Paraconsistent Research Articles

Explicit analyses of proof/refutation interaction for constructible falsity and Heyting–Brouwer logic

Abstract Nelson’s logic of constructible falsity $\textsf{N}$ and Rauszer’s Heyting–Brouwer logic $\textsf{HB}$ are well-known cases of extensions of intuitionistic logic $\textsf{Int}$ enriched with novel connectives. Wansing has suggested that Gödel’s provability interpretation of $\textsf{Int}$ can be extended to these systems by pairing the category of formal proofs with a distinct category of formal refutations. In this paper, we extend the framework of Artemov’s justification logic to provide explicit analyses of $\textsf{N}$ and $\textsf{HB}$ (and the dual-intuitionistic logic $\textsf{DualInt}$) that respect a distinction between proofs and refutations. The application distinguishes the categories by reinterpreting the agents of multiple-agent justification logic as devices that operate exclusively on one or the other category. The analyses reveal that differences between $\textsf{N}$ and $\textsf{HB}$ can be reduced to competing interaction principles characterizing the coordination between proofs and refutations. We conclude by reappraising some of the unusual features of $\textsf{HB}$ in light of the explicit analysis of $\textsf{HB}$.

The temporality of uncertainty in decision-making and treatment of severe brain injury.

The aim of the study is to investigate how time and uncertainties of clinical action and decision-making plays out in the practical work of early neurorehabilitation in order to present new analytical ways to understand the underlying logics and dynamic social processes that take place during professional treatment of patients with severe acquired brain injury. Drawing on ethnographic fieldwork in a Danish neuro-intensive step-down unit (NISU) specialising in early neurorehabilitation, we found that negotiation of futures takes place in the modern ICU in the present by strategically building upon past experiences. We have argued that the clinical programme therefore cannot be understood only from a "here and now perspective", since the early neurorehabilitation practice is embedded in overlapping temporalities of the past, the present, and desired futures. The study discusses the underlying logics-often hidden or unnoticed-that impact clinical practice of early neurorehabilitation, in what we have termed a logic of clinical reenactment, a logic of future negotiation and a logic of paradox.

Contradictions and falling bridges: what was Wittgenstein’s reply to Turing?

ABSTRACT In this paper, I offer a close reading of Wittgenstein's remarks on inconsistency, mostly as they appear in the Lectures on the Foundations of Mathematics. I focus especially on an objection to Wittgenstein's view given by Alan Turing, who attended the lectures, the so called ‘falling bridges’-objection. Wittgenstein's position is that if contradictions arise in some practice of language, they are not necessarily fatal to that practice nor necessitate a revision of that practice. If we then assume that we have adopted a paraconsistent logic, Wittgenstein's answer to Turing is that if we run into trouble building our bridge, it is either because we have made a calculation mistake or our calculus does not actually describe the phenomenon it is intended to model. The possibility of either kind of error is not particular to contradictions nor to inconsistency, and thus contradictions do not have any special status as a thing to be avoided.

Ordered pair semantics and negation in LP

In this note, I present a modified semantic framework for the multi-valued paraconsistent logic LP, which allows for a straightforward preservation of a significant classical intuition about negation, namely that the negation operator reverses truth-value.

A Game Theoretical Semantics for Logics of Nonsense

Logics of non-sense allow a third truth value to express propositions that are \emph{nonsense}. These logics are ideal formalisms to understand how errors are handled in programs and how they propagate throughout the programs once they appear. In this paper, we give a Hintikkan game semantics for logics of non-sense and prove its correctness. We also discuss how a known solution method in game theory, the iterated elimination of strictly dominated strategies, relates to semantic games for logics of nonsense. Finally, we extend the logics of nonsense only by means of semantic games, developing a new logic of nonsense, and propose a new game semantics for Priest's Logic of Paradox.

Sequent-Type Calculi for Systems of Nonmonotonic Paraconsistent Logics

Paraconsistent logics constitute an important class of formalisms dealing with non-trivial reasoning from inconsistent premisses. In this paper, we introduce uniform axiomatisations for a family of nonmonotonic paraconsistent logics based on minimal inconsistency in terms of sequent-type proof systems. The latter are prominent and widely-used forms of calculi well-suited for analysing proof search. In particular, we provide sequent-type calculi for Priest's three-valued minimally inconsistent logic of paradox, and for four-valued paraconsistent inference relations due to Arieli and Avron. Our calculi follow the sequent method first introduced in the context of nonmonotonic reasoning by Bonatti and Olivetti, whose distinguishing feature is the use of a so-called rejection calculus for axiomatising invalid formulas. In fact, we present a general method to obtain sequent systems for any many-valued logic based on minimal inconsistency, yielding the calculi for the logics of Priest and of Arieli and Avron as special instances.

