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A note on Grigoriev and Zaitsev's system CNL$^2_4$

Hitoshi Omori, Jonas R. B. Arenhart

TL;DR

The paper investigates the four-valued logic $CNL^2_4$ introduced by Grigoriev & Zaitsev, examining how a Haackian two-valued perspective can render its connectives and truth values intelligible to classical logicians. It identifies a robust Carnapian underdetermination, showing at least four compatible readings of the same deductive system that reinterpret truth, falsity, and negation in distinct ways. Through Dunn semantics, it constructs a two-valued representation that clarifies the semantic core and demonstrates functional completeness, ensuring a definable classical negation inside the system. The work highlights the interplay between meaning variance and deductive equivalence, arguing that the Haackian approach preserves classical intelligibility while acknowledging genuine interpretive flexibility surrounding negation and related connectives.

Abstract

The present article examines a system of four-valued logic recently introduced by Oleg Grigoriev and Dmitry Zaitsev. In particular, besides other interesting results, we will clarify the connection of this system to related systems developed by Paul Ruet and Norihiro Kamide. By doing so, we discuss two philosophical problems that arise from making such connections quite explicit: first, there is an issue with how to make intelligible the meaning of the connectives and the nature of the truth values involved in the many-valued setting employed -- what we have called `the Haackian theme'. We argue that this can be done in a satisfactory way, when seen according to the classicist's light. Second, and related to the first problem, there is a complication arising from the fact that the proof system advanced may be made sense of by advancing at least four such different and incompatible readings -- a sharpening of the so-called `Carnap problem'. We make explicit how the problems connect with each other precisely and argue that what results is a kind of underdetermination by the deductive apparatus for the system.

A note on Grigoriev and Zaitsev's system CNL$^2_4$

TL;DR

The paper investigates the four-valued logic introduced by Grigoriev & Zaitsev, examining how a Haackian two-valued perspective can render its connectives and truth values intelligible to classical logicians. It identifies a robust Carnapian underdetermination, showing at least four compatible readings of the same deductive system that reinterpret truth, falsity, and negation in distinct ways. Through Dunn semantics, it constructs a two-valued representation that clarifies the semantic core and demonstrates functional completeness, ensuring a definable classical negation inside the system. The work highlights the interplay between meaning variance and deductive equivalence, arguing that the Haackian approach preserves classical intelligibility while acknowledging genuine interpretive flexibility surrounding negation and related connectives.

Abstract

The present article examines a system of four-valued logic recently introduced by Oleg Grigoriev and Dmitry Zaitsev. In particular, besides other interesting results, we will clarify the connection of this system to related systems developed by Paul Ruet and Norihiro Kamide. By doing so, we discuss two philosophical problems that arise from making such connections quite explicit: first, there is an issue with how to make intelligible the meaning of the connectives and the nature of the truth values involved in the many-valued setting employed -- what we have called `the Haackian theme'. We argue that this can be done in a satisfactory way, when seen according to the classicist's light. Second, and related to the first problem, there is a complication arising from the fact that the proof system advanced may be made sense of by advancing at least four such different and incompatible readings -- a sharpening of the so-called `Carnap problem'. We make explicit how the problems connect with each other precisely and argue that what results is a kind of underdetermination by the deductive apparatus for the system.
Paper Structure (17 sections, 11 theorems, 2 equations)

This paper contains 17 sections, 11 theorems, 2 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic observations
  4. An alternative proof system
  5. Functional completeness
  6. A few more results
  7. Carnapian theme: Four interpretations of one truth table
  8. Option 1
  9. Option 2
  10. Option 3
  11. Option 4
  12. A summary of the four options
  13. Haackian theme: another representation for classicists
  14. Reflections
  15. Meaning of the connectives
  16. ...and 2 more sections

Key Result

Theorem 1

For all $\Gamma\cup \{ A \}\subseteq \mathsf{Form}$, $\Gamma\vdash A$ only if $\Gamma\models_{\bf CNL^2_4}A$.

Theorems & Definitions (28)

  • Definition 1: Grigoriev & Zaitsev
  • Definition 2
  • Remark 3
  • Definition 4
  • Theorem 1: Soundness
  • Definition 5
  • Lemma 1: Lindenbaum
  • Definition 6
  • Remark 7
  • Lemma 2
  • ...and 18 more