Showing papers on "Paraconsistent logic published in 2012"

An ensemble design of intrusion detection system for handling uncertainty using Neutrosophic Logic Classifier

Abstract: In the real world it is a routine that one must deal with uncertainty when security is concerned. Intrusion detection systems offer a new challenge in handling uncertainty due to imprecise knowledge in classifying the normal or abnormal behaviour patterns. In this paper we have introduced an emerging approach for intrusion detection system using Neutrosophic Logic Classifier which is an extension/combination of the fuzzy logic, intuitionistic logic, paraconsistent logic, and the three-valued logics that use an indeterminate value. It is capable of handling fuzzy, vague, incomplete and inconsistent information under one framework. Using this new approach there is an increase in detection rate and the significant decrease in false alarm rate. The proposed method tripartitions the dataset into normal, abnormal and indeterministic based on the degree of membership of truthness, degree of membership of indeterminacy and degree of membership of falsity. The proposed method was tested up on KDD Cup 99 dataset. The Neutrosophic Logic Classifier generates the Neutrosophic rules to determine the intrusion in progress. Improvised genetic algorithm is adopted in order to detect the potential rules for performing better classification. This paper exhibits the efficiency of handling uncertainty in Intrusion detection precisely using Neutrosophic Logic Classifier based Intrusion detection System.

Transfinite cardinals in paraconsistent set theory

Abstract: This paper develops a (nontrivial) theory of cardinal numbers from a naive set comprehension principle, in a suitable paraconsistent logic. To underwrite cardinal arithmetic, the axiom of choice is proved. A new proof of Cantor’s theorem is provided, as well as a method for demonstrating the existence of large cardinals by way of a reflection theorem.

Why the Logical Hexagon

Abstract: The logical hexagon (or hexagon of opposition) is a strange, yet beautiful, highly symmetrical mathematical figure, mysteriously intertwining fundamental logical and geometrical features. It was discovered more or less at the same time (i.e. around 1950), independently, by a few scholars. It is the successor of an equally strange (but mathematically less impressive) structure, the “logical square” (or “square of opposition”), of which it is a much more general and powerful “relative”. The discovery of the former did not raise interest, neither among logicians, nor among philosophers of logic, whereas the latter played a very important theoretical role (both for logic and philosophy) for nearly two thousand years, before falling in disgrace in the first half of the twentieth century: it was, so to say, “sentenced to death” by the so-called analytical philosophers and logicians. Contrary to this, since 2004 a new, unexpected promising branch of mathematics (dealing with “oppositions”) has appeared, “oppositional geometry” (also called “n-opposition theory”, “NOT”), inside which the logical hexagon (as well as its predecessor, the logical square) is only one term of an infinite series of “logical bi-simplexes of dimension m”, itself just one term of the more general infinite series (of series) of the “logical poly-simplexes of dimension m”. In this paper we recall the main historical and the main theoretical elements of these neglected recent discoveries. After proposing some new results, among which the notion of “hybrid logical hexagon”, we show which strong reasons, inside oppositional geometry, make understand that the logical hexagon is in fact a very important and profound mathematical structure, destined to many future fruitful developments and probably bearer of a major epistemological paradigm change.

Paraconsistency: Logic and Applications

Abstract: A logic is called 'paraconsistent' if it rejects the rule called 'ex contradictione quodlibet', according to which any conclusion follows from inconsistent premises. While logicians have proposed many technically developed paraconsistent logical systems and contemporary philosophers like Graham Priest have advanced the view that some contradictions can be true, and advocated a paraconsistent logic to deal with them, until recent times these systems have been little understood by philosophers. This book presents a comprehensive overview on paraconsistent logical systems to change this situation. The book includes almost every major author currently working in the field. The papers are on the cutting edge of the literature some of which discuss current debates and others present important new ideas. The editors have avoided papers about technical details of paraconsistent logic, but instead concentrated upon works that discuss more "big picture" ideas. Different treatments of paradoxes takes centre stage in many of the papers, but also there are several papers on how to interpret paraconistent logic and some on how it can be applied to philosophy of mathematics, the philosophy of language, and metaphysics.

