Takahito Aoto Hiroakira Ono NON-UNIQUENESS OF NORMAL PROOFS FOR MINIMAL FORMULAS IN IMPLICATION-CONJUNCTION FRAGMENT OF BCK

A minimal formula of a given logic L is a formula which is provable in L and is not a non-trivial substitution instance of other provable formulas in L. In [5], Y. Komori asked whether normal proofs of minimal formulas are unique in the implicational fragments of natural deduction systems for the intuitionistic logic and the logic BCK. It was already shown that the answer is positive for BCK, while it is negative for the intuitionistic logic ([3],[8],[9]). The present paper shows normal proofs for minimal formulas are not necessarily unique for the implication-conjunction fragment of BCK. This result contrasts sharply with the uniqueness of normal proofs of balanced formulas for the implication-conjunction fragment of the intuitionistic logic ([2],[6]). 1 Uniqueness of normal proofs We will use ! and ^ for the implication and conjunction throughout the present paper. For a given logic L, we say that a formula A is minimal in L if and only if (1) A is provable in L and (2) if A is a substitution instance of other provable formula A 0 then the substitution is trivial, i.e. it is a renaming of propositional variables. Y. Komori put a question in [5] whether normal proofs of minimal formulas are unique in the natural deduction systems for the implicational fragments of the intuitionistic Appeared in: Bulletin of the Section of Logic, Vol. 23, No. 3, pp. 104{112, 1994.