Nontrivial single axiom schemata and their quasi-nontriviality of Leśniewski-Ishimoto's propositional ontology ${\mathbf{L}}_1$

Abstract

On March 8, 1995, was found the following \it nontrivial \rm single axiom-schema characteristic of Leśniewski-Ishimoto's propositional ontology ${\mathbf{L}}_1$ (Inoué, 1995b \cite{inoue16}). $$({\mathrm{A}}_{M8})\epsilon ab\wedge \epsilon cd.\supset .\epsilon aa\wedge \epsilon cc\wedge (\epsilon bc\supset .\epsilon ad\wedge \epsilon ba).$$ In this paper, we shall present the progress about the above axiom-schema from 1995. Here we shall give two criteria \it nontiriviality \rm and \it quasi-nontriviality \rm in order to distinguish two axiom schemata. As main results, among others, in §6 - §8, we shall give the simplified axiom schemata (${\mathrm{A}}_{S1}$), (${\mathrm{A}}_{S2}$), (${\mathrm{A}}_{S3N}$) and (${\mathrm{A}}_{S3Nd}$) based on (${\mathrm{A}}_{M8}$), their nontriviality and quasi-nontriviality. In §9 - §11, we shall give a lot of conjectures for nontrivial single axiom schemata for ${\mathbf{L}}_1$. We shall conclude this paper with summary and some remarks in §12.

37 pages. The section 12 of this paper, Summary and Remarks was mainly revised