the (purely technical) Section 1.

Logic and Ontology

Abstract: In view of the present state of development of non classical logic, especially of paraconsistent logic, a new stand regarding the relations between logic and ontology is defended In a parody of a dictum of Quine, my stand May be summarized as follows. To be is to be the value of a variable a specific language with a given underlying logic Yet my stand differs from Quine’s, because, among other reasons, I accept some first order heterodox logics as genuine alternatives to classical logic I also discuss some questions of non classical logic to substantiate my argument, and suggest that may position complements and extends some ideas advanced by L Apostel.

Measuring inconsistency in knowledge via quasi-classical models

Abstract: The language for describing inconsistency is underdeveloped. If a knowledgebase (a set of formulae) is inconsistent, we need more illuminating ways to say how inconsistent it is, or to say whether one knowledgebase is "more inconsistent" than another. To address this, we provide a general characterization of inconsistency, based on quasi-classical logic (a form of paraconsistent logic with a more expressive semantics than Belnap's four-valued logic, and unlike other paraconsistent logics, allows the connectives to appear to behave as classical connectives). We analyse inconsistent knowledge by considering the conflicts arising in the minimal quasi-classical models for that knowledge. This is used for a measure of coherence for each knowledgebase, and for a preference ordering, called the compromise relation, over knowledgebases. In this paper, we formalize this framework, and consider applications in managing heterogeneous sources of knowledge.

Ampliative Adaptive Logics and the Foundation of Logic-Based Approaches to Abduction

Abstract: In this paper, we propose a reconstruction of logic-based approaches to abductive reasoning in terms of ampliative adaptive logics. A main advantage of this reconstruction is that the resulting logics have a proof theory. As abductive reasoning is non-monotonic, the latter is necessarily dynamic (conclusions derived at some stage may at a later stage be rejected). The proof theory warrants, however, that the conclusions derived at a given stage are justified in view of the insight in the premises at that stage. Thus, it even leads to justified conclusions for undecidable fragments. Another advantage of the proposed logics is that they are much closer to natural reasoning than the existing systems. Usually, abduction is viewed as a form of “backward reasoning”. The search procedure by which this is realized (for instance, some form of linear linear resolution) is very different from the search procedures of human reasoners. The proposed logics treat abduction as a form of “forward reasoning” (Modus Ponens in the “wrong direction”). As a result, abductive steps are very natural, and are moreover nicely integrated with deductive steps. We present two new adaptive logics for abduction, and illustrate both with some examples from the history of the sciences (the discovery of Uranus and of Neptune). We also present some alternative systems that are better suited for non-creative forms of abductive reasoning.

S5 is a paraconsistent logic and so is first-order classical logic

Abstract: * Quoi? quand je dis " Nicole, apportez-moi mes pantoufles, et me donnez mon bonnet de nuit " , c'est de la prose? Par ma foi! Il y a plus de quarante ans que je dis de la prose sans que j'en susse rien, et je vous suisle plus obligé du monde de m'avoir appris cela. Molière, Le bourgeois gentilhomme

Epistemic logic: a survey

Abstract: Epistemic logic is the logic of knowledge: how do you reason about the question whether your silent admirer knows that you know that (s)he sent you an anonymous Valentine card? Is it harmful if, at a literature-exam you don't know the contents of a chapter? No, as long as you know that the examiner does not know that you do not know it. Knowing whether your neighbor knows that he regularly plays his radio so loudly that you wake up during the night, may help you to solve the problem in an appropriate way. In negotiations, it will harm you to let the other party know your `bottom-line', but it may be helpful to disclose other information about yourself, for example about some of your values. In this article, we will use examples and puzzles to give some avor of the eld and to demonstrate that the notion \it is known that" is meaningful and interesting for researchers in theoretical computer science, arti cial intelligence and game theory. The rst person who wrote about epistemic logic was the Swedish-Finnish philosopher G.H. von Wright in his book \An Essay in Modal Logic" [53]. His treatment is completely axiomatic, with no mention of possible semantics. Most philosophical work on epistemic logic following up on Von Wright's work has concentrated on defending certain axioms and denouncing others. However, the subject of epistemic logic only started to ourish after Kripke's invention of a semantics for modal logic in the early sixties. Kripke introduced a possible worlds semantics for modal logics. The name \possible world" is somewhat misleading, because, according to Hintikka [30], \applications to entire universes are scarcely found outside philosophers' speculations. The primary intended applications are to scenarios covering relatively small pieces of spacetime". In the context of epistemic logic, one can view worlds that are possible for a certain agent in a world as epistemic alternatives, that are compatible with the agent's information in that world. The precise de nitions will be given in Section 2. The rst full-length book about epistemic logic, Hintikka's \Knowledge and Belief" [29], applies these semantical ideas, although his de nitions are

