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

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 of a countable series of linearly ordered sentences s(0), s(1), s(2), ... , where each s(i) says: For each k>i, s(k) is false (or equivalently: For no k>i is s(k) true). We generalize Yablo's results along two dimensions. First, we study the behavior of generalized Yablo-series in which each sentence s(i) has the form: For Q k>i, s(k) is true, where Q is a generalized quantifier (e.g. no, most, every, etc). We show that under broad conditions all the sentences in the series must have the same truth value, and we derive a characterization of those values of Q for which the series is paradoxical. Second, we show that in the Strong Kleene trivalent logic Yablo's results are a special case of a more general fact: under certain conditions, any semantic phenomenon that involves self-reference can be emulated without self-reference. Various translation procedures that eliminate self-reference from a nonquantificational language are defined and characterized. An Appendix sketches an extension to quantificational languages, as well as a new argument that Yablo's paradox and the translations we offer do not involve selfreference. It was traditionally thought that self-reference is a crucial ingredient of semantic paradoxes. However 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 of a countable series of linearly ordered sentences s(0), s(1), s(2), ... , where each s(i) says: For each k>i, s(k) is false (or equivalently: For no k>i is s(k) true). The present paper has two goals. First, we study the behavior of generalized Yablo-series in which each sentence s(i) has the form: For Q k>i, s(k) is true, where Q is a generalized quantifier (for instance no, most, every, infinitely many, all but finitely many, etc). We show that under broad conditions all the sentences in the series must have the same truth value, and we derive from this 'Uniformity Property' a characterization of those values of Q for which the series is paradoxical. Second, we show that in the Strong Kleene trivalent logic Yablo's result is a special case of a much more general fact: under certain conditions, any semantic phenomenon that involves self-reference can be emulated without self-reference (this can be seen as a generalization of Cook 2004). Specifically, we provide a translation procedure that associates to each sentence s of a non-quantificational language an infinite series of quantificational sentences h0(s), h1(s), ... , where each s(i) has the form: For Q k>i, [s]k (with [s]k a certain modification of s). The procedure at work in Yablo's paradox can thus be extended to yield a general elimination of self-reference. Importantly, however, the elimination procedure only works in full generality for certain values of Q in essence, the quantifiers infinitely many and all but finitely many. 1 I thank the following for discussion of various stages of this work: Denis Bonnay, Serge Bozon, Paul Egré, Marcus Kracht, Tony Martin, Benjamin Spector, Albert Visser, Ede Zimmermann, an anonymous referee, as well as audiences at IHPST, UCLA, U. of Frankfurt, U. of Amsterdam, and Sinn und Bedeutung 2005. Special thanks are due to Albert Visser, who commented on this paper at a French-Dutch logic meeting in Amsterdam (PALMYR, June '05), and to Denis Bonnay, who corrected several errors and greatly simplified the proof of (the special case of) the Uniformity Property. 2 A shorter and less technical version of the present research can be found in Schlenker 2006. [I realized after finishing the present paper that G. Schlesinger published in Analysis 27. 6 (1967) a 4-page article entitled ‘Elimination of Self-Reference’, which reaches different conclusions.]