Modal logics in the vicinity of S1
The authors define prenormal logics and show that certain well-known nonnormal logics are Lewis versions of corresponding prenormal logics. Determination and decidability are shown. Cresswell logics are introduced and characterized. The formal apparatus consists in a combination of relational and neighborhood semantics. The paper is a far-reaching contribution to the ongoing discussion around \(S1\). A logic is called classical, if it permits replacement of logically equivalent formulas. A classical logic is called prenormal, if it provides the schema \[ \square \mathbb{T} \to\bigl(\square (A\to B) \to (\square A\to \square B) \bigr). \] A Lewis version of a logic is the smallest logic that extends that logic and contains \(\square \mathbb{T}\). A Cresswell logic is a prenormal logic closed under the Cresswell rules \[ A_0 \vee \cdots \vee A_{n-1}/(\square A_0 \& \cdots \& A_{n-1}) \to \square\mathbb{T}. \]
combination of relational and neighborhood semantics
scientific article; zbMATH DE number 935222
scientific article