Can the Classical Logician Avoid the Revenge Paradoxes?

Abstract

Most work on the semantic paradoxes within classical logic has centred around what I call 'linguistic' accounts of the paradoxes: they attribute to sentences or utterances of sentences some property that is supposed to explain their paradoxical or non-paradoxical status. 'No proposition' views are paradigm examples of linguistic theories, although practically all accounts of the paradoxes subscribe to some kind of linguistic theory. This paper shows that linguistic accounts of the paradoxes endorsing classical logic are subject to a particularly acute form of the revenge paradox: that there is no exhaustive classification of sentences into 'good' and 'bad' such that the T-schema holds when restricted to the 'good' sentences unless it is also possible to prove some 'bad' sentences. The foundations for an alternative classical non-linguistic approach is outlined which is not subject to the same kinds of problems. Although revenge paradoxes of different strengths can be formulated, they are found to be indeterminate at higher orders and not inconsistent.

Against Disquotation

“…The view that, for any w that means something, the corresponding instance of M is true, but the L-instance of M is not true because L doesn't mean anything, is beyond the scope of this paper. Note, however, that weakening M to "if 'w' means something, then 'w' is true if and only if w', combined with MT, is vulnerable to the revenge argument from Bacon [2015]: the theory, if consistent, will prove that some of its theorems don't mean anything. 17 There is a very different kind of INSTANTIATION-denying response, unlike the one that we have been considering so far, which does not involve uttering L (or accepting any sentence while simultaneously holding that the sentence means nothing that is the case).…”

Radical Anti‐disquotationalism

“…I have argued elsewhere that a principle of this form is the correct way to capture the restriction on disquotational reasoning about truth (Bacon []).…”

“…In other work (Bacon ) I have spelled out what I take to be the general form of this paradox for the classical logician. There I also offered an alternative revenge‐free approach that still falls within the paradigm of trying to offer an analysis of the ‘exceptions’ to the disquotational principles.…”

Model-theoretic semantics and revenge paradoxes

“…3Revenge paradoxes also exist for classical theories of truth (see e.g. Bacon 2015): my arguments for MT-revenge generalize to classical theories as well, but I will not explicitly discuss classical approaches, in the interest of space.…”