Intermediate logics with the same disjunctionless fragment as intuitionistic logic (Q1096618)

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.