Showing papers on "Paraconsistent logic published in 2020"

The Logical Must: Wittgenstein on Logic

Abstract: Preface Introduction I. Kant on logic II. Naturalizing Kant on logic III. The Tractatus IV. Naturalizing the Tractatus V. Rule-following and logic VI. But isn't logic special?! VII. Naturalizing the logical must Conclusion Bibliography Index

The securitization audience in theologico-political perspective: Giorgio Agamben, doxological acclamations, and paraconsistent logic:

Abstract: Over the past two decades, securitization theory has developed into a robust literature of cases and critiques. The vast majority of the attention paid to securitization has been to the securitizin...

Kripke-Style Models for Logics of Evidence and Truth

Abstract: In this paper, we propose Kripke-style models for the logics of evidence and truth LETJ and LETF. These logics extend, respectively, Nelson’s logic N4 and the logic of first-degree entailment (FDE) with a classicality operator ∘ that recovers classical logic for formulas in its scope. According to the intended interpretation here proposed, these models represent a database that receives information as time passes, and such information can be positive, negative, non-reliable, or reliable, while a formula ∘A means that the information about A, either positive or negative, is reliable. This proposal is in line with the interpretation of N4 and FDE as information-based logics, but adds to the four scenarios expressed by them two new scenarios: reliable (or conclusive) information (i) for the truth and (ii) for the falsity of a given proposition.

The Normal and Self-extensional Extension of Dunn–Belnap Logic

Abstract: A logic $$\mathbf{L}$$ is called self-extensional if it allows to replace occurrences of a formula by occurrences of an $$\mathbf{L}$$ -equivalent one in the context of claims about logical consequence and logical validity. It is known that no three-valued paraconsistent logic which has an implication can be self-extensional. In this paper we show that in contrast, the famous Dunn–Belnap four-valued logic has (up to the choice of the primitive connectives) exactly one self-extensional four-valued extension which has an implication. We also investigate the main properties of this logic, determine the expressive power of its language (in the four-valued context), and provide a cut-free Gentzen-type proof system for it.

The Naïve Conception

Abstract: The chapter discusses the naive conception of set and criticizes attempts to rehabilitate it by modifying the logic of set theory. The focus is on the proposal that the Naive Comprehension Schema – which formally captures the thesis that every condition determines a set – is to be saved by adopting a paraconsistent logic. Three strategies for doing so are distinguished: the material strategy, the relevant strategy and the model-theoretic strategy. It is shown that these strategies lead to set theories that are either too weak or ad hoc or give up on the idea that sets are genuinely extensional entities.

Metatheory and dialetheism

Abstract: Given a formal language, a metalanguage is a language which can express — amongst other things — statements about it and its properties. And a metatheory is a theory couched in that language concerning how some of those notions behave. Two such notions that have been of particular interest to modern logicians — for obvious reasons — are truth and validity. These notions are, however, notoriously deeply entangled in paradox. A standard move is to take the metalanguage to be distinct from the language in question, and so avoid the paradoxes. One of the attractions of a dialetheic approach to the paradoxes of self-reference is that this move may be avoided. One may have a language with the expressive power to talk about — among other things — itself, and a theory in that language about how notions such as truth and validity for that language behave. The contradictions delivered by these notions are forthcoming, but they are quarantined by the use of a paraconsistent logic. The point of this paper is to discuss this project, the extent to which it has been successful, and the places where issues still remain.

Adaptive Fregean Set Theory

Abstract: This paper defines provably non-trivial theories that characterize Frege’s notion of a set, taking into account that the notion is inconsistent. By choosing an adaptive underlying logic, consistent sets behave classically notwithstanding the presence of inconsistent sets. Some of the theories have a full-blown presumably consistent set theory T as a subtheory, provided T is indeed consistent. An unexpected feature is the presence of classical negation within the language.

