Field on Revenge ∗

In a series of recent papers, 1 Hartry Field has proposed a novel class of solutions to the semantic paradoxes, and argued that the new solutions are 'revenge-immune'. He has argued, in particular, that by building on a sufficiently expressive language one can get a language which is able to express its own semantic theory, including its own truth predicates and any intelligible determinacy predicates. The purpose of this note is to argue that the plausibility of Field's revenge-immunity claim depends crucially on the status of higher-order languages. We show that by availing oneself of higher-order resources one can give an explicit characterization of the key semantic notion underlying Field's proposal, and note that inconsistency would ensue if the languages under discussion were expressive enough to capture this notion. For the sake of concreteness, we shall consider the version of the proposal developed in Field (2003a) and further elucidated in Field's contribution to the present volume. Start with a standard first-order language L containing the set-theoretic primitives '∈' and 'Set(.. .)', together with an arbitrary selection of non-set-theoretic predicates. On its intended interpretation, the range of L's quantifiers includes all sets, '∈' expresses * Thanks to Hartry Field, Hannes Leitgeb and Graham Priest for their many helpful comments. Special thanks are due to Vann McGee. † PDW acknowledges support from the British Academy and EPSRC Research Grant (EP/C531485/1).