Multi-agent Justification Logic: communication and evidence elimination

Abstract: This paper presents a logic combining Dynamic Epistemic Logic, a framework for reasoning about multi-agent communication, with a new multi-agent version of Justification Logic, a framework for reasoning about evidence and justification. This novel combination incorporates a new kind of multi-agent evidence elimination that cleanly meshes with the multi-agent communications from Dynamic Epistemic Logic, resulting in a system for reasoning about multi-agent communication and evidence elimination for groups of interacting rational agents.

Treatment of Uncertainties with Algorithms of the Paraconsistent Annotated Logic

Abstract: The method presented in this work is based on the fundamental concepts of Paraconsistent Annotated Logic with annotation of 2 values (PAL2v). The PAL2v is a non-classic Logics which admits contradiction and in this paper we perform a study using mathematical interpretation in its representative lattice. This studies result in algorithms and equations give an effective treatment on signals of information that represent situations found in uncertainty knowledge database. From the obtained equations, algorithms are elaborated to be utilized in computation models of the uncertainty treatment Systems. We presented some results that were obtained of analyses done with one of the algorithms that compose the paraconsistent analyzing system of logical signals with the PAL2v Logic. The paraconsistent reasoning system built according to the PAL2v methodology notions reveals itself to be more efficient than the traditional ones, because it gets to offer an appropriate treatment to contradictory information.

The dynamic turn in quantum logic

Abstract: In this paper we show how ideas coming from two areas of research in logic can reinforce each other. The first such line of inquiry concerns the “dynamic turn” in logic and especially the formalisms inspired by Propositional Dynamic Logic (PDL); while the second line concerns research into the logical foundations of Quantum Physics, and in particular the area known as Operational Quantum Logic, as developed by Jauch and Piron (Helve Phys Acta 42:842–848, 1969), Piron (Foundations of Quantum Physics, 1976). By bringing these areas together we explain the basic ingredients of Dynamic Quantum Logic, a new direction of research in the logical foundations of physics.

Real Analysis in Paraconsistent Logic

Abstract: This paper begins an analysis of the real line using an inconsistency-tolerant (paraconsistent) logic. We show that basic field and compactness properties hold, by way of novel proofs that make no use of consistency-reliant inferences; some techniques from constructive analysis are used instead. While no inconsistencies are found in the algebraic operations on the real number field, prospects for other non-trivializing contradictions are left open.

The New Rising of the Square of Opposition

Abstract: In this paper I relate the story about the new rising of the square of opposition: how I got in touch with it and started to develop new ideas and to organize world congresses on the topic with subsequent publications My first contact with the square was in connection with Slater’s criticisms of paraconsistent logic Then by looking for an intuitive basis for paraconsistent negation, I was led to reconstruct S5 as a paraconsistent logic considering ¬□ as a paraconsistent negation Making the connection between ¬□ and the O-corner of the square of opposition, I developed a paraconsistent star and hexagon of opposition and then a polyhedron of opposition, as a general framework to understand relations between modalities en negations I also proposed the generalization of the theory of oppositions to polytomy After having developed all this work I have begun to promote interdisciplinary world events on the square of opposition

Natural Deduction for Dual-intuitionistic Logic

Abstract: We present a natural deduction system for dual-intuitionistic logic. Its distinctive feature is that it is a single-premise multiple-conclusions system. Its relationships with the natural deduction systems for intuitionistic and classical logic are discussed.

Paraconsistent hybrid theories

Abstract: We consider the problem of reasoning from inconsistent hybrid theories, i.e., combinations of a structural part given by a classical first order theory (e.g., an ontology) and a rules part as a set of declarative logic program rules (under answer-set semantics). Paraconsistent reasoning is achieved by defining an appropriate semantics, so-called paraconsistent semi-equilibrium model semantics for such hybrid theories. Appropriateness of the semantics is established with respect to desirable properties attesting design objectives, such us to generalize the underlying semantics in case of consistency, as well as to generalize existing paraconsistent semantics for the individual parts. A complexity analysis of corresponding reasoning tasks complements these results.