A Family of Strict/Tolerant Logics

Strict/tolerant logic, ST, evaluates the premises and the consequences of its consequence relation differently, with the premises held to stricter standards while consequences are treated more tolerantly. More specifically, ST is a three-valued logic with left sides of sequents understood as if in Kleene’s Strong Three Valued Logic, and right sides as if in Priest’s Logic of Paradox. Surprisingly, this hybrid validates the same sequents that classical logic does. A version of this result has been extended to meta, metameta, … consequence levels in Barrio et al. (2019). In my earlier paper Fitting (2019) I showed that the original ideas behind ST are, in fact, much more general than first appeared, and an infinite family of many valued logics have Strict/Tolerant counterparts. This family includes both Kleene’s and Priest’s logic individually, as well as first degree entailment. For instance, for both the Kleene and the Priest logic, the corresponding strict/tolerant logic is six-valued, but with differing sets of strictly and tolerantly designated truth values. The present paper extends that generalization in two directions. We examine a reverse notion, of Tolerant/Strict logics, which exist for the same structures that were investigated in Fitting (2019). And we show that the generalization extends through the meta, metameta, … consequence levels for the same infinite family of many valued logics. Finally we close with remarks on the status of cut and related rules, which can actually be rather nuanced. Throughout, the aim is not the philosophical applications of the Strict/Tolerant idea, but the determination of how general a phenomenon it is.

Back to the Logic of Paradox

ABSTRACT The author once again offers a definition of paradox that fundamentally differs from a Hegelian notion of contradiction. In a paradox, thought should logically identify and comprehend the mental reproduction of the subject of thought and the object of thought as the nonidentical thought of an unthinkable being. This is why Bibler argues for the emergence of a new logic as the logic of the beginnings of logic, a dialogue of logics whose appearance is due to changing social conditions and a transformation happened in human activity itself. From joint labor, activity becomes universal labor directed at the subject, at which point thinking turns on itself and paradoxes emerge.

Twist-Valued Models for Three-Valued Paraconsistent Set Theory

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 Löwe 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.

A lattice of the paracomplete calculi


 
 
 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. 
 
 

Many-valuedness from a universal logic perspective


 
 
 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.
 
 

Metatheory and dialetheism


 
 
 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.
 
 

The evidence approach to paraconsistency versus the paraconsistent approach to evidence

In this paper, we analyze the epistemic approach to paraconsistency. This approach is advanced as an alternative to dialetheism on what concerns interpreting paraconsistency and contradictions; instead of having to accept that there are true contradictions (as dialetheists argue), it is suggested that we may understand such situations as involving only conflicting evidence, which restricts contradictions to a notion of evidence weaker than truth. In this paper, we first distinguish two conflicting programs entangled in the proposal: (1) interpreting paraconsistency in general through the notion of evidence, and (2) modeling reasoning with evidence by using paraconsistent logic. The first part of the program, we argue, does not succeed, on the grounds that it does not lead to a uniform proposal to the understanding of paraconsistency, and fails to engage with dialetheism in a legitimate dispute about interpretation of paraconsistency. Also, when seen through the lights of the second kind of approach, a ‘logic as modeling’ approach, weaknesses of dealing with evidence through paraconsistency come to light, basically because evidence does not seem to suggest the need of a paraconsistent treatment. As a result, one can neither approach paraconsistency in general through evidence, nor approach evidence with the use of paraconsistent logics.

A paraconsistent many-valued similarity method for multi-attribute decision making

In this paper, we introduce a method for resolving decision problems concerning multiple criteria in relation to a finite set of decision alternatives. This approach makes use of paraconsistent logic, Pavelka style fuzzy logic and many-valued similarity. To demonstrate the robustness of the method, two data sets, one on the performance of five mobile phone operators in Ghana and the other, a numerical example have been analysed and the rankings compared correspondingly with those of three existing dominant Multi-Attribute Decision Making (MADM) approaches, namely Elimination and Choice Translating Reality II (ELECTRE II); Preference Ranking Organisation MeTHod for Enrichment Evaluation (PROMETHEE I and II) and Technique for Order Preference by Similarity to Ideal Solution (TOPSIS). Apart from providing a ranking that is similar to these three famous outranking methods, the novel approach has the edge over them due to its ability to relatively handle large size decision problems - decision problems with numerous criteria and alternatives - without much difficulty.

Extensions of paraconsistent weak Kleene logic

Abstract Paraconsistent weak Kleene ($\textrm{PWK}$) logic is the $3$-valued logic based on the weak Kleene matrices and with two designated values. In this paper, we investigate the poset of prevarieties of generalized involutive bisemilattices, focussing in particular on the order ideal generated by Α$\textrm{lg} (\textrm{PWK})$. Applying to this poset a general result by Alexej Pynko, we prove that, exactly like Priest’s logic of paradox, $\textrm{PWK}$ has only one proper nontrivial extension apart from classical logic: $\textrm{PWK}_{\textrm{E}}\textrm{,}$ PWK logic plus explosion. This $6$-valued logic, unlike $\textrm{PWK} $, fails to be paraconsistent. We describe its consequence relation via a variable inclusion criterion and identify its Suszko-reduced models.

