Showing papers on "Paraconsistent logic published in 2005"

Normal Natural Deduction Proofs (in classical logic)

Abstract: We provide a theoretical framework that allows the direct search for natural deduction proofs in some non-classical logics, namely, intuitionistic sentential and predicate logic, but also in the modal logic S4. The framework uses so-called intercalation calculi to build up broad search spaces from which normal proofs can be extracted, if a proof exists at all. This claim is supported by completeness proofs establishing in a purely semantic way normal form theorems for the above logics. Logical restrictions on the search spaces are briefly discussed in the last section together with some heuristics for structuring a more efficient search. Our paper is a companion piece to [15], where classical logic was treated.

Dual Intuitionistic Logic and a Variety of Negations: The Logic of Scientific Research

Abstract: We consider a logic which is semantically dual (in some precise sense of the term) to intuitionistic. This logic can be labeled as “falsification logic”: it embodies the Popperian methodology of scientific discovery. Whereas intuitionistic logic deals with constructive truth and non-constructive falsity, and Nelson's logic takes both truth and falsity as constructive notions, in the falsification logic truth is essentially non-constructive as opposed to falsity that is conceived constructively. We also briefly clarify the relationships of our falsification logic to some other logical systems.

Query answering in normal logic programs under uncertainty

Abstract: We present a simple, yet general top-down query answering procedure for normal logic programs over lattices and bilattices, where functions may appear in the rule bodies. Its interest relies on the fact that many approaches to paraconsistency and uncertainty in logic programs with or without non-monotonic negation are based on bilattices or lattices, respectively.

The Class of Extensions of Nelson's Paraconsistent Logic

Abstract: The article is devoted to the systematic study of the lattice eN4⊥ consisting of logics extending N4⊥. The logic N4⊥ is obtained from paraconsistent Nelson logic N4 by adding the new constant ⊥ and axioms ⊥ → p, p → ∼ ⊥. We study interrelations between eN4⊥ and the lattice of superintuitionistic logics. Distinguish in eN4⊥ basic subclasses of explosive logics, normal logics, logics of general form and study how they are relate.

From Consequence Operator to Universal Logic: A Survey of General Abstract Logic

Abstract: We present an overview of the different frameworks and structures that have been proposed during the last century in order to develop a general theory of logics. This includes Tarski’s consequence operator, logical matrices, Hertz’s Satzsysteme, Gentzen’s sequent calculus, Suszko’s abstract logic, algebraic logic, da Costa’s theory of valuation and universal logic itself.

On a Problem of da Costa

Abstract: The two main founders of paraconsistent logic, Stanis law Jaśkowski and Newton da Costa, built their systems on distinct grounds. Starting from different projects, they used different tools and ultimately designed quite different calculi to attend their needs. How successful were their enterprises? Here we discuss the problem of defining paraconsistent logics following the original instructions laid down by da Costa. We present a new approach to P, the first full solution —proposed by Antonio Mario Sette— to the problem of da Costa, and argue in favor of yet another solution we shall study here: the logic P. Both P and P constitute maximal 3-valued paraconsistent fragments of classical logic. Constructive completeness proofs are here presented for both logics. 1 Requisites to paraconsistent calculi When proposing the first paraconsistent propositional system, in 1948, Jaśkowski expected it to enjoy the following properties (see [17]): Jas1 when applied to inconsistent systems it should not always entail their trivialization; Jas2 it should be rich enough to enable practical inferences; Jas3 it should have an intuitive justification. A few years later, in 1963, da Costa would independently tackle a similar problem, this time proposing a whole hierarchy of paraconsistent propositional calculi, known as Cn, for 0 < n < ω. His requisites to these calculi were the following (see [12]): NdC1 in these calculi the principle of non-contradiction, in the form ¬(A∧¬A), should not be a valid schema; NdC2 from two contradictory formulae, A and ¬A, it would not in general be possible to deduce an arbitrary formula B; NdC3 it should be simple to extend these calculi to corresponding predicate calculi (with or without equality); NdC4 they should contain the most part of the schemata and rules of the classical propositional calculus which do not interfere with the first conditions.

