H. B. Smith on modality: A logical reconstruction
type: Item
instance of: scholarly article
title: H. B. Smith on modality: A logical reconstruction
zbMATH Open document ID: 0637.03016
DOI: 10.1007/BF00431182
author: Tore Langholm
published in: Journal of Philosophical Logic
publication date: 1987-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 03B45
Mathematics Subject Classification ID: 03A05
zbMATH DE Number: 4037818
zbMATH Keywords: modal logic
zbMATH Keywords: semantics
reviewed by: Wolfgang Lenzen
MaRDI profile type: Publication
cites work: Q3867808
cites work: Q5801329
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description: scientific article; zbMATH DE number 4037818