Showing papers on "Paraconsistent logic published in 2016"

What Is an Inconsistent Truth Table

Abstract: Do truth tables—the ordinary sort that we use in teaching and explaining basic propositional logic—require an assumption of consistency for their construction? In this essay we show that truth tables can be built in a consistency-independent paraconsistent setting, without any appeal to classical logic. This is evidence for a more general claim—that when we write down the orthodox semantic clauses for a logic, whatever logic we presuppose in the background will be the logic that appears in the foreground. Rather than any one logic being privileged, then, on this count partisans across the logical spectrum are in relatively similar dialectical positions.

From Paraconsistent Logic to Dialetheic Logic

Abstract: The only condition for a logic to be paraconsistent is to invalidate the so-called explosion. However, the understanding of the only connective involved in the explosion, namely negation, is not shared among paraconsistentists. By returning to the modern origin of paraconsistent logic, this paper proposes an account of negation, and explores some of its implications. These will be followed by a consideration on underlying logics for dialetheic theories, especially those following the suggestion of Laura Goodship. More specifically, I will introduce a special kind of paraconsistent logic, called dialetheic logic, and present a new system of paraconsistent logic, which is dialetheic, by expanding the Logic of Paradox of Graham Priest. The new logic is obtained by combining connectives from different traditions of paraconsistency, and has some distinctive features such as its propositional fragment being Post complete. The logic is presented in a Hilbert-style calculus, and the soundness and completeness results are established.

An Interpretable Logical Theory: The case of Compensatory Fuzzy Logic

Abstract: This paper aims to formally define a concept of interpretability according to natural language of a logical theory, and show advances for demonstrating that the logic system called Compensatory Fuzzy Logic is interpretable. A logical theory is interpretable according to natural language if the calculus based upon the elements of this logical theory can be understood in natural language and vice versa. We present conditions necessary for a logical theory to be called interpretable, especially Compensatory Fuzzy Logic.

Explosion and the normativity of logic

Abstract: Logic has traditionally been construed as a normative discipline; it sets forth standards of correct reasoning. Explosion is a valid principle of classical logic. It states that an inconsistent set of propositions entails any proposition whatsoever. However, ordinary agents presumably do -- occasionally, at least -- have inconsistent belief sets. Yet it is false that such agents may, let alone ought to, believe any proposition they please. Therefore, our logic should not recognize explosion as a logical law. Call this the 'normative argument against explosion'. Arguments of this type play -- implicitly or explicitly -- a central role in motivating paraconsistent logics. Branden Fitelson (2008), in a throwaway remark, has conjectured that there is no plausible 'bridge principle' articulating the normative link between logic and reasoning capable of supporting such arguments. This paper offers a critical evaluation of Fitelson's conjecture, and hence of normative arguments for paraconsistency and the conceptions of logic's normative status on which they repose. It is argued that Fitelson’s conjecture turns out to be correct: normative arguments for paraconsistency probably fail.

Two Genuine 3-Valued Paraconsistent Logics

Abstract: In this paper we present two genuine three-valued paraconsistent logics, i.e. logics obeying neither \(p, \lnot p \vdash q\) nor \(\vdash \lnot (p \wedge \lnot p)\). We study their basic properties and their relations with other paraconsistent logics, in particular da Costa’s paraconsistent logics C1 and its extension \(C1+\).

Paraconsistent analysis network applied in the treatment of Raman spectroscopy data to support medical diagnosis of skin cancer

Abstract: Paraconsistent logic (PL) is a type of non-classical logic that accepts contradiction as a fundamental concept and has produced valuable results in the analysis of uncertainties. In this work, algorithms based on a type of PL-paraconsistent annotated logic of two values (PAL2v)-are interconnected into a network of paraconsistent analysis (PANnet). PANnet was applied to a dataset comprising 146 Raman spectra of skin tissue biopsy fragments of which 30 spectra were determined to represent normal skin tissue (N), 96 were determined to represent tissue with basal cell carcinoma, and 19 were determined to be tissue with melanoma (MEL). In this database, paraconsistent analysis was able to correctly discriminate 136 out of a total of 145 fragments, obtaining a 93.793 % correct diagnostic accuracy. The application of PAL2v in the analysis of Raman spectroscopy signals produces better discrimination of cells than conventional statistical processes and presents a good graphical overview through its associated lattice structure. The technique of PAL2v-based data processing can be fundamental in the development of a computational tool dedicated to support the diagnosis of skin cancer using Raman spectroscopy.

