From real analysis to the sorites paradox via Reverse Mathematics

TL;DR

This paper analyzes the logical foundations of the sorites paradox using reverse mathematics, linking classical and non-constructive principles to different formulations of the paradox.

Contribution

It provides a reverse mathematical classification of various sorites paradoxes, connecting them to foundational principles like ACA₀ and WKL₀.

Findings

The classical sorites relies on H"older's Representation Theorem.

The continuous sorites depends on arithmetical comprehension (ACA₀).

The covering sorites depends on the Heine-Borel Theorem and WKL₀.

Abstract

This paper presents a reverse mathematical analysis of several forms of the sorites paradox. We first illustrate how traditional formulations are reliant on H\"older'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 K\"onig'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.