
Authors:
Szabolcs MIKULAS
Title:Gabbaystyle calculi
Pages:5060
File:bibtex
Abstract ( + )
It is well known that there are logics, e.g., classical firstorder logic with n variables, L_{n}, that do not have strongly sound and complete Hilbertstyle inference systems. However, some of the logics of the above kind have weakly sound and complete Gabbaystyle inference systems. In this paper we give complete and sound Gabbaystyle inference systems for some logics, e.g., for L_{n} with and without equality, by applying representation theorems for the corresponding class of algebras.

Authors:
Jacek K.KABZINSKI
Title:On some equivalentially indistinguishable consequences
Pages:6165
File:bibtex
Abstract ( + )
I came across different implicational consequences with the identically, equivalential reducts, i.e. equivalentially indistinguishable consequences, on the occasion of studying the equivalential fragment of the threevalued logic of Lukasiewicz. It appeared that this fragment is coincides with the equivalential fragment of the well known threevalued logic of Heyting. The concept of equivalential indistinguishability was already mentioned. There I also presented some new examples of equivalentially indistinguihable consequences. The present paper provides another example of equivalentially indistinguishable consequences; these are the below stated implicational consequences C_{i} and C_{+i}.

Authors:
Jacek K.KABZINSKI
Title:Abelian groups and identity connective
Pages:6671
File:bibtex
Abstract ( + )
In 1937 M.H.Stone showed that natural semantics for one of the most important classical consequences  classical equivalential consequence  is two element Abelian group, more generally, the class of Abelian groups of the second order, i.e. groups wherein the converse element for a given element is the same element (given element is a converse element for itself). Due to Stone's results the wellknown criterium of Lesniewski concerning the classical equivalential calculus has got highly sound justification. This paper finds a consequence for which Abelian groups constitue a semantics in a similar way.

Authors:
Adam KOLANY
Title:On the logic of lie
Pages:7279
File:bibtex
Abstract ( + )
B.Majcher introduces an interpretation for his language, obtaining in this way a logic which will be denoted as L and called the Logic of Lie here. In this paper, we give an axiomatization of L and show its representation in the Classical Propositional Calculus.

Authors:
Olga AMBAS
Title:On Lukasiewicz and Schmitt threevalued logics
Pages:8084
File:bibtex
Abstract ( + )
Schmitt introduced a resolution method for the threevalued logic L_{3}. Our aim in this paper is to show that this method is in fact a resolution method for threevalued Lukasiewicz logic L_{3}.

Authors:
Burghard HERRMANN
Title:Algebraizability and Beth's theorem for equivalential logics
Pages:8588
File:bibtex
Abstract ( + )
We provide a new approach to algebraizable logics in the sense of Blok and Pigozzi. These are finitely equivalential logics with equationally definable truth. Theorem 1 states that the truth predicate in the reduced matrices for an equivalential logic is implicitly definable iff it is explicitly definable by a conjunction of equations. This is an improved instance of Beth's Theorem for L_{ω1ω}. We obtain a finite conjunction of equations even if the reduced matrices form a proper L_{ω1ω} class. We easily obtain the main results on algebraizable logics. Furthermore, a logic is algebraizable iff the restricted Leibniz operator is injective and orderpreserving and a certain theory is finitely based. The third condition was missing in a conjecture of Blok and Pigozzi. All results can be straightforward generalized to nonfinitary structural consequence relations.