Paraconsistent Research Articles

Literal and Controllable Paraconsistency

The principle of explosion asserts that any formula can be derived from any pair of other contradictory formulas. Paraconsistent logic is typically regarded as a logic in which the universal validity of this principle is questioned. Therefore, a key point is determining when the validity can be considered universal to classify a logic as paraconsistent. A pertinent example to illustrate this point is the calculus CB1 that admits the principle but only for negated formulas, i.e., from any set {α, ∼α} any other formula follows if and only if α is of the form ∼γ. Another example is Sette’s calculus P1, which is paraconsistent at the level of variables but not complex formulas. Both serve as compelling examples of the so-called borderline cases. In this paper, we examine several calculi expected to be paraconsistent at the level of literals. It means that a pair of formulas, α and ∼α, can yield any β if, and only if α is neither a propositional variable nor is its iterated negation. Furthermore, it is assumed that in some calculi presented here, β must adhere to specific restrictions. Once these conditions are satisfied, we refer to calculus as paraconsistent in a “controllable manner”.

Intuitionistic Implication and Logics of Formal Inconsistency

Logics of Formal Inconsistency (LFI for short) are a class of paraconsistent logics that validate the principle of gentle explosion, meaning that any formula can be derived from the set of formulas: ∘α, α and ∼α. A unique feature of LFI is the use of the symbol ‘∘’ to represent notions of consistency at the object-language level. These logics are simple in essence, built upon all the axiom schemas of positive classical logic, axioms for negation and the so-called ‘consistency operator’ ∘, with the only inference rule being detachment. In this paper, we propose an alternative foundation for LFI, which is the positive fragment of intuitionistic propositional logic. We present bi-valuational ‘Loparić-like’ semantics for the resulting logics and discuss their potential extensions.

Basic Framework of Apophatic Metaphysics (Negative Theology) Toward Islamic Higher Education System: Analytical Study at Perguruan Tinggi Keagamaan Islam Negeri (Ptkin) In Indonesia

The construction of the metaphysics of divinity that exists in the Islamic higher education system, especially in Perguruan Tinggi Keagamaan Islam Negeri (PTKIN) - in the form of universities) in Indonesia has been finalized and established. It turns out that there was a logic of metaphysics of divinity that was different from the mainstreaming of the established metaphysics, namely metaphysics (apophatic theology). Based on this reality, this study focused on the construction on the concept of apophatic theology that grew and developed in the Islamic higher education system in PTKIN (namely UIN Sumatera Utara, UIN Malang, and UIN Jember). The research problem in this study was how does the construction of apophatic theology in the Islamic higher education system develop at UIN Sumatera utara, UIN Malang, and UIN Jember? In order to analytically answer this question, this study used a qualitative approach with data collection techniques; such as in-depth interviews, participatory observation, and documentation. The data analysis used interactive data analysis from Miles, Hubermann, and Saldana. This study found that apophatic theology was positioned as a discourse of profetization of divine values expressed through a negative style. The theological construction basically "rejects" divine knowledge with the logic of negation and paradox. This apophatic theology in the Islamic higher education system builds an axiological framework that more oriented towards the dimensions of humanity and divinity; or the profane dimension and transcendental.

How Classical, Paracomplete and Paraconsistent Logicians (Dis-)Agree

This paper develops an account of disagreement and agreement in logic in terms of rules of acceptance, rejection, and suspension of judgement. Acceptance, rejection, and suspension in logic are thereby taken to be doxastic attitudes resulting from, respectively, assenting, dissenting, or refraining from assenting and dissenting to arguments or propositions in light of their logical validity/invalidity. Disagreement between advocates of different logics is characterized as a form of doxastic noncotenability. A full account of agreement in logic does not only require doxastic cotenability between two logicians. It is also necessary that they share the reasons that ground their respective doxastic attitudes. These notions of disagreement and agreement will be applied to disagreements/agreements between advocates of three different logical systems: classical logic, the paracomplete logic K3, and the paraconsistent logic LP. In particular, it will be discussed which doxastic attitudes those logicians ought to have with regard to a proposition expressed by the Liar sentence. In the last part of the paper, it will be examined in what sense a disagreement in logic can be understood as a genuine disagreement, even if no single neutral intertheoretic concept of validity is available that is shared by all logicians.

