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Showing papers on "Paraconsistent logic published in 2010"


Book
15 Jun 2010
TL;DR: This is it, the uncertainty treatment using paraconsistent logic introducingParaconsistent artificial neural netwo that will be your best choice for better reading book.
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65 citations


Journal ArticleDOI
TL;DR: The present paper is the first semantical and proof-theoretical study of bounded constructive linear-time temporal logics containing either intuitionistic or strong negation.

40 citations


Journal ArticleDOI
TL;DR: A method for measuring inconsistency based on the number of formulas needed for deriving a contradiction is introduced and the relationships to previously considered methods based on probability measures are discussed.

39 citations


Journal ArticleDOI
Zach Weber1
TL;DR: The main result is that the proposal, in the context of an independently motivated formalization of naive set theory, leads to triviality.
Abstract: The naive set theory problem is to begin with a full comprehension axiom, and to find a logic strong enough to prove theorems, but weak enough not to prove everything. This paper considers the sub-problem of expressing extensional identity and the subset relation in paraconsistent, relevant solutions, in light of a recent proposal from Beall, Brady, Hazen, Priest and Restall [4]. The main result is that the proposal, in the context of an independently motivated formalization of naive set theory, leads to triviality.

39 citations


Proceedings ArticleDOI
06 Jun 2010
TL;DR: The results establish NOT NULL constraints as an effective mechanism to balance the expressiveness and tractability of consequence relations, and to control the degree by which the existing classical theory of data dependencies can be soundly approximated in practice.
Abstract: We study functional and multivalued dependencies over SQL tables with NOT NULL constraints. Under a no-information interpretation of null values we develop tools for reasoning. We further show that in the absence of NOT NULL constraints the associated implication problem is equivalent to that in propositional fragments of Priest's paraconsistent Logic of Paradox. Subsequently, we extend the equivalence to Boolean dependencies and to the presence of NOT NULL constraints using Schaerf and Cadoli's S-3 logics where S corresponds to the set of attributes declared NOT NULL. The findings also apply to Codd's interpretation "value at present unknown" utilizing a weak possible world semantics. Our results establish NOT NULL constraints as an effective mechanism to balance the expressiveness and tractability of consequence relations, and to control the degree by which the existing classical theory of data dependencies can be soundly approximated in practice.

30 citations


Journal ArticleDOI
TL;DR: In this paper, the authors make a further attempt to establish Lupasco's concepts as significant contributions to the history and philosophy of logic, in line with the work of Godel, general relativity, and the ontological turn in philosophy.
Abstract: The advent of quantum mechanics in the early 20 th Century had profound consequences for science and mathematics, for philosophy (Schrodinger), and for logic (von Neumann). In 1968, Putnam wrote that quantum mechanics required a revolution in our understanding of logic per se. However, applications of quantum logics have been little explored outside the quantum domain. Dummett saw some implications of quantum logic for truth, but few philosophers applied similar intuitions to epistemology or ontology. Logic remained a truth-functional ’science’ of correct propositional reasoning. Starting in 1935, the Franco-Romanian thinker Stephane Lupasco described a logical system based on the inherent dialectics of energy and accordingly expressed in and applicable to complex real processes at higher levels of reality. Unfortunately, Lupasco’s fifteen major publications in French went unrecognized by mainstream logic and philosophy, and unnoticed outside a Francophone intellectual community, albeit with some translations into other Romance languages. In English, summaries of Lupasco’s logic appeared ca. 2000, but the first major treatment and extension of his system was published in 2008 (see Brenner 2008). This paper is a further attempt to establish Lupasco’s concepts as significant contributions to the history and philosophy of logic, in line with the work of Godel, general relativity, and the ontological turn in philosophy.

29 citations


Journal ArticleDOI
TL;DR: In this paper, a one-one correspondence between evidence couples and evidence matrices that holds in all injective MV-algebras is proved, based on a related study of Perny and Tsoukias, and the obtained logic is Pavelka style fuzzy sentential logic.

