A comment on Bermudez concerning the definability of identity

Abstract

José Luis Bermúdez's article ‘Indistinguishable elements and mathematical structuralism’ (2007) is a contribution to the debate about the ‘identity problem for structuralism’ (see Burgess 1999 and Keränen 2001) and includes comments on Ladyman 2005 and Ketland 2006.1 Bermúdez writes that Ketland 2006‘contains some puzzling claims’ and, in particular, that one specific result (given without proof) ‘seems badly wrong’. My comment is simply that the result that Bermúdez disputes is correct (as are the others). Below, a proof is given. After all, results concerning the definability of identity in relational structures may be of independent philosophical and logical interest.2 Bermúdez writes, Ketland's paper contains some puzzling claims about the first-order definability of identity. He claims that ‘one can show that if the identity relation is first-order definable in a structure at all, then it is defined by the indiscernibility formula’ (Ketland 2006: 307), where the identity formula is essentially Quine's trick for eliminating the identity predicate. This seems badly wrong, since Quine's trick only applies to structures with finite signatures. ... any structure which contains the identity predicate and has an infinite signature is a counter-example to his claim. (Bermúdez 2007: 113, n. 2)