From real analysis to the sorites paradox via Reverse Mathematics

TL;DR

The paper investigates the sorites paradox through reverse mathematics, showing that the continuous form hinges on the existence of suprema (equivalent to $ACA_0$) and the covering form relies on compactness via the Heine–Borel principle (equivalent to $WKL_0$). It grounds these results in measurement theory, using representation theorems to map vague predicates to real-valued scales, and employs recursive counterexamples to illuminate resolutions and philosophical implications. The work distinguishes two distinct paradoxes by their logical strength and connects them to foundational debates in vagueness (supervaluationism, epistemicism, constructivism). Overall, it provides a principled framework for understanding how the continuum and its mathematical properties drive paradoxical reasoning and when such paradoxes can be resolved by weakening or altering underlying assumptions.

Abstract

This paper presents a reverse mathematical analysis of several forms of the sorites paradox. We first illustrate how traditional formulations are reliant on Hölder's Representation Theorem for ordered Archimedean groups. While this is provable in RCA$_0$, we also consider two forms of the sorites which rest on non-constructive principles: the continuous sorites of Weber & Colyvan (2010) and a variant we refer to as the covering sorites. We show in the setting of second-order arithmetic that the former depends on the existence of suprema and thus on arithmetical comprehension (ACA$_0$) while the latter depends on the Heine-Borel Theorem and thus on Weak König's Lemma (WKL$_0$). We finally illustrate how recursive counterexamples to these principles provide resolutions to the corresponding paradoxes which can be contrasted with supervaluationist, epistemicist, and constructivist approaches.

Theorems & Definitions (16)

• Theorem 1
• Proposition 2
• proof
• Theorem 3
• Proposition 4
• proof
• Proposition 5
• Theorem 6
• proof
• Theorem 7