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A natural deduction and its corresponding sequent calculus for positive contraction-less relevant logic

100%

Ilić M.

Reports on Mathematical Logic

2017

tom Vol. 52

101--132

We give a normalizing system of natural deduction for positive contraction-less relevant logic RW+⁰ . The specific characteristic of our calculus is that it has a simple translational relationship to a particular sequent calculus for RW+⁰, such that normal natural deduction derivations correspond to cut-free sequent calculus derivations and vice versa. By translations from natural deduction to sequent calculus derivations, and back, together with cut-elimination, we obtain an indirect proof of the normalization.

Two Infinite Sequences of Pre-Maximal Extensions of the Relevant Logic E

80%

Typańska-Czajka L.

Bulletin of the Section of Logic

2019

tom 48

nr 1

The only maximal extension of the logic of relevant entailment E is the classical logic CL. A logic L ⊆ [E,CL] called pre-maximal if and only if L is a coatom in the interval [E,CL]. We present two denumerable infinite sequences of premaximal extensions of the logic E. Note that for the relevant logic R there exist exactly three pre-maximal logics, i.e. coatoms in the interval [R,CL].

Variable Sharing in Substructural Logics: an Algebraic Characterization

80%

Badia G.

Bulletin of the Section of Logic

2018

tom 47

nr 2

We characterize the non-trivial substructural logics having the variable sharing property as well as its strong version. To this end, we find the algebraic counterparts over varieties of these logical properties.

Relevant generalization starts here (and here = 2)

80%

Zaitsev D. , Grigoriev O.

Logic and Logical Philosophy

2010

tom 19

nr 4

329–340

There is a productive and suggestive approach in philosophical logic based on the idea of generalized truth values. This idea, which stems essentially from the pioneering works by J.M. Dunn, N. Belnap, and which has recently been developed further by Y. Shramko and H. Wansing, is closely connected to the power-setting formation on the base of some initial truth values. Having a set of generalized truth values, one can introduce fundamental logical notions, more specifically, the ones of logical operations and logical entailment. This can be done in two different ways. According to the first one, advanced by M. Dunn, N. Belnap, Y. Shramko and H. Wansing, one defines on the given set of generalized truth values a specific ordering relation (or even several such relations) called the logical order(s), and then interprets logical connectives as well as the entailment relation(s) via this ordering(s). In particular, the negation connective is determined then by the inversion of the logical order. But there is also another method grounded on the notion of a quasi-field of sets, considered by Białynicki-Birula and Rasiowa. The key point of this approach consists in defining an operation of quasi-complement via the very specific function g and then interpreting entailment just through the relation of set-inclusion between generalized truth values. In this paper, we will give a constructive proof of the claim that, for any finite set V with cardinality greater or equal 2, there exists a representation of a quasi-field of sets isomorphic to de Morgan lattice. In particular, it means that we offer a special procedure, which allows to make our negation de Morgan and our logic relevant.

CZYM JEST PLURALIZM LOGICZNY? (STANOWISKO J.C. BEALLA I GREGA RESTALLA)

41%

Czernecka-Rej B.

Roczniki Filozoficzne

2013

tom 61

nr 1

5-22

C. Beall and Greg Restall are advocates of a comprehensive pluralist approach to logic, which they call Logical Pluralism (LP). According to LP, there is not one correct logic, but many equally acceptable logical systems. The authors share Tarski’s conviction and follow the mainstream in thinking about logic as the discipline that investigates the notion of logical consequence. LP is the pluralism about logical consequence – a pluralist maintains that there is more than one relation of logical consequence. According to LP, classical, intuitionistic and relevant logics are not rivals, but they all are equally correct, they all count as genuine logics. The purpose of this paper is to present some remarks concerning J.C. Beall’s and Greg Restall’s exposition of LP. At the beginning, the definition of the relation of logical consequence, which is central to their proposal, is shown. According to Beall and Restall, argument is valid if, and only if, in every case when the premisses are true, then the conclusion is, too. They argue that by considering different types of cases the logical pluralist obtains different logics. The paper — apart from presenting LP — also gives a critical discussion of this approach. It seems, that the thesis of LP is far from being clear. It is even unclear what exactly LP is and where is stops. It is unclear what “equally good”, “equally correct”, “equally true” mean. It is not clear, how to explain, in scope of logic, that the system of logic, is a model of real logical connections.