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3 Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

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Some Propositional Calculus Oppositional with Respect to the Intuitionistic Propositional Calculus

Bryll G., Kaptur M.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

2005

Vol. 10

17--23

A fragmentary system of the classical propositional calculus, in which the law C N N αα is valid instead of the law CαN N α, is presented.

About Two Different Ways of Characterization of Łuksiewicz's Three-valued Modal Propositional Calculus

Bryll G., Kaptur M.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

2005

Vol. 10

25-31

In this article we consider two systems of Lukasiewicz's three- valued modal propositional calculus. One of them is the system based on such primary terms as the disjunction (A), negation (N) and necessity (L), whereas the second is based on such primary terms as the implication (C), negation (N) and definitively improved by modal necessity terms. The both systems are definitively equivalent.

The Proof of Certain Tarski's Theorem about the Existence of One-element Base for Axiomatizable Systems of Propositional Calculus

Kaptur M.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

2003

Vol. 9

51--59

In [1] the following theorem relating the existence of one-element base for spacious class of axiomatizable propositional calculus has been given: Theorem 1. System L, as well as each axiomatizable system propositional calculus, contains sentences "CpCqp" and „CpCqCCpCqrr" (or "CpCqCCpCqrCsr"), possesses the base consisting of only one sentence 1. In Postscript added to the English translation of publication [1] 2 the outline of proof of the above theorem, found by R. McKenzie, has been given. Author of the article advises to give the full proof of Theorem 1, because the outline contained in Postscript does not contain essential reasonings for the proof.