How to eliminate self-reference: a précis

We provide a systematic recipe for eliminating self-reference from a simple language in which semantic paradoxes (whether purely logical or empirical) can be expressed. We start from a non-quantificational language L which contains a truth predicate and sentence names, and we associate to each sentence F of L an infinite series of translations h0(F), h1(F), ..., stated in a quantificational language L*. Under certain conditions, we show that none of the translations is self-referential, but that any one of them perfectly mirrors the semantic behavior of the original. The result, which can be seen as a generalization of recent work by Yablo (1993, 2004) and Cook (2004), shows that under certain conditions self-reference is not essential to any of the semantic phenomena that can be obtained in a simple language. 1 Is Self-Reference Semantically Eliminable? The Liar (This sentence is not true) is paradoxical by virtue of its self-reference. If Tr is the truth predicate, the Liar can be seen as a sentence ¬Tr(s) named by a constant s (something we will represent as a pair , with the convention that the term s denotes the formula ¬Tr(s)). It used to be thought that self-reference is always a crucial ingredient of semantic paradoxes. Yablo (1993, 2004) showed that this was not so; he constructed an infinite series of sentences none of which is self-referential but which, taken together, yield a paradox. In its simplest form, Yablo's paradox consists of an infinite set of linearly ordered sentences, each of which claims that all the sentences following it are false. Using the notation we just introduced, the series can be represented as the set {i → ¬Tr(s(k))>: i≥0}, where for each integer i, s(i) is intended to name the formula ∀k (k>i → ¬Tr(s(k)) [when the intended denotation is clear, we will often use s(i) in the metalanguage to refer to the formula that the term s(i) is supposed to denote, i.e. ∀k (k>i → ¬Tr(s(k))]. We may then reason as follows: If all sentences in the series are false, we obtain an obvious contradiction because what s(0) asserts should be true. If some sentence, say s(i), is true, it must be the case that for all k>i (and hence in particular for all k>i+1), s(k) is false. But this should suffice to make s(i+1) true, which again yields a contradiction. Thus no bivalent valuation can be found for Yablo's series (Yablo 2004 discusses other versions of his paradox, a point to which we return below). There has been considerable debate to determine whether Yablo's result involves some 'concealed' self-reference (see Leitgeb 2002 for a particularly clear discussion). In the present paper we will assume that it does not, and we will ask instead how far Yablo's result can be generalized. Thus we will ask whether self-reference ever plays a crucial semantic role, be it in the production of paradoxes or of other semantic phenomena. We will show that to the extent that Yablo's sentences are not self-referential, self-reference can be systematically eliminated from a simple language in which both logical and empirical paradoxes can be expressed (for brevity we will henceforth omit the condition to the extent 1I wish to thank the following for helpful discussions: Denis Bonnay, Serge Bozon, Paul Egré, Marcus Kracht, Tony Martin, Benjamin Spector, Albert Visser, Steve Yablo, Ede Zimmermann, and audiences at IHPST, UCLA, U. of Amsterdam, U. of Frankfurt, ECAP'05, and Sinn und Bedeutung '05. Special thanks to Denis Bonnay and Albert Visser for extremely helpful corrections and suggestions, and to Paul Egré for comments on a first draft of the present paper. The author gratefully acknowledges the financial support of the American Council of Learned Societies (‘Ryskamp Fellowship’) and of UCLA. A longer and more technical version of this analysis is offered in Schlenker 2006. 2 By self-reference, we mean direct or indirect self-reference (this broad notion is sometimes called 'circularity' in the literature). It has long been known that paradoxes can be produced with indirect self-reference consider for instance a pair of sentences a and b, where a says: b is true, and where b says: a is false.