Logics without Craig's interpolation property

Abstract

It has been shown by Craig, Schtitte and Gabbay that Craig’s interpolation theorem holds or predicate logics LK, LJ, LJ/ (a), LJ/ (b) and LJ+ (a)+ (b) (cf. [1]), where (a) yx--A(x) --xA(x), (b) -PV--P. in this paper, we shall show that there exist some superclassical predicate logics where Craig’s interpolation theorem does not hold and that Craig’s interpolation theorem does not hold for any intermediate propositional logic on sliceq for 3__<nw. 1. Superclassical predicate logics. The definition o predicate logics is obtained by excluding the condition 1) LJLLK from Definition 1.1 in Ono [4]. By a superclassical predicate logic (SCPL), we mean a predicate logic including the predicate logic LK. We denot the set of formulas valid in any model whose dmain has n elements by L(n). We can easily see that L(n) is a SCPL and LK (n)