Showing papers on "Paraconsistent logic published in 2011"

Independence-friendly logic : a game-theoretic approach

Abstract: Bringing together over twenty years of research, this book gives a complete overview of independence-friendly logic. It emphasizes the game-theoretical approach to logic, according to which logical concepts such as truth and falsity are best understood via the notion of semantic games. The book pushes the paradigm of game-theoretical semantics further than the current literature by showing how mixed strategies and equilibria can be used to analyze independence-friendly formulas on finite models. The book is suitable for graduate students and advanced undergraduates who have taken a course on first-order logic. It contains a primer of the necessary background in game theory, numerous examples and full proofs.

Multiple-conclusion lp and default classicality

Abstract: Philosophical applications of familiar paracomplete and paraconsistent logics often rely on an idea of ‘default classicality’. With respect to the paraconsistent logic LP (the dual of Strong Kleene or K3), such ‘default classicality’ is standardly cashed out via an LP-based nonmonotonic logic due to Priest (1991, 2006a). In this paper, I offer an alternative approach via a monotonic multiple-conclusion version of LP.

Ideal Paraconsistent Logics

Abstract: We define in precise terms the basic properties that an `ideal propositional paraconsistent logic' is expected to have, and investigate the relations between them. This leads to a precise characterization of ideal propositional paraconsistent logics. We show that every three-valued paraconsistent logic which is contained in classical logic, and has a proper implication connective, is ideal. Then we show that for every n > 2 there exists an extensive family of ideal n-valued logics, each one of which is not equivalent to any k-valued logic with k < n.

Dialogues as a Dynamic Framework for Logic

Abstract: Dialogical logic is a game-theoretical approach to logic. Logic is studied with the help of certain games, which can be thought of as idealized argumentations. Two players, the Proponent, who puts forward the initial thesis and tries to defend it, and the Opponent, who tries to attack the Proponent’s thesis, alternately utter argumentative moves according to certain rules. For a long time the dialogical approach had been worked out only for classical and intuitionistic logic. The seven papers of this dissertation show that this narrowness was uncalled for. The initial paper presents an overview and serves as an introduction to the other papers. Those papers are related by one central theme. As each of them presents dialogical formulations of a different non-classical logic, they show that dialogical logic constitutes a powerful and flexible general framework for the development and study of various logical formalisms and combinations thereof. As such it is especially attractive to logical pluralists that reject the idea of “the single correct logic”. The collection contains treatments of free logic, modal logic, relevance logic, connexive logic, linear logic, and multi-valued logic.

Maximal and Premaximal Paraconsistency in the Framework of Three-Valued Semantics

Abstract: Maximality is a desirable property of paraconsistent logics, motivated by the aspiration to tolerate inconsistencies, but at the same time retain from classical logic as much as possible. In this paper we introduce the strongest possible notion of maximal paraconsistency, and investigate it in the context of logics that are based on deterministic or non-deterministic three-valued matrices. We show that all reasonable paraconsistent logics based on three-valued deterministic matrices are maximal in our strong sense. This applies to practically all three-valued paraconsistent logics that have been considered in the literature, including a large family of logics which were developed by da Costa's school. Then we show that in contrast, paraconsistent logics based on three-valued properly nondeterministic matrices are not maximal, except for a few special cases (which are fully characterized). However, these non-deterministic matrices are useful for representing in a clear and concise way the vast variety of the (deterministic) three-valued maximally paraconsistent matrices. The corresponding weaker notion of maximality, called premaximal paraconsistency, captures the "core" of maximal paraconsistency of all possible paraconsistent determinizations of a non-deterministic matrix, thus representing what is really essential for their maximal paraconsistency.

A Survey of Paraconsistent Logics

Abstract: A survey of paraconsistent logics that are prominent representatives of the different approaches that have been followed to develop paraconsistent logics is provided. The paraconsistent logics that will be discussed are an enrichment of Priest's logic LP, the logic RM3 from the school of relevance logic, da Costa's logics Cn, Jaskowski's logic D2, and Subrahmanian's logics Ptau. A deontic logic based on the first of these logics will be discussed as well. Moreover, some proposed adaptations of the AGM theory of belief revision to paraconsistent logics will be mentioned.

Revisiting information hiding: reflections on classical and nonclassical modularity

Abstract: What is modularity?Which kind of modularity should developers strive for? Despite decades of research on modularity, these basic questions have no definite answer. We submit that the common understanding of modularity, and in particular its notion of information hiding, is deeply rooted in classical logic. We analyze how classical modularity, based on classical logic, fails to address the needs of developers of large software systems, and encourage researchers to explore alternative visions of modularity, based on nonclassical logics, and henceforth called nonclassical modularity.