Modality across different logics

Abstract In this paper, we deal with the problem of putting together modal worlds that operate in different logic systems. When evaluating a modal sentence $\Box \varphi $, we argue that it is not sufficient to inspect the truth of $\varphi $ in accessed worlds (possibly in different logics). Instead, ways of transferring more subtle semantic information between logical systems must be established. Thus, we will introduce modal structures that accommodate communication between logic systems by fixing a common lattice $L$ that contains as sublattices the semantics operating in each world. The value of a formula $\Box \varphi $ in a world with lattice $L^{\prime}$ will be defined in terms of the values of $\varphi $ in accessible worlds relativized to $L^{\prime}$ using the common order of $L$. We will investigate natural instances where formulas $\varphi $ can be said to be necessary/possible even though all the accessible world falsify $\varphi $. Further, we will discuss frames that characterize dynamic relations between logic systems: classically increasing, classically decreasing and dialectic frames. Finally, we formalize the semantics of considering worlds operating in classical logic or logic of paradox, exemplifying the kind of issue one should face in this kind of formalization.

Identifying roles of formulas in inconsistency under Priest's minimally inconsistent logic of paradox

It has been increasingly recognized that identifying roles of formulas of a knowledge base in the inconsistency of that base can help us better look inside the inconsistency. However, there are few approaches to identifying such roles of formulas from a perspective of models in some paraconsistent logic, one of typical tools used to characterize inconsistency in semantics. In this paper, we characterize the role of each formula in the inconsistency arising in a knowledge base from informational as well as causal aspects in the framework of Priest's minimally inconsistent logic of paradox. At first, we identify the causal responsibility of a formula for the inconsistency based on the counterfactual dependence of the inconsistency on the formula under some contingency in semantics. Then we incorporate the change on semantic information in the framework of causal responsibility to develop the informational responsibility of a formula for the inconsistency to capture the contribution made by the formula for the inconsistent information. This incorporation makes the informational responsibility interpretable from the point of view of causality, and capable of catching the role of a formula in inconsistent information concisely. In addition, we propose notions of naive and quasi naive responsibilities as two auxiliaries to describe special relations between inconsistency and formulas in semantic sense. Some intuitive and interesting properties of the two kinds of responsibilities are also discussed.

Probabilistic Semantics and Calculi for Multi-valued and Paraconsistent Logics

Probabilistic Semantics and Calculi for Multi-valued and Paraconsistent Logics

Truth diagrams for some non-classical and modal logics

This paper examines truth diagrams for some non-classical, modal and dynamic logics. Truth diagrams are diagrammatic and visual ways to represent logical truth akin to truth tables, developed by Peter C.-H. Cheng. Currently, it is only given for classical propositional logic. In this paper, we establish truth diagrams for Priest's Logic of Paradox, Belnap–Dunn's Four-Valued Logic, MacColl's Connexive Logic, Bochvar–Halldén's Logic of Non-Sense, Carnielli–Coniglio's logic of formal inconsistency as well as classical modal logic and its dynamic extension to shed light on the semantic behaviour of some non-classical and modal logics.

