Yaroslav Petrukhin

University of Lodz · Center for Philosophy of Nature

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...

In this paper, we are going to combine two trends: non-deterministic logic and bi-facial logic of generalized truth values. Non-deterministic matrices allowed Avron, Ben-Naim, and Konikowska introduce a modification to Belnap and Dunn’s logic which may deal with the situation when the sources of information may give it about not only atomic formula...

In this paper, we introduce the notions of connexive and bi-intuitionistic multilattices and develop on their base the algebraic semantics for Kamide, Shramko, and Wansing’s connexive and bi-intuitionistic multilattice logics which were previously known in the form of sequent calculi and Kripke semantics. We prove that these logics are sound and co...

Jaśkowski's discussive (discursive) logic D2 is historically one of the first para-consistent logics, i.e., logics that '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 tradi...

We present a uniform characterisation of three-valued logics by means of bisequent calculus (BSC). It is a generalised form of sequent calculus (SC) where rules operate on the ordered pairs of ordinary sequents. BSC may be treated as the weakest kind of system in the rich family of generalised SC operating on items being some collections of ordinar...

In the paper we analyse the problem of axiomatizing the minimal variant of discussive logic denoted as D0. Our aim is to give its axiomatization that would correspond to a known axiomatization of the original discussive logic D2. The considered system is minimal in a class of discussive logics. It is defined similarly, as Jaśkowski’s logic D2 but w...

In this paper, we introduce provability multilattice logic PMLn and multilattice arithmetic MPAn which extends first-order multilattice logic with equality by multilattice versions of Peano axioms. We show that PMLn has the provability interpretation with respect to MPAn and prove the arithmetic completeness theorem for it. We formulate PMLn in the...

In this paper, we introduce two new logics which are combinations of Nelson's paraconsistent logic N4 of constructive falsity with Visser's basic and formal propositional logics BPL and FPL. BPL and FPL can be embedded by Gödel's translation to modal logic K4 and provability logic GL, respectively. They have the disjunction property and FPL can be...

The paper's novelty is in combining two comparatively new fields of research: non-transitive logic and the proof method of correspondence analysis. To be more detailed, in this paper the latter is adapted to Weir's non-transitive trivalent logic NC3. As a result, for each binary extension of NC3, we present a sound and complete Lemmon-style natural...

We consider certain infectious logics (Sfde, dSfde, K3w, and PWK) and several their non-infectious modifications, including two new logics, reformulate previously constructed natural deduction systems for them (or present such systems from scratch for the case of new logics) in way such that the proof of normalisation theorem becomes possible for t...

In the paper, we introduce multilattice versions of the basic congruent and monotonic modal logics. In the case of congruent and monotonic ones, we also study their extensions by Gödel's rule. We formulate these logics in the form of sequent calculi and prove syntactic embedding theorems (as a consequence we obtain cut admissibility and decidabilit...

The aim of the paper is to present some non-standard modalities (such as non-contingency, contingency, essence and accident) based on S5-models in a framework of cut-free hypersequent calculi. We also study negated modalities, i.e. negated necessity and negated possibility, which produce paraconsistent and paracomplete negations respectively. As a...

In this paper, we consider a set of quite interesting three-and four-valued logics and prove the normalisation theorem for their natural deduction formulations. Among the logics in question are the Logic of Paradox, First Degree Entailment, Strong Kleene logic, and some of their implicative extensions, including RM3 and RM3⊃. Also, we present a det...

In this paper, we introduce a new four-valued logic which may be viewed as a variation on the theme of Kubyshkina and Zaitsev's Logic of Rational Agent LRA [16]. We call our logic LIRA (Logic of Internal Rational Agency). In contrast to LRA, it has three designated values instead of one and a different interpretation of truth values, the same as in...

In the paper, we apply Kooi and Tamminga’s correspondence analysis (that has been previously applied to some notable three- and four-valued logics) to some conventional and functionally incomplete fragments of classical propositional logic. In particular, the paper deals with the implication, disjunction, and negation fragments. Additionally, we co...

In this paper, we consider modal multilattices with Tarski, Ku-ratowski, and Halmos closure and interior operators as well as the corresponding logics which are multilattice versions of the modal logics MNT4, S4, and S5, respectively. The former modal multilattice logic is a new one. The latter two modal multilattice logics have been already mentio...

In this paper, we present correspondence analysis for the well-known four-valued logic First Degree Entailment (FDE). Correspondence analysis is Kooi and Tamminga’s technique for finding adequate natural deduction systems for all the truth-functional unary and binary extensions of an arbitrary functionally incomplete many-valued logic. In particula...

In this paper, we study logical systems which represent entailment relations of two kinds. We extend the approach of finding 'exactly true' and 'non-falsity' versions of four-valued logics that emerged in series of recent works [Pietz & Rivieccio (2013). Nothing but the truth. Journal of Philosophical Logic, 42(1), 125-135; Shramko (2019). Dual-Bel...

In this paper, we consider non-associative generalisations of Hájek's logics BL and psBL. As it was shown by Cignoli, Esteva, Godo, and Torrens, the former is the logic of continuous t-norms and their residua. Botur introduced logic naBL which is the logic of non-associative continuous t-norms and their residua. Thus, naBL can be viewed as a non-as...

