Nontrivial single axiom schemata and their quasi-nontriviality of Leśniewski-Ishimoto's propositional ontology ${\mathbf{L}}_{\mathbf{1}}$

On March 8, 1995, was found the following \it nontrivial \rm single axiom-schema characteristic of Leśniewski-Ishimoto’s propositional ontology ${\mathbf{L}}_{\mathbf{1}}$ (Inoué, 1995b \cite{inoue16}). $$({\mathrm{A}}_{\mathrm{M}8})\epsilon ab\wedge \epsilon cd\mathrm{.}\supset \mathrm{.}\epsilon aa\wedge \epsilon cc\wedge (\epsilon bc\supset \mathrm{.}\epsilon ad\wedge \epsilon ba)\mathrm{.}$$ 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 (${\mathrm{A}}_{\mathrm{S}1}$), (${\mathrm{A}}_{\mathrm{S}2}$), (${\mathrm{A}}_{\mathrm{S}3\mathrm{N}}$) and (${\mathrm{A}}_{\mathrm{S}3\mathrm{N}\mathrm{d}}$) based on (${\mathrm{A}}_{\mathrm{M}8}$), their nontriviality and quasi-nontriviality. In §9 - §11, we shall give a lot of conjectures for nontrivial single axiom schemata for ${\mathbf{L}}_{\mathbf{1}}$. We shall conclude this paper with summary and some remarks in §12.