Full Idea
There is no condition in a first-order language for a predicate to express identity, rather than indiscernibility within the resources of the language. Leibniz's Law is statable in a second-order language, so identity can be uniquely characterised.
Gist of Idea
Identity can only be characterised in a second-order language
Source
Harold Noonan (Identity [2009], §2)
Book Reference
'Stanford Online Encyclopaedia of Philosophy', ed/tr. Stanford University [plato.stanford.edu], p.4
A Reaction
The point is that first-order languages only refer to all objects, but you need to refer to all properties to include Leibniz's Law. Quine's 'Identity, Ostension and Hypostasis' is the source of this idea.
Related Ideas
Idea 16014 It is controversial whether only 'numerical identity' allows two things to be counted as one [Noonan]
Idea 11095 We should just identify any items which are indiscernible within a given discourse [Quine]