23 citations


Book ChapterDOI
01 Jan 2010
TL;DR: This chapter argues that the paraconsistent logic SbV, or subvaluationism, is no less conservative than SpV nor more so, and that both logics offer equally compelling theoretical approaches to vagueness.
Abstract: Paraconsistent responses to vagueness are often thought to represent a revision of logical theory that is too radical to be defensible. The paracomplete logic of supervaluationism, SpV, is not only taken to be more conservative but is also commonly said to 'preserve classical logic'. This chapter argues that this is wrong on both counts. The paraconsistent logic SbV, or subvaluationism, is no less conservative than SpV nor more so. In the end both logics offer equally compelling theoretical approaches to vagueness. Each approach is also equally objectionable, with neither providing an adequate account of vagueness, but this criticism arises from a feature shared by each approach that is independent of their paracompleteness or paraconsistency per se. For all that has been said, a paraconsistent approach, and the associated recourse to truth-value gluts, remains a contender in accounting for vagueness.

16 citations



Proceedings Article
09 May 2010
TL;DR: This paper introduces the strongest possible notion of maximal paraconsistency, and investigates it in the context of logics that are based on deterministic or non-deterministic three-valued matrices.
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 first show that most of the logics that are based on properly non-deterministic three-valued matrices are not maximally paraconsistent. Then we show that in contrast, in the deterministic case all the natural three-valued paraconsistent logics are maximal. This includes well-known three-valued paraconsistent logics like P1, LP, J3, PAC and SRM3, as well as any extension of them obtained by enriching their languages with extra three-valued connectives.

13 citations


Journal ArticleDOI
01 May 2010
TL;DR: This work proves adjunct-elimination results for Context Logic applied to trees for multi-holed Context Logic, following previous results by Lozes for Separation Logic and Ambient Logic.
Abstract: We study adjunct-elimination results for Context Logic applied to trees, following previous results by Lozes for Separation Logic and Ambient Logic. In fact, it is not possible to prove such elimination results for the original single-holed formulation of Context Logic. Instead, we prove our results for multi-holed Context Logic.

Journal Article
TL;DR: A newParaconsistent description logic, PALC, is obtained from the description logic ALC by adding a paraconsistent negation, and some theorems for embedding PALC into ACL are proved, and PALC is shown to be decidable.
Abstract: Inconsistency handling is of growing importance in Knowl- edge Representation since inconsistencies may frequently occur in an open world. Paraconsistent (or inconsistency-tolerant) description logics have been studied by several researchers to cope with such inconsis- tencies. In this paper, a new paraconsistent description logic, PALC, is obtained from the description logic ALC by adding a paraconsistent negation. Some theorems for embedding PALC into ACL are proved, and PALC is shown to be decidable. A tableau calculus for PALC is introduced, and the completeness theorem for this calculus is proved.

Journal ArticleDOI
23 Mar 2010-Health
TL;DR: In this work it is presented a mathematical foundation of fuzzy logic (with connectives and inference rules) and then the application of fuzzy reasoning to the study of a clinical case and its relation with probabilistic logic is explored.
Abstract: Fuzzy logic is a logical calculus which operates with many truth values (while classical logic works with the two values of true and false). Since fuzzy logic considers the truth of scientific statements like something softened, it is fruitfully applied to the study of biological phenomena, biology is indeed considered the field of complexity, uncertainty and vagueness. In this paper fuzzy logic is successfully applied to the clinical diagnosis of a patient who suffers from different diseases bound by a complex causal chain. In this work it is presented a mathematical foundation of fuzzy logic (with connectives and inference rules) and then the application of fuzzy reasoning to the study of a clinical case. Probabilistic logic is widely considered the unique logical calculus useful in clinical diagnosis, thus the usefulness of fuzzy logic and its relation with probabilistic logic is here explored. The presentation of the case is supplied with all the features necessary to affect a clinical diagnosis: physical exam, anamnesis and tests.

Proceedings ArticleDOI
11 Jul 2010
TL;DR: This paper introduces a new, strong notion of maximal paraconsistency, which is based on possible extensions of the consequence relation of a logic, and investigates this notion in the framework of finite-valued paraconsistent logics.
Abstract: Maximality is a desirable property of paraconsistent logics, motivated by the aspiration to tolerate inconsistencies, but at the same time retain as much as possible from classical logic. In this paper we introduce a new, strong notion of maximal paraconsistency, which is based on possible extensions of the consequence relation of a logic. We investigate this notion in the framework of finite-valued paraconsistent logics, and show that for every $n>2$ there exists an extensive family of $n$-valued logics, each of which is maximally paraconsistent in our sense, is partial to classical logic, and is not equivalent to any $k$-valued logic with $k

