Paraconsistent Negation Research Articles

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.

An Extended Paradefinite Logic Combining Conflation, Paraconsistent Negation, Classical Negation, and Classical Implication: How to Construct Nice Gentzen-type Sequent Calculi

An Extended Paradefinite Logic Combining Conflation, Paraconsistent Negation, Classical Negation, and Classical Implication: How to Construct Nice Gentzen-type Sequent Calculi

Mortensen Logics

Mortensen introduced a connexive logic commonly known as 'M3V'. M3V is obtained by adding a special conditional to LP. Among its most notable features, besides its being connexive, M3V is negation-inconsistent and it validates the negation of every conditional. But Mortensen has also studied and applied extensively other non-connexive logics, for example, closed set logic, CSL, and a variant of Sette's logic, identified and called 'P2' by Marcos. In this paper, we analyze and compare systematically the connexive variants of CSL and P2, obtained by adding the M3V conditional to them. Our main observations are two. First, that the inconsistency of M3V is exacerbated in the connexive variant of closed set logic, while it is attenuated in the connexive variant of the Sette-like P2. Second, that the M3V conditional is, unlike other conditionals, "connexively stable", meaning that it remains connexive when combined with the main paraconsistent negations.

La negación paraconsistente a la luz del realismo lógico

In this paper I analyze some of the negation operators from the family of paraconsistent logics from the perspective of logical realism. I start by taking as cases two cases of study: the LP negation operator along with its metaphysical interpretation stated as dialetheism and the FDE negation with the metaphysical interpretation proposed by Beall (2016). I use this operators to draw a comparison with the different level of metaphysical commitments in some other paraconsistent negations. I discuss how LP negation a robust metaphysical commitment such as dialetheism relies on a leading role of logical negation while in Beall’s interpretation the search for a metaphysically balanced picture entails the disappearance of logical negation as a substantive notion. I the argue that it is possible to obtain an intermediate alternative, with a four-valued paraconsistent logic SF, which is compatible with a metaphysically balanced picture as Beall’s but it recovers a substantive notion of logical negation as a fundamental connective.

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.

Kripke-Completeness and Cut-elimination Theorems for Intuitionistic Paradefinite Logics With and Without Quasi-Explosion

Two intuitionistic paradefinite logics N4C and N4C+ are introduced as Gentzen-type sequent calculi. These logics are regarded as a combination of Nelson’s paraconsistent four-valued logic N4 and Wansing’s basic constructive connexive logic C. The proposed logics are also regarded as intuitionistic variants of Arieli, Avron, and Zamansky’s ideal paraconistent four-valued logic 4CC. The logic N4C has no quasi-explosion axiom that represents a relationship between conflation and paraconsistent negation, but the logic N4C+ has this axiom. The Kripke-completeness and cut-elimination theorems for N4C and N4C+ are proved.

Obtaining infinitely many degrees of inconsistency by adding a strictly paraconsistent negation to classical logic

This paper is devoted to a consequence relation combining the negation of Classical Logic ($$\mathbf {CL}$$) and a paraconsistent negation based on Graham Priest’s Logic of Paradox ($$\mathbf {LP}$$). We give a number of natural desiderata for a logic $$\mathbf {L}$$ that combines both negations. They are motivated by a particular property-theoretic perspective on paraconsistency and are all about warranting that the combining logic has the same characteristics as the combined logics, without giving up on the radically paraconsistent nature of the paraconsistent negation. We devise the logic $$\mathbf {CLP}$$ by means of an axiomatization and three equivalent semantical characterizations (a non-deterministic semantics, an infinite-valued set-theoretic semantics and an infinite-valued semantics with integer numbers as values). By showing that this logic is maximally paraconsistent, we prove that $$\mathbf {CLP}$$ is the only logic satisfying all postulated desiderata. Finally we show how the logic’s infinite-valued semantics permits defining different types of entailment relations.

SOME MODEL-THEORETIC RESULTS ON THE 3-VALUED PARACONSISTENT FIRST-ORDER LOGIC QCIORE

AbstractThe 3-valued paraconsistent logic Ciore was developed by Carnielli, Marcos and de Amo under the name LFI2, in the study of inconsistent databases from the point of view of logics of formal inconsistency (LFIs). They also considered a first-order version of Ciore called LFI2*. The logic Ciore enjoys extreme features concerning propagation and retropropagation of the consistency operator: a formula is consistent if and only if some of its subformulas is consistent. In addition, Ciore is algebraizable in the sense of Blok and Pigozzi. On the other hand, the logic LFI2* satisfies a somewhat counter-intuitive property: the universal and the existential quantifier are inter-definable by means of the paraconsistent negation, as it happens in classical first-order logic with respect to the classical negation. This feature seems to be unnatural, given that both quantifiers have the classical meaning in LFI2*, and that this logic does not satisfy the De Morgan laws with respect to its paraconsistent negation. The first goal of the present article is to introduce a first-order version of Ciore (which we call QCiore) preserving the spirit of Ciore, that is, without introducing unexpected relationships between the quantifiers. The second goal of the article is to adapt to QCiore the partial structures semantics for the first-order paraconsistent logic LPT1 introduced by Coniglio and Silvestrini, which generalizes the semantic notion of quasi-truth considered by Mikeberg, da Costa and Chuaqui. Finally, some important results of classical Model Theory are obtained for this logic, such as Robinson’s joint consistency theorem, amalgamation and interpolation. Although we focus on QCiore, this framework can be adapted to other 3-valued first-order LFIs.

