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An interpolation theorem in many-valued logic

A many-valued logic version of the Craig-Lyndon interpolation theorem has been given by Gill [1] and Miyama [2]. The former dealt with three-valued logic and the latter generalized it to M-valued logic with M ≥ 3. The purpose of this paper is to improve the form of Miyama's version of the interpolation theorem. The system used in this paper is a many-valued analogue (Takahashi [4], Rousseau [3]) of Gentzen's logical calculus LK. Let T = {1,…, M} be the set of truth values. An M-tuple (Γ1,…, ΓM) of sets of formulas is called a sequent, which is regarded as valid if for any valuation (in canonical sense) there is a truth value μ Є T such that the set Γμ contains a formula of the value μ with respect to the valuation. (In the next section, and thereafter, we change the format of the sequent for typographical reasons.) Miyama's result is as follows (in representative form): (I) If a sequent ({A}, ∅,…, ∅, {B}) is valid, then there is a formula D such that (i) every predicate or propositional variable occurring in D occurs in A and B, and (ii) the sequents {{A}, ∅,…, ∅, {D}) and (D}, ∅,…, ∅, {B}) are both valid. What shall be proved in this paper is the following (in representative form): (II) If a sequent ({A1}, {A2}, …, {AM}) is valid, then there is a formula D such that (i) every predicate or propositional variable occurring in D occurs in at least two of the formulas A1,…, AM, and (ii) the following M sequents are valid: ({A1},{D},…,{D}),({D},{A2},…,{D}),…,({D},{D},…,{AM}). Clearly the former can be obtained as a corollary of the latter.