Nontrivial single axiom schemata and their quasi-nontriviality of Le\'{s}niewski-Ishimoto's propositional ontology $\bf L_1$

Abstract

On March 8, 1995, was found the following \it nontrivial \rm single axiom-schema characteristic of Le\'{s}niewski-Ishimoto's propositional ontology $\bf L_1$ (Inou\'{e}, 1995b \cite{inoue16}). $$(\mathrm{A_{M8})} \enspace \epsilon ab \wedge \epsilon cd . \supset . \epsilon aa \wedge \epsilon cc \wedge (\epsilon bc \supset . \epsilon ad \wedge \epsilon ba).$$ 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 \S 6 - \S 8, we shall give the simplified axiom schemata ($\rm A_{S1}$), ($\rm A_{S2}$), ($\rm A_{S3N}$) and ($\rm A_{S3Nd}$) based on ($\mathrm{A_{M8}}$), their nontriviality and quasi-nontriviality. In \S 9 - \S 11, we shall give a lot of conjectures for nontrivial single axiom schemata for $\bf L_1$. We shall conclude this paper with summary and some remarks in \S 12.