The Disjunction Property of the Logics with Axioms of Only One Variable

The logics treated here are the intermediate propositional logics of the form LJ + Ni(a), where Ni(a)’s are axioms of only one variable studied in Nishimura [6]. As is easily seen, the problem of the disjunction property of LJ+Ni(a) starts from i = 10 and the case of odd i can be trivially neglected. The case of i = 10 was solved by Kreisel and Putnam [5] and other cases except i = 14 were solved by Anderson [1]. Finally Wroński [7] proved that the logics LJ + Ni(a) (i = 10, 12, 14, . . .) have the disjunction property. The method of Kreisel and Putnam is syntactical, but those of Anderson and Wroński are semantical. Here we give a syntactical proof of the disjunction property of the logics LJ + N4 m + 2(a) (m ≥ 2) 1. Preliminaries A logic L has the disjunction property if and only if for any formulas A and B, A ∨ B ∈ L implies either A ∈ L or B ∈ L. In Gödel [3], it is proved that the intuitionistic propositional logic has the disjunction property. In this paper, we study the disjuction property of logics with axioms of only one variable. By WFF , we mean the set of all the formulas. For the intuitionistic logic, we use the system LJ presented in Gentzen [2], and for the terminology concerning the system, we mainly follow Kleene [4]. The axioms of only one variable have been studied in detail in Nishimura [6]. Instead of his Ni(a), we use a little modified axioms as follows. Definition 1.1. The basic formulas Ni(a)’s are: