Showing papers on "Paraconsistent logic published in 2001"

A logic for uncertain probabilities

Abstract: We first describe a metric for uncertain probabilities called opinion, and subsequently a set of logical operators that can be used for logical reasoning with uncertain propositions. This framework which is called subjective logic uses elements from the Dempster-Shafer belief theory and we show that it is compatible with binary logic and probability calculus.

A Taxonomy of C-systems

Abstract: A thorough investigation of the foundations of paraconsistent logics. Relations between logical principles are formally studied, a novel notion of consistency is introduced, the logics of formal inconsistency, and the subclasses of C-systems and dC-systems are defined and studied. An enormous variety of paraconsistent logics in the literature is shown to constitute C-systems.

A logic for uncertain probabilities

Abstract: We first describe a metric for uncertain probabilities called opinion, and subsequently a set of logical operators that can be used for logical reasoning with uncertain propositions. This framework which is called subjective logic uses elements from the Dempster-Shafer belief theory and we show that it is compatible with binary logic and probability calculus.

The Evolution of Reason: Logic as a Branch of Biology

Abstract: 1. The biology of logic 2. The evolutionary derivation of life-history strategy theory 3. The evolutionary derivation of decision logic 4. The evolutionary derivation of inductive logic (Part I) 5. The evolutionary derivation of deductive logic 6. The evolutionary derivation of inductive logic (Part II) 7. The evolutionary derivation of mathematics 8. Broadening the evolutionary base of classical logic 9. The evolutionary derivation of nonclassical logic 10. Radical reductionism in logic 11. Toward a unified science of reason Appendix: formal theory.

PANDORA: a multi-agent system using paraconsistent logic

Abstract: This work is part of the Multicheck Project that defines architecture of cognitive and independents agents for the automatic treatment of handwritten Brazilian bank checks. The concept of autonomous agents allows us to organize the application knowledge and brings several own benefits to the approach. The choice of this approach is supported in a triple hypothesis. First, the nature of the problem in question allows decomposition in well-defined tasks, and each of them can be encapsulated in an independent agent. Second, the natural capability of interaction of the agents makes the check treatment process more robust, solving situations apparently difficult. Third, the natural parallelism between the agents can contribute to implement an application with high performance.

Combinators for Paraconsistent Attitudes

Abstract: In order to analyse the semantics of natural language sentences a translation into a partial type logic using lexical and logical combinators is presented. The sentences cover a fragment of English with propositional attitudes like knowledge, belief and assertion. A combinator is a closed term of the lambda calculus possibly containing lexical and/or logical constants. Such combinators seem promising from both a cognitive and computational point of view. There is approximately one lexical combinator for each word, but just eleven logical combinators for the present fragment. The partiality is only used for embedded sentences expressing propositional attitudes, thereby allowing for inconsistency without explosion (also called paraconsistency), and is based on a few key equalities for the connectives giving four truth values (truth, falsehood, and undefinedness with negative and positive polarity; only the first truth value is designated, i.e. yields the logical truths).

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.

A Semantic Tableau Version of First-Order Quasi-Classical Logic

Abstract: Quasi-classical logic (QC logic) allows the derivation of non-trivial classical inferences from inconsistent information. A paraconsistent, or non-trivializable, 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" at the object level so that important proof rules such as modus tollens, modus ponens, and disjunctive syllogism hold. Here we develop QC logic by presenting a semantic tableau version for first-order QC logic.

Essays on Non-Classical Logic

Abstract: Fine-Grained Theories of Time (P Blackburn) Revision Sequences and Computers with an Infinite Amount of Time (B Lowe) On Frege's Nightmare: A Combination of Intuitionistic, Free and Paraconsistent Logics (S Rahman) Truthmakers, Entailment and Necessity (S Read) Global Definability in Basic Modal Logic (M de Rijke & H Sturm) Ackermann's Implication for Typefree Logic (K Robering) Why Dialogical Logic? (H Ruckert) Semantics for Constructive Negations (Y Shramko) Recent Trends in Paraconsistent Logic (M Urchs) Obligations, Authorities, and History Dependence (H Wansing).

A Natural Deduction System for Intuitionistic Fuzzy Logic

Abstract: Intuitionistic fuzzy logic IF was introduced by Takeuti and Titani This logic coincides with the first-order Godel logic based on the real unit interval [0,1] as set of truth-values We present a natural deduction system NIF for IF NIF is defined by suitably translating a first-order extension of Avron’s hypersequent calculus for Godel logic Soundness, completeness and normal form theorems for NIF are provided

Paraconsistency: Towards a tentative interpretation

Abstract: In this expository paper, we examine some philosophical and technical issues brought by paraconsistency (such as, motivations for developing a paraconsistent logic, the nature of this logic, and its application to set theory). We also suggest a way of accommodating these issues by considering some problems in the philosophy of logic from a new perspective.

The pleasures of anticipation: enriching intuitionistic logic

Abstract: We explore a relation we call ‘anticipation’ between formulas, where AanticipatesB (according to some logic) just in case B is a consequence (according to that logic, presumed to support some distinguished implicational connective →) of the formula A→B We are especially interested in the case in which the logic is intuitionistic (propositional) logic and are much concerned with an extension of that logic with a new connective, written as “a”, governed by rules which guarantee that for any formula B, aB is the (logically) strongest formula anticipating B The investigation of this new logic, which we call ILa, will confront us on several occasions with some of the finer points in the theory of rules and with issues in the philosophy of logic arising from the proposed explication of the existence of a connective (with prescribed logical behaviour) in terms of the conservative extension of a favoured logic by the addition of such a connective Other points of interest include the provision of a Kripke semantics with respect to which ILa is demonstrably sound, deployed to establish certain unprovability results as well as to forge connections with C Rauszer's logic of dual intuitionistic negation and dual intuitionistic implication, and the isolation of two relations (between formulas), head-implication and head-linkage, which, though trivial in the setting of classical logic, are of considerable significance in the intuitionistic context

On Frege's Nightmare: A Combination of Intuitionistic, Free and Paraconsistent Logics

Abstract: In this paper I present a dialogical formulation of free logic which allows a straightforward combination of paraconsistent and intuitionistic logic-I call this combination Frege's Nightmare. The ideas behind this combination can be expressed in few words: in an argumentation, it sometimes makes sense to restrict the use and introduction of singular terms in the context of quantification to a formal use of those terms. That is, the Proponent is allowed to use a constant for a defence (of an existential quantifier) or an attack (on a universal quantifier) iff this constant has been explicitly conceded by the Opponent. When the Opponent concedes any constant occurring in an atomic formula he concedes tertium non-datur (for this formula) to. This yield a free-logic which combines classical (for propositions with singular terms for realities) with intuitionistic logic (for propositions with singular terms for fictions). This extended free-logic can be also combined with paraconsistent logic in such a way that contradictory objects can be included in the domain of fictions. The idea is here to combine the concept of formal use of constants in free logics and that of the formal use of elementary negations in paraconsistent logics.

Soft constraints and heuristic constraint correction in entity-relationship modelling

Abstract: We expose paraconsistent logic with regard to its potential to contribute to the foundations of databases. We do so from a historical perspective, starting at the ancient inception and arriving at the contemporary use of logic as a computational device. We show that an understanding of the logic foundations of databases in terms of paraconsistency is adequate. It avoids absurd connotations of the ex contradictione quodlibet principle, which in fact never applies in databases. We interpret datalog, its origins and some of its extensions by negation and abduction, in terms of paraconsistency. We propose a procedural definition of paraconsistency and show that many well-known query answering procedures comply with it.

