Showing papers on "Paraconsistent logic published in 2013"

The Paraconsistent Logic of Quantum Superpositions

Abstract: Physical superpositions exist both in classical and in quantum physics. However, what is exactly meant by ‘superposition’ in each case is extremely different. In this paper we discuss some of the multiple interpretations which exist in the literature regarding superpositions in quantum mechanics. We argue that all these interpretations have something in common: they all attempt to avoid ‘contradiction’. We argue in this paper, in favor of the importance of developing a new interpretation of superpositions which takes into account contradic- tion, as a key element of the formal structure of the theory, “right from the start”. In order to show the feasibility of our interpretational project we present an outline of a paraconsistent approach to quantum superpositions which attempts to account for the contradictory properties present in general within quantum superpositions. This approach must not be understood as a closed formal and conceptual scheme but rather as a first step towards a different type of understanding regarding quantum superpositions.

Assertion, Denial and Non-classical Theories

Abstract: In this paper I urge friends of truth-value gaps and truth-value gluts—proponents of paracomplete and paraconsistent logics—to consider theories not merely as sets of sentences, but as pairs of sets of sentences, or what I call ‘bitheories,’ which keep track not only of what holds according to the theory, but also what fails to hold according to the theory. I explain the connection between bitheories, sequents, and the speech acts of assertion and denial. I illustrate the usefulness of bitheories by showing how they make available a technique for characterising different theories while abstracting away from logical vocabulary such as connectives or quantifiers, thereby making theoretical commitments independent of the choice of this or that particular non-classical logic. Examples discussed include theories of numbers, classes and truth. In the latter two cases, the bitheoretical perspective brings to light some heretofore unconsidered puzzles for friends of naive theories of classes and truth.

Paraconsistent OWL and related logics

Abstract: The Web Ontology Language OWL is currently the most prominent formalism for representing ontologies in Semantic Web applications. OWL is based on description logics, and automated reasoners are used to infer knowledge implicitly present in OWL ontologies. However, because typical description logics obey the classical principle of explosion, reasoning over inconsistent ontologies is impossible in OWL. This is so despite the fact that inconsistencies are bound to occur in many realistic cases, e.g., when multiple ontologies are merged or when ontologies are created by machine learning or data mining tools.In this paper, we present four-valued paraconsistent description logics which can reason over inconsistencies. We focus on logics corresponding to OWL DL and its profiles. We present the logic $\mathcal {SROIQ}4$, showing that it is both sound relative to classical $\mathcal {SROIQ}$ and that its embedding into $\mathcal {SROIQ}$ is consequence preserving. We also examine paraconsistent varieties of $\mathcal{EL}^{++}$, DL-Lite, and Horn-DLs. The general framework described here has the distinct advantage of allowing classical reasoners to draw sound but nontrivial conclusions from even inconsistent knowledge bases. Truth-value gaps and gluts can also be selectively eliminated from models by inserting additional axioms into knowledge bases. If gaps but not gluts are eliminated, additional classical conclusions can be drawn without affecting paraconsistency.

An Inconsistency-Adaptive Deontic Logic for Normative Conflicts

Abstract: We present the inconsistency-adaptive deontic logic DP r , a nonmonotonic logic for dealing with conflicts between normative statements. On the one hand, this logic does not lead to explosion in view of normative conflicts such as O A ∧ O ∼A, O A ∧ P ∼A or even O A ∧ ∼O A. On the other hand, DP r still verifies all intuitively reliable inferences valid in Standard Deontic Logic (SDL). DP r interprets a given premise set ‘as normally as possible’ with respect to SDL. Whereas some SDL-rules are verified unconditionally by DP r , others are verified conditionally. The latter are applicable unless they rely on formulas that turn out to behave inconsistently in view of the premises. This dynamic process is mirrored by the proof theory of DP r .

A simple approach towards recapturing consistent theories in paraconsistent settings

Abstract: . I believe that, for reasons elaborated elsewhere (Beall, 2009; Priest, 2006a, 2006b), the logic LP (Asenjo, 1966; Asenjo & Tamburino, 1975; Priest, 1979) is roughly right as far as logic goes.1 But logic cannot go everywhere; we need to provide nonlogical axioms to specify our (axiomatic) theories. This is uncontroversial, but it has also been the source of discomfort forLP-based theorists, particularly with respect to true mathematical theories which we take to be consistent. My example, throughout, is arithmetic; but the more general case is also considered.

A Note on Freedom from Detachment in the Logic of Paradox

Abstract: We shed light on an old problem by showing that the logic LP cannot dene a binary connective obeying detachment in the sense that every valuation satisfying ' and ' also satises , except trivially We derive this as a corollary of a more general result concerning variable-sharing

Decision Making for Inconsistent Expert Judgments Using Negative Probabilities

Abstract: In this paper we provide a simple random-variable example of inconsistent information, and analyze it using three different approaches: Bayesian, quantum-like, and negative probabilities. We then show that, at least for this particular example, both the Bayesian and the quantum-like approaches have less normative power than the negative probabilities one.

