In this contribution we shall characterize matrix consequence operation determined by a direct product and an ultraproduct of a family of logical matrices. As an application we shall describe finite consequence operations with the help of ultrapowers.

The aim of this paper is to show that the operations of forming direct products and submatrices suffice to construct exhaustive semantics for all structural strengthenings of the consequence determined by a given class of logical matrices.

In the paper some consequence operations generated by ukasiewicz's matrices are examined.

The paper is concerned with the problem of characterization of strengthenings of the so-called Lukasiewicz-like sentential calculi. The calculi under consideration are determined byn-valued Lukasiewicz matrices (n>2,n finite) with superdesignated logical values. In general. Lukasiewicz-like sentential calculi are not implicative in the sense of [7]. Despite of this fact, in our considerations we use matrices analogous toS-algebras of Rasiowa. The main result of the paper says that the degree of maximality of anyn-valued Lukasiewicz-like sentential calculus is finite and equal to (...) the degree of maximality of the correspondingn-valued Lukasiewicz calculus. (shrink)

The paper is concerned with the problem of characterization of strengthenings of the so-called Łukasiewicz-like sentential calculi. The calculi under consideration are determined by n-valued Łukasiewicz matrices with superdesignated logical values. In general, Łukasiewicz-like sentential calculi are not implicative in the sense of [7]. Despite of this fact, in our considerations we use matrices analogous to S-algebras of Rasiowa. The main result of the paper says that the degree of maximality of any n-valued Łukasiewicz-like sentential calculus is finite and equal (...) to the degree of maximality of the corresponding n-valued Łukasiewicz calculus. (shrink)

The class Matr(C) of all matrices for a prepositional logic (, C) is investigated. The paper contains general results with no special reference to particular logics. The main theorem (Th. (5.1)) which gives the algebraic characterization of the class Matr(C) states the following. Assume C to be the consequence operation on a prepositional language induced by a class K of matrices. Let m be a regular cardinal not less than the cardinality of C. Then Matr (C) is the least class (...) of matrices containing K and closed under m-reduced products, submatrices, matrix homomorphisms, and matrix homomorphic counter-images. (shrink)

Two characterizations are given of those structural consequence operations on a propositional language which can be defined via proofs from a finite number of polynomial rules.

Prior to investigating the degree of maximality of a propositional logic, as is the notion introduced by Ryszard Wojcicki [8], it is important to determine all structural consequence stronger than the logic considered. This characterization of all structural strengthenings was performed while establishing the degree of maximality for Lukasiewicz and Lukasiewicz-like logics [8], [4], [9], [5]. In these papers the authors used matrix consequences together with certain results concerning this notion and a representation theorem for Lukasiewicz algebras as given by (...) R. S. Grigolia [2]. (shrink)

First, we prove some theorem characterizing the niteness of degree of maximality of a strongly nite consequence operation and establish some results following from it. Second, we deal with the notion of strongly nite model property which was proposed by R. Wojcicki in [9]. The results presented in the paper came out after series of discussions which the author held with Prof. R. Wojcicki.

In this note we investigate the notions of re ection and projection of structural sentential calculi as introduced by R. Wojcicki . This note was read at the Autumn School on Strongly Finite Sentential Calculi organized by Section of Logic, Polish Academy of Sciences, Institute of Philosophy and Sociology, Ustronie , November 1978.