Logics of Formal Inconsistency enriched with replacement: an algebraic and modal account

Abstract: It is customary to expect from a logical system that it can be algebraizable, in the sense that an algebraic companion of the deductive machinery can always be found. Since the inception of da Costa's paraconsistent calculi $C_n$, algebraic equivalents for such systems have been sought. It is known, however, that these systems are not self-extensional (i.e., they do not satisfy the replacement property). More than this, they are not algebraizable in the sense of Blok-Pigozzi. The same negative results hold for several systems of the hierarchy of paraconsistent logics known as Logics of Formal Inconsistency (LFIs). Because of this, several systems belonging to this class of logics are only characterizable by semantics of a non-deterministic nature. This paper offers a solution for two open problems in the domain of paraconsistency, in particular connected to algebraization of LFIs, by extending with rules several LFIs weaker than $C_1$ , thus obtaining the replacement property (that is, such LFIs turn out to be self-extensional). Moreover, these logics become algebraizable in the standard Lindenbaum-Tarski's sense by a suitable variety of Boolean algebras extended with additional operations. The weakest LFI satisfying replacement presented here is called RmbC, which is obtained from the basic LFI called mbC. Some axiomatic extensions of RmbC are also studied. In addition, a neighborhood semantics is defined for such systems. It is shown that RmbC can be defined within the minimal bimodal non-normal logic E+E defined by the fusion of the non-normal modal logic E with itself. Finally, the framework is extended to first-order languages. RQmbC, the quantified extension of RmbC, is shown to be sound and complete w.r.t. the proposed algebraic semantics.

On the Historical Transformations of the Square of Opposition as Semiotic Object

Abstract: In this paper, we would show how the logical object “square of opposition”, viewed as semiotic object (articulated in textual or/and diagrammatic code), has been historically transformed since its appearance in Aristotle’s texts until the works of Vasiliev. These transformations were accompanied each time with a new understanding and interpretation of Aristotle’s original text and, in the last case, with a transformation of its geometric configuration. The initial textual codification of the theory of opposition in Aristotle’s works is transformed into a diagrammatic one, based on a new “reading” of the Aristotelian text by the medieval scholars that altered the semantics of the O form. Further, based on the medieval “Neo-Aristotelian” reading, the logicians of the nineteenth century suggest new diagrammatic representations, based on new interpretations of quantification of judgements within the algebraic and the functional logical traditions. In all these interpretations, the original square configuration remains invariant. However, Nikolai A. Vasiliev marks a turning point in history. He explicitly attacks the established logical tradition and suggests a new alternation of semantics of the O form, based on Aristotelian concepts that were neglected in the Aristotelian tradition of logic, notably the concept of indefinite judgement. This leads to a configurational transformation of the “square” of opposition into a “triangle”, where the points standing for the O and I forms are contracted into one point, the M(I, O) form that now stands for particular judgement with altered semantics. The new transformation goes beyond the Aristotelian logic paradigm to a new “Non-Aristotelian” logic (and associated ontology), i.e. to paraconsistent logic, although the argumentation used in support of it is phrased in (Neo-)Aristotelian style and the context of discovery is foundational (analogical to Lobachevsky’s research on the axiomatics of geometry). It establishes a bifurcation (proliferation) point in the development of logic. No unique logic is recognized, but different logics concerning different domains (ontologies, respectively). One branch of logic remains to be the “Neo-Aristotelian” one, while the new logic is “Non-Aristotelian”.

A Note on Fernández–Coniglio’s Hierarchy of Paraconsistent Systems

Abstract: A logic is called explosive if its consequence relation validates the so-called principle of ex contradictione sequitur quodlibet. A logic is called paraconsistent so long as it is not explosive. Sette’s calculus P 1 is widely recognized as one of the most important paraconsistent calculi. It is not surprising then that the calculus was a starting point for many research studies on paraconsistency. Fernandez–Coniglio’s hierarchy of paraconsistent systems is a good example of such an approach. The hierarchy is presented in Newton da Costa’s style. Therefore, the law of non-contradiction plays the main role in its negative axioms. The principle of ex contradictione sequitur quodlibet has been marginalized: it does not play any leading role in the hierarchy. The objective of this paper is to present an alternative axiomatization for the hierarchy. The main idea behind it is to focus explicitly on the (in)validity of the principle of ex contradictione sequitur quodlibet. This makes the hierarchy less complex and more transparent, especially from the viewpoint of paraconsistency.

