ORIGINAL PAPER How to eliminate self-reference: a précis

Abstract We provide a systematic recipe for eliminating self-reference from a sim-ple 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 languageL∗. 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 byYablo (1993,Analysis, 53, 251– 252; 2004, Self-reference, CSLI) and Cook (2004, Journal of Symbolic Logic, 69(3), 767–774), shows that under certain conditions self-reference is not essential to any of the semantic phenomena that can be obtained in a simple language. Keywords Paradox · Semantics · Self-reference · Yablo’s paradox 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 〈s,¬Tr(s)〉, with the convention that the term s denotes the formula ¬Tr(s)). It used to be thought that self-reference1 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 1 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 fals