INEFFABILITY AND REVENGE CHRIS SCAMBLER Abstract. In recent work Philip Welch has proven the existence of ‘ineffable liars’ for Hartry Field’s theory of truth. These are offered as liar-like sentences that escape classification in Field’s transfinite hierarchy of determinateness operators. In this article I present a slightly more general characterization of the ineffability phenomenon, and discuss its philosophical significance. I show the ineffable sentences to be less ‘liar-like’ than they appear in Welch’s presentation. I also point to some open technical problems whose resolution would greatly clarify the philosophical issues raised by the ineffability phenomenon. §1. Introduction. In [2] and [4], Hartry Field sets out a paracomplete solu- tion to the paradoxes of truth that he claims is revenge immune. Part of the basis for this claim is his theory’s provision of a transfinite hierarchy of definable determinacy operators D ˙ α . 1 DA is intended to mean that A is determinately true; DDA (or D 2 A) that A is determinately determinately true; and so on. These D α s are associated with theorems to the effect that certain ‘intuitively paradoxical’ sentences are α-indeterminate for some α. The standard liar Q 0 , for example, is equivalent to its own untruth (¬T Q 0 ), which implies (in Field’s logic) that it isn’t determinate (¬DQ 0 ) by elementary reasoning. In fact one can argue in general that where Q α is equivalent to its own α-indeterminacy, expressed by ¬D α T Q α , we always have ¬D α+1 Q α , that is, that Q α is α + 1- indeterminate. The Q α s are a natural class of paradoxical sentences, and the arguments for their α + 1-indeterminacy are simple. One can create more and more complex paradoxical sentences working in Field’s object language by e.g iterated com- binations of variants on Curry and liar sentences, using the D operator. It is natural to wonder whether every ‘intuitively paradoxical’ sentence in Field’s lan- guage can be shown to be α-indeterminate (for some α) in this way, though it should be said that even formulating this idea precisely is not a straightforward matter. (There will be some discussion on this below.) In recent work Philip Welch has turned his attention to these issues and derived some interesting and philosophically noteworthy results. In particular he has shown that, over any given (standard) ground model M , there will be sentences W in the language of Field’s theory whose “ultimate value” is 1 2 in the Field This is the final draft of a paper due to appear in the Review of Symbolic Logic. 1 I use ˙ α in the paper to denote a notation for the ordinal α in a given formal language. Generally I may leave of the dots when notation systems aren’t explicitly relevant. 1