H. B. Smith on modality: A logical reconstruction
This paper offers a reconstruction of some ideas of H. B. Smith on the logic of (alethic) modalities developed in the mid-30's. Smith's main assumptions are (A) that any two positive modalities \(\Gamma\), \(\Delta\) (i.e. any strings of modal operators \(\square\), \(\diamond)\) are comparable in the sense that either \(\Gamma\) p entails \(\Delta\) p or vice versa; and (B) that no two different modalities are equivalent in the sense of \(\Gamma\) entailing \(\Delta\) and \(\Delta\) entailing \(\Gamma\). It is shown that (A) and (B) can be realized within a non-normal system (called \(K+SM)\) which is obtained by adding to the (normal) basic system K the axiom schemas \(p\supset \square^ n\diamond p\) (n\(\geq 0)\). (The non-normality is due to the fact that the rule of necessitation is restricted to the theorems of K; if necessitation were unrestrictedly valid in \(K+SM\), the resulting system would be equivalent to S5). A semantics is developed with respect to which \(K+SM\) can be proved to be complete.
scientific article; zbMATH DE number 4037818