Restricted Rules of Inference and Paraconsistency

Abstract: In this paper, we study two companions to a logic, viz., the left variable inclusion companion and the restricted rules companion, their nature and interrelations, especially in connection with paraconsistency. A sufficient condition for the two companions to coincide has also been proved. Two new logical systems - Intuitionistic Paraconsistent Weak Kleene logic (IPWK) and Paraconsistent Pre-Rough logic (PPRL) - are presented here as examples of logics of left variable inclusion. IPWK is the left variable inclusion companion of Intuitionistic Propositional logic (IPC) and is also the restricted rules companion of it. PPRL, on the other hand, is the left variable inclusion companion of Pre-Rough logic (PRL) but differs from the restricted rules companion of it. We have discussed algebraic semantics for these logics in terms of Plonka sums. This amounts to introducing a contaminating truth value, intended to denote a state of indeterminacy.

Many-valuedness from a universal logic perspective

Abstract: We start by presenting various ways to define and to talk about many-valued logic(s). We make the distinction between on the one hand the class of many-valued logics and on the other hand what we call “many-valuedness”: the meta-theory of many-valued logics and the related meta-theoretical framework that is useful for the study of any logical systems. We point out that universal logic, considered as a general theory of logical systems, can be seen as an extension of many-valuedness. After a short story of many-valuedness, stressing that it is present since the beginning of the history of logic in Ancient Greece, we discuss the distinction between dichotomy and polytomy and the possible reduction to bivalence. We then examine the relations between singularity and universality and the connection of many-valuedness with the universe of logical systems. In particular, we have a look at the interrelationship between modal logic, 3-valued logic and paraconsistent logic. We go on by dealing with philosophical aspects and discussing the applications of many-valuedness. We end with some personal recollections regarding Alexander Karpenko, from our first meeting in Ghent, Belgium in 1997, up to our last meeting in Saint Petersburg, Russia in 2016.

Process and Subprocess Studies to Implement the Paraconsistent Artificial Neural Networks for Decision-Making

Abstract: Since the time of Aristotle’s thinking of logic to being a tool for orderly think, has maintained its importance until the present times. So soon, studies of the non-classical logical calls (Abe in 4th International Workshop on Soft Computing Applications. IEEE, pp. 11–18, 2010 [1]) have become a powerful tool as an aid in the making of decisions. The Paraconsistent logic calls attention to the clarity of containing provisions contrary to some of the basic principles of Aristotelian logic, such as the principle of contradiction. In this article, the use of technology allows proposing a structured organization in the process for use of Paraconsistent artificial neural networks. This process aims to be a facilitator in supporting the construction of the decision support (Abe in 4th International Workshop on Soft Computing Applications. IEEE, pp. 11–18, 2010 [1]) with the announcements for project recount in the function point analysis technique.

A Semi-lattice of Four-valued Literal-paraconsistent-paracomplete Logics

Abstract: In this paper, we consider the class of four-valued literal-paraconsistent-paracomplete logics constructed by combination of isomorphs of classical logic CPC. These logics form a 10-element upper semi-lattice with respect to the functional embeddinig one logic into another. The mechanism of variation of paraconsistency and paracompleteness properties in logics is demonstrated on the example of two four-element lattices included in the upper semi-lattice. Functional properties and sets of tautologies of corresponding literal-paraconsistent-paracomplete matrices are investigated. Among the considered matrices there are the matrix of Puga and da Costa's logic V and the matrix of paranormal logic P1I1, which is the part of a sequence of paranormal matrices proposed by V. Fernandez.

A classical-logic view of a paraconsistent logic

Abstract: This paper is concerned with the first-order paraconsistent logic LPQ$^{\supset,\mathsf{F}}$. A sequent-style natural deduction proof system for this logic is presented and, for this proof system, both a model-theoretic justification and a logical justification by means of an embedding into first-order classical logic is given. For no logic that is essentially the same as LPQ$^{\supset,\mathsf{F}}$, a natural deduction proof system is currently available in the literature. The given embedding provides both a classical-logic explanation of this logic and a logical justification of its proof system. The major properties of LPQ$^{\supset,\mathsf{F}}$ are also treated.