A Hybrid Linear Logic for Constrained Transition Systems

Abstract: Linear implication can represent state transitions, but real transition systems operate under temporal, stochastic or probabilistic constraints that are not directly representable in ordinary linear logic. We propose a general modal extension of intuitionistic linear logic where logical truth is indexed by constraints and hybrid connectives combine constraint reasoning with logical reasoning. The logic has a focused cut-free sequent calculus that can be used to internalize the rules of particular constrained transition systems; we illustrate this with an adequate encoding of the synchronous stochastic pi-calculus.

Hallden's Logic of Nonsense and Its Expansions in View of Logics of Formal Inconsistency

Abstract: The paper explores the connection between Soren Hallden's logic of nonsense (and its expansions) and Logics of Formal Inconsistency (LFIs), one of the main traditions in paraconsistency. Although not much attention has been payed by paraconsistentists, Hallden's logic can be nicely connected to LFIs. The main result of the paper is a reformulation of an expansion of logic of nonsense studied by Lennart Aqvist and Krister Segerberg in the light of LFIs. More specifically, we present a three-valued semantics and a Hilbert-style system, and prove soundness and completeness results. We also observe some definability results related to the consistency and just-true operators, and briefly discuss an 'interpretation' of truth values of the three-valued semantics in view of the recent work of Graham Priest on plurivalent semantics.

Inconsistency-adaptive dialogical logic

Abstract: Even when inconsistencies are present in our premise set, we can sensibly distinguish between good and bad arguments relying on these premises. In making this distinction, the inconsistency-adaptive approach of Batens strikes a particularly nice balance between inconsistency-tolerance and inferential strength. In this paper, we use the machinery of Batens’ approach to extend the paraconsistent approach to dialogical logic as developed by Rahman and Carnielli. In bringing these frameworks closer together, we obtain a dynamic mechanism for the systematic study of dialogues in which two parties exchange arguments over a central claim, in the possible presence of inconsistencies.

On the Methodology of Paraconsistent Logic

Abstract: The present note contains a critical discussion of the methodology of paraconsistent logic in general and “the central optimisation problem of paraconsistent logics” in particular. It is argued that there exist several reasons not to consider classical logic as the reference logic for developing systems of paraconsistent logic, and it is suggested to weaken a certain maximality condition that may be seen as essential for “optimisation”, which is a methodology in the tradition of Newton da Costa. It is argued that the guiding motivation for the development of paraconsistent logics should be neither epistemological nor ontological, but informational. Moreover, it is pointed out that there are other notions of maximality and other methodologies. A methodology due to Graham Priest and Richard Routley and another methodology that focuses on a minimal shrinkage of expressiveness relative to a given reference logic are considered in some detail.

Two aspects of relevance in structured argumentation: minimality and paraconsistency

Abstract: This paper studies two issues concerning relevance in structured argumentation in the context of the ASPIC+ framework, arising from the combined use of strict and defeasible inference rules. One issue arises if the strict inference rules correspond to classical logic. A longstanding problem is how the trivialising effect of the classical Ex Falso principle can be avoided while satisfying consistency and closure postulates. In this paper, this problem is solved by disallowing chaining of strict rules, resulting in a variant of the ASPIC+ framework called ASPIC*, and then disallowing the application of strict rules to inconsistent sets of formulas. Thus in effect Rescher & Manor's paraconsistent notion of weak consequence is embedded in ASPIC*. Another issue is minimality of arguments. If arguments can apply defeasible inference rules, then they cannot be required to have subset-minimal premises, since defeasible rules based on more information may well make an argument stronger. In this paper instead minimality is required of applications of strict rules throughout an argument. It is shown that under some plausible assumptions this does not affect the set of conclusions. In addition, circular arguments are in the new ASPIC* framework excluded in a way that satisfies closure and consistency postulates and that generates finitary argumentation frameworks if the knowledge base and set of defeasible rules are finite. For the latter result the exclusion of chaining of strict rules is essential. Finally, the combined results of this paper are shown to be a proper extension of classical-logic argumentation with preferences and defeasible rules.

