New Directions in Paraconsistent Logic

This paper contains a concise introduction to a few central features of inconsistency-adaptive logics. The focus is on the aim of the program, on logics that may be useful with respect to applications, and on insights that are central for judging the importance of the research goals and the adequacy of results. Given the nature of adaptive logics, the paper may be read as a peculiar introduction to defeasible reasoning.

We investigate the notion of contradiction taking as a central point the idea of a round square. After discussing the question of images of contradiction, related to the contest Picturing Contradiction, we explain why from the point of view of the theory of opposition, a round square is not a contradiction. We then draw a parallel between different kinds of oppositions and different kinds of negations. We explain why from this perspective, when we have a paraconsistent negation \(\lnot \), the formulas p and \(\lnot p\) cannot be considered as forming a contradiction. We finally introduce the notions of paranormal negation and opposition which may catch the concept of a round square.

The aim of this text is to present the philosophical motivations for the Logics of Formal Inconsistency (LFIs), along with some relevant technical results. The text is divided into two main parts (besides a short introduction). In Sect. 3.2, we present and discuss philosophical issues related to paraconsistency in general, and especially to logics of formal inconsistency. We argue that there are two basic and philosophically legitimate approaches to paraconsistency that depend on whether the contradictions are understood ontologically or epistemologically. LFIs are suitable to both options, but we emphasize the epistemological interpretation of contradictions. The main argument depends on the duality between paraconsistency and paracompleteness. In a few words, the idea is as follows: just as excluded middle may be rejected by intuitionistic logic due to epistemological reasons, explosion may also be rejected by paraconsistent logics due to epistemological reasons. In Sect. 3.3, some formal systems and a few basic technical results about them are presented.

Three-valued matrices provide the simplest semantic framework for introducing paraconsistent logics. This paper is a comprehensive study of the main properties of propositional paraconsistent three-valued logics in general, and of the most important such logics in particular. For each logic in the latter group, we also provide a corresponding cut-free Gentzen-type system.

After describing the two formulations of the principle of non contradiction in modern logic \(T \vdash \lnot (p \wedge \lnot p)\) (NC) and \(T, p, \lnot p \vdash q\) (EC) and explaining that three-valued matrices can be used to easily prove their independence, we investigate the possibilities to construct strong paraconsistent negations, i.e., for which neither (NC) nor (EC) holds, using three-valued logical matrices.

The main purpose of the paper is to connect some kind of dialetheism to the use of complex truth values, with new definitions of basic truth-functional connectives that allow for p, \(\backslash \)not p to both be true. ‘True’ is interpreted as \(\left| p \right| = 1\), ‘False’ as \(\left| p \right| = 0\); other values are dispensed with. New definitions of basic truth-functional connectives then allow for “p and not p” to be true. A propositional logic is discussed with the set of connectives including negation, conjunction, disjunction, implication, concordance, discordance, complementary, and equivalence. The authors introduce truth values of propositions, which belong to a subset E, of an uncountable semi-ring F and valuations of propositions, which can be obtained from truth values with the help of a function \(V:E \rightarrow \left[ {0,1} \right] \) satisfying simple properties. Finally, a paraconsistent Boolean logic is introduced.

This paper presents a three-valued paraconsistent logic obtained from some algebra-valued model of set theory. Soundness and completeness theorems are established. The logic has been compared with other three-valued paraconsistent logics.

In this reviewing paper, we recall the main results of our papers [24, 31] where we introduced two paraconsistent semantics for Pavelka style fuzzy logic. Each logic formula \(\alpha \) is associated with a \(2 \times 2\) matrix called evidence matrix. The two semantics are consistent if they are seen from ‘outside’; the structure of the set of the evidence matrices \({{\textit{M}}}\) is an MV-algebra and there is nothing paraconsistent there. However, seen from ‘inside,’ that is, in the construction of a single evidence matrix paraconsistency comes in, truth and falsehood are not each others complements and there is also contradiction and lack of information (unknown) involved. Moreover, we discuss the possible applications of the two logics in real-world phenomena.

The present paper concerns Jaśkowski-like discussive logics which arise by modification of Jaśkowski’s original translation of discussive conjunction. In each case, we indicate the smallest modal logic defining a given Jaśkowski-like discussive logic.

