Intuitionism and the Modal Logic of Vagueness

Susanne Bobzien,Ian Rumfitt

doi:10.1007/s10992-019-09507-x

Abstract

Intuitionistic logic provides an elegant solution to the Sorites Paradox. Its acceptance has been hampered by two factors. First, the lack of an accepted semantics for languages containing vague terms has led even philosophers sympathetic to intuitionism to complain that no explanation has been given of why intuitionistic logic is the correct logic for such languages. Second, switching from classical to intuitionistic logic, while it may help with the Sorites, does not appear to offer any advantages when dealing with the so-called paradoxes of higher-order vagueness. We offer a proposal that makes strides on both issues. We argue that the intuitionist’s characteristic rejection of any third alethic value alongside true and false is best elaborated by taking the normal modal system S4M to be the sentential logic of the operator ‘it is clearly the case that’. S4M opens the way to an account of higher-order vagueness which avoids the paradoxes that have been thought to infect the notion. S4M is one of the modal counterparts of the intuitionistic sentential calculus (IPC) and we use this fact to explain why IPC is the correct sentential logic to use when reasoning with vague statements. We also show that our key results go through in an intuitionistic version of S4M. Finally, we deploy our analysis to reply to Timothy Williamson’s objections to intuitionistic treatments of vagueness.

Highlights

Intuitionistic logic provides an elegant solution to the Sorites Paradox
The lack of an accepted semantics for languages containing vague terms has led even philosophers sympathetic to intuitionism to complain that no explanation has been given of why intuitionistic logic is the correct logic for such languages
In this paper we offer a proposal that makes strides on both issues: higher-order vagueness and the question why intuitionistic logic is the correct logic for vague languages

Summary

Wright’s Argument for Intuitionism as the Logic of Vague Statements

Crispin Wright has drawn attention to a version of the Sorites Paradox that is especially stark and difficult to solve (see [38, 39, 41] and especially [40]). The correct response, Wright claims, is to refrain both from asserting and from denying ‘Some red tube in the sequence is immediately followed by a non-red tube’. For many of those predicates, it will be possible to construct a Sorites sequence To any such predicate, Wright’s argument will apply: on pain of contradicting clear truths, we must deny that there is no sharp cut-off point, whilst resisting the unsupported assertion that there is such a cut-off. Wright’s argument will apply: on pain of contradicting clear truths, we must deny that there is no sharp cut-off point, whilst resisting the unsupported assertion that there is such a cut-off This position is stable only if there are restrictions on the elimination of double negations; so we have an argument in favour of a logic which (like intuitionistic logic) imposes such restrictions. Will be on the thesis that IPC is the correct sentential logic of vagueness

Dummett’s Challenge

The Modal Logic of Vagueness

Consequence for Vague Statements

An Intuitionistic Modal Logic of Clarity