Computer-Aided Searching for a Tabular Many-Valued Discussive Logic—Matrices

Abstract In the paper, we tackle the matter of non-classical logics, in particular, paraconsistent ones, for which not every formula follows in general from inconsistent premisses. Our benchmark is Jaśkowski’s logic, modeled with the help of discussion. The second key origin of this paper is the matter of being tabular, i.e. being adequately expressible by finitely many finite matrices. We analyse Jaśkowski’s non-tabular discussive (discursive) logic $ \textbf {D}_{2}$, one of the first paraconsistent logics, from the perspective of a trivalent tabular logic. We are motivated to step down from the ongoing modal $ \textbf {S5}$-perspective of developing $ \textbf {D}_{2}$ both by certain mysteries that have been surrounding it and by gaps in Jaśkowski’s arguments contra the multivalent tabular perspective. Although Jaśkowski’s idea to use $ \textbf {S5}$ in order to define $ \textbf {D}_{2}$ is very attractive since it allows one to benefit from the tools and results of modal logic, it also gives a ‘non-direct’ formulation and, as it appeared later, is superfluous with respect to what is meant to be achieved since one can define the very same logic but using modal logics weaker than S5. It is also known, due to Kotas, that discussive logic is not finite-valued. So, in light of Kotas’s result that $ \textbf {D}_{2}$ is non-tabular, we propose to associate it with a few dozen discussive formulae that Jaśkowski unequivocally and illustratively suggests to be its theorems or non-theorems rather than with axioms of its modern axiomatizations (one of which Kotas employs) in order to be capable of performing a computer-aided brute-force search for suitable trivalent matrices in the cases of one and two designated values. As a result, we find trivalent matrices with two designated values that might be dubbed ‘discussive’ because they meet Jaśkowski’s suggestion to validate and invalidate the litmus theorems and non-theorems, respectively, despite the fact that none of them validates all the negation axioms in the modern axiomatizations of $ \textbf {D}_{2}$. The matrices found are then analysed along with highlighting the ones that were previously mentioned in the literature. We conclude the paper with a comparative analysis of Omori’s results and a test of Karpenko’s hypothesis.

Meet-Combination of Consequence Systems

We extend meet-combination of logics for capturing the consequences that are common to both logics. With this purpose in mind we define meet-combination of consequence systems. This notion has the advantage of accommodating different ways of presenting the semantics and the deductive calculi. We consider consequence systems generated by a matrix semantics and consequence systems generated by Hilbert calculi. The meet-combination of consequence systems generated by matrix semantics is the consequence system generated by their product. On the other hand, the meet-combination of consequence systems generated by Hilbert calculi is the consequence system generated by their interconnection. We investigate preservation of several properties. Capitalizing on these results we show that interconnection provides an axiomatization for the product. Illustrations are given for intuitionistic and modal logics, Łukasiewicz logic and some paraconsistent logics.

Paraconsistency in Non-Fregean Framework

AbstractA non-Fregean framework aims to provide a formal tool for reasoning about semantic denotations of sentences and their interactions. Extending a logic to its non-Fregean version involves introducing a new connective $$\equiv $$ ≡ that allows to separate denotations of sentences from their logical values. Intuitively, $$\equiv $$ ≡ combines two sentences $$\varphi $$ φ and $$\psi $$ ψ into a true one whenever $$\varphi $$ φ and $$\psi $$ ψ have the same semantic correlates, describe the same situations, or have the same content or meaning. The paper aims to compare non-Fregean paraconsistent Grzegorczyk’s logics (Logic of Descriptions $$\textsf{LD}$$ LD , Logic of Descriptions with Suszko’s Axioms $$\textsf{LDS}$$ LDS , Logic of Equimeaning $$\textsf{LDE}$$ LDE ) with non-Fregean versions of certain well-known paraconsistent logics (Jaśkowski’s Discussive Logic $$\textsf{D}_2$$ D 2 , Logic of Paradox $$\textsf{LP}$$ LP , Logics of Formal Inconsistency $$\textsf{LFI}{1}$$ LFI 1 and $$\textsf{LFI}{2}$$ LFI 2 ). We prove that Grzegorczyk’s logics are either weaker than or incomparable to non-Fregean extensions of $$\textsf{LP}$$ LP , $$\textsf{LFI}{1}$$ LFI 1 , $$\textsf{LFI}{2}$$ LFI 2 . Furthermore, we show that non-Fregean extensions of $$\textsf{LP}$$ LP , $$\textsf{LFI}{1}$$ LFI 1 , $$\textsf{LFI}{2}$$ LFI 2 , and $$\textsf{D}_2$$ D 2 are more expressive than their original counterparts. Our results highlight that the non-Fregean connective $$\equiv $$ ≡ can serve as a tool for expressing various properties of the ontology underlying the logics under consideration.