Some topological properties of paraconsistent models

Abstract: In this work, we investigate the relationship between paraconsistent semantics and some well-known topological spaces such as connected and continuous spaces. We also discuss homotopies as truth preserving operations in paraconsistent topological models.

From paraconsistent three-valued logics to multiple-source epistemic logic

Abstract: Several interpretations can be given to the third truth value in three-valued logics. Here, we consider the case when it refers to the epistemic notion of contradictory, or both true and false at the same time. We study several paraconsistent three-valued logics that carry this concern and show that they can be translated into a fragment of a simple epistemic logic where modalities can only appear in front of literals. This logic is unusual in the sense that necessity modalities distribute over disjunctions instead of conjunctions. An equivalent translation into a fragment of KD modal logic can be obtained by exchanging the role of possibility and necessity modalities, highlighting the perfect symmetry between three-valued logics of contradiction and three-valued logics of incomplete information.

An Approach to Human-Level Commonsense Reasoning

Abstract: Commonsense reasoning has proven exceedingly difficult both to model and to implement in artificial reasoning systems. This paper discusses some of the features of human reasoning that may account for this difficulty, surveys a number of reasoning systems and formalisms, and offers an outline of active logic, a non-classical paraconsistent logic that may be of some use in implementing commonsense reasoning.

If A, Then B: How the World Discovered Logic

Abstract: PrefaceIntroduction: What Is Logic?1 The Dawn of Logic2 Aristotle: Greatest of the Greek Logicians3 Aristotle's System: The Logic of Classification4 Chrysippus and the Stoics: A World of Interlocking Structures5 Logic Versus Antilogic: The Laws of Contradiction and Excluded Middle6 Logical Fanatics7 Will the Future Resemble the Past? Inductive Logic and Scientific Method8 Rhetorical Frauds and Sophistical Ploys: Ten Classic Tricks9 Symbolic Logic and the Digital Future10 Faith and the Limits of Logic: The Last Unanswered QuestionAppendix: Further FallaciesNotesBibliographyIndex

A Routley–Meyer Semantics for Gödel 3-Valued Logic and Its Paraconsistent Counterpart

Abstract: Routley–Meyer semantics (RM-semantics) is defined for Godel 3-valued logic G3 and some logics related to it among which a paraconsistent one differing only from G3 in the interpretation of negation is to be remarked. The logics are defined in the Hilbert-style way and also by means of proof-theoretical and semantical consequence relations. The RM-semantics is defined upon the models for Routley and Meyer’s basic positive logic B+, the weakest positive RM-semantics. In this way, it is to be expected that the models defined can be adapted to other related many-valued logics.

Exploring Paraconsistency in Degree-Preserving Fuzzy Logics

Abstract: Paraconsistent logics are specially tailored to deal with inconsistency, while fuzzy logics primarily deal with graded truth and vagueness. In the last decade, mathematical fuzzy logic has been developed as a discipline studying formal many-valued systems arising from fuzzy set theory. In this paper we study to what extent dierent systems of fuzzy logic are paraconsistent, identifying which families of fuzzy logics satisfy interesting paraconsistency properties.

Recent Work in Relevant Logic

Abstract: But if A and B are utterly irrelevant to one another, many feel reluctant to call these inferences acceptable. Similarly for the validity of the corresponding material implications, often called ‘paradoxes’ of material implication. Relevant logic can be seen as the attempt to avoid these ‘paradoxes’. Relevant logic has a long history. Key early works include Anderson and Belnap 1962; 1963; 1975, and many important results appear in Routley et al. 1982. Those looking for a short introduction to relevant logics might look at Mares 2012 or Priest 2008. For a more detailed but still accessible introduction, there’s Dunn and Restall 2002; Mares 2004b; Priest 2008 and Read 1988. The aim of this article is to survey some of the most important work in the eld in the past ten years, in a way that I hope will be of interest to a philosophical audience. Much of this recent work has been of a formal nature. I will try to outline these technical developments, and convey something of their importance, with the minimum of technical jargon. A good deal of this recent technical work concerns how quanti ers should work in relevant logic. This is the topic of §2. §3 describes other advances in the recent technical literature. In §4, I discuss several recent attempts to give a philosophical interpretation of the most prominent semantics for relevant logic (the Routley-Meyer ternary relation semantics), and highlight some problems. These three sections may be read independently of one another. Those unfamiliar with the ternary relation semantics might like to read the introduction to §4, or consult Mares 2012, before going further.

