Self-extensional three-valued paraconsistent logics have no implication

A b s t r a c t. It is known that many-valued paraconsistent logics are useful for expressing uncertain and inconsistency-tolerant reasoning in a wide range of Computer Science. Some four-valued and sixteen-valued logics have especially been well-studied. Some four-valued logics are not so ne-grained, and some sixteen-valued logics are enough ne-grained, but rather complex. In this paper, a natural eight-valued paraconsistent logic rather than four-valued and sixteen-valued logics is introduced as a Gentzen-type sequent calculus. This eight-valued logic is enough ne-grained and simpler than sixteen-valued logic. A triplet valuation semantics is introduced for this logic, and the completeness theorem for this semantics is proved. The cut-elimination theorem for this logic is proved, and this logic is shown to be decidable.

Many-valued logics in general and 3-valued logic in particular is an old subject which had its beginning in the work of Łukasiewicz [Łuk]. Recently there is a revived interest in this topic, both for its own sake (see, for example, [Ho]), and also because of its potential applications in several areas of computer science, such as proving correctness of programs [Jo], knowledge bases [CP] and artificial intelligence [Tu]. There are, however, a huge number of 3-valued systems which logicians have studied throughout the years. The motivation behind them and their properties are not always clear, and their proof theory is frequently not well developed. This state of affairs makes both the use of 3-valued logics and doing fruitful research on them rather difficult.Our first goal in this work is, accordingly, to identify and characterize a class of 3-valued logics which might be called natural. For this we use the general framework for characterizing and investigating logics which we have developed in [Av1]. Not many 3-valued logics appear as natural within this framework, but it turns out that those that do include some of the best known ones. These include the 3-valued logics of Łukasiewicz, Kleene and Sobociński, the logic LPF used in the VDM project, the logic RM3 from the relevance family and the paraconsistent 3-valued logic of [dCA]. Our presentation provides justifications for the introduction of certain connectives in these logics which are often regarded as ad hoc. It also shows that they are all closely related to each other. It is shown, for example, that Łukasiewicz 3-valued logic and RM3 (the strongest logic in the family of relevance logics) are in a strong sense dual to each other, and that both are derivable by the same general construction from, respectively, Kleene 3-valued logic and the 3-valued paraconsistent logic.

It is known that many-valued paraconsistent logics are useful for expressing uncertain and inconsistency-tolerant reasoning in a wide range of Computer Science. Some four-valued and sixteen-valued paraconsistent logics have especially been well-studied. Some four-valued logics are not so fine-grained, and some sixteen-valued logics are enough fine-grained, but rather complex. In this paper, a natural eight-valued paraconsistent logic in between four-valued and sixteen-valued logics is introduced as a Gentzen-type sequent calculus. A triplet valuation semantics is introduced for this logic, and the completeness theorem for this semantics is proved. The cut-elimination theorem for this logic is proved, and this logic is shown to be decidable.

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

Every truth-functional three-valued propositional logic can be conservatively translated into the modal logic S5. We prove this claim constructively in two steps. First, we define a Translation Manual that converts any propositional formula of any three-valued logic into a modal formula. Second, we show that for every S5-model there is an equivalent three-valued valuation and vice versa. In general, our Translation Manual gives rise to translations that are exponentially longer than their originals. This fact raises the question whether there are three-valued logics for which there is a shorter translation into S5. The answer is affirmative: we present an elegant linear translation of the Logic of Paradox and of Strong Three-valued Logic into S5.

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.