Medical equipment classification: method and decision-making support based on paraconsistent annotated logic.

Abstract: As technology evolves, the role of medical equipment in the healthcare system, as well as technology management, becomes more important. Although the existence of large databases containing management information is currently common, extracting useful information from them is still difficult. A useful tool for identification of frequently failing equipment, which increases maintenance cost and downtime, would be the classification according to the corrective maintenance data. Nevertheless, establishment of classes may create inconsistencies, since an item may be close to two classes by the same extent. Paraconsistent logic might help solve this problem, as it allows the existence of inconsistent (contradictory) information without trivialization. In this paper, a methodology for medical equipment classification based on the ABC analysis of corrective maintenance data is presented, and complemented with a paraconsistent annotated logic analysis, which may enable the decision maker to take into consideration alerts created by the identification of inconsistencies and indeterminacies in the classification.

Natural deduction system in paraconsistent setting: proof search for PCont

Abstract: This paper continues a systematic approach to build natural deduction calculi and corresponding proof procedures for non-classical logics. Our attention is now paid to the framework of paraconsistent logics. These logics are used, in particular, for reasoning about systems where paradoxes do not lead to the `deductive explosion', i.e., where formulae of the type `A follows from false', for any A, are not valid. We formulate the natural deduction system for the logic PCont, explain its main concepts, define a proof searching technique and illustrate it by examples. The presentation is accompanied by demonstrating the correctness of these developments.

Towards a logic of argumentation

Abstract: Starting from a typology of argumentative forms proposed in linguistics by Apotheloz, and observing that the four basic forms can be organized in a square of oppositions, we present a logical language, somewhat inspired from generalized possibilistic logic, where these basic forms can be expressed. We further analyze the interplay between the formulas of this language by means of two hexagons of oppositions. We then outline the inference machinery underlying this logic, and discuss its interest for argumentation.

How Peircean was the “‘Fregean’ revolution” in logic?

Abstract: The historiography of logic conceives of a Fregean revolution in which modern mathematical logic (also called symbolic logic) has replaced Aristotelian logic. The preeminent expositors of this conception are Jean van Heijenoort (1912-1986) and Donald Angus Gillies. The innovations and characteristics that comprise mathematical logic and distinguish it from Aristotelian logic, according to this conception, created ex nihlo by Gottlob Frege (1848-1925) in his Begriffsschrift of 1879, and with Bertrand Russell (1872-1970) as its chief This position likewise understands the algebraic logic of Augustus De Morgan (1806-1871), George Boole (1815-1864), Charles Sanders Peirce (1838-1914), and Ernst Schr\"oder (1841-1902) as belonging to the Aristotelian tradition. The "Booleans" are understood, from this vantage point, to merely have rewritten Aristotelian syllogistic in algebraic guise. The most detailed listing and elaboration of Frege's innovations, and the characteristics that distinguish mathematical logic from Aristotelian logic, were set forth by van Heijenoort. I consider each of the elements of van Heijenoort's list and note the extent to which Peirce had also developed each of these aspects of logic. I also consider the extent to which Peirce and Frege were aware of, and may have influenced, one another's logical writings.

The ubiquity of conservative translations

Abstract: We study the notion of conservative translation between logics introduced by (Feitosa & D’Ottaviano2001). We show that classical propositional logic (CPC) is universal in the sense that every finitary consequence relation over a countable set of formulas can be conservatively translated into CPC. The translation is computable if the consequence relation is decidable. More generally, we show that one can take instead of CPC a broad class of logics (extensions of a certain fragment of full Lambek calculus FL) including most nonclassical logics studied in the literature, hence in a sense, (almost) any two reasonable deductive systems can be conservatively translated into each other. We also provide some counterexamples, in particular the paraconsistent logic LP is not universal.

