University of UtrechtINTERPOLATION IN A FRAGMENT OF INTUITIONISTIC PROPOSITIONAL LOGIC

Let NNIL (No Nestings of Implication to the Left) be the fragment of IpL (intuitionistic propositional logic) in which the antecedent of any implication is always prime. The following strong interpolation theorem is proved: if IpL }-A+B and A or B is in NNIL, then there is an interpolant I in KNIT The proof consists in constructing I from a proof of A+B in a sequent calculus system by means of a variant of a method devised by K. Schutte. This settles a question posed by A. Visser.

H calculus for IpL, §3 consists of three lemmata about elementary fragments, §4 contains Schutte's method to prove the interpolation theorem for IpL, which is used subsequently to show interpolation for all elementary fragments, in §5 we discuss the consequences of not adding the constants T and 1 to the fragments. §6 is rather tentative: it reports on unsuccessful attempts to prove interpolation for some fragments containing the connectives H and 1.5. Acknowled eg ments. The author is indebted to M.H. Lob, who pointed out to him an error in a previous version of theorem 4.5. §2. Preliminaries. 2.2. The derivation system. We use the following sequent. calculus, denoted by SC: The proofs are standard (as for related systems, e.g in [S62] and [T75]). Note that the Subformula Property only holds in the following version: if B occurs in a cut-free derivation of °F l-A, then B=1 or B is a subformula of F,A; the addition B=1 is made necessary by the inwhi ch 1 is eliminated.
The following consequence is important in the context of this paper: 2.3. SC* is SC with (vL) and (-L) replaced by:

Lemma. F F A if and only if F F* A.
Proof. We write F Fn A for: '11' F A has a derivation with length at most n'; idem for F F*n A. With induction over n one-easily proves: (1) if F, AvB,A Fn'C, then F, A Fn C; (2) if F, AvB,B, Fri C, then; ',:B; I- if I' Fn C, then F,AFnC; if r F*n C, then IF, 0 F*n C.
We turn to the 'if part of the lemma. Assume i.e. F *.n A . for some n; we show F Fn A with induction over n. If n=1 then IF F* A is an axiom, hence F F A; if n>1 and F F* A is (an axiom or) the conclusion of (vL)* w or a rule, different from, L)*, then the result directly follows from the induction hypothesis (using that every instance of (vL)* is an instance of (vL)). If IF F* A is the conclusion of (-aL)*, when the premises :are, of the form F', Finally we prove the 'only if part, with, induction over, the length of aderivation.of F I-A.
Assume I l-A, so I' Fn A for some n. If n=1 then F F A is an axiom, hence 11'F* A; if n> 1 and F I-Ais (an-axiom-or,)-the conclusion.of-an. instance of -(vL)* or a rule different from (vL), (-L), then the result directly. follows from the induction hypothesis. There are three cases left: i) F F-A is the conclusion of (vL) with premises of the form F', BvC, B I-A and r', BvC, C F-A where r' := r -(BvC}: apply (1) Proof. i) Formula induction; using the following equivalences: The method to obtain the interpolant I for r,A F A can be rendered as follows: (iPl) We explain this notation with an example. We show (i) -(iv) of (2), writing I for I(A1-->A2,f+).
by (9)  To make life simpler, we considered T and .l as constants which are present in every fragment. If we do not choose to do so, we have to be slightly more careful, as we shall now explain.
In this last section we consider fragments containing and and sketch some attempts to prove interpolation. So [E-a] is the only new fragment. We conjecture that interpolation holds, but a proof has not been found. We sketch two approaches. (1), plus the following rules: Unfortunately, this extension of Schiitte's method may (by (i-,L2)) introduce -, in the definition of an interpolant for A F B with A,B in some fragment containing but not Closer inspection learns that (i-1L2) is only needed in fragments containing -1 or -a, so interpolation holds in the rather trivial fragments [-], [-,n], [-,v] and [-,n,v]. For the other fragments, the question arises: which fragments containing -and -4 satisfy interpolation? 6.5. Uniform interpolation. In classical logic, the left and right'variant are equivalent. Uniform interpolation holds for classical propositional logic, but not for classical predicate logic (see [H63]),. and hence not for intuitionistic predicate logic.