Tolerance and degrees of truth

TL;DR

This paper demonstrates that both fuzzy and strict-tolerant approaches to vagueness can be unified under a three-valued logical framework, simplifying the understanding of vagueness and its logical consequences.

Contribution

It shows that three-valued logic suffices for modeling vagueness, eliminating the need for continuum many values in Smith's approach, and unifies different logical theories of vagueness.

Findings

Three-valued logic captures Smith's core desiderata.

Continuum many values are unnecessary for solving the jolt problem.

A three-valued framework internalizes Smith's closeness principle.

Abstract

This paper explores the relations between two logical approaches to vagueness: on the one hand the fuzzy approach defended by Smith (2008), and on the other the strict-tolerant approach defended by Cobreros, Egr\'e, Ripley and van Rooij (2012). Although the former approach uses continuum many values and the latter implicitly four, we show that both approaches can be subsumed under a common three-valued framework. In particular, we defend the claim that Smith's continuum many values are not needed to solve what Smith calls `the jolt problem', and we show that they are not needed for his account of logical consequence either. Not only are three values enough to satisfy Smith's central desiderata, but they also allow us to internalize Smith's closeness principle in the form of a tolerance principle at the object-language. The reduction, we argue, matters for the justification of many-valuedness in an adequate theory of vague language.