01 Jan 2010
TL;DR: An alternative proof of the Kripke-completeness theorem for N4 is obtained by combining both the syntactical and semantical embedding theorems, and the completeness, cut-elimination and decidability theorem can uniformly be obtained from these embeddingTheorems.
Abstract: It is known that a syntactical embedding theorem of Nelson’s paraconsistent logic N4 into the positive intuitionistic logic LJ is useful to show the cut-elimination and decidability theorems for N4. In this paper, a semantical embedding theorem of N4 into LJ is shown. An alternative proof of the Kripke-completeness theorem for N4 is obtained by combining both the syntactical and semantical embedding theorems. Thus, the completeness, cut-elimination and decidability theorems can uniformly be obtained from these embedding theorems. A singleconsequence Kripke semantics for N4 is also addressed based on a modication of the semantical embedding theorem.

Journal ArticleDOI
TL;DR: A new computation model, the paraconsistent Turing machine, is defined, which better approaches quantum computing features and defines complexity classes for such models, and establishes some relationships with classical complexity classes.
Abstract: We describe a method to axiomatize computations in deterministic Turing machines (TMs). When applied to computations in non-deterministic TMs, this method may produce contradictory (and therefore trivial) theories, considering classical logic as the underlying logic. By substituting in such theories the underlying logic by a paraconsistent logic we define a new computation model, the paraconsistent Turing machine. This model allows a partial simulation of superposed states of quantum computing. Such a feature allows the definition of paraconsistent algorithms which solve (with some restrictions) the well-known Deutsch's and Deutsch-Jozsa problems. This first model of computation, however, does not adequately represent the notions of entangled states and relative phase, which are key features in quantum computing. In this way, a more sharpened model of paraconsistent TMs is defined, which better approaches quantum computing features. Finally, we define complexity classes for such models, and establish some relationships with classical complexity classes.

Proceedings Article
01 Jan 2010
TL;DR: The embedding and decidability results indicate that the validity, satisfiability, and modelchecking problems of PCTL are shown to be decidable and can be reused by the existing CTL-based algorithms for validity, Satisfiability and model-checking.
Abstract: 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 modelchecking 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.

Journal ArticleDOI
TL;DR: In this paper, the authors consider the logical system of Royce as a development of the idealist version and argue that it has the potential to account both for modern logical systems and for the place of agency and purpose in logic.
Abstract: At the center of the rise of modern logic in the twentieth century was an unquestioned commitment to the idea that inclusive disjunction and negation were the fundamental logical operations. Lost in the development of this logic was an alternative starting point proposed by earlier idealist logicians taking exclusive disjunction and negation as fundamental. This paper considers the logical system of Josiah Royce as a development of the idealist version and argues that it has the potential to account both for modern logical systems and for the place of agency and purpose in logic. I will briefly present the late twentieth century effort to settle on a single concept of disjunction in logic in order to reveal the issues at work. This late debate over the nature of disjunction follows an earlier one in which the leading British idealists were challenged by a variety of philosophers who sought to separate formal systems from conscious agency. Once I have set out the debate, I present a reformulation of Royce’s System ∑ that shows how it takes up the conception of disjunction as it developed in idealist logic and uses it to reconnect agency and formal logic.

Journal ArticleDOI
TL;DR: The notion of strong equivalence between two disjunctive logic programs under the G 3 -stable model semantics, also called the p-stable semantics, was studied in this paper, where some particular cases of testing strong equivalences between programs can be reduced to checking whether a formula is a theorem in some paraconsistent logic, or in some cases in classical logic.

Book
24 Aug 2010
TL;DR: In this article, the concept of logical consequence is examined in a model-theoretic and proof theoretic setting, based on Tarski's characterization of the concept.
Abstract: "The Concept of Logical Consequence" is a critical evaluation of the model-theoretic and proof-theoretic characterizations of logical consequence that proceeds from Alfred Tarski's characterization of the informal concept of logical consequence. This study evaluates and expands upon ideas set forth in Tarski's 1936 article on logical consequence, and appeals to his 1935 article on truth. Classical logic, as well as extensions and deviations are considered. Issues in the philosophy of logic such as the nature of logical constants, the philosophical significance of completeness, and the metaphysical and epistemological implications of logic are discussed in the context of the examination of the concept of logical consequence.

