A note on the reverse mathematics of the sorites

Abstract

Sorites is an ancient piece of paradoxical reasoning pertaining to sets with the following properties: (Supervenience) elements of the set are mapped into some set of "attributes", (Tolerance) if an element has a given attribute then so are the elements in some vicinity of this element, and (Connectedness) such vicinities can be arranged into pairwise overlapping finite chains connecting two elements with different attributes. Obviously, if Superveneince is assumed, then (1) Tolerance implies lack of Connectedness, and (2) Connectedness implies lack of Tolerance. Using a very general but precise definition of "vicinity", Dzhafarov and Dzhafarov (2010) offered two formalizations of these mutual contrapositions. Mathematically, the formalizations are equally valid, but in this paper, we offer a different basis by which to compare them. Namely, we show that the formalizations have different proof-theoretic strengths when measured in the framework of reverse mathematics: the formalization of (1) is provable in $\mathsf{RCA}_0$, while the formalization of (2) is equivalent to $\mathsf{ACA}_0$ over $\mathsf{RCA}_0$. Thus, in a certain precise sense, the approach of (1) is more constructive than that of (2).