Paraconsistent Probabilities: Consistency, Contradictions and Bayes’ Theorem

Abstract: This paper represents the first steps towards constructing a paraconsistent theory of probability based on the Logics of Formal Inconsistency (LFIs). We show that LFIs encode very naturally an extension of the notion of probability able to express sophisticated probabilistic reasoning under contradictions employing appropriate notions of conditional probability and paraconsistent updating, via a version of Bayes’ theorem for conditionalization. We argue that the dissimilarity between the notions of inconsistency and contradiction, one of the pillars of LFIs, plays a central role in our extended notion of probability. Some critical historical and conceptual points about probability theory are also reviewed.

Nicephorus Blemmydes on the Holy Trinity and the Paraconsistent Notion of Numbers: A Logical Analysis of a Byzantine Approach to the Filioque

Abstract: Abstract The paper deals with the most controversial - in the modern scholarly discussion - episode within the Byzantine polemics on the Filioque, Nicephorus Blemmydes‘ acknowledgement of proceeding of the Spirit through the Son providing that the Son be considered as generated through the Spirit. The logical intuition behind this theological idea is explicated in the terms of paraconsistent logic and especially of a kind of paraconsistent numbers called by the author “pseudo-natural numbers”. Such numbers could not be interpreted via the notion of ordered pair. Instead, they imply a known (first described by Emil Post in 1941) but still little studied logical connective ternary exclusive OR.

Paraconsistent Double Negation That Can Simulate Classical Negation

Abstract: A new classical paraconsistent logic (CP), which is a variant of Nelson's paraconsistent four-valued logic, is introduced as a Gentzen-type sequent calculus. The logic CP can simulate the classical negation in classical logic by paraconsistent double negation in CP. Some theorems for syntactically and semantically embedding CP into a Gentzen-type sequent calculus LK for classical logic and vice versa are proved. The cut-elimination and completeness theorems for CP are also shown using these embedding theorems.

Logical Formalization and the Formation of Logic(s)

Abstract: The project of logic as a theoretical tool useful for the sciences and humanities involves, as a crucial step, logical formalization – the conversion of sentences of natural language to formulas of a formal language. But what do we do, exactly, when we do logical formalization? What are the criteria of adequacy of the conversion? In how far is logic normative? The paper offers answers to these central (but surprisingly rather neglected) questions and shows that getting a proper grasp on the process of formalization is important for understanding the nature of logic. The key point is that logic as a theoretical tool manages to consolidate our linguistic – in particular argumentative – practices by means of attaining a specific sort of reflective equilibrium. The paper provides a detailed discussion of the answers to the above questions implied by this understanding of logic.

The very idea of a substructural approach to paradox

Abstract: This paper aims to call into question the customary division of logically revisionary responses to the truth-theoretic paradoxes into those that are “substructural” and those that are “(fully) structural.” I proceed by examining, as a case study, Beall’s recent proposal based on the paraconsistent logic LP. Beall formulates his response to paradox in terms of a consequence relation that obeys all standard structural rules, though at the price of the language’s lacking a detaching conditional. I argue that the same response to paradox can be given using a consequence relation that preserves detachment rules for a conditional, though at the price of restricting structural rules. The question “Is paradox being blocked by invoking a substructural consequence relation?” is thus ill-posed. The lesson of this example, I conclude, is that there is no useful explication of the idea of a substructural approach to paradox.