We develop logics and translations for inconsistency-tolerant model checking that can be used to verify systems having inconsistencies. Paraconsistent linear-time temporal logic (pLTL), paraconsistent computation tree logic (pCTL), and paraconsistent full computation tree logic (pCTL\(^*\)) are introduced. These are extensions of standard linear-time temporal logic (LTL), standard computation tree logic (CTL), and standard full computation tree logic (CTL\(^*\)), respectively. These novel logics can be applied when handling inconsistency-tolerant temporal reasoning. They are also regarded as four-valued temporal logics that extend Belnap and Dunn’s four-valued logic. Translations from pLTL into LTL, pCTL into CTL, and pCTL\(^*\) into CTL\(^*\), are defined, and these are used to prove the theorems for embedding pLTL into LTL, pCTL into CTL, and pCTL\(^*\) into CTL\(^*\). These embedding theorems allow the standard LTL-, CTL-, and CTL\(^*\)-based model checking algorithms to be used for verifying inconsistent systems that are modeled and specified by pLTL, pCTL, and pCTL\(^*\). Some illustrative examples for inconsistency-tolerant model checking are presented based on the proposed logics and translations.

In the paper we explore the idea of describing Pawlak’s rough sets using three-valued logic, whereby the value t corresponds to the positive region of a set, the value f — to the negative region, and the undefined value u — to the border of the set. Due to the properties of the above regions in rough set theory, the semantics of the logic is described using a non-deterministic matrix (Nmatrix). With the strong semantics, where only the value t is treated as designated, the above logic is a “common denominator” for Kleene and Łukasiewicz 3-valued logics, which represent its two different “determinizations”. In turn, the weak semantics—where both t and u are treated as designated—represents such a “common denominator” for two major 3-valued paraconsistent logics.

Статья посвящена анализу нескольких классов паранепротиворечивых и параполных логик на примере трехзначных и четырехзначных логик, сохраняющих классические истинностные значения. Результаты, представленные в статье, можно разделить на три группы. 
 Первая группа результатов касается паранепротиворечивых логик. Показано, что все трехзначные подлинно паранепротиворечивые логики являются логиками формальной противоречивости, а все трехзначные логики формальной противоречивости, расширяющие позитивный фрагмент классической логики, являются языковыми вариантами подлинно паранепротиворечивых логик. Приводятся примеры четырехзначных подлинно паранепротиворечивых логик, которые не являются логиками формальной противоречивости. Найден ряд необходимых условий, которым должны соответствовать четырехзначные матрицы подлинно паранепротиворечивых логик, чтобы задаваемые этими матрицами логики не являлись логиками формальной противоречивости. Вторая группа результатов касается параполных логик. Показано, что все трехзначные подлинно параполные логики являются логиками формальной неопределенности, а все трехзначные логики формальной неопределенности, расширяющие дуально-позитивный фрагмент классической логики, являются языковыми вариантами подлинно параполных логик. Приводятся примеры четырехзначных подлинно параполных логик, которые не являются логиками формальной неопределенности. Найден ряд необходимых условий, которым должны соответствовать четырехзначные матрицы подлинно параполных логик, чтобы задаваемые этими матрицами логики не являлись логиками формальной неопределенности. Третья группа результатов касается паранормальных логик. Приводится ряд необходимых условий, которым должны соответствовать четырехзначные матрицы подлинно паранормальных логик, чтобы они не являлись ни логиками формальной противоречивости, ни логиками формальной неопределенности. Для паранормальных расширений Белнапа даются также необходимые и достаточные условия, для того чтобы они были максимальными подлинно паранормальными логиками и в то же время не являлись ни логиками формальной противоречивости, ни логиками формальной неопределенности.
 

There are several three-valued logical systems that form a scattered landscape, even if all reasonable connectives in three-valued logics can be derived from a few of them. Most papers on this subject neglect the issue of the relevance of such logics in relation with the intended meaning of the third truth-value. Here, we focus on the case where the third truth-value means unknown, as suggested by Kleene. Under such an understanding, we show that any truth-qualified formula in a large range of three-valued logics can be translated into KD as a modal formula of depth 1, with modalities in front of literals only, while preserving all tautologies and inference rules of the original three-valued logic. This simple information logic is a two-tiered classical propositional logic with simple semantics in terms of epistemic states understood as subsets of classical interpretations. We study in particular the translations of Kleene, Gödel, ᴌukasiewicz and Nelson logics. We show that Priest’s logic of paradox, closely connected to Kleene’s, can also be translated into our modal setting, simply by exchanging the modalities possible and necessary. Our work enables the precise expressive power of three-valued logics to be laid bare for the purpose of uncertainty management.

