Vagueness and the Connectives

TL;DR

This paper explores how vagueness in natural language suggests weakening classical logic, comparing orthologic and intuitionistic logic as potential non-classical bases to better model linguistic phenomena.

Contribution

It analyzes the suitability of orthologic and intuitionistic logic as foundational frameworks to account for vagueness and epistemic modals in natural language.

Findings

Orthologic may not fully capture vagueness.

Intuitionistic logic offers a promising alternative.

A weaker logic might better accommodate linguistic non-classicality.

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.