Nothingness and Paraconsistent Logic

This paper explores the concept of “Nothingness” and its connection to Graham Priest’s paraconsistent logic, with a critical focus on Heidegger's ontological perspective. Heidegger argues that logic and ontology are incompatible, and truth extends beyond mere propositions, tied to the indescribable experience of “Nothing.” He contends that logical rules are not essential for ontological truth, leading to two conceptions of truth: fundamental and propositional. The study delves into this profound examination, considering the implications for understanding truth and the limitations of logic in grasping the elusive aspects of existence.

Rules of Explosion and Excluded Middle: Constructing a Unified Single-Succedent Gentzen-Style Framework for Classical, Paradefinite, Paraconsistent, and Paracomplete Logics

Rules of Explosion and Excluded Middle: Constructing a Unified Single-Succedent Gentzen-Style Framework for Classical, Paradefinite, Paraconsistent, and Paracomplete Logics

Symmetric and conflated intuitionistic logics

Abstract Two new propositional non-classical logics, referred to as symmetric intuitionistic logic (SIL) and conflated intuitionistic logic (CIL), are introduced as indexed and non-indexed Gentzen-style sequent calculi. SIL is regarded as a natural hybrid logic combining intuitionistic and dual-intuitionistic logics, whereas CIL is regarded as a variant of intuitionistic paraconsistent logic with conflation and without paraconsistent negation. The cut-elimination theorems for SIL and CIL are proved. CIL is shown to be conservative over SIL.

Paraconsistent logics as a Way of Expressing Objective Contradictions in Science

The article focuses on the problems associated with the need to express in the language of science in a consistent way the movement and various kinds of changes taking place in the objective world by developing logics that would be tolerant of logical contradictions. The paper presents a brief history of the creation of paraconsistent (paraconsistent) logics, in which logical contradictions turned out to be quite permissible. It is noted that the priority in their development belongs to Russian scientists N. A. Vasiliev and I. E. Orlov, as well as Polish philosophers and logicians Y. Lukasevich and S. Yaskovsky; that since the seventies of the last century, the development of paraconsistent logics has assumed an international character; that interest in these logics is not abating at the present time. The possibilities of using paraconsistent logics presented in the works of foreign and domestic philosophers and logicians for the formalization of dialectics are discussed. The positive role of these logics is shown in solving many paradoxes in the foundations of classical propositional calculation and predicate logic, as well as in order to limit undesirable consequences when using IT technologies related to the processing of inconsistent or contradictory information. It is concluded that the use of paranoherent logics in the construction of models of individual fragments of dialectics contributes to its development as logic and as ontology, and indicates that the apparatus of non-classical logics as a whole is a very powerful means of studying and explaining many problems of theoretical cognition.

Visibilizing and managing paradox: redefining the role of non-executive directors

PurposeThis paper aims to discuss the work of non-executive directors (NEDs) as inherently paradoxical. Paradox refers to the presence of persistent contradictions between interdependent forces. Those persistent tensions are explored, and approaches are indicated to stimulate the adaptive use of paradoxes as forces of innovation and renewal.Design/methodology/approachThis conceptual approach can be read as an invitation for corporate governance scholars to embrace the logic of paradox to expand the understanding of this topic. Paradox is not conceptualized as an alternative to dominant structural views, including board composition, but as a complementary conceptual perspective, a meta-theoretical lens to shed light on the tensions inherent to governance.FindingsThe authors propose that paradox theory offers a fresh conceptual lens to study the role of NEDs. This approach may help NEDs to turn tensions and paradoxes visible to develop a rich understanding of their work, as well as helping them navigate the complexities of organizing, a process rich in inherent paradoxicality.Originality/valueOrganizational paradox theory is a bourgeoning field of study, but the conceptual lens of paradox has still been underexplored in the study of corporate governance.

