Some modal logics based on a three-valued logic
type: Item
instance of: scholarly article
title: Some modal logics based on a three-valued logic
zbMATH Open document ID: 0665.03013
author: Osamu Morikawa
published in: Notre Dame Journal of Formal Logic
publication date: 1989-01-01T00:00:00Z
review text: In this paper three-valued versions of K, T, S4, and S5 are constructed. The language has propositional variables, a family of propositional connectives \(F_ i\), \(1\leq i\leq \beta\) \((F_ i\) is \(\alpha_ i\)-ary), and the modal operator \(\square\). A \(3\)-K\({}_ 3\) model is a triple (W,R,\(\phi)\) in which \(R\subseteq W\times W\) and \(\phi\) is a valuation (if p is a variable and \(s\in W\) then \(\phi\) (p,s)\(\in {\mathbb{T}}=\{1,2,3\})\). \(\phi\) is extended to all formulas using the \(rule:\) \(\phi\) (\(\square A,s)=1\) if \(\forall t\in R(s)\) \(\phi (A,t)=1\), \(=2\) if \(\exists t\in R(s)\) \(\phi (A,t)=2\), and \(=3\) otherwise. In \(3\)-K\({}_ 2\) models \(\phi\) (\(\square A,s)\) is the same, otherwise it is 3. Definitions of \(T\)-K\({}_ i,..\). models involve corresponding properties of R. A matrix is a triple of finite sets of formulas: \(<Q_ 1,Q_ 2,Q_ 3>\). It is called \(3\)-L\({}_ j\) valid in a \(3\)-L\({}_ j\) model (W,R,\(\phi)\) iff \(\forall s\in W\) \(\exists i\in {\mathbb{T}}\) \(\exists A\in Q_ i\) \(\phi (A,s)=i\). The author describes ``matrix calculi'' \(3\)- L\({}_ j\) and proves completeness theorems for them.
Mathematics Subject Classification ID: 03B45
Mathematics Subject Classification ID: 03B50
zbMATH DE Number: 4087632
zbMATH Keywords: three-valued modal logics
zbMATH Keywords: three-valued versions of K, T, S4, and S5
zbMATH Keywords: models
zbMATH Keywords: matrix calculi
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1305/ndjfl/1093635000
OpenAlex ID: W1983858044
DOI: 10.1305/NDJFL/1093635000
description: scientific article; zbMATH DE number 4087632