Three-valued Paraconsistent Logics Research Articles

Consistency and Decidability in Some Paraconsistent Arithmetics


 
 
 The standard style of argument used to prove that a theory is unde- cidable relies on certain consistency assumptions, usually that some fragment or other is negation consistent. In a non-paraconsistent set- ting, this amounts to an assumption that the theory is non-trivial, but these diverge when theories are couched in paraconsistent logics. Furthermore, there are general methods for constructing inconsistent models of arithmetic from consistent models, and the theories of such inconsistent models seem likely to differ in terms of complexity. In this paper, I begin to explore this terrain, working, particularly, in incon- sistent theories of arithmetic couched in three-valued paraconsistent logics which have strong (i.e. detaching) conditionals. 
 
 

Axiomatisations of the Genuine Three-Valued Paraconsistent Logics $$\mathbf {L3A_G}$$ and $$\mathbf {L3B_G}$$

Genuine Paraconsistent logics $$\mathbf {L3A}$$ and $$\mathbf {L3B}$$ were defined in 2016 by Beziau et al, including only three logical connectives, namely, negation disjunction and conjunction. Afterwards in 2017 Hernandez-Tello et al, provide implications for both logics and define the logics $$\mathbf {L3A_G}$$ and $$\mathbf {L3B_G}$$ . In this work we continue the study of these logics, providing sound and complete Hilbert-type axiomatic systems for each logic. We prove among other properties that $$\mathbf {L3A_G}$$ and $$\mathbf {L3B_G}$$ satisfy a restricted version of the Substitution Theorem, and that both of them are maximal with respect to Classical Propositional Logic. To conclude we make some comparisons between $$\mathbf {L3A_G}$$ and $$\mathbf {L3B_G}$$ and among other logics, for instance $${\mathbf {Int}}$$ and some $${\mathbf {LFI}}$$ s.

On the strongest three-valued paraconsistent logic contained in classical logic and its dual

Abstract $\textrm{LP}^{\mathbin{\supset },{\mathsf{F}}}$ is a three-valued paraconsistent propositional logic that is essentially the same as J3. It has the most properties that have been proposed as desirable properties of a reasonable paraconsistent propositional logic. However, it follows easily from already published results that there are exactly 8192 different three-valued paraconsistent propositional logics that have the properties concerned. In this paper, properties concerning the logical equivalence relation of a logic are used to distinguish $\textrm{LP}^{\mathbin{\supset },{\mathsf{F}}}$ from the others. As one of the bonuses of focusing on the logical equivalence relation, it is found that only 32 of the 8192 logics have a logical equivalence relation that satisfies the identity, annihilation, idempotent and commutative laws for conjunction and disjunction. For most properties of $\textrm{LP}^{\mathbin{\supset },{\mathsf{F}}}$ that have been proposed as desirable properties of a reasonable paraconsistent propositional logic, its paracomplete analogue has a comparable property. In this paper, properties concerning the logical equivalence relation of a logic are also used to distinguish the paracomplete analogue of $\textrm{LP}^{\mathbin{\supset },{\mathsf{F}}}$ from the other three-valued paracomplete propositional logics with those comparable properties.

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.

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

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.

N-valued maximal paraconsistent matrices

ABSTRACTThe articles Maximality and Refutability Skura [(2004). Maximality and refutability. Notre Dame Journal of Formal Logic, 45, 65–72] and Three-valued Maximal Paraconsistent Logics Skura and Tuziak [(2005). Three-valued maximal paraconsistent logics. In Logika (Vol. 23). Wydawnictwo Uniwersytetu Wrocławskiego] introduced a simple method of proving maximality (in the two distinguished senses) of a given paraconsistent matrix. This method stemmed from the so-called refutation calculus, where the focus in on rejecting rather than accepting formulas. The article A Generalisation of a Refutation-related Method in Paraconsistent Logics Trybus [(2018). A generalisation of a refutation-related method in paraconsistent logics. Logic and Logical Philosophy, 27(2). doi:10.12775/LLP.2018.002] was a first step towards generalising the method. In it, a number of 3-valued paraconsistent matrices were shown maximal. In this article we extend these results to cover a number of n-valued (n>2) paraconsistent matrices using the same method.

