Thanks to the long weekend I finally found some time to write up a first draft of a paper that I've meant to write for quite a while. (I know, spending the long weekend doing research is a tad pathetic and geeky, but I really wouldn't have time to do this some other time). It's about a thing that seems to have been settled - the definability of identity. Below is the abstract. The full version of the paper is here.
It is a commonplace remark that the identity relation, even though not expressible in a first-order language without identity with classical set-theoretic semantics, can be defined in a language without identity, as soon as we admit second-order, set-theoretically interpreted quantifiers binding predicate variables that range over all subsets of the domain. However, there are fairly simple and intuitive higher-order languages with set-theoretic semantics (where the variables range over all subsets of the domain) in which the identity relation is not definable. The point is that the definability of identity in higher-order languages not only depends on what variables range over, but also is sensitive to how predication is construed.
Comments
Just follow the link to the full paper, it's not too long, just a few pages. The idea is quite simple (btw, it's been accepted and is forthcoming in the Australasian Journal of Logic).