On Discourses Addressed by Infidel Logicians

Abstract: We here attempt to address certain criticisms of the philosophical import of the so-called Brazilian approach to paraconsistency by providing some epistemic elucidations of the whole enterprise of the logics of formal inconsistency. In the course of this discussion, we substantiate the view that difficulties in reasoning under contradictions in both the Buddhist and the Aristotelian traditions can be accommodated within the precepts of the Brazilian school of paraconsistency.

Modeling and Verifying Inconsistency-Tolerant Temporal Reasoning with Hierarchical Information: Dealing with Students' Learning Processes

Abstract: In this paper, we propose a formal method for modeling and verifying inconsistency-tolerant temporal reasoning with hierarchical information. For this purpose, a temporal logic called sequential paraconsistent computation tree logic (SPCTL) is obtained from the well-known computation tree logic (CTL) by adding a paraconsistent negation connective and some sequence modal operators. SPCTL can appropriately represent both, inconsistency-tolerant reasoning by the paraconsistent negation connective, and hierarchical information by the sequence modal operators. The validity, satisfiability and model-checking problems of SPCTL are shown to be EXPTIME-complete, deterministic EXPTIME-complete and deterministic PTIME-complete, respectively. Some new illustrative examples for students' learning processes are presented using SPCTL.

Paraconsistency and Plausible Argumentation in Generative Grammar: A Case Study

Abstract: While the analytical philosophy of science regards inconsistent theories as disastrous, Chomsky allows for the temporary tolerance of inconsistency between the hypotheses and the data. However, in linguistics there seem to be several types of inconsistency. The present paper aims at the development of a novel metatheoretical framework which provides tools for the representation and evaluation of inconsistencies in linguistic theories. The metatheoretical model relies on a system of paraconsistent logic and distinguishes between strong and weak inconsistency. Strong inconsistency is destructive in that it leads to logical chaos. In contrast, weak inconsistency may be constructive, because it is capable of accounting for the simultaneous presence of seemingly incompatible structures. However, paraconsistent logic cannot grasp the dynamism of the emergence and resolution of weak inconsistencies. Therefore, the metatheoretical approach is extended to plausible argumentation. The workability of this metatheoretical model is tested with the help of a detailed case study on an analysis of discontinuous constituents in Government-Binding Theory.

Making Sense of Paraconsistent Logic: The Nature of Logic, Classical Logic and Paraconsistent Logic

Abstract: Max Cresswell and Hilary Putnam seem to hold the view, often shared by classical logicians, that paraconsistent logic has not been made sense of, despite its well-developed mathematics. In this paper, I examine the nature of logic in order to understand what it means to make sense of logic. I then show that, just as one can make sense of non-normal modal logics (as Cresswell demonstrates), we can make ‘sense’ of paraconsistent logic. Finally, I turn the tables on classical logicians and ask what sense can be made of explosive reasoning. While I acknowledge a bias on this issue, it is not clear that even classical logicians can answer this question.

The Unexpected Applicability of Paraconsistent Logic: A Chomskyan Route to Dialetheism

Abstract: Paraconsistent logics are characterized by rejection of ex falso quodlibet, the principle of explosion, which states that from a contradiction, anything can be derived. Strikingly these logics have found a wide range of application, despite the misgivings of philosophers as prominent as Lewis and Putnam. Such applications, I will argue, are of significant philosophical interest. They suggest ways to employ these logics in philosophical and scientific theories. To this end I will sketch out a ‘naturalized semantic dialetheism’ following Priest’s early suggestion that the principles governing human natural language may well be inconsistent. There will be a significant deviation from Priest’s work, namely, the assumption of a broadly Chomskyan picture of semantics. This allows us to explain natural language inconsistency tolerance without commitment to contentious views in formal logic.

Formalizing Inconsistency-Tolerant Relevant Human Reasoning: A Decidable Paraconsistent Relevant Logic with Constructible Falsity

Abstract: Formalizing inconsistency-tolerant relevant human reasoning in a philosophically plausible logic is useful for modeling sophisticated agents similar to human. For this aim, the positive fragment of the logic RW of contraction-less relevant implication is extended with the addition of a Para consistent negation connective similar to the strong negation connective in Nelson's Para consistent four-valued logic N4. This extended para-consistent relevant logic is called RWP, and it has the property of constructible falsity which is known to be useful for representing inexact predicates. A Gentzen-type sequent calculus SRWP for RWP is introduced, and the decidability and cut-elimination theorems for SRWP are proved. An extended Routley-Meyer semantics is introduced for RWP, and the completeness theorem with respect to this semantics is proved.

