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LINGUISTIX&LOGIK, Tony Marmo's blog
Monday, 4 July 2005

Topic: Interconnections

The Elimination of Self-Reference
(Generalized Yablo-Series and the Theory of Truth)


By Phillippe Schlenker

Although it was traditionally thought that self-reference is a crucial ingredient of semantic paradoxes, Yablo showed that this is not so by displaying an infinite series of non-referential sentences which, taken together, are paradoxical (e.g. Yablo 2004). We generalize Yablo's result along two dimensions.
1. First, we investigate the behavior of Yablo-style series of the form {<s(i), [Qk: k> i] f[(s(k)) k≥i ]>: i≥0}, where for each i s( i) is a term that denotes the sentence [Qk: k> i] f[(s(k)) k≥i] ] (for some generalized quantifier Q and for some (fixed) truth function f). We show that for any n-valued compositional semantics and for any quantifier Q that satisfies certain natural properties, all the sentences in the series must have the same value. We derive a characterization of those values of Q for which the series is paradoxical in a natural trivalent logic.
2. Second, we show that in the Strong Kleene trivalent logic, Yablo's results are a special case of a much more general phenomenon: given certain assumptions, any semantic phenomenon that involves self-reference can be reproduced without self-reference (Cook 2004 proves a special case of this result, which only applies to logical paradoxes).

Specifically, we can associate to each pair <s, F> of a formula F named by a term s in a language L' a series of translations {<s( i), [Qk: k> i] [F] k>: i≥0} (where [F] kis a certain modification of F) in a quantificational language L* in such a way that
(i) none of the translations are self-referential,
(ii) in any fixed point I* of L*, all the translations of a given formula of L have the same value according to I*, and
(iii) there is a correspondence between the fixed points of L' and the fixed points of L* which ensures that the translations really do have the same semantic behavior as the sentences they translate.

We give a characterization of those generalized quantifiers Q which can be used in the translation.
Source: Online Papers in Philosophy

Posted by Tony Marmo at 07:15 BST
Updated: Monday, 4 July 2005 07:24 BST

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