A Reply to Beall and Colyvan

Abstract: Beall and Colyvan (2001) extend the debate over paraconsistent approaches to the Sorites Paradox, offering additional argument for and additional argument against pursuing such an approach. Additional argument for a paraconsistent approach comes from considerations of simplicity which stress, amongst other things, uniformity of approach. This, coupled with the assumption that the paradoxes of self-reference are best dealt with by a paraconsistent approach, gives weight to the thought that, all other things being equal, one should pursue paraconsistent solutions to Sorites paradoxes as well. However, as Beall and Colyvan themselves note (see their footnote 5), all other things may not be equal. One does not need to look far to find independent arguments for paracompleteness -presupposition failure, reference failure, future contingents, etc. If any such argument is found compelling then similar considerations to those adduced by Beall and Colyvan will push in the opposite direction to paraconsistency, namely towards paracompleteness. On this plausible view paraconsistent logics are simply not able to 'do it all' and so the question re-arises as to whether a paraconsistent approach to the Sorites is superior. (Interestingly, if the Sorites Paradox could be shown to be 'of a kind' with the paradoxes of self-reference, then a more specific uniformity argument could be mounted in favour of a paraconsistent approach to paradoxes 'of this kind'.) In the end, considerations of uniformity may simply amount to an argument in favour of an even weaker paralogical approach-that is, one that is both paraconsistent and paracomplete.1

Paraconsistency and dialogue logic critical examination and further explorations

Abstract: The first part of this paper presents asympathetic and critical examination of the approachof Shahid Rahman and Walter Carnielli, as presented intheir paper “The Dialogical Approach toParaconsistency”. In the second part, possibleextensions are presented and evaluated: (a) top-downanalysis of a dialogue situation versus bottom-up, (b)the specific role of ambiguities and how to deal withthem, and (c) the problem of common knowledge andbackground knowledge in dialogues. In the third part,I claim that dialogue logic is the best-suitedinstrument to analyse paradoxes of the Sorites type.All these considerations lead to philosophicallyrelevant observations concerning principles of charityon the one hand, and compactness on the other.

Historical and computational aspects of paraconsistency in view of the logic foundation of databases

Abstract: Modern information systems must respect certain restrictions in order to guarantee the proper and desired functionality Semantic constraints help to prevent inconsistencies in the stored data resulting from faulty updates Security constraints are to maintain integrity, secrecy and availability over updates and over queries In this paper we design an unifying framework for both kinds of constraints in order to study interactions between them We view a distributed information system as a multi-agent system in which all components of the system are seen as agents We present a temporal and epistemic logic for the defined framework and show in an example how security constraints and semantic constraints can be expressed in this framework

Classical and constructive logic

Abstract: Logic can be described as the science of reasoning. Some of the central questions that logicians try to address are: “What constitutes a logical argument?” “What does it mean to say that a certain statement is a logical consequence of another?” These questions are extremely broad, and to make any kind of progress we need to narrow our focus. In everyday life, we use different modes of reasoning in different contexts. We can reason about our experiences, and try to determine causal relations between different types of events; this forms the basis of scientific inquiry. We can reason probabilistically, and try to determine the “odds” that the Pirates will win the World Series; or we can employ subjunctive reasoning, and wonder what would have happened had Bill Clinton lost the election. We can reason about events occuring in time, or space; we can reason about knowledge, and belief; or we can reason about moral responsibility, and ethical behavior. We can even try to reason about properties that are vague and imprecise, or try to draw “reasonable” conclusions from vague or incomplete data. For the most part, in this course we will be interested in one very specific kind of reasoning, namely, the kind that is appropriate for reasoning about abstract mathematical data like numbers, sets, functions, trees, sequences (lists), graphs, and so on. Consider the statement “every even number

Bibel's matrix connection method in paraconsistent logic: general concepts and implementation

Abstract: Bibel's matrix connection method is an alternative to resolution for the mechanized proof of logical statements. Bibel's method was originally defined for classical logic. In this work, an adaptation of the method for annotated propositional logic is given, followed by a simple case study. Some implementation details are also presented.