In da Costa and de Ronde (Found Phys 43:845–858, 2013), Newton da Costa together with the author of this paper argued in favor of the possibility to consider quantum superpositions in terms of a paraconsistent approach. We claimed that, even though most interpretations of Quantum Mechanics (QM) attempt to escape contradictions, there are many hints that indicate it could be worth while to engage in a research of this kind. Recently, Arenhart and Krause (New dimensions of the square of opposition, Philosophia Verlag, Munich, 2014; Logique et Analyse, 2014; The Road to Universal Logic (volume II), Springer, 2014) have raised several arguments against this approach and claimed that—taking into account the square of opposition—quantum superpositions are better understood in terms of contrariety propositions rather than contradictory propositions. In de Ronde ( Los Alamos 2014) we defended the Paraconsistent Approach to Quantum Superpositions (PAQS) and provided arguments in favor of its development. In the present paper we attempt to analyze the meaning of modality, potentiality, and contradiction in QM, and provide further arguments of why the PAQS is better suited, than the Contrariety Approach to Quantum Superpositions (CAQS) proposed by Arenhart and Krause, to face the interpretational questions that quantum technology is forcing us to consider.

This paper deals with a relativized notion of inconsistency, which turns out to be equivalent to a non-explosive consequence under certain sets of axiomatization in a propositional language. The paper also shows that several existing paraconsistent systems fall under this characterization.

Vladimir Voevodsky in his Univalent Foundations Project writes that univalent foundations can be used both for constructive and for non-constructive mathematics. The last is of extreme interest since this project would be understood in a sense that this means an opportunity to extend univalent approach on non-classical mathematics. In general, Univalent Foundations Project allows the exploitation of the structures on homotopy types instead of structures on sets or structures on categories as in case of set-level mathematics or category-level mathematics. Non-classical mathematics should be respectively considered either as non-classical set-level mathematics or as non-classical category-level (toposes-level) mathematics. Since it is possible to directly formalize the world of homotopy types using in particular Martin-Lof type systems then the task is to pass to non-classical type systems e.g. da Costa paraconsistent type systems in order to formalize the world of non-classical homotopy types. Taking into account that the univalent model takes values in the homotopy category associated with a given set theory and to construct this model one usually first chooses a locally cartesian closed model category (in the sense of homotopy theory) then trying to extend this scheme for a case of non-classical set theories (e.g. paraconsistent ones) we need to evaluate respective non-classical homotopy types not in cartesian closed categories but in more suitable ones. In any case it seems that such Non-Classical Univalent Foundations Project should be directly developed according to Logical Pluralism paradigma and and it seems that it is difficult to find counter-argument of logical or mathematical character against such an opportunity except the globality and complexity of a such enterprise.

The aim of this paper is to show how to simply define paraconsistent tableau systems by liberalization of construction of complete tableaus. The presented notions allow us to list all tableau inconsistencies that appear in a complete tableau. Then we can easily choose these inconsistencies that are effects of interactions between premises and a conclusion, simultaneously excluding other inconsistencies. A general technique we describe is presented here for the case of Propositional Logic, as the simplest one, but it can be easily extended to more complex cases. In other words, a kind of paraconsistent consequence relation is being studied here, and a simple tableau system is shown to exist that captures that consequence relation.

Some insights were gained from the study of inconsistency-adaptive logics. The aim of the present paper is to put some of these insights to work for the study of logics of formal inconsistency. The focus of attention is application contexts of the aforementioned logics and their theoretical properties in as far as they are relevant for applications. As the questions discussed are difficult but important, a serious attempt was made to make the paper concise but transparent.

In this paper I lay some of the groundwork for a naturalistic, empirically oriented view of logic, attributing the special status of our knowledge of logic to the power of stipulation and expressing the stipulations that constitute the vocabulary of formal logic by rules of inference. The stipulation hypothesis does nothing by itself to explain the usefulness of logic. However, though I do not argue for it here, I believe the selective adoption and application of stipulations can. My concern here is with an issue that has already received a good bit of attention: it seems that we are free to make whatever stipulations we care to make, but we also know that logical stipulations must be carefully constrained, to avoid trivialization, as well as subtler impositions on the already established inferential practices to which we apply our logical vocabulary. I propose three increasingly stringent criteria that fully conservative extensions of a language should meet, and apply them to evaluate three symmetrical, multiple-conclusion logics. A new result, proven first for classical multiple-conclusion logics and then modified and extended to all reflexive, monotonic, and transitive consequence relations, undergirds the focus on proof-theoretic approaches to the consequence relation I adopt here.

