In what sense is the no-no paradox a paradox?

Abstract

Cook regards Sorenson's so-called 'the no-no paradox' as only a kind of 'meta-paradox' or 'quasi-paradox' because the symmetry principle that Sorenson imposes on the paradox is meta-theoretic. He rebuilds this paradox at the objectlanguage level by replacing the symmetry principle with some 'background principles governing the truth predicate'. He thus argues that the no-no paradox is a 'new type of paradox' in that its paradoxicality depends on these principles. This paper shows that any theory (not necessarily meeting Cook's background principles) is inconsistent with the T-schema instances for the no-no sentences, plus the T-schema instance for a Curry sentence associated with the symmetry of the no-no sentences. It turns out that the no-no paradox still depends on the problematic instances of the T-schema in a way that the liar paradox does. What distinguishes the no-no paradox is the T-schema instance for the above Curry sentence, which encodes Sorensen's symmetry principle at the object-language level.

Paradoxicality in Kripke’s theory of truth

“…The main thought underpinning the accounts of paradoxicality we will consider is what we shall call "the fixed-point conception of paradoxicality". 3 According to this, a statement is paradoxical if, and only if, there is no fixed-point at which it obtains a classical truth-value, where 'classical truth-value' is meant to exclude statements that are neither-true-nor-false and statements 1 Just to give the reader an idea, we can identify (i) the naive conception of paradoxicality (see Cook (2011) and Hsiung (2021)); (ii) the conception of paradoxicality as non-normalizability (cf. Prawitz (1965) and Tennant (1982)); (iii) the revision-theoretic conception (cf.…”