On Logical Relativity

Abstract: One logic or many? I say—many. Or rather, I say there is one logic for each way of specifying the class of all possible circumstances, or models, i.e., all ways of interpreting a given language. But because there is no unique way of doing this, I say there is no unique logic except in a relative sense. Indeed, given any two competing logical theories T1 and T2 (in the same language) one could always consider their common core, T, and settle on that theory. So, given any language L, one could settle on the minimal logic T0 corresponding to the common core shared by all competitors. That would be a way of resisting relativism, as long as one is willing to redraw the bounds of logic accordingly. However, such a minimal theory T0 may be empty if the syntax of L contains no special ingredients the interpretation of which is independent of the specification of the relevant Lmodels. And generally —I argue— this is indeed the case.

A first-order, four valued, weakly paraconsistent logic and its relation to rough sets semantics

Abstract: A first order four-valued logic, called DDT, is presented in the paper as an extension of Belnap’s logic using a weak negation and establishing an appropriate semantic for the predicate calculus. The logic uses a simple algebraic structure, that is the smallest non trivial interlaced bilattice on the four truth values, thus resulting in a boolean algebra on the set of truth values. The logic is a language for reasoning under uncertainty, enabling to capture hesitation due either to inconsistent or incomplete information, while keeping a clear distinction between these epistemic states. The logic was originally developed for preference modelling purposes (for which a brief account is given in the paper). The paper demonstrates and discusses the equivalence between the semantics of this logic and of rough sets semantics. On this basis, this papers presents the possibility of inducing rules from examples, that can be integrated in systems whose inference is expressed in the above logic. Such an approach enhances the potentialities of the use of rough sets in classification, reasoning and decision support.

Handbook of the Logic of Argument and Inference: The Turn Towards the Practical

Abstract: Preface. List of Authors. Logic and The Practical Turn (J. Woods, R.H. Johnson, D.M. Gabbay, Hans Jurgen Ohlbach). Standard Logics as Theories of Argument and Inference: Deduction (J. Woods). Standard Logics as Theories of Argument and Inference: Induction (J. Woods). Internal Critique: A logic is not a Theory of Reasoning and a Theory of Reasoning is not a logic (G. Harman). Standard Logic as a Model of Reasoning: The Empirical Critique (D.N. Perkins). A Framework for Intersubjective Accountability: Dialogical Logic (E.M. Barth). Interrrogative Logic (J. Hintikka, I. Halonen, A. Mutanen). Informal Logic and the Reconfiguration of Logic (R.H. Johnson, J.A. Blair). Probability Logic (J. Williamson). Philosophical Incidence of Logic Programming (L.M. Pereira). Formal Approaches to Practical Reasoning: A Survey (D.M. Gabbay, J. Woods). Index.

A Paraconsistent Theory Of Belief Revision

Abstract: This paper presents a theory of belief revision that allows people to come tobelieve in contradictions. The AGM theory of belief revision takes revision,in part, to be consistency maintenance. The present theory replacesconsistency with a weaker property called “coherence”. In addition to herbelief set, we take a set of statements that she rejects. These two sets arecoherent if they do not overlap. On this theory, belief revision maintains coherence.

Inconsistency and the Empirical Sciences

Abstract: What role does, or should, inconsistency play in the empirical sciences? This is the question that I will address in this essay. The question is hardly a new one, but the development of modern formal paraconsistent logics has a profound impact on the subject. Paraconsistent logicians have realised that their subject has important implications for the empirical sciences and the philosophy thereof,1 but discussions of the applications of paraconsistent logic have focused largely on non-empirical areas, such as semantics and metaphysics. It therefore seems appropriate to address the question directly.2

