Schlenker, P. 2005. "The Elimination of Self-Reference (Generalized Yablo-Series and the Theory of Truth)". Ms.,  UCLA & IJN (approx. 38 pages)


[Full paper in pdf]   

Abstract: Although it was traditionally thought that self-reference is a crucial ingredient of semantic paradoxes, Yablo (1993, 2004) showed that this was not so by displaying an infinite series of sentences none of which is self-referential but which, taken together, are paradoxical. Yablo's paradox consists in a series  {<s(i), [∀k: k>i] ¬Tr(s(k))>: i≥0}, where Tr is the truth predicate and where for each number i the term s(i) denotes the sentence [∀k: k>i] ¬Tr(s(k)). We generalize Yablo's result along two dimensions.
1. First, we investigate the general behavior of the series {<s(i), [Qk: k>i] f[(s(k))_k≥i]>: i≥0}, where Q is a generalized quantifier and where f is some fixed truth function. We show that under broad conditions all the sentences in the series must have the same value, and we derive a characterization of those values of Q for which the series is paradoxical.                
2. Second, we show that in the Strong Kleene trivalent logic, Yablo's results are a special case of a much more general phenomenon: under certain conditions,  any semantic phenomenon that involves self-reference can be reproduced without self-reference. This is shown by way of a translation which associates to each pair <s, F> in a non-quantificational language L' (where the term s denotes the sentence F) an infinite series of translations {<s(i), [Qk: k>i] [F]_k>: i≥0} in a quantificational language L*(where [F]_k is a modification of F). We provide a characterization of those values of Q for which the translation goes through, and we discuss various extensions of the translation procedure. The paper, which generalizes recent results by Cook (2004), shows that under certain conditions self-reference is not essential to any semantic phenomena.