Modal logics \(\mathsf{K45}\), \(\mathsf{KB4}\), \(\mathsf{KD45}\) and \(\mathsf{S5}\) are of particular interest in knowledge representation, especially in the context of knowledge and belief modelling. Pietruszczak showed that these logics are curious for another reason, namely for the fact that their Kripke-style semantics can be simplified. A s...

The goal of this paper is to propose correspondence analysis as a technique for generating the so-called erotetic (i.e. pertaining to the logic of questions) calculi which constitute the method of Socratic proofs by Andrzej Wi´sniewskiWi´sniewski. As we explain in the paper, in order to successfully design an erotetic calculus one needs invertible...

Shramko [(2016). Truth, falsehood, information and beyond: The American plan generalized. In K. Bimbo (Ed.), J. Michael Dunn on information based logics, outstanding contributions to logic (pp. 191–212). Dordrecht: Springer] formulated multilattice logic and the algebraic completeness theorem for it. However, the proof has not been presented. In th...

In this paper, we present a logic MMLS5n which is a combination of multilattice logic and modal logic S5. MMLS5n is an extension of Kamide and Shramko’s modal multilattice logic which is a multilattice analogue of S4. We present a cut-free hypersequent calculus for MMLS5n in the spirit of Restall’s one for S5 and develop a Kripke semantics for MMLS...

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 obt...

Pietruszczak (Bull Sect Log 38(3/4):163-171, 2009. https://doi.org/10.12775/ LLP.2009.013) proved that the normal logics K45, KB4 (= KB5), KD45 are determined by suitable classes of simplified Kripke frames of the form W, A, where A ⊆ W. In this paper, we extend this result. Firstly, we show that a modal logic is determined by a class composed of s...

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.

In this paper, we combine the concept of natural deduction and the concept of three-valued natural logic. In particular, we use a semantic definition of the concept of natural logic presented by N. Tomova. By using the correspondence analysis given by B. Kooi and A. Tamminga, we present a syntactical counterpart of the semantic definition in questi...

The paper deals with a problem posed by Mathieu Vidal to provide a formal representation for defective conditional in mathematics Vidal, M. [(2014). The defective conditional in mathematics. Journal of Applied Non-Classical Logics, 24(1–2), 169–179]. The key feature of defective conditional is that its truth-value is indeterminate if its antecedent...

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 ha...

N.D. Belnap formulated a relevant logic called FDE (First Degree Entailment) which avoids so-called paradoxes of classical entailment: “any proposition follows from a contradiction” and “a tautology follows from any proposition”. FDE deals with formulas which have an implication as the main connective and its antecedent as well as consequent that...

Using the method of correspondence analysis, Tamminga obtains sound and complete natural deduction systems for all the unary and binary truth-functional extensions of Kleene’s strong three-valued logic K3 . In this paper, we extend Tamminga’s result by presenting an original finite, sound and complete proof-searching technique for all the truth-fun...

In this paper, we introduce the notion of dual Post’s negation and an infinite class of Dual Post’s finitely-valued logics which differ from Post’s ones with respect to the definitions of negation and the sets of designated truth values. We present adequate natural deduction systems for all Post’s k-valued (k⩾3\documentclass[12pt]{minimal} \usepack...

The natural deduction systems for the three-valued nonsense logics Z and E are presented in the paper.

In this paper, we present sound and complete natural deduction systems for Fitting’s four-valued generalizations of Kleene’s three-valued regular logics.

In this paper, we examine Kubyshkina & Zaitsev's Logic of Rational Agent (LRA) from a proof-theoretic point of view. We present three natural deduction systems for LRA which differ from Kubyshkina & Zaitsev's axiomatization of LRA. Moreover, we introduce a general method for axiomatizing LRA's unary and binary truth-functional extensions via natura...

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...

A Gentzen-style natural deduction system for the propositional fragment of three-valued Heyting’s logic is presented.

The development of recursion theory motivated Kleene to create regular three-valued logics. Taking his inspiration from the computerscience, Fitting later continued to investigate regular three-valued logicsand defined them as monotonic ones. Afterwards, Komendantskaya provedthat there are four regular three-valued logics and in the three-valued ca...

IIn this paper, I consider a family of three-valued regular logics: the well-known strong and weak S.C. Kleene’s logics and two intermediate logics, where one was discovered by M. Fitting and the other one by E. Komendantskaya. All these systems were originally presented in the semantical way and based on the theory of recursion. However, the proof...

In this paper natural deduction systems for four-valued logic FDE (first degree entailment) and its extensions are constructed. At that B. Kooi and A. Tamminga’s method of correspondence analysis is used. All possible four-valued unary (⋆) and binary (o) propositional connectives which could be added to FDE are considered. Then FDE is extended by B...

In this paper natural deduction systems for four-valued logic $FDE$ (first degree entailment) and its extensions are constructed. At that B. Kooi and A. Tamminga’s method of correspondence analysis is used. All possible four-valued unary $\star$ and binary $\circ $ propositional connectives which could be added to $FDE$ are considered. Then $FDE$ i...