SUBSTRUCTURAL INQUISITIVE LOGICS

AbstractThis paper shows that any propositional logic that extends a basic substructural logic BSL (a weak, nondistributive, nonassociative, and noncommutative version of Full Lambek logic with a paraconsistent negation) can be enriched with questions in the style of inquisitive semantics and logic. We introduce a relational semantic framework for substructural logics that enables us to define the notion of an inquisitive extension of λ, denoted as ${\lambda ^?}$, for any logic λ that is at least as strong as BSL. A general theory of these “inquisitive extensions” is worked out. In particular, it is shown how to axiomatize ${\lambda ^?}$, given the axiomatization of λ. Furthermore, the general theory is applied to some prominent logical systems in the class: classical logic Cl, intuitionistic logic Int, and t-norm based fuzzy logics, including for example Łukasiewicz fuzzy logic Ł. For the inquisitive extensions of these logics, axiomatization is provided and a suitable semantics found.

Paraconsistency, self-extensionality, modality

Abstract Paraconsistent logics are logics that, in contrast to classical and intuitionistic logic, do not trivialize inconsistent theories. In this paper we take a paraconsistent view on two famous modal logics: B and S5. We use for this a well-known general method for turning modal logics to paraconsistent logics by defining a new (paraconsistent) negation as $\neg \varphi =_{Def} \sim \Box \varphi$ (where $\sim$ is the classical negation). We show that while that makes both B and S5 members of the well-studied family of paraconsistent C-systems, they differ from most other C-systems in having the important replacement property (which means that equivalence of formulas implies their congruence). We further show that B is a very robust C-system in the sense that almost any axiom which has been considered in the context of C-systems is either already a theorem of B or its addition to B leads to a logic that is no longer paraconsistent. There is exactly one notable exception, and the result of adding this exception to B leads to the other logic studied here, S5.

Method for Combining Paraconsistency and Probability in Temporal Reasoning

Computation tree logic (CTL) is known to be one of the most useful temporal logics for verifying concurrent systems by model checking technologies. However, CTL is not sufficient for handling inconsistency-tolerant and probabilistic accounts of concurrent systems. In this paper, a paraconsistent (or inconsistency-tolerant) probabilistic computation tree logic (PpCTL) is derived from an existing probabilistic computation tree logic (pCTL) by adding a paraconsistent negation connective. A theorem for embedding PpCTL into pCTL is proven, thereby indicating that we can reuse existing pCTL-based model checking algorithms. A relative decidability theorem for PpCTL, wherein the decidability of pCTL implies that of PpCTL, is proven using this embedding theorem. Some illustrative examples involving the use of PpCTL are also presented.

A decidable paraconsistent relevant logic: Gentzen system and Routley‐Meyer semantics

In this paper, the positive fragment of the logic of contraction‐less relevant implication is extended with the addition of a paraconsistent negation connective similar to the strong negation connective in Nelson's paraconsistent four‐valued logic . This extended relevant logic is called , and it has the property of constructible falsity which is known to be a characteristic property of . A Gentzen‐type sequent calculus for is introduced, and the cut‐elimination and decidability theorems for are proved. Two extended Routley‐Meyer semantics are introduced for , and the completeness theorems with respect to these semantics are proved.

Liberating Paraconsistency from Contradiction

In this paper we propose to take seriously the claim that at least some kinds of paraconsistent negations are subcontrariety forming operators. We shall argue that from an intuitive point of view, by considering paraconsistent negations as formalizing that particular kind of opposition, one needs not worry with issues about the meaning of true contradictions and the like, given that “true contradictions” are not involved in these paraconsistent logics. Our strategy will consist in showing that, on the one hand, the natural translation for subcontrariety in formal languages is not a contradiction in natural language, and on the other, translating alleged cases of contradiction in natural language to paraconsistent formal systems works only provided we transform them into a subcontrariety. Transforming contradictions into subcontrariety shall provide for an intuitive interpretation for paraconsistent negation, which we also discuss here. By putting all those pieces together, we hope a clearer sense of paraconsistency can be made, one which may liberate us from the need to tame contradictions.