Contradictions and Rationality: An Analysis of Two Biblical Cases

Abstract: In this chapter, I want to read particular biblical cases as a means for examining two philosophical problems; namely, how to deal logically with contradictions and the consequences that follow from the alternatives examined regarding the notion of rationality. Considering a perspective according to which rationality is closely connected to a certain idea of logicity, my aim is to show that, on the one hand, in the story of Susanna and the Elders, Daniel’s belief in Susanna’s chastity can be understood as supported by a reasoning that is grounded on classical logic and thus fits perfectly into the classical model of rationality. The narrations of Jesus’s resurrection on the other hand would require a different approach both to contradictions and to the relationship between contradictions and rationality. Otherwise, there is a risk of either shaking up important parts of the Christian creed—such as the idea that God has a salvific plan that is made possible precisely by the resurrection of Jesus—or accepting that believing in the resurrection is irrational. It is worthy to take into account that, for fideists, accepting a religious belief as irrational is no problem because they hold that religious beliefs are a question of faith, and faith is either independent of or adversarial toward reason in the sense that it is not founded, nor can it be founded, on arguments. Something different happens with those who adhere to more rationalistic traditions like natural theology, which attempts to give a rational support to the belief of the existence of God. The Christian believer to which I refer in this work stands on the latter, more rationalistic side. With regard to the rationality of the belief in the resurrection, I contend that in order to be considered rational, one who believes in the resurrection could support that belief with a reasoning grounded on some form of paraconsistent logic. I focus here on da Costa’s C1 system and on the LFI1 system of Carnielli, Marcos and de Amo, but other paraconsistent systems could also be used. In the last section of the chapter, I present some problems that emerge when a perspective like the one proposed here is adopted.

Mathematical logic: mathematics of logic or logic of mathematics

Abstract: This brief historical survey is written from a logical point of view. It is a rational reconstruction of the genesis of some interrelations between formal logic and mathematics. We examine how mathematical logic was conceived: as the abstract mathematics of logic or as the logic of mathematical practice.

Leibnizian Identity and Paraconsistent Logic

Abstract: The standard Leibnizian view of identity allows for substitutivity of identicals and validates transitivity of identity within classical semantics. However, in a series of works, Graham Priest argu...

A lattice of the paracomplete calculi

Abstract: Paracomplete logic is intended to cope with the problem of vagueness, or uncertain and incomplete data. It deals with the situation when some propositions and their negations are allowed to be simultaneously false, which is obviously impossible in the classical and many non--classical propositional logics. In paracomplete logic, such classical laws as tertium non datur or consequentia mirabilis are not generally accepted. This implies that the logic is defined negatively. In this paper, we introduce a family of the paracomplete calculi that will be defined in a Hilbert-style formalization. We propose the so-called bi--valuational semantics and prove the key metatheorems for the calculi. We also discuss a generalization of the paracomplete calculus $QD^{1}$ to the hierarchy of related calculi.

Analysis of the displacement of terrestrial mobile robots in corridors using paraconsistent annotated evidential logic e{\tau}

Abstract: This article proposes an algorithm for a servo motor that controls the movement of an autonomous terrestrial mobile robot using Paraconsistent Logic. The design process of mechatronic systems guided the robot construction phases. The project intends to monitor the robot through its sensors that send positioning signals to the microcontroller. The signals are adjusted by an embedded technology interface maintained in the concepts of Paraconsistent Annotated Logic acting directly on the servo steering motor. The electric signals sent to the servo motor were analyzed, and it indicates that the algorithm paraconsistent can contribute to the increase of precision of movements of servo motors.

Is God Paraconsistent

Abstract: In this paper we examine what is the relation between God and Paraconsistency. We first start by some general considerations about paraconsistent negations and paraconsistent things. We then examine in which sense the “thing” called “God” is paraconsistent or not. Finally, we look the other way round and discuss the divinity of paraconsistency.

A Note on FDE “All the Way Up”

Abstract: A very natural and philosophically important subclassical logic is FDE (for first-degree entailment). This account of logical consequence can be seen as going beyond the standard two-valued account (of “just true” and “just false”) to a four-valued account (adding the additional values of “both true and false” and “neither true nor false”). A natural question arises: What account of logical consequence arises from considering further (positive) combinations of such values? A partial answer was given by Priest in 2014; Shramko and Wansing had also given a partial result some years earlier, although in a different (more algebraic) context. In this note we generalize Priest’s (and indirectly Shramko and Wansing’s) result to show that even if one considers ordinal-many (positive) combinations of the previous values, for any ordinal, the resulting consequence relation (the resulting logic) remains FDE.

