Some modal logics based on a three-valued logic
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.
three-valued versions of K, T, S4, and S5
scientific article; zbMATH DE number 4087632