A Modality Called ‘Negation’:

I propose a comprehensive account of negation as a modal operator, vindicating a moderate logical pluralism. Negation is taken as a quantifier on worlds, restricted by an accessibility relation encoding the basic concept of compatibility. This latter captures the core meaning of the operator. While some candidate negations are then ruled out as violating plausible constraints on compatibility, different specifications of the notion of world support different logical conducts for (the admissible) negations. The approach unifies in a philosophically motivated picture the following results: nothing can be called a negation properly if it does not satisfy (Minimal) Contraposition and Double Negation Introduction; the pair consisting of two split or Galois negations encodes a distinction without a difference; some paraconsistent negations also fail to count as real negations, but others may; intuitionistic negation qualifies as real negation, and classical Boolean negation does as well, to the extent that constructivist and paraconsistent doubts on it do not turn on the basic concept of compatibility but rather on the interpretation of worlds

Inconsistency-tolerant temporal reasoning with hierarchical information

A formal method is proposed for modeling and verifying inconsistency-tolerant temporal reasoning with hierarchical information. For this purpose, temporal logic called sequential paraconsistent computation tree logic (SPCTL) is obtained from computation tree logic by adding a paraconsistent negation connective and some sequence modal operators. SPCTL can appropriately represent both inconsistency-tolerant reasoning by the paraconsistent negation connective and hierarchical information by the sequence modal operators. The validity, satisfiability, and model-checking problems of SPCTL are shown to be EXPTIME-complete, deterministic EXPTIME-complete, and deterministic PTIME-complete, respectively. Illustrative examples for inconsistency-tolerant temporal reasoning with hierarchical information are presented using the proposed SPCTL.

Paraconsistent Computation Tree Logic

It is known that paraconsistent logical systems are more appropriate for inconsistency-tolerant and uncertainty reasoning than other types of logical systems. In this paper, a paraconsistent computation tree logic, PCTL, is obtained by adding paraconsistent negation to the standard computation tree logic CTL. PCTL can be used to appropriately formalize inconsistency-tolerant temporal reasoning. A theorem for embedding PCTL into CTL is proved. The validity, satisfiability, and model-checking problems of PCTL are shown to be decidable. The embedding and decidability results indicate that we can reuse the existing CTL-based algorithms for validity, satisfiability, and model-checking. An illustrative example of medical reasoning involving the use of PCTL is presented.

Embedding-based approaches to paraconsistent and temporal description logics

In this article, some paraconsistent and temporal description logics are studied based on an embedding-based proof method. A new paraconsistent description logic, PALC, is obtained from the description logic ALC by adding a paraconsistent negation. Two temporal description logics, XALC and BALCl, are introduced by combining and modifying ALC and Prior’s tomorrow tense logic. XALC has the next-time operator, and BALCl has some restricted versions of the next-time, any-time and some-time operators, in which the time domain is bounded by a positive integer l. Some theorems for embedding PALC, XALC and BALCl into ALC are proved in a uniform way, and PALC, XALC and BALCl are shown to be decidable. Three tableau calculi for PALC, XALC and BALCl are introduced, and the completeness theorems for these calculi are proved.

A Paraconsistent Linear-time Temporal Logic

Inconsistency-tolerant reasoning and paraconsistent logic are of growing importance not only in Knowledge Representation, AI and other areas of Computer Science, but also in Philosophical Logic. In this paper, a new logic, paraconsistent linear-time temporal logic (PLTL), is obtained semantically from the linear-time temporal logic LTL by adding a paraconsistent negation. Some theorems for embedding PLTL into LTL are proved, and PLTL is shown to be decidable. A Gentzentype sequent calculus PLTω for PLTL is introduced, and the completeness and cut-elimination theorems for this calculus are proved. In addition, a display calculus δPLTω for PLTL is defined.

Priestley Duality for Paraconsistent Nelson’s Logic

The variety of $${{\bf N4}^\perp}$$ -lattices provides an algebraic semantics for the logic $${{\bf N4}^\perp}$$ , a version of Nelson's logic combining paraconsistent strong negation and explosive intuitionistic negation. In this paper we construct the Priestley duality for the category of $${{\bf N4}^\perp}$$ -lattices and their homomorphisms. The obtained duality naturally extends the Priestley duality for Nelson algebras constructed by R. Cignoli and A. Sendlewski.

Proof Systems Combining Classical and Paraconsistent Negations

New propositional and first-order paraconsistent logics (called L ω and FL ω , respectively) are introduced as Gentzen-type sequent calculi with classical and paraconsistent negations. The embedding theorems of L ω and FL ω into propositional (first-order, respectively) classical logic are shown, and the completeness theorems with respect to simple semantics for L ω and FL ω are proved. The cut-elimination theorems for L ω and FL ω are shown using both syntactical ways via the embedding theorems and semantical ways via the completeness theorems.