Extensions of the \(\aleph_ 0\)-valued Łukasiewicz propositional logic
type: Item
instance of: scholarly article
title: Extensions of the \(\aleph_ 0\)-valued Łukasiewicz propositional logic
zbMATH Open document ID: 0789.03020
author: M. G. Beavers
published in: Notre Dame Journal of Formal Logic
publication date: 1993-12-06T00:00:00Z
review text: \textit{A. Rose's} genus [J. Lond. Math. Soc. 27, 92-102 (1952; Zbl 0048.244)] and \textit{Y. Komori's} \(S_ n^ \omega\) [Nagoya Math. J. 84, 119- 133 (1981; Zbl 0482.03007)] play important roles in the study of extensions of the \(\aleph_ 0\)-valued Łukasiewicz propositional logic. Nevertheless connections between Rose and Komori had not been made clear. In this paper the author clarifies the connections. The set of formulas valid in an algebra \(A\) is denoted by \(\text{Taut}(A)\). The set of formulas which are theorems of the logic \(L\) is denoted \(\text{Th}(L)\). \(S_ n\) denotes the well-known \(n\)-valued Łukasiewicz matrix. \({\mathbf{\L}}_{\aleph_ 0}\) denotes the \(\aleph_ 0\)-valued Łukasiewicz propositional logic. Let \(G_ \varphi\) denote the genus of a formula \(\varphi\). The main theorem states that if \(\varphi\) is any formula and \(G_ \varphi= \langle B;C\rangle\), then \([\text{Th}({\mathbf{\L}}_{\aleph_ 0}+\varphi)=\bigcap_{i\in C}\text{Taut}(S_ i)\cap\bigcap_{i\in B}\text{Taut}(S_ i^ \omega)]\).
Mathematics Subject Classification ID: 03B50
zbMATH DE Number: 469219
zbMATH Keywords: extensions of the \(\aleph_ 0\)-valued Łukasiewicz propositional logic
zbMATH Keywords: genus of a formula
reviewed by: Yuichi Komori
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1305/ndjfl/1093634656
OpenAlex ID: W2128755340
DOI: 10.1305/NDJFL/1093634656
description: scientific article; zbMATH DE number 469219