Deduction Normalization Theorem for Sette’s Logic and Its Modifications

In this paper we formulate natural deduction systems for Sette’s three-valued paraconsistent logic P1 and some related logics. For presented calculi we prove the soundness, completeness, and normalization theorems.

A Generalisation of a Refutation-related Method in Paraconsistent Logics

This article describes a refutation method of proving maximality of three-valued paraconsistent logics. After outlining the philosophical background related to paraconsistent logics and the refutation approach to modern logic, we briefly describe how these two areas meet in the case of maximal paraconsistent logics. We focus on a method of proving maximality introduced in [34] and [37] that has the benefit of being simple and effective. We show how the method works on a number of examples, thus emphasising the fact that it provides a unifying approach to the search for maximal paraconsistent logics. Finally, we show how the method can be generalised to cover a wide range of paraconsistent logics. We also conduct a small experimental setting that confirms the theoretical results.

Self-Extensional Three-Valued Paraconsistent Logics

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, there is exactly one self-extensional three-valued paraconsistent logic in the language of \(\{\lnot ,\wedge ,\vee \}\) for which \(\vee \) is a disjunction, and \(\wedge \) is a conjunction. We also investigate the main properties of this logic, determine the expressive power of its language (in the three-valued context), and provide a cut-free Gentzen-type proof system for it.

Self-extensional three-valued paraconsistent logics have no implication

Self-extensional three-valued paraconsistent logics have no implication

Maximal and Premaximal Paraconsistency in the Framework of Three-Valued Semantics

Maximality is a desirable property of paraconsistent logics, motivated by the aspiration to tolerate inconsistencies, but at the same time retain from classical logic as much as possible. In this paper we introduce the strongest possible notion of maximal paraconsistency, and investigate it in the context of logics that are based on deterministic or non-deterministic three-valued matrices. We show that all reasonable paraconsistent logics based on three-valued deterministic matrices are maximal in our strong sense. This applies to practically all three-valued paraconsistent logics that have been considered in the literature, including a large family of logics which were developed by da Costa's school. Then we show that in contrast, paraconsistent logics based on three-valued properly nondeterministic matrices are not maximal, except for a few special cases (which are fully characterized). However, these non-deterministic matrices are useful for representing in a clear and concise way the vast variety of the (deterministic) three-valued maximally paraconsistent matrices. The corresponding weaker notion of maximality, called premaximal paraconsistency, captures the "core" of maximal paraconsistency of all possible paraconsistent determinizations of a non-deterministic matrix, thus representing what is really essential for their maximal paraconsistency.

Jaśkowski's criterion and three-valued paraconsistent logics

A survey is given of three-valued paraconsistent propositional logics connected with Jaśkowski’s criterion for constructing paraconsistent logics. Several problems are raised and four new matrix three-valued paraconsistent logics are suggested.

Extensions of Hałkowska-Zajac's three-valued paraconsistent logic

As it was proved in [4, Sect. 3], the poset of extensions of the propositional logic defined by a class of logical matrices with equationally-definable set of distinguished values is a retract, under a Galois connection, of the poset of subprevarieties of the prevariety generated by the class of the underlying algebras of the defining matrices. In the present paper we apply this general result to the three-valued paraconsistent logic proposed by Halkowska–Zajac [2]. Studying corresponding prevarieties, we prove that extensions of the logic involved form a four-element chain, the only proper consistent extensions being the least non-paraconsistent extension of it and the classical logic. RID=""ID="" Mathematics Subject Classification (2000): 03B50, 03B53, 03G10 RID=""ID="" Key words or phrases: Many-valued logic – Paraconsistent logic – Extension – Prevariety – Distributive lattice