Dyrda and Prucnal gave a Hilbert-style axiomatization for the \-fragment of classical propositional logic. Their proof of completeness follows a different approach to the standard one proving the completeness of classical propositional logic. In this note, we present an alternative proof of Dyrda and Prucnal’s result following the standard arguments which prove the completeness of classical propositional logic.

Dyrda and Prucnal gave a Hilbert-style axiomatization for the {∧,∨}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{\wedge,\vee \}$$\end{document}-fragment of classical propositional logic. Their proof of completeness follows a different approach to the standard one proving the completeness of classical propositional logic. In this note, we present an alternative proof of Dyrda and Prucnal’s result following the standard arguments which prove the completeness of classical propositional logic.

The paper is conceived as a first study on the Suszko operator. The purpose of this paper is to indicate the existence of close relations holding between the properties of the Suszko operator and the structural properties of the model class for various sentential logics. The emphasis is put on generality both of the results and methods of tackling the problems that arise in the theory of this operator. The attempt is made here to develop the theory for non-protoalgebraic logics.