An axiomatic approach toCG′3 logic

AbstractIn memoriam José Arrazola Ramírez (1962–2018) The logic $\textbf{G}^{\prime}_3$ was introduced by Osorio et al. in 2008; it is a three-valued logic, closely related to the paraconsistent logic $\textbf{CG}^{\prime}_3$ introduced by Osorio et al. in 2014. The logic $\textbf{CG}^{\prime}_3$ is defined in terms of a multi-valued semantics and has the property that each theorem in $\textbf{G}^{\prime}_3$ is a theorem in $\textbf{CG}^{\prime}_3$. Kripke-type semantics has been given to $\textbf{CG}^{\prime}_3$ in two different ways by Borja et al. in 2016. In this work, we continue the study of $\textbf{CG}^{\prime}_3$, obtaining a Hilbert-type axiomatic system and proving a soundness and completeness theorem for this logic.

Measuring Inconsistency in Generalized Propositional Logic

Consistency is one of the key concepts of logic; logicians have put a great deal of effort into proving the consistency of many logics. Understanding what causes inconsistency is also important; some logicians have developed paraconsistent logics that, unlike classical logics, allow some contradictions without making all formulas provable. Another direction of research studies inconsistency by measuring the amount of inconsistency of sets of formulas. While the initial attempt in 1978 was too ambitious in trying to do this for first-order logic, this research got a substantial boost when an inconsistency measure was proposed for propositional logic in 2002. Since then, researchers in logic and artificial intelligence (AI systems need the capability to deal with inconsistency) have made many interesting proposals and found related issues. Almost all of this work has been done for propositional logic. The purpose of this paper is to extend inconsistency measures to logics that also contain operators, such as modal operators. We use the terminology “generalized propositional logic” for such logics. We show how to extend propositional inconsistency measures to sets of formulas in any such generalized propositional logic. Examples are used to illustrate how various modal operators, including spatial and tense operators, fit into this framework. We also show that the addition of operators leads to a weak type of inconsistency. In all cases, the calculations for several inconsistency measures are given.

When Curry met Abel

AbstractBased on his Inclosure Schema and the Principle of Uniform Solution (PUS), Priest has argued that Curry’s paradox belongs to a different family of paradoxes than the Liar. Pleitz (2015, The Logica Yearbook 2014, pp. 233–248) argued that Curry’s paradox shares the same structure as the other paradoxes and proposed a scheme of which the Inclosure Schema is a particular case and he criticizes Priest’s position by pointing out that applying the PUS implies the use of a paraconsistent logic that does not validate Contraction, but that this can hardly seen as uniform. In this paper, we will develop some further reasons to defend Pleitz’ thesis that Curry’s paradox belongs to the same family as the rest of the self-referential paradoxes & using the idea that conditionals are generalized negations. However, we will not follow Pleitz in considering doubtful that there is a uniform solution for the paradoxes in a paraconsistent spirit. We will argue that the paraconsistent strategies can be seen as special cases of the strategy of restricting Detachment and that the latter uniformly blocks all the connective-involving self-referential paradoxes, including Curry’s.

A BRIDGE BETWEEN Q-WORLDS

AbstractQuantum set theory (QST) and topos quantum theory (TQT) are two long running projects in the mathematical foundations of quantum mechanics (QM) that share a great deal of conceptual and technical affinity. Most pertinently, both approaches attempt to resolve some of the conceptual difficulties surrounding QM by reformulating parts of the theory inside of nonclassical mathematical universes, albeit with very different internal logics. We call such mathematical universes, together with those mathematical and logical structures within them that are pertinent to the physical interpretation, ‘Q-worlds’. Here, we provide a unifying framework that allows us to (i) better understand the relationship between different Q-worlds, and (ii) define a general method for transferring concepts and results between TQT and QST, thereby significantly increasing the expressive power of both approaches. Along the way, we develop a novel connection to paraconsistent logic and introduce a new class of structures that have significant implications for recent work on paraconsistent set theory.

On Logic of Strictly-Deontic Modalities. A Semantic and Tableau Approach

Standard deontic logic (SDL) is defined on the basis of possible world semantics and is a logic of alethic-deontic modalities rather than deontic modalities alone. The interpretation of the concepts of obligation and permission comes down exclusively to the logical value that a sentence adopts for the accessible deontic alternatives. Here, we set forth a different approach, this being a logic which additionally takes into consideration whether sentences stand in relation to the normative system or to the system of values under which we predicate the deontic qualifications. By taking this aspect into account, we arrive at a logical system which preserves laws proper to a deontic logic but where the standard paradoxes of deontic logic do not arise. It is a logic of strictly-deontic modalities DR.