
Authors:
Antonio M. SETTE and Elias H. ALVES
Title:On the equivalence between some systems of nonclassical logic
Pages:6872
File:bibtex
Abstract ( + )
Loparic and da Costa define three systems of propositional logic, called β_{0}, β_{1} and β_{2}. In β_{0} neither the principle of excluded middle nor the principle of noncontradiction is valid, in general. System β_{1} is an extension of β_{0}, where the principle of noncontradiction is valid, but the principle of excludedmiddle is not. System β_{2} is also an extension of β_{0}, where the principle of excluded middle is valid, but the principle of noncontradiction is not. Systems such as β_{2} are called paraconsistent systems. Systems such as β_{1} are called by Loparic and da Costa paracomplete systems. Loparic and da Costa mentioned that system β_{2} is equivalent to system P1, introduced by Sette. On the other hand, Sette and Carnielli study a system, called I1, which is, according to them, weaklyintuitionistic, that is, where the law of excluded middle cannot be proved. (This corresponds to the notion of paracompleteness of Loparic and da Costa.) According to Sette and Carnielli, system I1 is a counterpart of the paraconsistent calculus P1. We will show, here, that system β_{2} is, in fact, equivalent to P1. In addition, we will show that β_{1} is equivalent to I1.

Authors:
Roma J. ADILLON and Ventura VERDU
Title:A Gentzen system equivalent to the BCKlogic
Pages:7380
File:bibtex
Abstract ( + )
The sequent calculus L_{BCK} is obtained by deleting the contraction rule and the introduction rules of the connectives of meet, join and negation from the sequent calculus for the Intuitionistic Propositional logic, LJ. In this paper we show that the Gentzen system G_{BCK}, naturally associated with L_{BCK}, is equivalent to the Hilbertstyle logic BCK, in the sense that G_{BCK} and BCK are interpretable one another, the interpretations being essentially inverse in each other. This result is a strengthening of the one obtained by H. Ono and Y. Komori, which concerned only the derivable sequents of L_{BCK} and the theorems of BCK. We obtain, as a corollary of this result, that the quasivariety of BCKalgebras is the equivalent quasivariety semantics of G_{BCK}.

Authors:
Piotr LUKOWSKI
Title:Modal interpretation of HeytingBrouwer logic
Pages:8083
File:bibtex
Abstract ( + )
Long before the presentations of semantics for intuitionistic logic and modal system S4, Goedel as the first interpreted the intuitionism in S4. The intuitionistic logic is however "only one half" of the HeytingBrouwer logic. This paper introduces the enlarged translation interpreting HB logic in some modal system for which, S4 is "one half", only.

Authors:
Alexander YASHIN
Title:On Novikov's approach to the notion of a new intuitionistic connective: two negative examples
Pages:8488
File:bibtex
Abstract ( + )
About 35 years ago P. Novikov suggested the socalled syntactic approach to the notion of a new intuitionistic logical connective. In this paper we shall consider two examples of additional connectives, which were intensively studied but, however, are useless in the sense of P.Novikov's definition.

Authors:
JeanYves BEZIAU
Title:Identity, structure and logic
Pages:8994
File:bibtex
Abstract ( + )
The concept of identity is a central idea of thought. We will see how modern mathematics can give a precise analysis of this concept and how logic can deal with this analysis. We will define three kinds of identity: the Bourbaki identity, the logical identity and the diagonal identity (in short B, l, didentity respectively) and study the connections between them.

Authors:
Andrzej INDRZEJCZAK
Title:Cutfree sequent calculus for S5
Pages:95102
File:bibtex
Abstract ( + )
We aim at an exposition of some nonstandard cutfree Gentzen formalization for S5, called DSC (double sequent calculus). DSC operates on two types of sequents instead of one, and shifting of wffs from one side of sequent to another is regulated by special rules and subject to some restrictions. Despite of this apparent inconvenience it seems to be simpler than other, known Gentzenstyle systems for S5. The number of additional, formal machinery is kept in reasonable bounds and proofs are quite simple. A byproduct of the completeness theorem for this calculus is an automated proofsearch procedure.

Authors:
Janusz CZELAKOWSKI
Title:Filtered subdirect products
Pages:103
File:bibtex
Abstract ( + )
The operation P_{f}^{S} of forming filtered subdirect products is examined. It is shown that for any class K of models, P_{f}^{S} is the least quasivariety which contains K.