Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Non-distributive relatives of ETL and NFL

Daniil Kozhemiachenko

Abstract

In this paper, we devise non-distributive relatives of Exactly True Logic (ETL) by Pietz and Riveccio and its dual (NFL) Non-Falsity Logic by Shramko, Zaitsev and Belikov. We consider two pre-orders which are algebraic counterparts of the ETL's and NFL's entailment relations on the De Morgan lattice $\mathbf{4}$. We generalise these pre-orders and determine which distributive properties that hold on $\mathbf{4}$ are not forced by either of the pre-orders. We then construct relatives of ETL and NFL but lack such distributive properties. For these logics, we also devise a truth table semantics which uses non-distributive lattice $\mathbf{M3}$ as their lattice of truth values. We also provide analytic tableaux systems that work with sequents of the form $φ\vdashχ$. We also prove the correctness and completeness results for these proof systems and provide a neat generalisation for non-distributive ETL- and NFL-like logics built over a certain family of non-distributive modular lattices.

Non-distributive relatives of ETL and NFL

Abstract

In this paper, we devise non-distributive relatives of Exactly True Logic (ETL) by Pietz and Riveccio and its dual (NFL) Non-Falsity Logic by Shramko, Zaitsev and Belikov. We consider two pre-orders which are algebraic counterparts of the ETL's and NFL's entailment relations on the De Morgan lattice . We generalise these pre-orders and determine which distributive properties that hold on are not forced by either of the pre-orders. We then construct relatives of ETL and NFL but lack such distributive properties. For these logics, we also devise a truth table semantics which uses non-distributive lattice as their lattice of truth values. We also provide analytic tableaux systems that work with sequents of the form . We also prove the correctness and completeness results for these proof systems and provide a neat generalisation for non-distributive ETL- and NFL-like logics built over a certain family of non-distributive modular lattices.
Paper Structure (17 sections, 20 theorems, 39 equations, 3 figures)

This paper contains 17 sections, 20 theorems, 39 equations, 3 figures.

Table of Contents

  1. Introduction
  2. FDE and its relatives
  3. Non-distributive logics
  4. Entailment as lattice pre-order
  5. Plan of the paper
  6. Distributive properties and pre-orders on bounded lattices
  7. Semantics for non-distributive relatives of ETL and NFL
  8. Analytic tableaux
  9. Motivation
  10. A unified tableau calculus for $\mathrm{ETL}_{\mathbf{M3}}$ and $\mathrm{NFL}_{\mathbf{M3}}$
  11. Correctness and completeness
  12. Generalisation
  13. Tuning the tableaux
  14. Separating $\mathbf{Mn}$ and $\mathbf{Mn+1}$
  15. $\mathbf{M\omega}$
  16. ...and 2 more sections

Key Result

Lemma 2.1

For any matrix $\mathrm{ETL}_{\mathfrak{L}}$ over a bounded lattice $\mathfrak{L}$ the following holds.

Figures (3)

  • Figure 1: $\mathbf{Mn}$ lattices. From left to right: the three-element lattice, 4, M3, etc. In general, $\mathbf{Mn}$ lattice contains $n$ elements on its middle level.
  • Figure 2: A tableaux showing that there is no valuation $v$ such that $v((p\vee q)\wedge r)=\mathbf{T}$ but $v(p\vee(q\wedge r))\in\{\mathbf{B},\mathbf{0},\mathbf{N}\}$. All branches are closed.
  • Figure 3: A tableau refuting $(p\wedge\neg p)\vee(q\wedge\neg q)\nvdash r$. The frownie shows a complete open branch.

Theorems & Definitions (37)

  • Definition 2.1: ETL- and NFL-like logics
  • Lemma 2.1
  • Lemma 2.1.1
  • proof
  • Lemma 2.1.2
  • Lemma 2.1.3
  • proof
  • Lemma 2.1.4
  • proof
  • Lemma 2.1.5
  • ...and 27 more