Intermediate logics with the same disjunctionless fragment as intuitionistic logic
type: Item
instance of: scholarly article
title: Intermediate logics with the same disjunctionless fragment as intuitionistic logic
zbMATH Open document ID: 0634.03021
DOI: 10.1007/BF00373276
author: Pierluigi Minari
published in: Studia Logica
publication date: 1986-01-01T00:00:00Z
review text: An intermediate logic M is any consistent propositional logic containing intuitionistic logic I and closed under substitution and modus ponens. The author defines a sequence of formulae J as follows: start with \(\neg p\vee \neg \neg p\) and at any stage use the next sentential variable q and the formula A of the previous stage to form (q\(\to A)\vee (\neg q\to A)\). It is shown that for evry intermediate M, if M and I have the same disjunctionless fragment, then no member of J is an M-theorem. The main part of the paper is then given over to an algebraic proof of the converse: if M fails to prove any member of J, then M and I have the same disjunctionless fragment. Finally, the author gives a Kripke characterization theorem for the logic \(I+J(n):\) each \(I+J(n)\) is the logic determined by a certain class of finite principal posets.
Mathematics Subject Classification ID: 03B55
zbMATH DE Number: 4031645
zbMATH Keywords: intuitionistic logic
zbMATH Keywords: intermediate logic
zbMATH Keywords: disjunctionless fragment
zbMATH Keywords: Kripke characterization theorem
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4031645