Three-element nonfinitely axiomatizable matrices

There are exactly two nonfinitely axiomatizable algebraic matrices with one binary connective o such thatx(yz) is a tautology of . This answers a question asked by W. Rautenberg in [2], P. Wojtylak in [8] and W. Dziobiak in [1]. Since every 2-element matrix can be finitely axiomatized ([3]), the matrices presented here are of the smallest possible size and in some sense are the simplest possible.Presented byWolfgang Rautenberg     

Finite structural axiomatization of every finite-valued propositional calculus

In [2] A. Wroski proved that there is a strongly finite consequence C which is not finitely based i.e. for every consequence C ⁺ determined by a finite set of standard rules C C ⁺. In this paper it will be proved that for every strongly finite consequence C there is a consequence C ⁺ determined by a finite set of structural rules such that C(Ø)=C ⁺(Ø) and = (where , are consequences obtained by adding to the rules of C, C ⁺ respectively the rule of substitution). Moreover it will be shown that under certain assumptions C=C ⁺.     

Social perspective-taking and adjustment in emotionally disturbed,learning-disabled,and normal children

Preadolescent emotionally disturbed, learning-disabled, and normal boys were compared on social perspective-taking and behavioral measures to examine possible contributions of social cognitive deficits to children's adjustment problems. Antisocial-prosocial and withdrawn-gregarious behavior dimensions were studied through subscales derived from teacher ratings. Results indicated that across all groups, high perspective-taking was associated with significantly less withdrawal than was low perspective-taking; within groups, this finding was significant only for the emotionally disturbed boys. Contrary to theoretical assumptions, antisocial behavior was not significantly related to perspective-taking across the sample. Among emotionally disturbed boys, relatively higher affective perspectivetaking was significantly correlated with higher antisocial behavior. This positive correlation for the emotionally disturbed group was significantly different from the nonsignificant negative correlation between antisocial behavior and perspective-taking among normals. Findings for learning-disabled boys were intermediate. between results for emotionally disturbed and normal boys on both perspectivetaking and behavioral measures, and the learning-disabled group generally did not differ significantly from either other group. Theoretical and clinical implications of the findings are discussed The authors acknowledge the invaluable assistance and cooperation of many teachers and principals in Jefferson County School District in Colorado, especially Ms. Norma Sobesky, coordinator of the program for Significantly Impaired Emotionally and Behaviorally Disturbed Children, and Ms. Beth Young, coordinator of the program for Perceptual and Communicative Disorders.     

Strong completeness with respect to finite kripke models

We prove that each intermediate or normal modal logic is strongly complete with respect to a class of finite Kripke frames iff it is tabular, i.e. the respective variety of pseudo-Boolean or modal algebras, corresponding to it, is generated by a finite algebra. The author wishes to thank the Editors for calling his attention to the fact that the result of this paper concerning the intermediate logics was announced earlier by A. Wroński at the conference “Logical calculi”, Wrocław, October 5–7, 1975, though without proof. Wroński's result has not been published.     

On the {↔, ∼} -reduct of the intuitionistic consequence operation

The intuitionistic consequence operation restricted to the language with (equivalence) and (negation) as the only connectives is axiomatized by means of a finite set of sequential rules of inference.To the memory of Professor Roman Suszko     

On finite approximability of ψ-intermediate logics

The aim of this note is to show (Theorem 1.6) that in each of the cases: = {, }, or {, , }, or {, , } there are uncountably many -intermediate logics which are not finitely approximable. This result together with the results known in literature allow us to conclude (Theorem 2.2) that for each : either all -intermediate logics are finitely approximate or there are uncountably many of them which lack the property.