A Paraconsistent Linear-time Temporal Logic

Abstract: Inconsistency-tolerant reasoning and paraconsistent logic are of growing importance not only in Knowledge Representation, AI and other areas of Computer Science, but also in Philosophical Logic. In this paper, a new logic, paraconsistent linear-time temporal logic (PLTL), is obtained semantically from the linear-time temporal logic LTL by adding a paraconsistent negation. Some theorems for embedding PLTL into LTL are proved, and PLTL is shown to be decidable. A Gentzentype sequent calculus PLTω for PLTL is introduced, and the completeness and cut-elimination theorems for this calculus are proved. In addition, a display calculus δPLTω for PLTL is defined.

Paraconsistent vagueness: a positive argument

Abstract: Paraconsistent approaches have received little attention in the literature on vagueness (at least compared to other proposals). The reason seems to be that many philosophers have found the idea that a contradiction might be true (or that a sentence and its negation might both be true) hard to swallow. Even advocates of paraconsistency on vagueness do not look very convinced when they consider this fact; since they seem to have spent more time arguing that paraconsistent theories are at least as good as their paracomplete counterparts, than giving positive reasons to believe on a particular paraconsistent proposal. But it sometimes happens that the weakness of a theory turns out to be its mayor ally, and this is what (I claim) happens in a particular paraconsistent proposal known as subvaluationism. In order to make room for truth-value gluts subvaluationism needs to endorse a notion of logical consequence that is, in some sense, weaker than standard notions of consequence. But this weakness allows the subvaluationist theory to accommodate higher-order vagueness in a way that it is not available to other theories of vagueness (such as, for example, its paracomplete counterpart, supervaluationism).

Paraconsistent Algorithm Extractor of Contradiction Effects - Paraextrctr ctr

Abstract: Nowadays networks of analyses based in non-classic logics are used with success in the treatment of uncertainties The characteristic of accepting the contradiction in his structure is the main cause of the methodologies based in Paraconsistent Logic is ideals for applications in systems of analyses and decision making In this work we presented an algorithm based in Paraconsistent logic capable to extract in a gradual way the effects of the contradiction in originated signals of information of uncertain knowledge database The Algorithm Paraconsistent Extractor of Contradiction effects - Paraextrctrctr is formed with base in fundamental concepts of the Paraconsistent Annotated Logic with annotation of two values (PAL2v) it can be applied in filters of networks of analyses of signal information where uncertain and contradictory signals can be present The process of extraction of the effect of the contradiction is always begun by the largest inconsistency degree among two signals that belong to the group that is in analysis In the end of the analysis it is found a consensus value In this work we presented numeric example and one example of application of the Paraextrctrctr in Load Profile Forecast used in support to decision of the operation in an Electric Power System, but his application potentiality is demonstrated in several fields of the Artificial Intelligence

Logic is not Logic

Abstract: In this paper we discuss the difference between

Some Observations on the Systems LFI1 and LFI1

Abstract: One of the well-known systems of para consistent logic called {\bf LFI1} is designed to be a base system in constructing evolutionary databases. This system {\bf LFI1} is proved to be a 3-valued logic and also maximal relative to classical logic enriched with inconsistency operator in an obvious manner. The present paper aims to examine the system {\bf LFI1} from the viewpoint of Belnap's 4-valued logic. More concretely, we develop the Belnapian 4-valued system of Logics of Formal Inconsistency (LFIs) which can be seen as a natural generalization of {\bf LFI1}. As a consequence, from the viewpoint of the Belnapian logic, we obtain a formalization which contains the notion of ``normality'' instead of the constant $\perp$. On the other hand, from the viewpoint of LFIs, we lose the maximality but might be able to cope with more data in constructing databases. This is because the fourth value of the Belnapian matrix corresponds to incomplete data which cannot be dealt with in {\bf LFI1}. Our results contain an axiomatization of the Belnapian LFI, a characterization of ``normality'' in the system, and a translation result between the existing Belnapian system and the system we introduce.

Is logic in the mind or in the world

Abstract: The paper presents an outline of a unified answer to five questions concerning logic: (1) Is logic in the mind or in the world? (2) Does logic need a foundation? What is the main obstacle to a foundation for logic? Can it be overcome? (3) How does logic work? What does logical form represent? Are logical constants referential? (4) Is there a criterion of logicality? (5) What is the relation between logic and mathematics?