In Philosophical Logic, the Liar Paradox has been used to motivate the introduction of both truth value gaps and truth value gluts. Moreover, in the light of "revenge Liar" arguments, also higher-order combinations of generalized truth values have been suggested to account for so-called hyper-contradictions. In the present paper, Graham Priest's treatment of generalized truth values is scrutinized and compared with another strategy of generalizing the set of classical truth values and defining an entailment relation on the resulting sets of higher-order values. This method is based on the concept of a multilattice. If the method is applied to the set of truth values of Belnap's "useful four-valued logic", one obtains a trilattice, and, more generally, structures here called Belnap-trilattices. As in Priest's case, it is shown that the generalized truth values motivated by hyper-contradictions have no effect on the logic. Whereas Priest's construction in terms of designated truth values always results in his Logic of Paradox, the present construction in terms of truth and falsity orderings always results in First Degree Entailment. However, it is observed that applying the multilattice-approach to Priest's initial set of truth values leads to an interesting algebraic structure of a "bi-and-a-half" lattice which determines seven-valued logics different from Priest's Logic of Paradox.

This paper explores allowing truth value assignments to be undetermined or "partial" (no truth values) and overdetermined or "inconsistent" (both truth values), thus returning to an investigation of the four-valued semantics that I initiated in the sixties. I examine some natural consequence relations and show how they are related to existing logics, including Łukasiewicz's three-valued logic, Kleene's three-valued logic, Anderson and Belnap's (first-degree) relevant entailments, Priest's "Logic of Paradox", and the first-degree fragment of the Dunn-McCall system "R-mingle". None of these systems have nested implications, and I investigate twelve natural extensions containing nested implications, all of which can be viewed as coming from natural variations on Kripke's semantics for intuitionistic logic. Many of these logics exist antecedently in the literature, in particular Nelson's "constructible falsity".

The significance of three-valued logics partly depends on the interpretation of the third truth-value. When it refers to the idea of unknown, we have shown that a number of three-valued logics, especially Kleene, Łukasiewicz, and Nelson, can be encoded in a simple fragment of the modal logic KD, called MEL, containing only modal formulas without nesting. This is the logic of possibility theory, the semantics of which can be expressed in terms of all-or-nothing possibility distributions representing an agent’s epistemic state. Here we show that this formalism can also encode some three-valued paraconsistent logics, like Priest, Jaśkowski, and Sobociński’s, where the third truth-value represents the idea of contradiction. The idea is just to change the designated truth-values used for their translations. We show that all these translations into modal logic are very close in spirit to Avron’s early work expressing natural three-valued logics using hypersequents. Our work unifies a number of existing formalisms and the translation also highlights the perfect symmetry between three-valued logics of contradiction and three-valued logics of incomplete information, which corresponds to a swapping of modalities in MEL.

The aim of this paper is technically to study Belnap's four‐valued sentential logic (see [2]). First, we obtain a Gentzen‐style axiomatization of this logic that contains no structural rules while all they are still admissible in the Gentzen system what is proved with using some algebraic tools. Further, the mentioned logic is proved to be the least closure operator on the set of {Λ, V, ⌝}‐formulas satisfying Tarski's conditions for classical conjunction and disjunction together with De Morgan's laws for negation. It is also proved that Belnap's logic is the only sentential logic satisfying the above‐mentioned conditions together with Anderson‐Belnap's Variable‐Sharing Property. Finally, we obtain a finite Hilbert‐style axiomatization of this logic. As a consequence, we obtain a finite Hilbert‐style axiomatization of Priest's logic of paradox (see [12]).

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.