Paraconsistent dynamics

Abstract: It has been an open question whether or not we can define a belief revision operation that is distinct from simple belief expansion using paraconsistent logic. In this paper, we investigate the possibility of meeting the challenge of defining a belief revision operation using the resources made available by the study of dynamic epistemic logic in the presence of paraconsistent logic. We will show that it is possible to define dynamic operations of belief revision in a paraconsistent setting.

On a paraconsistentization functor in the category of consequence structures

Abstract: This paper is an attempt to solve the following problem: given a logic, how to turn it into a paraconsistent one? In other words, given a logic in which ex falso quodlibet holds, how to convert it into a logic not satisfying this principle? We use a framework provided by category theory in order to define a category of consequence structures. Then, we propose a functor to transform a logic not able to deal with contradictions into a paraconsistent one. Moreover, we study the case of paraconsistentization of propositional classical logic.

Determination of Operating Parameters and Performance Analysis of Computer Networks with Paraconsistent Annotated Evidential Logic Eτ

Abstract: Computer networks have two important characteristics: the vast diversity of connecting devices and a great variability of the physical distribution of equipments. Therefore, the performance analysis of a specific network based on absolute references or third parties may not be applicable in all circumstances, especially in highly complex and heterogeneous networks. Indeed, it carries a high degree of uncertainty, and the classical logic may not be appropriate to deal it. This paper aims to parameterize and evaluate the operating elements of heterogeneous networks, from the analysis of representative attributes, based on concepts of Paraconsistent Annotated Evidential Logic Eτ.

Paraconsistency in hybrid logic

Abstract: As in standard knowledge bases, hybrid knowledge bases (i.e., sets of information specified by hybrid formulas) may contain inconsistencies arising from different sources, namely from the many mechanisms used to collect relevant information. Being a fact, rather than a queer anomaly, inconsistency also needs to be addressed in the context of hybrid logic applications. This paper introduces a paraconsistent version of hybrid logic which is able to accommodate inconsistencies at local points without implying global failure. A main feature of the resulting logic, crucial to our approach, is the fact that every hybrid formula has an equivalent formula in negation normal form. The paper also provides a measure to quantify the inconsistency of a hybrid knowledge base, useful as a possible basis for comparing knowledge bases. Finally, the concepts of extrinsic and intrinsic inconsistency of a theory are discussed.

Understanding Negation Implicationally in the Relevant Logic R

Abstract: A star-free relational semantics for relevant logic is presented together with a sound and complete sequent proof theory (display calculus). It is an extension of the dualist approach to negation regarded as modality, according to which de Morgan negation in relevant logic is better understood as the confusion of two negative modalities. The present work shows a way to define them in terms of implication and a new connective, co-implication, which is modeled by respective ternary relations. The defined negations are confused by a special constraint on ternary relation, called the generalized star postulate, which implies definability of the Routley star in the frame. The resultant logic is shown to be equivalent to the well-known relevant logic R. Thus it can be seen as a reconstruction of R in the dualist framework.

The Limits of Logic : Higher-Order Logic and the Löwenheim-Skolem Theorem

Abstract: Contents: Is Second-Order Logic Logic?: Beyond first-order logic: the historical interplay between mathematical logic and axiomatic set theory, Gregory H. Moore Which logic is the right logic?, Leslie H. Tharp On second-order logic, George S. Boolos Second-order languages and mathematical practice, Stewart Shapiro What are logical notions?, Alfred Tarski A curious inference, George Boolos The rationalist conception of logic, Steven J. Wagner A critical appraisal of second-order logic, Ignacio JanA(c) Who's afraid of higher-order logic?, Peter Simons. Ontological Reduction, Intended Interpretations and the LA wenheim-Skolem Theorems: Ontological reduction, Leslie H. Tharp Intended models and the LA wenheim-Skolem theorem, Virginia Klenk Categoricity, John Corcoran Skolem's paradox and constructivism, Charles McCarty and Neil Tennant Second-order logic, foundations and rules, Stewart Shapiro. Plural Quantification: To be is to be a value of a variable (or to be some values of some variables), George Boolos Nominalist Platonism, George Boolos Second-order logic still wild, Michael D. Resnick. Philosophy of Set Theory: Kreisel, the continuum hypothesis, and second-order set theory, Thomas Weston Skolem and the LA wenheim-Skolem theorem: a case study of the philosophical significance of mathematical results, Alexander George Skolem and the skeptic, Paul Benacerraf Skolem and the skeptic, Crispin Wright Predication versus membership in the distinction between logic as language and logic as calculus, Nino B. Cocchiarella Logicism, the continuum and anti-realism, Peter Clark Name index.

