In  Scott asked if there was a proof theoretic proof of his interpolation theorem. The purpose of this paper is to provide such a proof, working with the first order system LK of Gentzen . Our method is an extension of the one in Maehara  for Craig's interpolation theorem. We will also sketch the original model theoretic proof and show how Scott used his result to obtain a definability theorem of Svenonius .
A language for LK contains the usual logical symbols: , ∧, ∨, ⊃, ∀, ∃; countably many free variables a0, a1, … and bound variables x0, x1, …; and some or all of the following nonlogical symbols: n-ary predicates ; n-ary functions ; and individual constants c0, c1, …. Semiterms are defined as follows: (1) Free variables, bound variables and individual constants are semiterms. (2) If f is an n-ary function and s1 …, sn are semiterms, then f(s1 …, sn), is a semiterm. A term is a semiterm that does not contain a bound variable. Formulas are defined as follows: (1) If R is an n-ary predicate and t1 …, tn are terms, then R(t1 …, tn) is a formula. (2) If A and B are formulas, then A, A ∧ B, A ∨ B and A ⊃ B are formulas. (3) If A(t) is a formula which has zero or more occurrences of the term t, and if x is a bound variable not contained in A(t), then ∀xA(x) and ∃xA(x) are formulas where A(x) is obtained from A(t) by substituting x for t at all indicated places.