The big picture is my big picture, as I see it, based on a lifetime of research into logic. We will cover a reasonably wide range of topics, with some level of author focus. The paper builds on my earlier work [9, 11, 19] (with Rush). Indeed, it can also be seen as an update of the approach to logic taken in [9]. We start with the issue of what logic is about, identifying two inference concepts, one of meaning containment (a connective) and one of deductive argument in general (a rule). Examining the other connectives, we point out the difference between disjunction, as understood in proof-theoretic systems, as opposed to that understood in standard semantics, and show why distribution is not an instance of meaning containment. Negation is judged as being incompletely captured, due to the non-recursive nature of deductive systems in general, but with Boolean negation being the intended concept. We then focus on the logic MC of meaning containment, setting out its axiomatization, content semantics and metavaluation. Quantification is added in a standard way, based on the connectives. We finally deal with applications, focusing on set theory and arithmetic.

In this paper I describe how several notions and constructions in topos logic can be dualized, giving rise to complement-toposes with their paraconsistent internal logic, instead of the usual standard toposes with their intuitionistic logic.

In this work, we consider a well-known and well-studied system of paraconsistent logic which is due to Newton da Costa, and present a topological semantics for it.

In this paper, we argue in the light of visual perceptual experience in favour of the fact that true contradictions are very much perceivable and happen to be an inherent part of the real world. We first describe the phenomenon of perception of two contrary types of brightness illusions, termed as the brightness–contrast and the brightness assimilation type illusions. Next, we present a model of brightness induction which can envisage the above-mentioned contradictions in visual brightness perception. The proposed model, called DDOG (Difference of Difference of Gaussians) is based on two aspects. First, two Difference of Gaussians (DOG) functions acting in opposition in two complementary channels, Magno & Parvo, in the central visual pathway and second, a two-pass model of attentive vision. Although the Oriented Difference of Gaussian (ODOG) model of Blakeslee et al. (Vis Res 45:607–615, 2005) can already account for most of these types of illusions, our model is significantly simpler, more consistent than ODOG and biologically more plausible as a neurocomputational model for explaining brightness contradictions in the brain.

Dialetheism is the view according to which some contradictions (i.e., statements of the form, A and not-A) are true. In this paper, I discuss three strategies to block dialetheism: (i) Contradictions cannot be true because some theories of truth preclude them from emerging. (ii) Contradictions cannot be true because we cannot see what it is like to perceive them. Although that does not undercut the possibility that there are true contradictions that we cannot perceive, it makes their introduction a genuine cost. (iii) Contradictions cannot be true because if they were, we would end up sliding down into believing that everything is true (trivialism). Even if the dialetheist is not committed to that slippery slope, it is crucial that the dialetheist establishes that trivialism is unacceptable; but it is not clear how that could be done successfully. Graham Priest has considered these strategies (in his Doubt Truth to be a Liar), but I argue that none of his responses successfully block them.

A semantic relation of being permitted by a set of possible worlds is defined and analysed. We call it “permittance”. The domain of permittance comprises declarative sentences/formulas. A paraconsistent consequence relation which is both permittance-preserving and truth-preserving is characterized. An application of the introduced concepts in the analysis of question raising is presented.

The main purpose of this article is to discuss and clarify philosophical issues in connection with the librationst set theory £. We take technical results in and upon £ for granted though make references to them as that is useful. We defend the view that £ is neither an inconsistent nor a contradictory system and point out that it is neither paraconsistent nor dialetheist; in contrast we consider £ a bialethic and paracoherent theory.

The catuṣkoṭi (Greek: tetralemma; English: four corners) is a venerable principle of Indian logic, which has been central to important aspects of reasoning in the Buddhist tradition. What, exactly, it is, and how it is applied, are, however, moot—though one thing that does seem clear is that it has been applied in different ways at different times and by different people. Of course, Indian logicians did not incorporate the various interpretations of the principle in anything like a theory of validity in the modern Western sense; but the tools of modern non-classical logic show exactly how to do this. The tools are those of the paraconsistent logic of First Degree Entailment and some of its modifications.

An alternative semantic framework is proposed in the following to reconstruct and make sense of “Eastern logics”: a Question-Answer Semantics (thereafter: QAS), including a set of questions-answers and a finite number of ensuing non-Fregean logical values. Thus, meaning is provided by yes-no answers to corresponding questions about relevant properties. These logical values help to show that the saptabhaṅgī (and its dual, viz., the Buddhist Mādhyamaka catuṣkoṭi) is not a many-valued paraconsistent logic but, rather, a one-valued proto-logic: a constructive machinery that serves as a formal theory of judgment, rather than a Tarskian-like theory of consequence. Such an explanatory model of contradiction assumes a deep redefinition of logical values.