Turing Machine Halting Problem, Russell’s Paradox and Gödel Incompleteness Theorem

The Turing Machine Halting Problem is a major problem in computer theory, Russell’s Paradox is the root of the Third Mathematical Crisis, and the Gödel Incompleteness Theorem is a major discovery in modern logic. The three have had a profound impact on the development of science and have attracted the attention of scientific and philosophical circles. However, since the Gödel Incompleteness Theorem was put forward, the scientific and philosophical significance of its proof has been questioned; in particular, Wittgenstein regards it as a certain logical paradox, and Russell’s Paradox has not yet been settled. This paper makes a detailed analysis of the three based on the view of dialectical infinity. The author notes that the Principle of Comprehension based on the view of actual infinity is the root of Russell’s Paradox. The Turing Machine Halting Problem shows that it is impossible to make an actual-infinite ultimate judgment of the constantly generated infinite world, but the philosophical significance of the Gödel Incompleteness Theorem is that our understanding of the world is essentially potentially infinite. At the end of the article, the author raises several questions about the proof of the Gödel Incompleteness Theorem, finds out the specific paradox form in the proof, points out the high consistency of its proof method and Russell’s Paradox, which strongly supports Wittgenstein’s view. The author points out that the philosophical basis of the proof of the Gödel Incompleteness Theorem is the idea of actual infinity, the proof of the theorem is based on a logically invalid circular formula, the contradiction of the proof originates from the Gödel formula itself, and cannot be attributed to the incompleteness of the system, so the proof is wrong. Therefore, the conclusion of this paper is that the world is constantly developing and changing, and our human understanding of the world is essentially a potential infinite, that is, the world is Aristotelian, not Platonic.

On Paracomplete Versions of Jaśkowski's Discussive Logic

Jaśkowski's discussive (discursive) logic D2 is historically one of the first paraconsistent logics, i.e., logics which 'tolerate' contradictions. Following Jaśkowski's idea to define his discussive logic by means of the modal logic S5 via special translation functions between discussive and modal languages, and supporting at the same time the tradition of paracomplete logics being the counterpart of paraconsistent ones, we present a paracomplete discussive logic D2p.

Wansing's bi-intuitionistic logic: semantics, extension and unilateralisation

ABSTRACT The well-known algebraic semantics and topological semantics for intuitionistic logic ( Int ) is here extended to Wansing's bi-intuitionistic logic ( 2 Int ). The logic 2 Int is also characterised by a quasi-twist structure semantics, which leads to an alternative topological characterisation of 2 Int . Later, notions of Fregean negation and of unilateralisation are proposed. The logic 2 Int is extended with a ‘Fregean negation’ connective ∼ , obtaining 2 In t ∼ , and it is showed that the logic N 4 ⋆ (an extension of Nelson's paraconsistent logic) results to be the unilateralisation of 2 In t ∼ via ∼ . The logic 2 In t ∼ is also characterised by a Kripke-style semantics, a twist structure semantics and a topological semantics.

Paraconsistent logic and query answering in inconsistent databases

ABSTRACT This paper concerns the paraconsistent logic LPQ⊃,F and an application of it in the area of relational database theory. The notions of a relational database, a query applicable to a relational database, and a consistent answer to a query with respect to a possibly inconsistent relational database are considered from the perspective of this logic. This perspective enables among other things the definition of a consistent answer to a query with respect to a possibly inconsistent database without resort to database repairs. In an earlier paper, LPQ⊃ ,F is presented with a sequent-style natural deduction proof system. In this paper, a sequent calculus proof system is presented instead because such proof systems are generally considered more suitable as the basis of proof search procedures than natural deduction proof systems and proof search procedures can serve as the core of algorithms for computing consistent answers to queries.