01 Jan 2010
TL;DR: In this paper, the authors bring the two perspectives together in a light and preliminary manner, and make a distinction between mathematical logic and philosophical logic, arguing that mathematical logic should still be the hallmark of logic at a meta-level.
Abstract: If logic is the general study of a priori valid reasoning, then where is the paradigmatic area where we see this reasoning in its full glory? To some, this is clearly mathematics, where precision is relentless, and strings of inferences are taken to impressive lengths. But on another view, the highest form of reasoning is displayed in the ordinary world of common sense – say, when engaging in conversation about something that matters, where pure information is deeply intertwined with evaluation and goals, and where, crucially, we are surrounded by further agents like us that we must interact with. On the first view, to simplify things a bit, logic is about mathematical proof and related processes like computation, making mathematical logic and foundations of mathematics the heart of the field. Agency is not even needed, and no human aspects are modeled. On the second view (frankly speaking: my own), logic is about interactive agency and all that entails, making philosophical logic and much more equally central to the discipline. The purpose of this brief note is to bring the two perspectives together – though admittedly, only in a light and preliminary manner. But before I do, let me make sure that I am not setting up the wrong debate. First, from the viewpoint of agency, there is no competition. Mathematics is an important special form of human cognitive behaviour – and the fact that it has developed historically out of our daily social planning abilities does not detract from its power and importance. Any general logic of agency must come to terms with our mathematical activities. Moreover, one can even grant that agenda contraction and restriction to a subdomain can be a winning move in terms of scientific progress: the more specialized concerns of the foundations of mathematics have had immense benefits for logic in general. Also, a distinction needs to be kept in mind here. It might well be that mathematical logic should still be the hallmark of logic at a meta-level, in

Book ChapterDOI
01 Jan 2010
TL;DR: The logical analysis of generality leads to the perspective of induction as the inverse of deduction and forms the basis for an analysis of various logical frameworks for reasoning about generality and for traversing the space of possible hypotheses.
Abstract: One hypothesis is more general than another one if it covers all instances that are also covered by the later one. The former hypothesis is called a generalization of the later, and the later a specialization of the former. When using logical formulae as hypotheses, the generality relation coincides with the notion of logical entailment, which implies that the generality relation can be analyzed from a logical perspective. The logical analysis of generality, which is pursued in this entry, leads to the perspective of induction as the inverse of deduction. This forms the basis for an analysis of various logical frameworks for reasoning about generality and for traversing the space of possible hypotheses. Many of these frameworks (such as for instance, θ-subsumption) are employed in the field inductive logic programming and are introduced below.

Book ChapterDOI
01 Jan 2010
TL;DR: Some of the traditional and majorizability interpretations, including the recent bounded interpretations, are discussed, and the main theoretical techniques behind proof mining are focused on.
Abstract: In the last fifteen years, the traditional proof interpretations of modified realizability and functional (dialectica) interpretation in finite-type arithmetic have been adapted by taking into account majorizability considerations. One of such adaptations, the monotone functional interpretation of Ulrich Kohlenbach, has been at the center of a vigorous program in applied proof theory dubbed proof mining. We discuss some of the traditional and majorizability interpretations, including the recent bounded interpretations, and focus on the main theoretical techniques behind proof mining.

Journal ArticleDOI
TL;DR: Two new first-order paraconsistent logics with De Morgan-type negations and co-implication, called symmetricParaconsistent logic (SPL) and dual paraconsistant logic (DPL), are introduced as Gentzen-type sequent calculi, and the completeness theorems with respect to these semantics are proved.
Abstract: Two new first-order paraconsistent logics with De Morgan-type negations and co-implication, called symmetric paraconsistent logic (SPL) and dual paraconsistent logic (DPL), are introduced as Gentzen-type sequent calculi. The logic SPL is symmetric in the sense that the rule of contraposition is admissible in cut-free SPL. By using this symmetry property, a simpler cut-free sequent calculus for SPL is obtained. The logic DPL is not symmetric, but it has the duality principle. Simple semantics for SPL and DPL are introduced, and the completeness theorems with respect to these semantics are proved. The cut-elimination theorems for SPL and DPL are proved in two ways: One is a syntactical way which is based on the embedding theorems of SPL and DPL into Gentzen’s LK, and the other is a semantical way which is based on the completeness theorems.

