Abstract
Given an intermediate prepositional logic L, denote by L −dits disjuctionless fragment. We introduce an infinite sequence {J n}n⩾1 of propositional formulas, and prove:
(1)For any L: L −d=I −d (I=intuitionistic logic) if and only if J n∉ L for every n ⩾ 1.
Since it turns out that L∩{J n} n⩾1 = Ø for any L having the disjunction property, we obtain as a corollary that L −d = I −d for every L with d.p. (cf. open problem 7.19 of [5]). Algebraic semantic is used in the proof of the “if” part of (1). In the last section of the paper we provide a characterization in Kripke's semantic for the logics J n =I+ +J n (n ⩾ 1).
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Minari, P. Intermediate logics with the same disjunctionless fragment as intuitionistic logic. Stud Logica 45, 207–222 (1986). https://doi.org/10.1007/BF00373276
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DOI: https://doi.org/10.1007/BF00373276