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Vagueness and the Connectives

Wesley H. Holliday

TL;DR

This paper investigates which non-classical base logic best accommodates two sources of non-classicality in natural language: epistemic modals and vagueness. It surveys candidate logics—orthologic, fundamental logic, and compatibility logic—and develops relational fixpoint semantics to test their behavior on Sorites-style arguments. By constructing symmetric and pseudosymmetric Sorites models, the author shows orthologic can render Sorites consistent via non-distributivity, while fundamental logic preserves genuine expressive power through excluded middle in a targeted way, suggesting it as a natural weaker base than orthologic for combining vagueness with modality. The work clarifies the trade-offs between distributivity, double negation, and excluded middle, and highlights a path for future comparison with other vagueness theories such as supervaluationism and epistemicism.

Abstract

Challenges to classical logic have emerged from several sources. According to recent work, the behavior of epistemic modals in natural language motivates weakening classical logic to orthologic, a logic originally discovered by Birkhoff and von Neumann in the study of quantum mechanics. In this paper, we consider a different tradition of thinking that the behavior of vague predicates in natural language motivates weakening classical logic to intuitionistic logic or even giving up some intuitionistic principles. We focus in particular on Fine's recent approach to vagueness. Our main question is: what is a natural non-classical base logic to which to retreat in light of both the non-classicality emerging from epistemic modals and the non-classicality emerging from vagueness? We first consider whether orthologic itself might be the answer. We then discuss whether accommodating the non-classicality emerging from epistemic modals and vagueness might point in the direction of a weaker system of fundamental logic.

Vagueness and the Connectives

TL;DR

This paper investigates which non-classical base logic best accommodates two sources of non-classicality in natural language: epistemic modals and vagueness. It surveys candidate logics—orthologic, fundamental logic, and compatibility logic—and develops relational fixpoint semantics to test their behavior on Sorites-style arguments. By constructing symmetric and pseudosymmetric Sorites models, the author shows orthologic can render Sorites consistent via non-distributivity, while fundamental logic preserves genuine expressive power through excluded middle in a targeted way, suggesting it as a natural weaker base than orthologic for combining vagueness with modality. The work clarifies the trade-offs between distributivity, double negation, and excluded middle, and highlights a path for future comparison with other vagueness theories such as supervaluationism and epistemicism.

Abstract

Challenges to classical logic have emerged from several sources. According to recent work, the behavior of epistemic modals in natural language motivates weakening classical logic to orthologic, a logic originally discovered by Birkhoff and von Neumann in the study of quantum mechanics. In this paper, we consider a different tradition of thinking that the behavior of vague predicates in natural language motivates weakening classical logic to intuitionistic logic or even giving up some intuitionistic principles. We focus in particular on Fine's recent approach to vagueness. Our main question is: what is a natural non-classical base logic to which to retreat in light of both the non-classicality emerging from epistemic modals and the non-classicality emerging from vagueness? We first consider whether orthologic itself might be the answer. We then discuss whether accommodating the non-classicality emerging from epistemic modals and vagueness might point in the direction of a weaker system of fundamental logic.

Paper Structure

This paper contains 8 sections, 15 theorems, 17 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Logics
  3. Relational semantics for ortho and fundamental logic
  4. Relational semantics for compatibility logic
  5. The symmetric Sorites model
  6. A pseudosymmetric Sorites model
  7. Discussion
  8. Conclusion

Key Result

lemma thmcounterlemma

In a relational frame $(X,\vartriangleleft)$, $x\vartriangleleft y$ iff $x$ does not reject any proposition that $y$ accepts.

Figures (2)

  • Figure 1: The symmetric Sorites model for $n=4$ and $\delta=1$.
  • Figure 2: The pseudosymmetric Sorites model for $n=4$ and $\delta=1$. We indicate $x\vartriangleleft y$ by an arrow from $y$ to $x$. The arrow from $k$ to the dashed rectangle indicates arrows from $k$ to all states inside the rectangle. An edge with no arrow head indicates arrows in both direction.

Theorems & Definitions (43)

  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • lemma thmcounterlemma
  • proof
  • definition thmcounterdefinition
  • lemma thmcounterlemma
  • definition thmcounterdefinition
  • ...and 33 more