A Logic of Comparative Support: Qualitative Conditional Probability Relations Representable by Popper Functions

Abstract: This article explicates a logic of comparative support relations of the form “conclusion C1 is supported by premises P1 at least as strongly as conclusion C2 is supported by premises P2”, abbreviated “C1|P1 ≽ C2|P2”. This logic is a generalization of the logic of comparative conditional probability first developed by B. O. Koopman (1940). The version of the logic on offer here contains several innovations. Its axioms will not presuppose deductive logic. Rather, classical logical entailment will fall out as a special case of comparative support. Furthermore, these comparative support relations turn out to be the qualitative conditional probabilistic analogs of the conditional probability functions known as Popper functions. As a result, this logic can capture all of the features of Bayesian confirmation theory. Thus, this logic of comparative support provides a foundation for the notion of evidential support that underlies both classical deductive logic and probabilistic inductive logic. Along the way I'll provide background on the nature of the Popper functions, including a particularly spare axiomatization of the Popper functions that won’t presuppose deductive logic.

Game theoretical semantics for some non-classical logics

Abstract: Paraconsistent logics are the formal systems in which absurdities do not trivialise the logic. In this paper, we give Hintikka-style game theoretical semantics for a variety of paraconsistent and non-classical logics. For this purpose, we consider Priest’s Logic of Paradox, Dunn’s First-Degree Entailment, Routleys’ Relevant Logics, McCall’s Connexive Logic and Belnap’s four-valued logic. We also present a game theoretical characterisation of a translation between Logic of Paradox/Kleene’s K3 and S5. We underline how non-classical logics require different verification games and prove the correctness theorems of their respective game theoretical semantics. This allows us to observe that paraconsistent logics break the classical bidirectional connection between winning strategies and truth values.

On a “most telling” argument for paraconsistent logic

Abstract: Priest and others have presented their “most telling” argument for paraconsistent logic: that only paraconsistent logics allow non-trivial inconsistent theories. This is a very prevalent argument; occurring as it does in the work of many relevant and more generally paraconsistent logicians. However this argument can be shown to be unsuccessful. There is a crucial ambiguity in the notion of non-triviality. Disambiguated the most telling reason for paraconsistent logics is either question-begging or mistaken. This highlights an important confusion about the role of logic in our development of our theories of the world. Does logic chart good reasoning or our commitments? We also consider another abductive argument for paraconsistent logics which also is shown to fail.

In defence of a logic for ‘because’

Abstract: The present author developed a calculus for the logic of ‘because’. In a recent paper in this journal, it has been claimed that the central inference rules for the logic are invalid and that the intuition upon which the rules are based is not accounted for. This note criticises these arguments and presents an independent argument in favour of the rules used in the logic.

Novel method to characterize upper-limb movements based on paraconsistent logic and myoelectric signals

Abstract: This paper presents a novel method that investigates the use of Paraconsistent Artificial Neural Network (PANN) and upper-limb electromyography signals for classification of movements, due to their intrinsic ability to deal with imprecise, inconsistent and paracomplete data. The preliminary study presents promising results in terms of processing time and accuracy. The average classification accuracy for the developed paraconsistent logic method was 76,0±9,1% for 17 distinguish movements and a classification average processing time of 14 ms per movement.