Routley semantics for answer sets

Abstract: We present an alternative model theory for answer sets based on the possible worlds semantics proposed by Routley (1974) as a framework for the propositional logics of Fitch and Nelson By introducing a falsity constant or second negation into Routley models, we show how paraconsistent as well as ordinary answer sets can be represented via a simple minimality condition on models This means we can define a paraconsistent version of equilibrium logic, or paraconsistent answer sets (PAS) for propositional theories The underlying logic of PAS is denoted by N9 We characterise it axiomatically and algebraically, showing it to be the least conservative extension of the logic of here-and-there with strong negation In addition, we show that N9 captures the strong equivalence of programs in the paraconsistent case and can thus serve as a useful mathematical foundation for PAS We end by showing that N9 has the Interpolation Property

Modality and Paraconsistency

Abstract: Paraconsistent logic was born in the vicinity of modal logic. Moreover, as every other non-classical logicians, paraconsistentists have very often flirted with modalities. The first known system of paraconsistent logic was in fact defined as a fragment of S5, in the late 40s. But a fragment of a modal system is not necessarily a modal system. I will show here, indeed, that Jaśkowski’s D2 is not a modal logic, in the contemporary usual meaning of the term. By contrast, I will also show, subsequently, that any non-degenerate normal modal system is inherently paraconsistent.

A Case for Paraconsistent Logic as Foundation of Future Information Systems.

Abstract: Logic links philosophy with computer science and is the acknowledged foundation of information systems. Since the large scale proliferation of the internet and the world wide web, however, a rush of new technologies is avalanching, in many cases without much consideration of a solid foundation that would be up to par with the rigor of the traditional logic fundament. Philosophy may help to question established foundations, especially in times of technological breakthroughs that seem to override such foundations. In particular, the intolerance associated with the consistency requirements of classical logic begs question of its legitimacy, in the face of ubiquitous inconsistency in virtually all information systems of sizable extent. Based on that, we propose to overcome classical logic foundations by adopting paraconsistency as a foundational concept for future information systems engineering (ISE).

Natural deduction systems for Nelson's paraconsistent logic and its neighbors

Abstract: Firstly, a natural deduction system in standard style is introduced for Nelson's para-consistent logic N4, and a normalization theorem is shown for this system. Secondly, a natural deduction system...

Psychologism in the Logic of John Stuart Mill: Mill on the Subject Matter and Foundations of Ratiocinative Logic

Abstract: This paper considers the question of whether Mill's account of the nature and justificatory foundations of deductive logic is psychologistic. Logical psychologism asserts the dependency of logic on psychology. Frequently, this dependency arises as a result of a metaphysical thesis asserting the psychological nature of the subject matter of logic. A study of Mill's System of Logic and his Examination reveals that Mill held an equivocal view of the subject matter of logic, sometimes treating it as a set of psychological processes and at other times as the objects of those processes. The consequences of each of these views upon the justificatory foundations of logic are explored. The paper concludes that, despite his providing logic with a prescriptive function, and despite his avoidance of conceptualism, Mill's theory fails to provide deductive logic with a justificatory foundation that is independent of psychology.

Socratic Proofs and Paraconsistency: A Case Study

Abstract: This paper develops a new proof method for two propositional paraconsistent logics: the propositional part of Batens' weak paraconsistent logic CLuN and Schutte's maximally paraconsistent logic Φv. Proofs are de.ned as certain sequences of questions. The method is grounded in Inferential Erotetic Logic.

Metareasoning for multi-agent epistemic logics

Abstract: We present an encoding of a sequent calculus for a multi-agent epistemic logic in Athena, an interactive theorem proving system for many-sorted first-order logic. We then use Athena as a metalanguage in order to reason about the multi-agent logic an as object language. This facilitates theorem proving in the multi-agent logic in several ways. First, it lets us marshal the highly efficient theorem provers for classical first-order logic that are integrated with Athena for the purpose of doing proofs in the multi-agent logic. Second, unlike model-theoretic embeddings of modal logics into classical first-order logic, our proofs are directly convertible into native epistemic logic proofs. Third, because we are able to quantify over propositions and agents, we get much of the generality and power of higher-order logic even though we are in a first-order setting. Finally, we are able to use Athena's versatile tactics for proof automation in the multi-agent logic. We illustrate by developing a tactic for solving the generalized version of the wise men problem.

Possible-translations algebraization for paraconsistent logics

Abstract: This note proposes a new notion of algebraizability, which we call possibletranslations algebraic semantics, based upon the newly developed possibletranslations semantics. This semantics is naturally adequate to obtain an algebraic interpretation for paraconsistent logics, and generalizes the well-known method of algebraization by W. Blok and D. Pigozzi. This generalization obtains algebraic semantics up to translations, applicable to several non-classical logics and particularly apt for paraconsistent logics, a philosophically relevant class of logics with growing importance for applications.

