On some properties of intermediate logics
Toshio Umezawa · Proceedings of the Japan Academy Series A Mathematical Sciences · 1959
In [1 I investigated inclusion and non-inclusion between certain intermediate predicate logics.The purpose of this note is to prove some properties of intermediate logics.We use in this note the same notations as in 1 without definitions.1. Interpretation of classical logic.THEOREM 1. LK and LMK are minimal in the set of all predicate logics which have the properties (I) and (II) respectively.( I For any K-provable sequent FoE, the sequent 7 7 -> 7 rE is provable.(II) For any K-provable sequent ->, the sequent is provable.PROOF.First we prove that (I) and (II) hold in LK and LMK respectively.For (I) we use as a deductive system of K-provable