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Nontrivial single axiom schemata and their quasi-nontriviality of Leśniewski-Ishimoto's propositional ontology $\bf L_1$

Takao Inoué, Tadayoshi Miwa

TL;DR

The paper addresses how to distinguish nontrivial single axiom schemata for Leśniewski-Ishimoto's propositional ontology $L_1$, focusing on the known schema $A_{M8}$. It introduces two criteria, nontriviality and quasi-nontriviality, and derives simplified schemata $A_{S1}$, $A_{S2}$, $A_{S3N}$, and $A_{S3Nd}$ from $A_{M8}$, proving their nontriviality and analyzing their quasi-nontrivial relationships, including a provable path $A_{S3}$ to deduce transitivity (Ax2) and exchangeability (Ax3). The work establishes structural results and furnishes conjectures on additional nontrivial single axiom schemata, while highlighting computational approaches to advance foundations of $L_1$ and the calculus of names. Overall, the paper clarifies how single axiom schemata can robustly characterize $L_1$ while offering a framework to explore a broad landscape of nontrivial axioms and their interrelations. The findings have implications for the foundational study of Lesniewski's system and its propositional ontology, with potential computational tools aiding further exploration.

Abstract

On March 8, 1995, was found the following \it nontrivial \rm single axiom-schema characteristic of Leśniewski-Ishimoto's propositional ontology $\bf L_1$ (Inoué, 1995b \cite{inoue16}). $$(\mathrm{A_{M8})} \enspace εab \wedge εcd . \supset . εaa \wedge εcc \wedge (εbc \supset . εad \wedge εba).$$ 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 ($\rm A_{S1}$), ($\rm A_{S2}$), ($\rm A_{S3N}$) and ($\rm A_{S3Nd}$) 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 $\bf L_1$. We shall conclude this paper with summary and some remarks in §12.

Nontrivial single axiom schemata and their quasi-nontriviality of Leśniewski-Ishimoto's propositional ontology $\bf L_1$

TL;DR

The paper addresses how to distinguish nontrivial single axiom schemata for Leśniewski-Ishimoto's propositional ontology , focusing on the known schema . It introduces two criteria, nontriviality and quasi-nontriviality, and derives simplified schemata , , , and from , proving their nontriviality and analyzing their quasi-nontrivial relationships, including a provable path to deduce transitivity (Ax2) and exchangeability (Ax3). The work establishes structural results and furnishes conjectures on additional nontrivial single axiom schemata, while highlighting computational approaches to advance foundations of and the calculus of names. Overall, the paper clarifies how single axiom schemata can robustly characterize while offering a framework to explore a broad landscape of nontrivial axioms and their interrelations. The findings have implications for the foundational study of Lesniewski's system and its propositional ontology, with potential computational tools aiding further exploration.

Abstract

On March 8, 1995, was found the following \it nontrivial \rm single axiom-schema characteristic of Leśniewski-Ishimoto's propositional ontology (Inoué, 1995b \cite{inoue16}). 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 (), (), () and () based on (), their nontriviality and quasi-nontriviality. In §9 - §11, we shall give a lot of conjectures for nontrivial single axiom schemata for . We shall conclude this paper with summary and some remarks in §12.
Paper Structure (12 sections, 13 theorems, 115 equations)

This paper contains 12 sections, 13 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. The definition of nontrivial axiom schemata
  3. The criterion, the quasi-nontiriviality
  4. Structure theorems for nontriviality and quasi-nontriviality
  5. A $\bf L_1$-provable formula to deduce transitivity (Ax2) and exchangeability (Ax3)
  6. Simplified axiom schemata $\mathrm{(A_{S1})}$, $\mathrm{(A_{S2})}$, $\mathrm{(A_{S3N})}$ and $\mathrm{(A_{S3Nd})}$ characteristic of $\bf L_1$
  7. Nontriviality of $\mathrm{(A_{S1})}$, $\mathrm{(A_{S2})}$, $\mathrm{(A_{S3N})}$ and $\mathrm{(A_{S3Nd})}$ with respect to $\mathrm{(A_{t})}$
  8. The quasi-nontrivialities
  9. Conjecture 1. $A_k$-type single axiom schemata
  10. Conjecture 2. Additional single axiom schemata, Part I
  11. Conjecture 3. Additional single axiom schemata, Part II
  12. Summary and Remarks

Key Result

Proposition 1.1

For any uniform substitution $\sigma$ for meta-name variables, we have

Theorems & Definitions (21)

  • Definition 1.1
  • Proposition 1.1
  • Proposition 1.2
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.1
  • Definition 2.3
  • Proposition 2.2
  • Definition 3.1
  • Proposition 3.1
  • ...and 11 more