Twist-Valued Models for Three-Valued Paraconsistent Set Theory

Abstract: We propose in this paper a family of algebraic models of ZFC based on the three-valued paraconsistent logic LPT0, a linguistic variant of da Costa and D’Ottaviano’s logic J3. The semantics is given by twist structures defined over complete Boolean agebras. The Boolean-valued models of ZFC are adapted to twist-valued models of an expansion of ZFC by adding a paraconsistent negation. This allows for inconsistent sets w satisfying ‘not (w = w)’, where ‘not’ stands for the paraconsistent negation. Finally, our framework is adapted to provide a class of twist-valued models generalizing Lowe and Tarafder’s model based on logic (PS 3,∗), showing that they are paraconsistent models of ZFC. The present approach offers more options for investigating independence results in paraconsistent set theory.

Research on Set Theory Based on Paraconsistent Logic

Abstract: Different from ZF axiomatic set theory, the paraconsistent set theory has changed the basic logic of set theory and selected paraconsistent logic which can accommodate or deal with contradictions, it effectively avoids the whole theory falling into a non-trivial dilemma when there are contradictions in set theory. In this paper, we first review the history and current situation of the praconsistent set theory; then, we give three kinds of paraconsistent logic which can be used to construct the praconsistent set theory among many kinds of paraconsistent logics. And then, we analyze the differences of methods of the paraconsistent set theory with strong or weak structure of paraconsistent logic and get different paraconsistent set theory. Finally, we verify that paraconsistent set theory is a new method to solve the paradox of set theor. The development of paraconsistent set theory can solve the difficulties in the development of set theory in a unique way, which is not only the extension of the application of paraconsistent logic, but also the new form and new trend of the development of set theory.

A Paraconsistent Conditional Logic

Abstract: We develop a paraconsistent logic by introducing new models for conditionals with acceptive and rejective selection functions which are variants of Chellas’ conditional models. The acceptance and rejection conditions are substituted for truth conditions of conditionals. The paraconsistent conditional logic is axiomatized by a sequent system $\mathcal {C}$ which is an extension of the Belnap-Dunn four-valued logic with a conditional operator. Some acceptive extensions of $\mathcal {C}$ are shown to be sound and complete. We also show the finite acceptive model property and decidability of these logics.

Four-Valued Logics of Truth, Nonfalsity, Exact Truth, and Material Equivalence

Abstract: The four-valued semantics of Belnap–Dunn logic, consisting of the truth values True, False, Neither, and Both, gives rise to several nonclassical logics depending on which feature of propositions we wish to preserve: truth, nonfalsity, or exact truth (truth and nonfalsity). Interpreting equality of truth values in this semantics as material equivalence of propositions, we can moreover see the equational consequence relation of this four-element algebra as a logic of material equivalence. In this paper, we axiomatize all combinations of these four-valued logics, for example, the logic of truth and exact truth or the logic of truth and material equivalence. These combined systems are consequence relations which allow us to express implications involving more than one of these features of propositions.

Introducing consistency in a dialogical framework for paraconsistent logic

Abstract: The Logics of Formal Inconsistency are logics tolerant to some amount of inconsistency but in which some versions of explosion still hold. The main result of this paper is to provide a reconstruction of two such logics in the dialogical framework. By doing so, we achieve two things. On the one hand, we provide a formal approach to argumentative situations in which some inconsistencies may occur while keeping the idea that there may still be situations in which some propositions are “safe” in the sense of immune to the contradictions. On the other hand, we open a new line of study on these logics, in the context of the game-theoretical approach to semantics born in the 1960s, with various interesting prospectives some of which are discussed at the end of this work.

Bayes Meets Entailment and Prediction: Commonsense Reasoning with Non-monotonicity, Paraconsistency and Predictive Accuracy.

Abstract: The recent success of Bayesian methods in neuroscience and artificial intelligence gives rise to the hypothesis that the brain is a Bayesian machine. Since logic and learning are both practices of the human brain, it leads to another hypothesis that there is a Bayesian interpretation underlying both logical reasoning and machine learning. In this paper, we introduce a generative model of logical consequence relations. It formalises the process of how the truth value of a sentence is probabilistically generated from the probability distribution over states of the world. We show that the generative model characterises a classical consequence relation, paraconsistent consequence relation and nonmonotonic consequence relation. In particular, the generative model gives a new consequence relation that outperforms them in reasoning with inconsistent knowledge. We also show that the generative model gives a new classification algorithm that outperforms several representative algorithms in predictive accuracy and complexity on the Kaggle Titanic dataset.