Hyperclassical Logic (A.K.A. If Logic) and its Implications for Logical Theory

Abstract: Let us assume that you are entrusted by UNESCO with an important task. You are asked to devise a universal logical language, a Begriffsschrift in Frege's sense, which is to serve the purposes of science, business and everyday life. What requirements should such a “conceptual notation” satisfy? There are undoubtedly many relevant desiderata, but here I am focusing on one unmistakable one. In order to be a viable lingua universalis, your language must in any case be capable of representing any possible configuration of dependence and independence between different variables. For if such a configuration is possible in principle, there is no guarantee that it might not one day show up among the natural, human or social phenomena we have to study. But how are dependencies and independencies between variables expressed in our familiar logical notation? Every logician worth his or her truth-table knows the answer. Dependencies between two variables are expressed by dependencies between the quantifiers to which they are bound. For instance, in the variable y depends on x, while in z depends on x but not on y, while u depends on both x and y. But how is the dependence of a quantifier on another one expressed in familiar logical languages? Obviously by occurring in its scope, indicated by the pair of parentheses following it (cf. here Hintikka [1997]). But the nesting of scopes is a transitive and antisymmetrical relation which allows branching only in one direction. Hence other kinds of structures of dependence and independence between variables are not representable in the received logical notation. Such previously inexpressible structures form the subject matter of what has been referred to as independence-friendly (IF) logic.

Paraconsistent Semantics for Extended Logic Programs.

Abstract: We introduce a declarative semantics for extended logic programs, and demonstrate its usefulness for reasoning with uncertainty. We show that this is a robust formalism that overcomes some drawbacks of related xpoint semantics for incomplete or inconsistent logic programs.

Paraconsistent logic program based safety verification for air traffic control

Abstract: It has become a crucial issue to assure the safety for air traffic control. We consider how some air traffic accidents can be avoided by verifying the safety for air traffic control logically. We propose a theoretical framework for a logical safety verification for air traffic control based on a paraconsistent logic program called an Extended Vector Annotated Logic Program with Strong Negation (EVALPSN for short). Compared to other kinds of safety verification such as safety verification for railway interlocking, the safety verification for air traffic control contains more uncertainty and sometimes has to deal with probabilistic datum. Therefore, we extend EVALPSN to probabilistic EVALPSN (P-EVALPSN for short) for dealing with the safety verification containing probability. We introduce the ideas of the safety verification based on both EVALPSN and P-EVALPSN, taking a simple example for landing clearance by air traffic controllers.

Three-Valued Logics for Inconsistency Handling

Abstract: While three-valued paraconsistent logic is a valuable framework for reasoning under inconsistency, the corresponding basic inference relation is too cautious and fails in discriminating in a fine-grained way the set of expected consequences of belief bases. To address both issues, we point out more refined inference relations. We analyze them from the logical and computational points of view and we compare them with respect to their relative cautiousness.

Paraconsistent Logic Programs

Abstract: We propose a framework which extends Antitonic Logic Programs [2] to an arbitrary complete bilattice of truth-values, where belief and doubt are explicitly represented. Based on Fitting's ideas, this framework allows a precise definition of important operators found in logic programming such as explicit negation and the default negation. In particular, it leads to a natural integration of explicit negation with the default negation through the coherence principle [19]. According to this principle, the explicit negation entails the default negation. We then define Coherent Answer Sets, and the Paraconsistent Well-founded Model semantics, generalizing paraconsistent semantics for logic programs (for instance, WFSXp [4]). Our framework is an extension of important classes of Antitonic Logic Programs, and is general enough to capture Probabilistic Deductive Databases, Possibilistic Logic Programming, Hybrid Probabilistic Logic Programs, and Fuzzy Logic Programming. Thus, we have a powerful mathematical formalism for dealing with default reasoning, paraconsistency, and uncertainty.

Logic and Reasoning

Abstract: This article describes logic and explains that it is not a theory of how humans reason. It presents three contemporary psychological theories based respectively on formal rules of inference akin to those of logic, on probabilistic considerations designed to replace logical principles, and on mental models that use the meanings of assertions to represent possibilities. It reviews seven empirical phenomena that elucidate these theories.

Logic: a misleading concept. A contradiction study toward agent's logic ontology

Abstract: The paper presents a fresh new comprehensive ideology on Neutrosophic Logic based on contradiction study in a broad sense: general critics on conventional logic by examining the essence of logic, fresh insights on logic definition based on Chinese philosophical survey, and a novel and genetic logic model as the elementary cell against Von Neumann oriented ones based on this novel definition. As for the logic definition, the paper illustrates that logic is rather a tradeoff between different factors than truth and false abstraction. It is stressed that the kernel of any intelligent system is exactly a contradiction model. The paper aims to solve the chaos of logic and exhibit the potential power of neutrosophy: a new branch of scientific philosophy.

In Defence of a Programme for Handling Inconsistencies

Abstract: This paper states and defends the philosophical programme underlying the Ghent approach to adaptive logics. Two central arguments are epistemic in nature, one logical. The underlying claim is that even people with rather classical views should see adaptive logics as the only sensible way to handle the inconsistencies that regularly arise in human knowledge, including scientific theories.

Inconsistency in Science: A Partial Perspective

Abstract: The prevalence of inconsistency in both our scientific and ‘everyday’ belief structures is something which is being increasingly recognised.1 In the world of scientific representations, Bohr’s theory, of course, is one of the more well known examples, described by Lakatos as ‘... sat like a baroque tower upon the Gothic base of classical electrodynamics’ (Lakatos 1970, 142; see also Brown 1992); others that have been put forward include the old quantum theory of black-body radiation, the Everett-Wheeler interpretation of quantum mechanics, Newtonian cosmology, the (early) theory of infinitesimals in calculus, the Dirac δ-function, Stokes’ analysis of pendulum motion and Michelson’s ‘single-ray’ analysis of the Michelson-Morley interferometer arrangement. The problem, of course, is how to accommodate this aspect of scientific practice given that within the framework of classical logic an inconsistent set of premises yields any well-formed statement as a consequence. The result is disastrous: the set of consequences of an inconsistent theory will explode into triviality and the theory is rendered useless. Another way of expressing this descent into logical anarchy which will be useful for our discussion to follow is to say that under classical logic the closure of any inconsistent set of sentences includes every sentence. It is this which lies behind Popper’s famous declaration that the acceptance of inconsistency ‘... would mean the complete breakdown of science’ since an inconsistent system is ultimately uninformative (Popper 1940, 408; 1972, 91–92).

Deontic Relevant Logic: A Strong Relevant Logic Approach to Removing Paradoxes from Deontic Logic

Abstract: In this paper, we propose a strong relevant logic approach to solve the problems of deontic logic paradoxes. Since the paradoxes in deontic logic have the same form as the paradoxes in traditional (weak) relevant logic, which have been rejected by our strong relevant logic, we show that a new family of logic, named deontic relevant logics, can be established by introducing deontic operators and relative axioms and inference rules into strong relevant logics such that those deontic logic paradoxes are rejected by deontic relevant logics.

Complexity Results for Paraconsistent Inference Relations.

Abstract: In this paper, the complexity of several paraconsistent inference relations, based on multivalued logics, is investigated. Many inference relations pointed out so far by Arieli and Avron, Besnard and Schaub, D’Ottaviano and da Costa, Frisch, Levesque, Priest are considered from the computational side. All these relations can be gathered into two categories: the basic ones stem directly from the notions of models within 3 or 4-valued logics, while the refined ones are based on notions of preferred models for such logics. Completing complexity results by Cadoli and Schaerf (centered on the basic relations), we show that the refined paraconsistent inference relations that have been defined in the framework of multivalued logics are highly intractable. Especially, we prove that the inference problems corresponding to these relations are Πp2complete, even in the simple case where the database is a CNF formula and the query is a propositional symbol.

On some remarkable relations between paraconsistent logics, modal logics, and ambiguity logics

Abstract: This paper concerns some connections between paraconsistent logics, modal logics (mainly S5), and Ambiguity Logic AL (Classical Logic applied to a language in which all letters are indexed and in which quantifiers over such indices are present). S5 may be defined from AL. Three kinds of connections will be illustrated. First, a paraconsistent logic A is presented that has the same expressive power as S5. Next, I consider the definition of paraconsistent logics from S5 and AL. Such definition is shown to work for some logics, for example Priest's LP. Other paraconsistent logics appear to withstand such definition, typically those that contain a detachable material implication. Finally, I show that some paraconsistent logics and inconsistency-adaptive logics serve exactly the same purpose as some modal logics and ampliative adaptive logics based on S5. However, they serve this purpose along very different roads and the logics cannot be defined from one another. The paper intends to open lines of research rather than pursuing them to the end. It also contains a poor person's semantics for S5 as well as a description of the simple but useful and powerful AL.