Paraconsistent Computation Tree Logic

Abstract: It is known that paraconsistent logical systems are more appropriate for inconsistency-tolerant and uncertainty reasoning than other types of logical systems. In this paper, a paraconsistent computation tree logic, PCTL, is obtained by adding paraconsistent negation to the standard computation tree logic CTL. PCTL can be used to appropriately formalize inconsistency-tolerant temporal reasoning. A theorem for embedding PCTL into CTL is proved. The validity, satisfiability, and model-checking problems of PCTL are shown to be decidable. The embedding and decidability results indicate that we can reuse the existing CTL-based algorithms for validity, satisfiability, and model-checking. An illustrative example of medical reasoning involving the use of PCTL is presented.

Paraconsistent Annotated Logic in Analysis of Physical Systems: Introducing the Paraquantum Factor of Quantization hψ

Abstract: We present in this paper an alternative of modeling physical systems through a non-Classical logic namely the Paraconsistent Logic (PL) whose main feature is the revocation of the principle of non-contradiction. The Paraconsistent Annotated Logic with annotation of two values (PAL2v) is a type of PL and has in its theoretical structure the main feature of dealing with contradictions offering flexibility in drawing conclusions. Several works about applications of PAL2v have shown that such logic is able to provide us with an adequate treatment to uncertainties. Based on the foundations of the PAL2v we presented the ParaQuantum logic (PQL) with the goal of performing analysis of signals from information sources which model physical systems. The formalization of the concepts of the logics PQL, that it is represented in a Lattice, requires the considering of Paraquantum logical states ψ which are propagated through variations of the evidence Degrees µ and λ which come out from measurements performed in Observable Variables in the physical world. When we analyze the lattice of the PQL, we obtain equations which quantify values of physical quantities from where we obtain the effects of propagation of the Paraquantum logical states ψ. In this paper, we introduce the Paraquantum Factor of quantization hψ whose value is associated with a special logical state on the lattice which is identified with the Planck constant h. We conclude through these studies that the Paraquantum Logical Model based on the ParaQuantum logics PQL can link the several fields of the physical sciences by means of quantization of values. It is an innovative approach of formulating natural phenomena.

Hans reichenbach's probability logic

Abstract: Publisher Summary Reichenbach states that an inductive logic cannot be built up entirely from logical principles independent of experience, but must develop out of the reasoning practiced and useful to the natural sciences. Inductive inference system needs to be built on some solid to guide scientific methodology. This chapter describes Reichenbach's reasons for stating the inverse approach for inductive logic. Instead of “a priori” foundation of inductive logic, Reichenbach's approach to induction is largely axiomatic. Reichenbach distinguishes deductive and mathematical logic from inductive logic. The former deals with the relations among tautologies, whereas the latter deals with truth in the sense of truth in reality. Deductive and mathematical logic are built on an axiomatic system. In contrast to the formal relations that are of interest in deductive logic, inductive logic is concerned with the determination of whether various relations among quantities are true in the world.

A paraconsistent multi-agent framework for dealing with normative conflicts

Abstract: In a multi-agent deontic setting, normative conflicts can take a variety of different logical forms. In this paper, we present a very general characterization of such conflicts, including both intra- and interagent normative conflicts, conflicts between groups of agents, conflicts between obligations and permissions, and conflicts between contradictory norms. In order to account for the consistent possibility of this wide variety of conflict-types, we present a paraconsistent deontic logic, i.e. a logic that invalidates the classical principle of non-contradiction. Next, we strengthen this logic within the adaptive logics framework for defeasible reasoning. The resulting inconsistency-adaptive deontic logic interprets a given set of norms 'as consistently as possible'.

Analysis of Physical Systems with Paraconsistent Annotated Logic: Introducing the Paraquantum Gamma Factor γ ψ

Abstract: In this paper we use a non-classical logic called ParaQuantum Logic (PQL) which is based on the foundations of the Paraconsistent Annotated logic with annotation of two values (PAL2v). The formalizations of the PQL concepts, which is represented by a lattice with four vertices, leads us to consider Paraquantum logical states ψ which are propagated by means of variations of the evidence Degrees extracted from measurements performed on the Observable Variables of the physical world. In this work we introduce the Paraquantum Gamma Factor γPψ which is an expansion factor on the PQL lattice that act in the physical world and is correlated with the Paraquantum Factor of quantization hψ whose value is associated with a special logical state on the lattice which is identified with the Planck constant h. Our studies show that the behavior of the Paraquantum Gamma Factor γPψ, at the time of reading the evidence Degrees through measurements of the Observable Variables in the physical world, is identical to that one of the Lorentz Factor γ used in the relativity theory. In the final part of this paper we present results about studies of expansion and contraction of the Paraquantum Logical Model which correlate the factors γPψ, and γ. By applying these correlation factors, the lattice of the PQL suitable for the universe understudy can be contracted or expanded, allowing the quantization model to cover the several study fields of physics.

