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Expansion by a new constant may change the finite axiomatization property of a matrix

Pałasińska K.

Czasopismo Techniczne. Nauki Podstawowe

2016

Y. 113, iss. 1-NP

131--136

We give an example of a finite matrix with the property that expanding its language with a constant changes its finite axiomatization property: in the language with one binary operation the tautologies of the matrix are finitely axiomatizable while in the expanded language they are not. The constant we add is not definable in the original language. The deductive system generated by this matrix is not algebraizable.

Podajemy przykład skończonej matrycy logicznej, która jest skończenie aksjomatyzowalna, ale po dodaniu stałej do sygnatury tej matrycy, własność ta się psuje. Dodawana stała nie jest definiowalna w języku matrycy, a operator konsekwencji wyznaczony przez tę matrycę nie jest algebraizowalny.

Towards a Behavioral Algebraic Theory of Logical Valuations

Caleiro C., Goncalves R.

Fundamenta Informaticae

2011

Vol. 106, nr 2-4

191-209

Logical matrices are widely accepted as the semantic structures that most naturally fit the traditional approach to algebraic logic. The behavioral approach to the algebraization of logics extends the applicability of the traditional methods of algebraic logic to a wider range of logical systems, possibly encompassingmany-sorted languages and non-truth-functional phenomena. However, as one needs to work with behavioral congruences, matrix semantics are unsuited to the behavioral setting. In [5], a promising version of algebraic valuation semantics was proposed in order to fill in this gap. Herein, we define the class of valuations that should be canonically associated to a logic, and we show, by means of new meaningful bridge results, how it is related to the behaviorally equivalent algebraic semantics of a behaviorally algebraizable logic.

The logic dual to Sobociński's n-valued logic

Górnicka A.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

2010

Vol. 15

35--42

In this paper, we describe the logic dual to n-valued Sobociński logic. According to the idea presented by Malinowski and Spasowski [1], we introduce the consequence dual to the consequence of n-valued Sobociński logic in two ways: by a logical matrix and by a set of rules of inference. Then we prove that both approaches are equivalent and the consequence is dual in Wójcicki sense (see [3]).