Axiomatic Rejection for the Propositional Fragment of Leśniewski's Ontology

Abstract

Deceased. This paper is dedicated to the memory of Professor Emeritus Arata Ishimoto. This paper is cited in Ishimoto [27].(3) For any formula A of L 1 , if A is provable in the Hilbert-type L 1 , then it is provable in the the Gentzen-type L 1 (tableau method), (4) Contradiction and Dichotomy theorems for the Hilbert-type L 1 hold,(5) No Hintikka formula is provable in the Hilbert-type L 1 , (6) The Hilbert-type L 1 is L-decidable with respect to HL 1 (i.e. the set of all formula of L 1 is the disjoint union of the set of all the theorem of the Hilbert-type L 1 and that of all the theorem of HL 1 ), where HL 1 is the axiomatic rejection with Hintikka formulas as axioms.Then we shall show that HL 1 has the same strength with HAR.