Many valued paraconsistent logic

Abstract: In contrast to most logics, in paraconsistent logic it is not true that everything followed from a contradiction. The semantics for one of the best known paraconsistent logics, LP, permits sentences to be both true and false; but at the same time, the semantic characterization of the logical particles is classical. We define the notion of "molecular logic", of which all finite valued variants of LP are a type. Generally paraconsistent logics do not contain extensional conditionals. Molecular logics of n values may be conservatively extended to standard many valued logics of 2/sup n/-1 values, in which it is easy to define extensional conditionals with the usual detachment rules. The extension, while paraconsistent relative to a negation satisfying standard conditions on the original n values, is not paraconsistent relative to a negation satisfying standard conditions on the 2/sup n/-1 values of the extension. We conclude that the logic LP and its many valued generalizations are paraconsistent because of expressive incompleteness.

Law and Logic

Abstract: By a system of law we shall mean — in this paper — any system of rules which has the purpose of regulating human action under certain conditions. Examples: A nation’s constitution, the traffic laws, club’s statutes, recipes in a cook-book, etc.

Vasiliev's paraconsistent logic interpreted by means of the dual role played by the double negation law

Abstract: Vasiliev's paraconsistent logic is interpreted by differentiating in a scientific theory among axiom-principles, methodological principles and mere guess. The three kinds of sentence fit the three kinds of Vasiliev's sentences when "S is A" is translated in ¬¬A→A, and accordingly the remaining two sentences. In this interpretation it is shown that the parallelism Vasiliev claimed to hold between his logic and Lobachevskii's non-Euclidean geometry may be formalised. Between the two formal paraconsistent systems previously suggested - i.e. Arruda's one and da Costa and Puga's one - the latter one fits the above interpretation..

Presentation of the course: Some propositional paraconsistent logics

Abstract: Paraconsistent logics are logics in which inconsistencies do not necessarily lead trivialization, i.e. logics which challenge the classical principle of pseudo-scotus, according to which anything follows from contradictory premises, ex contradictione quodlibet. Such principle is connected, in classical logic, with the idea that no contradiction is possible, or the principle of non-contradiction.1 The philosophical environment for the discussion of logics derogating the principle of non-contradiction took place with the emmerging non-classical logics in the beginnings of the 20th century [11]. N. A. Vaśilev [2] and J. Lukasiewicz [35] may be pointed out as the precursors of non-classical logics, already in 1910, by their proposed derogation of the basic laws of Aristotelian logic. By their attack of Aristotle’s defence of non-contradiction as the safest of the logical principles (Metaphisics, book Γ), they are also indicated as precursors of paraconsistent logics. However, sistematizations of paraconsistent calculi only appeared in the 50’s through the work on discussive logic by Stanislaw Jaskowski [29], in 1948, in Poland and on hierarchies of paraconsistentlogics, by the brazilian logician Newton C. A. da Costa [14] and [15], in 1954. Both authors worked independently, with different motivations, and only many years after Jaskowski’s premature death, which hindered the development of his ideas, da Costa took contact with discussive logic and expanded it to first and higher order logics [19] and [38]. (on the history of paraconsistent logic, see [3], [23] and [26]). Jaskowski’s main motivation was to develop a logic of discussion where different and consistent arguers may hold statements which contradict the argument of other arguers. Such a logic should be able to cope with the

What Should We Do With Traditional Logic

Abstract: There is a clash between some people's positive logical intuitions about traditional or Aristotelian logic and the assessment ofthat logic made by modem logic. In response to the clash, four sorts of reasons that might be given for referring one logic to the other are considered, but it is argued that none of them provides a decisive reason in favor of one rather than the other. A reformist and a radical response to the apparent inability to give reasons to prefer one logic to the other are considered and reasons given for preferring the radical response.