Embedding-based approaches to paraconsistent and temporal description logics*

Abstract: In this article, some paraconsistent and temporal description logics are studied based on an embedding-based proof method. A new paraconsistent description logic, PALC, is obtained from the description logic ALC by adding a paraconsistent negation. Two temporal description logics, XALC and BALCl, are introduced by combining and modifying ALC and Prior’s tomorrow tense logic. XALC has the next-time operator, and BALCl has some restricted versions of the next-time, any-time and some-time operators, in which the time domain is bounded by a positive integer l. Some theorems for embedding PALC, XALC and BALCl into ALC are proved in a uniform way, and PALC, XALC and BALCl are shown to be decidable. Three tableau calculi for PALC, XALC and BALCl are introduced, and the completeness theorems for these calculi are proved.

The ‘Galilean Style in Science’ and the Inconsistency of Linguistic Theorising

Abstract: Chomsky’s principle of epistemological tolerance says that in theoretical linguistics contradictions between the data and the hypotheses may be temporarily tolerated in order to protect the explanatory power of the theory. The paper raises the following problem: What kinds of contradictions may be tolerated between the data and the hypotheses in theoretical linguistics? First a model of paraconsistent logic is introduced which differentiates between week and strong contradiction. As a second step, a case study is carried out which exemplifies that the principle of epistemological tolerance may be interpreted as the tolerance of week contradiction. The third step of the argumentation focuses on another case study which exemplifies that the principle of epistemological tolerance must not be interpreted as the tolerance of strong contradiction. The reason for the latter insight is the unreliability and the uncertainty of introspective data. From this finding the author draws the conclusion that it is the integration of different data types that may lead to the improvement of current theoretical linguistics and that the integration of different data types requires a novel methodology which, for the time being, is not available.

Aristotle's Logic and Metaphysics

Abstract: Formalisation in higher order logic of parts of Aristotle’s logic and metaphysics. Created 2009/05/21 Last Change Date: 2012/01/23 21:40:02 Id: t028.doc,v 1.33 2012/01/23 21:40:02 rbj Exp http://www.rbjones.com/rbjpub/pp/doc/t028.pdf c © Roger Bishop Jones; Licenced under Gnu LGPL

A New–old Characterisation of Logical Knowledge

Abstract: We seek means of distinguishing logical knowledge from other kinds of knowledge, especially mathematics. The attempt is restricted to classical two-valued logic and assumes that the basic notion in logic is the proposition. First, we explain the distinction between the parts and the moments of a whole, and theories of ‘sortal terms’, two theories that will feature prominently. Second, we propose that logic comprises four ‘momental sectors’: the propositional and the functional calculi, the calculus of asserted propositions, and rules for (in)valid deduction, inference or substitution. Third, we elaborate on two neglected features of logic: the various modes of negating some part(s) of a proposition R, not only its ‘external’ negation not-R; and the assertion of R in the pair of propositions ‘it is (un)true that R’ belonging to the neglected logic of asserted propositions, which is usually left unstated. We also address the overlooked task of testing the asserted truth-value of R. Fourth, we locate logic a...

Paraconsistent reasoning for semantic web agents

Abstract: Description logics refer to a family of formalisms concentrated around concepts, roles and individuals. They are used in many multiagent and Semantic Web applications as a foundation for specifying knowledge bases and reasoning about them. Among them, one of the most important logics is $\mathcal{SROIQ}$, providing the logical foundation for the OWL 2 Web Ontology Language recommended by W3C in October 2009. In the current paper we address the problem of inconsistent knowledge. Inconsistencies may naturally appear in the considered application domains, for example as a result of fusing knowledge from distributed sources. We introduce a number of paraconsistent semantics for $\mathcal{SROIQ}$, including three-valued and four-valued semantics. The four-valued semantics reflects the well-known approach introduced in [5,4] and is considered here for comparison reasons only. We also study the relationship between the semantics and paraconsistent reasoning in $\mathcal{SROIQ}$ through a translation into the traditional two-valued semantics. Such a translation allows one to use existing tools and reasoners to deal with inconsistent knowledge.