Librationist Closures of the Paradoxes

Abstract: We present a semi-formal foundational theory of sorts, akin to sets, named librationism because of its way of dealing with paradoxes. Its semantics is related to Herzberger’s semi inductive approach, it is negation complete and free variables (noemata) name sorts. Librationism deals with paradoxes in a novel way related to paraconsistent dialetheic approaches, but we think of it as bialethic and parasistent. Classical logical theorems are retained, and none contradicted. Novel inferential principles make recourse to theoremhood and failure of theoremhood. Identity is introduced a la Leibniz-Russell, and librationism is highly non-extensional. Π 1 1 -comprehension with ordinary Bar-Induction is accounted for (to be lifted). Power sorts are generally paradoxical, and Cantor’s Theorem is blocked as a camouflaged premise is naturally discarded.

Information, Negation, and Paraconsistency

Abstract: This paper begins by arguing that a truth conditional approach to the semantics for relevant logic is unnatural. Rather, we should adopt an informational semantics. On this view, the indices in the model theory are not possible or impossible worlds, but are situations. A statement is not true or false at a situation; rather a situation can be said either to contain or fail to contain certain pieces of information. Valid inference, then, is seen as information preservation, not truth preservation. The distinction between truth and information gives us some freedom in our treatment of logic. For example, we may have a very classical theory of truth but a very non-classical theory of information. On the other hand, we may accept very non-classical theories of truth (such as dialetheism) together with an informational treatment of logic.

Facing inconsistency: Theories and our relations to them

Abstract: Classical logic is explosive in the face of contradiction, yet we find ourselves using inconsistent theories. Mark Colyvan, one of the prominent advocates of the indispensability argument for realism about mathematical objects, suggests that such use can be garnered to develop an argument for commitment to inconsistent objects and, because of that, a paraconsistent underlying logic. I argue to the contrary that it is open to a classical logician to make distinctions, also needed by the paraconsistent logician, which allow a more nuanced ranking of theories in which inconsistent theories can have different degrees of usefulness and productivity. Facing inconsistency does not force us to adopt an underlying paraconsistent logic. Moreover we will see that the argument to best explanation deployed by Colyvan in this context is unsuccessful. I suggest that Quinean approach which Colyvan champions will not lead to the revolutionary doctrines Colyvan endorses.

On deductive bases for paraconsistent answer set semantics

Abstract: It was proved by Odintsov and Pearce that the logic is a deductive base for paraconsistent answer set semantics (PAS) of logic programs with two kinds of negation. Here we describe the lattice of logics extending , characterise these logics via classes of -models, and prove that none of the proper extensions of is a deductive base for PAS.

Hegel’s Living Logic

Abstract: For Hegel, logic does not essentially consist of formal categories used to think about non-logical content. Rather, it consists of formal categories which are also themselves the content of logic. The idea that logic is its own form and its own content means that forms are used to think through other forms such that the same logical determination is a form in one context and a content in another. The generation of form and content out of one another—which precludes the need for the importation of external content into logic—is part of Hegel’s definition of the logical category of ‘life’ in his Science of Logic. A living logic is a logic that accounts for its own self by means of its own self. Through contrasting this idea of logic with formal logic, and logical life with natural life, this essay provides a snapshot of how Hegel views the activity of living, self-determining logic.

What trends in non-classical logic were anticipated by Nikolai Vasiliev?

Abstract: In this paper we discuss a question about the trends in non-classical logic that were exactly anticipated by Nikolai Vasiliev. We show the influence of Vasiliev’s Imaginary logic on paraconsistent logic. Metatheoretical relations between Vasiliev’s logical systems and many-valued predicate logics are established. We also make clear that Vasiliev has developed a sketch of original system of intensional logic and expressed certain ideas of modal and temporal logics.

A Paraconsistent and Substructural Conditional Logic

Abstract: I introduce and motivate a conditional logic based on the substructural system HL from Paoli (Substructural logics: a primer, Kluwer, Dordrecht, 2002) Its hallmark is the presence of three logical levels (each one of which contains its own conditional connective), linked to one another by means of appropriate distribution principles Such a theory brings about a twofold benefit: on the one hand, it yields a new classification of conditionals where the traditional dichotomies (indicative vs subjunctive, factual vs counterfactual) do not play a decisive role; on the other hand, it allows to retain suitable versions of both substitution of provable equivalents and simplification of disjunctive antecedents, while still keeping out such debatable principles as transitivity, monotonicity, and contraposition

FDE: A Logic of Clutters

Abstract: We uncover a ‘naturally occurring’ first degree system, AL of Articular Logic that is both relevant and paraconsistent. The principal semantic innovation is an informationally articulated, but nevertheless entirely classical representation of wffs as clutters (Clutters are also referred to as simple hypergraphs in some contexts.) on the power set of a set of possible states. The principal methodological novelty is the general observation that distinct classical representations of wffs can be selected and combined with redeployments of classical inference to accommodate particular inferential requirements such as paraconsistency and relevance.