The geometry of non-distributive logics

Abstract: In this paper we introduce a new natural deduction system for the logic of lattices, and a number of extensions of lattice logic with different negation connectives. We provide the class of natural deduction proofs with both a standard inductive definition and a global graph-theoretical criterion for correctness, and we show how normalisation in this system corresponds to cut elimination in the sequent calculus for lattice logic. This natural deduction system is inspired both by Shoesmith and Smiley's multiple conclusion systems for classical logic and Girard's proofnets for linear logic. © 2005, Association for Symbolic Logic.

Paraconsistent artificial neural network: an application in cephalometric analysis

Abstract: Paraconsistent artificial neural network (PANN) is a mathematical structure based on paraconsistent logic, which allows dealing with uncertainties and contradictions. In this paper we propose an application of PANN to analyze cephalometric measurements in order to support orthodontics diagnostics. Orthodontic and cephalometrical analysis taking into account several uncertainties and contradictions, ideal scenario to be treated by paraconsistent approach.

A framework for reasoning with rough sets

Abstract: Rough sets framework has two appealing aspects. First, it is a mathematical approach to deal with vague concepts. Second, rough set techniques can be used in data analysis to find patterns hidden in the data. The number of applications of rough sets to practical problems in different fields demonstrates the increasing interest in this framework and its applicability. This thesis proposes a language that caters for implicit definitions of rough sets obtained by combining different regions of other rough sets. In this way, concept approximations can be derived by taking into account domain knowledge. A declarative semantics for the language is also discussed. It is then shown that programs in the proposed language can be compiled to extended logic programs under the paraconsistent stable model semantics. The equivalence between the declarative semantics of the language and the declarative semantics of the compiled programs is proved. This transformation provides the computational basis for implementing our ideas. A query language for retrieving information about the concepts represented through the defined rough sets is also discussed. Several motivating applications are described. Finally, an extension of the proposed language with numerical measures is presented. This extension is motivated by the fact that numerical measures are an important aspect in data mining applications.

Knowledge creation and sharing: complex methods of inquiry and inconsistent theory

Abstract: Recent research and practice have led to the development of relatively complex methods for inquiry which can be applied by human analysts. However, it has appeared until recently that these could not be supported by software tools, since the limitations of traditional mathematical algorithms constrained their development. We suggest a model which lays the foundations for the development of software support, based on a paraconsistent approach. Some of the methods available to analysts are based on the SST (Strategic Systemic Thinking) framework. This framework recognizes contextual dependencies, and enables analysts to include, as part of their analytical resolutions, conclusions which are in themselves contradictory. Software support for this kind of thought process would have been impossible to achieve in using traditional mathematical models. Tools supporting analytical work have, in the past, fallen into one of three categories:- those which support data manipulation, those which provide support for process, and those which attempt to support analysis directly. Until recently, for complex analytical models such as the SST framework, only the first of these categories was realistically available. However, making use of developments in the field of paraconsistent logic, it is now possible to envisage development of tools in the second category – process support. Paraconsistent logic was developed to provide a framework for inconsistent but non-trivial theories. Since the early 20th century the field has become very fruitful. Many thousands of papers have been published and important applications in computer science, information theory and artificial intelligence owe their origins to insights gained from paraconsistency. The application suggested in this paper is very much in the spirit of this well established tradition. (Less)

An intelligent safety verification based on a paraconsistent logic program

Abstract: We introduce an intelligent safety verification method based on a paraconsistent logic program EVALPSN with a simple example for brewery pipeline valve control. The safety verification is carried out by paraconsistent logic programming called EVALPSN.

Comparison of deduction theorems in diverse logic systems

Abstract: Deduction theorem and its weak forms in classical mathematical logic system, Łukasiewicz logic system, Godel logic system, product logic system, and the fuzzy logic system ℒ* are discussed and compared. It is pointed out that the weak form of deduction theorem in ℒ* has a clear structure and can be employed to define the concept of consistency degrees of finite theories. Moreover, it is clarified that the negation operator of Godel type is too strong and is therefore unsuitable for establishing fuzzy logic systems.

On the Usefulness of Paraconsistent Logic

Abstract: In this paper, we examine some intuitive motivations to develop a para-consistent logic. These motivations are formally developed using semantic ideas, and we employ, in particular, bivaluations and truth-tables to characterise this logic. After discussing these ideas, we examine some applications of paraconsistent logic to various domains. With these motivations and applications in hand, the usefulness of paraconsistent logic becomes hard to deny.