**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.