New Waves in Philosophical Logic

Abstract: Series Editors' Preface Acknowledgements Notes on Contributors How Things Are Elsewhere W. Schwarz Information Change and First-Order Dynamic Logic B.Kooi Interpreting and Applying Proof Theories for Modal Logic F.Poggiolesi & G.Restall The Logic(s) of Modal Knowledge D.Cohnitz On Probabilistically Closed Languages H.Leitgeb Dogmatism, Probability and Logical Uncertainty B.Weatherson & D.Jehle Skepticism about Reasoning S.Roush, K.Allen & I.Herbert Lessons in Philosophy of Logic from Medieval Obligations C.D.Novaes How to Rule Out Things with Words: Strong Paraconsistency and the Algebra of Exclusion F.Berto Lessons from the Logic of Demonstratives G.Russell The Multitude View on Logic M.Eklund Index

Paraconsistency on the rocks of dialetheism

Abstract: Can one be a non-dialetheic paraconsistentist? I will show that on a standard model-theoretic approach to consequence the answer to this question depends on the philosophical motivation behind the models. If the models are interpretations of the formal language, the answer to the question is “No”, but if the models are representations of how things are, the answer is less clear. That different approaches to semantic characterisations of consequence come apart in this way demonstrates that attention should be paid, especially by paraconsistentists, to the motivations behind the theories.

From turner's logic of universal causation to the logic of GK

Abstract: Logic of knowledge and justified assumptions, also known as logic of grounded knowledge (GK), was proposed by Lin and Shoham as a general logic for nonmonotonic reasoning To date, it has been used to embed in it default logic, autoepistemic logic, and general logic programming under stable model semantics Besides showing the generality of GK as a logic for nonmonotonic reasoning, these embeddings shed light on the relationships among these other logics Along this line, we show that Turner's logic of universal causation can be naturally embedded into logic of GK as well

Logic as a Science and Logic as a Theory: Remarks on Frege, Russell and the Logocentric Predicament

Abstract: Since its publication in 1967, van Heijenoort’s paper, “Logic as Calculus and Logic as Language” has become a classic in the historiography of modern logic. According to van Heijenoort, the contrast between the two conceptions of logic provides the key to many philosophical issues underlying the entire classical period of modern logic, the period from Frege’s Begriffsschrift (1879) to the work of Herbrand, Godel and Tarski in the late 1920s and early 1930s. The present paper is a critical reflection on some aspects of van Heijenoort’s thesis. I concentrate on the case of Frege and Russell and the claim that their philosophies of logic are marked through and through by acceptance of the universalist conception of logic, which is an integral part of the view of logic as language. Using the so-called “Logocentric Predicament” (Henry M. Sheffer) as an illustration, I shall argue that the universalist conception does not have the consequences drawn from it by the van Heijenoort tradition. The crucial element here is that we draw a distinction between logic as a universal science and logic as a theory. According to both Frege and Russell, logic is first and foremost a universal science, which is concerned with the principles governing inferential transitions between propositions; but this in no way excludes the possibility of studying logic also as a theory, i.e., as an explicit formulation of (some) of these principles. Some aspects of this distinction will be discussed.

Ontology and the mathematization of the scientific enterprise

Abstract: In this basically expository paper we discuss the role of logic and mathematics in researches concerning the ontology of scientific theories, and we consider the particular case of quantum mechanics. We argue that systems of logic in general, and classical logic in particular, may contribute substantially with the ontology of any theory that has this logic in its base. In the case of quantum mechanics, however, from the point of view of philosophical discussions concerning identity and individuality, those contributions may not be welcome for a specific interpretation, and an alternative system of logic perhaps could be used instead of a classical system. In this sense, we argue that the logic and ontology of a scientific theory may be seen as mutually influencing each other. On the one hand, logic contributes to shape the general features of the ontology of a theory; on the other hand, the theory also puts constraints on the possible understanding of ontology and, respectively, on possible systems of logic that may be the underlying logic of the theory.