Vagueness, Truth and Permissive Consequence

Abstract

We say that a sentence A is a permissive consequence of a set of premises Γ whenever, if all the premises of Γ hold up to some standard, then A holds to some weaker standard. In this paper, we focus on a three-valued version of this notion, which we call strict-to-tolerant consequence, and discuss its fruitfulness toward a unified treatment of the paradoxes of vagueness and self-referential truth. For vagueness, st-consequence supports the principle of tolerance; for truth, it supports the requisit of transparency. Permissive consequence is non-transitive, however, but this feature is argued to be an essential component to the understanding of paradoxical reasoning in cases involving vagueness or self-reference.

Vagueness is a kind of conflation

“…The overall shape of the logic, though, will depend on the shape of the underlying logic taken to apply to unconflated or precise terms. When this logic is classical, the resulting logic for vagueness turns out to bear very close connections to the logics proposed in [6,8,22]; exactly how close these connections prove to be is a question for future work.…”

“…8 Rather than attempt a full picture, I only present those parts of the formal treatment that I will appeal to in Section 2, where I turn to vagueness. This treatment attempts to capture validity in a representational system exhibiting conflation; it says nothing about truth, reference, or the like.…”

Tolerant reasoning: nontransitive or nonmonotonic?

“…Following van Fraassen (1969), we give the following 17 Other pragmatic strategies can be proposed that would agree with the idea of interpreting a sentence strictly if possible. See Egré and Zehr (2016) for a proposal to compute pragmatic strengthening more locally, and for comparisons with the algorithm in Cobreros et al (2015b). Both approaches can answer the challenge raised by Alxatib et al, but they make different predictions in other specific cases.…”

“…Unfortunately, as discussed in Cobreros et al (2015b), this interpretation rule won't do for our purposes. Recall that according to Cobreros et al (2012), to communicate that John is a borderline case of a tall man one can say 'John is tall and not tall'.…”

“…Alxatib et al (2013) show that this wrongly predicts that a sentence like 'Adam is tall and not tall, or John is rich' entails that John is strictly rich, although it should not entail this and intuitively should mean that either Adam is borderline tall or John is strictly rich. In Cobreros et al (2015b) we responded by providing a more sophisticated pragmatic interpretation rule to strengthen the meaning of a sentence, which we review below. 17…”