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The Disjunction-Free Fragment of D2 is Three-Valued

Hitoshi Omori

TL;DR

The disjunction-free fragment of Ja\'skowski's discussive logic D2 in the language of classical logic is shown to be complete with respect to three- and four-valued semantics with a rather simple axiomatization of the disjunction-free fragment of D2.

Abstract

In this article, the disjunction-free fragment of Jaśkowski's discussive logic D2 in the language of classical logic is shown to be complete with respect to three- and four-valued semantics. As a byproduct, a rather simple axiomatization of the disjunction-free fragment of D2 is obtained. Some implications of this result are also discussed.

The Disjunction-Free Fragment of D2 is Three-Valued

TL;DR

The disjunction-free fragment of Ja\'skowski's discussive logic D2 in the language of classical logic is shown to be complete with respect to three- and four-valued semantics with a rather simple axiomatization of the disjunction-free fragment of D2.

Abstract

In this article, the disjunction-free fragment of Jaśkowski's discussive logic D2 in the language of classical logic is shown to be complete with respect to three- and four-valued semantics. As a byproduct, a rather simple axiomatization of the disjunction-free fragment of D2 is obtained. Some implications of this result are also discussed.
Paper Structure (15 sections, 12 theorems, 24 equations)

This paper contains 15 sections, 12 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. Semantics and proof theory
  3. Semantics for the disjunction-free fragment of D$_2$
  4. Proof system for the disjunction-free fragment of D$_2$
  5. Soundness and Completeness for the three-valued semantics
  6. Soundness
  7. Completeness
  8. The main result
  9. Reflections
  10. The language $\mathcal{L}_l$
  11. The language $\mathcal{L}$
  12. Discussive negation
  13. Disjunction
  14. An application
  15. Concluding remarks

Key Result

Proposition 10

For all $\Gamma\cup \{ A, B \} \subseteq \mathsf{Form}_r^-$, $\Gamma,A\vdash B$ iff $\Gamma\vdash A{\to_d} B$.

Theorems & Definitions (34)

  • Definition 1: D$_2^-$-model
  • Remark 2
  • Definition 3
  • Remark 4
  • Definition 5
  • Remark 6
  • Remark 7
  • Definition 8
  • Remark 9
  • Proposition 10
  • ...and 24 more