We consider the logics determined by the set of all natural implicative expansions of Kleene’s strong 3-valued matrix and select the class of all logics functionally equivalent to Łukasiewicz’s 3-valued logic Ł3. The concept of a “natural implicative matrix” is based upon the notion of a “natural conditional” defined in Tomova.

Equivalent overdetermined and underdetermined bivalent Belnap–Dunn type semantics for the logics determined by all natural implicative expansions of Kleene’s strong 3-valued matrix with only one designated value are provided.

Let MK3 I and MK3 II be Kleene's strong 3-valued matrix with only one and two designated values, respectively. Next, let MK3 G be defined exactly as MK3 I, except th...

ABSTRACTA conditional is natural if it fulfils the three following conditions. It coincides with the classical conditional when restricted to the classical values T and F; it satisfies the Modus Ponens; and it is assigned a designated value whenever the value assigned to its antecedent is less than or equal to the value assigned to its consequent. The aim of this paper is to provide a ‘bivalent’ Belnap-Dunn semantics for all natural implicative expansions of Kleene's strong 3-valued matrix with two (...) designated elements. (shrink)

This paper is a sequel to ‘Belnap-Dunn semantics for natural implicative expansions of Kleene's strong three-valued matrix with two designated values’, where a ‘bivalent’ Belnap-Dunn semantics is provided for all the expansions referred to in its title. The aim of the present paper is to carry out a parallel investigation for all natural implicative expansions of Kleene's strong 3-valued matrix now with only one designated value.

The present paper is a sequel to Robles et al. :349–374, 2020. https://doi.org/10.1007/s10849-019-09306-2). A class of implicative expansions of Kleene’s 3-valued logic functionally including Łukasiewicz’s logic Ł3 is defined. Several properties of this class and/or some of its subclasses are investigated. Properties contemplated include functional completeness for the 3-element set of truth-values, presence of natural conditionals, variable-sharing property and vsp-related properties.

Correspondence analysis is Kooi and Tamminga’s universal approach which generates in one go sound and complete natural deduction systems with independent inference rules for tabular extensions of many-valued functionally incomplete logics. Originally, this method was applied to Asenjo–Priest’s paraconsistent logic of paradox LP. As a result, one has natural deduction systems for all the logics obtainable from the basic three-valued connectives of LP -language) by the addition of unary and binary connectives. Tamminga has also applied this technique to the paracomplete (...) analogue of LP, strong Kleene logic \. In this paper, we generalize these results for the negative fragments of LP and \, respectively. Thus, the method of correspondence analysis works for the logics which have the same negations as LP or \, but either have different conjunctions or disjunctions or even don’t have them as well at all. Besides, we show that correspondence analyses for the negative fragments of \ and LP, respectively, are also suitable without any changes for the negative fragments of Heyting’s logic \ and its dual \ and LP). (shrink)

B. Kooi and A. Tamminga present a correspondence analysis for extensions of G. Priest’s logic of paradox. Each unary or binary extension is characterizable by a special operator and analyzable via a sound and complete natural deduction system. The present paper develops a sound and complete proof searching technique for the binary extensions of the logic of paradox.

The aim of this paper is to define the logical system Sm4 characterised by the degree of truth-preserving consequence relation defined on the ordered set of values of Smiley’s four-element matrix MSm4. The matrix MSm4 has been of considerable importance in the development of relevant logics and it is at the origin of bilattice logics. It will be shown that Sm4 is a most interesting paraconsistent logic which encloses a sound theory of logical necessity similar to that of Anderson and (...) Belnap’s logic of entailment E. Intuitively, Sm4 can be described as a four-valued expansion of the positive fragment of Lewis’ S5 or, alternatively, as a four-valued version of S5. (shrink)

The goal of this paper is to propose correspondence analysis as a technique for generating the so-called erotetic calculi which constitute the method of Socratic proofs by Andrzej Wiśniewski. As we explain in the paper, in order to successfully design an erotetic calculus one needs invertible sequent-calculus-style rules. For this reason, the proposed correspondence analysis resulting in invertible rules can constitute a new foundation for the method of Socratic proofs. Correspondence analysis is Kooi and Tamminga's technique for designing proof systems. (...) In this paper it is used to consider sequent calculi with non-branching, invertible rules for the negation fragment of classical propositional logic and its extensions by binary Boolean functions. (shrink)

Kooi and Tamminga's correspondence analysis is a technique for designing proof systems, mostly, natural deduction and sequent systems. In this paper it is used to generate sequent calculi with invertible rules, whose only branching rule is the rule of cut. The calculi pertain to classical propositional logic and any of its fragments that may be obtained from adding a set of rules characterizing a two-argument Boolean function to the negation fragment of classical propositional logic. The properties of soundness and completeness (...) of the calculi are demonstrated. The proof of completeness is conducted by Kalmár's method. Most of the presented sequent-calculus rules have been obtained automatically, by a rule-generating algorithm implemented in Python. Correctness of the algorithm is demonstrated. This automated approach allowed us to analyse thousands of possible rules' schemes, hundreds of rules corresponding to Boolean functions, and to nd dozens of those invertible. Interestingly, the analysis revealed that the presented proof-theoretic framework provides a syntactic characteristics of such an important semantic property as functional completeness. (shrink)

In the present paper, various features of the class of propositional literal paralogics are considered. Literal paralogics are logics in which the paraproperties such as paraconsistence, paracompleteness and paranormality, occur only at the level of literals; that is, formulas that are propositional letters or their iterated negations. We begin by analyzing Bochvar’s three-valued nonsense logic B3, which includes two isomorphs of the propositional classical logic CPC. The combination of these two ‘strong’ isomorphs leads to the construction of two famous paralogics (...) P1 and I1, which are functionally equivalent. Moreover, each of these logics is functionally equivalent to the fragment of logic B3 consisting of external formulas only. In conclusion, we structure a four-element lattice of three-valued paralogics with respect to the possession of paraproperties. (shrink)

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. (shrink)

How to say no less, no more about conditional than what is needed? From a logical analysis of necessary and sufficient conditions (Section 1), we argue that a stronger account of conditional can be obtained in two steps: firstly, by reminding its historical roots inside modal logic and set-theory (Section 2); secondly, by revising the meaning of logical values, thereby getting rid of the paradoxes of material implication whilst showing the bivalent roots of conditional as a speech-act based on affirmations (...) and rejections (Section 3). Finally, the two main inference rules for conditional, viz. Modus Ponens and Modus Tollens, are reassessed through a broader definition of logical consequence that encompasses both a normal relation of truth propagation and a weaker relation of falsity non-propagation from premises to conclusion (Section 3). (shrink)