Proceedings Article
22 Sep 2010
TL;DR: It is shown that the modified logic is classically sound and that its embedding into classical SROIQ is consequence preserving, and that inserting special axioms into a SRO IQ4 knowledge base allows additional nontrivial conclusions to be drawn, without affecting paraconsistency.
Abstract: The four-valued paraconsistent logic SROIQ4, originally presented by Ma and Hitzler, is extended to incorporate additional elements of SROIQ. It is shown that the modified logic is classically sound and that its embedding into classical SROIQ is consequence preserving. Furthermore, inserting special axioms into a SROIQ4 knowledge base allows additional nontrivial conclusions to be drawn, without affecting paraconsistency. It is also shown that the interaction of nominals and cardinality restrictions prevents some SROIQ4 knowledge bases from having models. For such knowledge bases, the logic remains explosive.

Book ChapterDOI
08 Sep 2010
TL;DR: A sensing system for an autonomous mobile robot that is based on the Paraconsistent Neural Network to inform the other robot components the position where there is an obstacle.
Abstract: This paper shows a sensing system for an autonomous mobile robot. The Sensing System is based on the Paraconsistent Neural Network. The type of artificial neural network used in this work is based on the Paraconsistent Evidential Logic - Eτ. The objective of the Sensing System is to inform the other robot components the position where there is an obstacle. The reached results have been satisfactory.

Book ChapterDOI
15 Jan 2010

Journal ArticleDOI
TL;DR: In this article, the authors analyse Beziau's anti-Slater move and show both its right intuitions and its technical limits, and suggest that Slater's criticism is much akin to a well-known one by Suszko (1975) against the conceivability of many-valued logics.
Abstract: In 1995 Slater argued both against Priest’s paraconsistent system LP (1979) and against paraconsistency in general, invoking the fundamental opposition relations ruling the classical logical square. Around 2002 Beziau constructed a double defence of paraconsistency (logical and philosophical), relying, in its philosophical part, on Sesmat’s (1951) and Blanche’s (1953) “logical hexagon”, a geometrical, conservative extension of the logical square, and proposing a new (tridimensional) “solid of opposition”, meant to shed new light on the point raised by Slater. By using n-opposition theory (NOT) we analyse Beziau’s anti-Slater move and show both its right intuitions and its technical limits. Moreover, we suggest that Slater’s criticism is much akin to a well-known one by Suszko (1975) against the conceivability of many-valued logics. This last criticism has been addressed by Malinowski (1990) and Shramko and Wansing (2005), who developed a family of tenable logical counter-examples to it: trans-Suszkian systems are radically many-valued. This family of new logics has some strange logical features, essentially: each system has more than one consequence operator. We show that a new, deeper part of the aforementioned geometry of logical oppositions (NOT), the “logical poly-simplexes of dimension m”, generates new logical-geometrical structures, essentially many-valued, which could be a very natural (and intuitive) geometrical counterpart to the “strange”, new, non-Suszkian logics of Malinowski, Shramko and Wansing. By a similar move, the geometry of opposition therefore sheds light both on the foundations of paraconsistent logics and on those of many-valued logics.

Journal ArticleDOI
TL;DR: In this article, a basic logic for application in physics dispensing with the Principle of Excluded Middle is proposed, which is based on the article "Matrix Based Logics for Application in Physics (RMQ) which appeared 2009".
Abstract: This article proposes a basic logic for application in physics dispensing with the Principle of Excluded Middle. It is based on the article “Matrix Based Logics for Application in Physics (RMQ) which appeared 2009. In his article with Stachow on the Principle of Excluded Middle in Quantum Logic (QL), Peter Mittelstaedt showed that for some suitable QLs, including their own, the Principle of Excluded Middle can be added without any harm for QL; where ‘without any harm for QL’ means that the basic desiderata and the basic results (theorems) of those QLs remain satised in the sense that they avoid the well known difficulties with commensurability and distributivity. In the following article I want to show that the basic desiderata and results (theorems) of RMQ (of avoiding the well-known difficulties with commensurability, distributivity, fusion and Bell’s inequalities) remain satised if by introducing a strong negation (or strong negation and disjunction) the resulting weak intuitionist system RMQI dispenses with the Principle of Excluded Middle; it becomes either invalid or not strictly valid.

Book ChapterDOI
28 Jun 2010
TL;DR: In this article, a number of paraconsistent semantics, including three-valued and four-valued semantics, are introduced for the description logic SROIQ, which is the logical foundation of OWL 2.
Abstract: We introduce a number of paraconsistent semantics, including three-valued and four-valued semantics, for the description logic SROIQ, which is the logical foundation of OWL 2. We then study the relationship between the semantics and paraconsistent reasoning in SROIQ w.r.t. some of them through a translation into the traditional semantics. We also present a formalization of rough concepts in SROIQ.