The separable axiomatization of the intermediate propositional systems $S_n $ of Gödel
Tsutomu Hosoi · Proceedings of the Japan Academy Series A Mathematical Sciences · 1966
In [3 GSdel introduced a series of many-valued propositional systems S, which is widely known and is quite frequently made use of when propositional systems are treated.And in our paper 6 we introduced two kinds of axiomatization for these S.But the separation theorem mentioned below does not hold on those axiomatized systems.Separation Theorem.A provable formula in the system can be proved using only the axioms for implication and those for the logical symbols actually appearing in the formula.We introduce, in this paper, another axiomatization for S and prove the separation theorem on them.1. Preliminaries.Definition 1.1.S is a many-valued propositional system, whose values ave integers 1, 2,..., n and w (w is regarded grea$ev han any positive integers), and whose sole designated value is 1.Logical opera$ions D, A, V, and ave defined in S as follows:v v 1 vz v otherwise, v v max (v, v.), v V v min (v, v.), VV.An extension of S is LC of Dummett 2, in which values are defined to be all the positive integers and w.