Modeling Paraconsistent Reasoning by Classical Logic

Abstract: We introduce a general method for paraconsistent reasoning in knowledge systems by classical second-order formulae. A standard technique for paraconsistent reasoning on inconsistent classical theories is by shifting to multiple-valued logics. We show how these multiple-valued theories can be "shifted back" to two-valued classical theories (through a polynomial transformation), and how preferential reasoning based on multiple-valued logic can be represented by classical circumscription-like axioms. By applying this process we manage to overcome the shortcoming of classical logic in properly handling inconsistent data, and provide new ways of implementing multiple-valued paraconsistent reasoning in knowledge systems. Standard multiple-valued reasoning can thus be performed through theorem provers for classical logic, and multiple-valued preferential reasoning can be implemented using algorithms for processing circumscriptive theories (such as DLS and SCAN).

Logic: A Misleading Concept. A Contradiction Study toward Agent's Logic

Abstract: The paper presents a fresh new comprehensive ideology on Neutrosophic Logic based on contradiction study in a broad sense: general critics on conventional logic by examining the essence of logic, fresh insights on logic definition based on Chinese philosophical survey, and a novel and genetic logic model as the elementary cell against Von Neumann oriented ones based on this novel definition. As for the logic definition, the paper illustrates that logic is rather a tradeoff between different factors than truth and false abstraction. It is stressed that the kernel of any intelligent system is exactly a contradiction model. The paper aims to solve the chaos of logic and exhibit the potential power of neutrosophy: a new branch of scientific philosophy.

Quantum Logic as a Fragment of Independence-Friendly Logic

Abstract: The working assumption of this paper is that noncommuting variables are irreducibly interdependent. The logic of such dependence relations is the author's independence-friendly (IF) logic, extended by adding to it sentence-initial contradictory negation ¬ over and above the dual (strong) negation ∼. Then in a Hilbert space ∼ turns out to express orthocomplementation. This can be extended to any logical space, which makes it possible to define the dimension of a logical space. The received Birkhoff and von Neumann “quantum logic” can be interpreted by taking their “disjunction” to be ¬(∼A & ∼B). Their logic can thus be mapped into a Boolean structure to which an additional operator ∼ has been added.

Paraconsistent Knowledge Bases and Many-Valued Logic

Abstract: Classical logic predicts that everything (thus nothing useful at all) follows from inconsistency. A paraconsistent logic is a logic where an inconsistency does not lead to such an explosion, and since in practice consistency is difficult to achieve there are many potential applications of paraconsistent logics in knowledge bases. We present a paraconsistent many-valued logic with a simple and new semantics for the logical operators. In particular we compare our approach with work based on bilattices. The adequacy of the logic is examined by a case study in the domain of medicine.

How to Reason Sensibly Yet Naturally from Inconsistencies

Abstract: This paper concerns problem solving in inconsistent contexts. It is usually taken for granted that inconsistencies are false, and I shall not challenge this view here.’ Resolving some inconsistency may constitute the very problem one tries to solve. Alternatively, one may realize that one is (and for some time will be) unable to resolve an inconsistency within some domain, but nevertheless aim at solving another problem within that domain. In cases like this, one faces two difficulties. The first is to distinguish between inferences that are sensible and those that are not. The second is to determine when a solution to the problem is acceptable. For a decision on the latter difficulty, mere derivability (by some appropriate logic) is not sufficient. It should also be plausible that the solution will remain derivable after the inconsistencies are resolved.

Paraconsistent Query Answering Systems

Abstract: Classical logic predicts that everything (thus nothing useful at all) follows from inconsistency. A paraconsistent logic is a logic where inconsistency does not lead to such an explosion, and since in practice consistency is difficult to achieve there are many potential applications of paraconsistent logics in query answering systems. We compare the paraconsistent and the non-monotonic solutions to the problem of contradictions. We propose a many-valued paraconsistent logic based on a simple notion of indeterminacy. In particular we describe the semantics of the logic using key equalities for the logical operators. We relate our approach to works on bilattices. We also discuss and provide formalizations of two case studies, notably the well-known example involving penguins and a more interesting example in the domain of medicine.