Equivalential Interpolation

By a consequence relation on a set L of formulas we understand a relation I — c p(L) x L satisfying the conditions called 'Overlap', 'Dilution', and 'Cut for Sets' at p.15 of [25]; we do not repeat the conditions here since we are simply fixing notation and the concept of a consequence relation is well known in any case. (The characterization in [25] amounts to that familiar from Tarski's work, except that there is no 'finitariness' restriction to the effect that when I I — A, for I c L, A c L, we must have I o I — A for some finite I o c I . The presence or absence of this condition makes no difference to anything that follows.) Each language L to be considered will be a sentential language whose formulas are built in the usual way by application of it-ary (primitive) connectives to it simpler formulas, starting with the simplest formulas — the propositional variables (or 'sentence letters') — not constructed with the aid of connectives. We assume, as usual, that there are countably many such variables, and they will be denoted by p, q, r, ... possibly with numerical subscripts. A consequence relation I- on such an L has the Unrestricted Interpolation Property when for any A, C c L with A I — C, there exists B c L with A I — B and B I — C, such that C is constructed only out of such propositional variables as occur both in A and in C. (Such a B is called an interpolant for A and C.) Note that we take the usual notational liberties here, writing "A I — C" (and the like) for "iAi I — C", "I, A I — C" to mean "I u iAi I — C", and "I — C" to mean "8 I — C". Further, we sometimes abbreviate the claim that A I — B and B I — C to "A I — B I — C", and when C is A itself, we always write this simply as "A — II— B"..
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