Slop Factor

On the Definability of Strong Negation in Bilateral Logics

Ryan Simonelli

arXiv:2607.27252v1 Primary: math.LO

Abstract

Recent work in bilateral logic has concerned itself with whether and to what extent strong negation, which "toggles" between assertion and denial, is definable in bilateral systems that lack it. This paper presents a number of results about the definability of strong negation in bilateral systems. First, I show that a constructive Nelson-style Sheffer stroke defines the strong negation of A as (A|(A|A))|A. The same stroke defines the constructive Nelson implication, thus providing a single-connective basis for the strong-negation-and-implication fragment of N4. Second, I provide an exhaustive characterization of the eight combinations of a single "aggregative" connective and a single Nelson-style connective that suffice to jointly define strong negation, with neither defining it individually. One such combination is the material conditional together with constructive co-implication. Third, I show that a constructive and connexive Wansing-style Sheffer stroke (recently referred to as "connexive exclusion") likewise defines the strong negation of A as (A|(A| A))|A and also defines connexive implication. Fourth, I show that, unlike with the Nelson-style connectives, there are no combinations of Wansing-style and aggregative connectives that jointly define strong negation if neither connective defines it individually.

Disclosure

“Pragmatics, Semantics, and Conceptual Roles. Routledge, New York, 2024. Norihiro Kamide and Heinrich Wansing. Proof theory of Nelson’s paraconsistent logic: A uniform perspective. Theoretical Computer Science, 415:1–38, 2012. 15 OpenAI’s ChatGPT (5.6 Sol) was used during the exploratory stages of this research to help search the space of possible connectives and relations among them, find relevant literature, and assist in the preliminary construction of the proofs of Propositions”

PDF page 17
Classification
Proof ideas or individual proof-step assistance
Multiplier
8
Verified

Structural counts

Pages 18 pdf
Theorems 0 source
Lemmas 0 source
Propositions 0 source
Corollaries 0 source
Definitions 0 source
Displayed equations 6 source
Bibliography entries 147 source
Appendix pages 0 estimated

Count notes

  • Source counts use the expanded primary TeX file Simonelli_-_On_the_Definability_of_Strong_Negation.tex.
  • Appendix pages include the first PDF page with an explicit Appendix heading through the final page.