Natural Deduction: An Introduction to Logic with Real Arguments, a Little History, and Some Humour

Abstract: Richard Arthur's Natural Deduction provides a wide-ranging introduction to logic. In lively and readable prose, Arthur presents a new approach to the study of logic, one that seeks to integrate methods of argument analysis developed in modern "informal logic" with natural deduction techniques. The dry bones of logic are given flesh by unusual attention to the history of the subject, from Pythagoras, the Stoics, and Indian Buddhist logic, through Lewis Carroll, Venn, and Boole, to Russell, Frege, and Monty Python.

Departing from classical logic: A logical analysis of Personal Construct Theory

Abstract: This article draws a parallel between personal construct theory and intuitionistic logic i, in order to account for Kelly's claim to have departed from classical logic. Assuming that different theoretical paradigms correspond to different logical languages, it is argued that the constructivist paradigm is linked to intuitionism. Similarities between some key syntactic and semantic features of i logic and the underlying logic of Kelly's theory are made explicit. The strengths and limitations of such an approach are discussed in light of issues emerging from clinical observation and from the philosophy of science.

First-Order da Costa Logic

Abstract: Priest (2009) formulates a propositional logic which, by employing the worldsemantics for intuitionist logic, has the same positive part but dualises the negation, to produce a paraconsistent logic which it calls `Da Costa Logic'. This paper extends matters to the first-order case. The paper establishes various connections between first order da Costa logic, da Costa's own C?, and classical logic. Tableau and natural deductions systems are provided and proved sound and complete.

Paraconsistency and Topological Semantics

Abstract: The well-studied notion of deductive explosion describes the situation where any formula can be deduced from an inconsistent set of formulas. Paraconsistent logic, on the other hand, is the umbrella term for logical systems where the logical consequence relation is not explosive. In this work, we investigate the relationship between some different topological spaces and paraconsistency.

Paraconsistency in Categories: Case of Relevance Logic

Abstract: Categorical-theoretic semantics for the relevance logic is proposed which is based on the construction of the topos of functors from a relevant algebra (considered as a preorder category endowed with the special endofunctors) in the category of sets Set. The completeness of the relevant system R of entailment is proved in respect to the semantic considered.

Constructive discursive logic with strong negation

Abstract: Jaskowski’s discursive logic (or discussive logic) is the first formal paraconsistent logic which is classified as a non-adjunctive system It is now recognized that discursive logic is not generally appropriate for paraconsistent reasoning To improve it in a constructive setting, we propose a constructive discursive logic with strong negation CDLSN based on Nelson’s constructive logic N In CDLSN , discursive negation is defined similar to intuitionistic negation and discursive implication is defined as material implication using discursive negation We give an axiomatic system and Kripke semantics with a completeness proof We also discuss some advantages of the proposed system over other paraconsistent systems

Paraconsistent Rough Description Logic.

Abstract: In this paper, we introduce a paraconsistent extension of Rough Description Logics which allows the representation of incomplete and contradictory concepts, as well as the lower and upper approximations of these kinds of concepts. Furthermore, we use the notions of approximations, which can be applied successively, to make restrictions or relaxations in the concept queries with the objective of decreasing or increasing the number of results of the queries, respectively.

Bayesianism as a pure logic of Inference

Abstract: Publisher Summary A connection forges between the idea of a graded probability and another new branch of mathematics, the discrete mathematics of combinations and permutations. Bayesians see their discipline less as a part of logic, which in common with most contemporary deductive logicians they regard as comprising just deductive logic, than of a general theory of rational belief and decision. The idea that there might be an intimate relationship between logic and probability, at any rate epistemic probability, is the subject of exploration and controversy for over three centuries. Both disciplines specify rules of valid non-domain-specific reasoning, and it would seem a reasonable question why one should be distinguished as logic and the other not. The two fundamental notions of modern deductive logic are (semantic) consistency and (semantic) logical consequence, which in classical logic at least are interdefinable.