Definability of Leibniz Equality

Equality in first-order logic is represented in the language by means of a logical symbol and so it is interpreted uniformly; the semantics says that its interpretation in any structure is the identity relation. This is the absolute notion of identity however, and in fact the only one deserving this name. But there is a related relation: the relation of indiscernibility. Given a structure for a first-order language L, two objects of its domain can be indiscernible relative to the properties expressible in L, without using the equality symbol, even without being the same. It is this last relation that interests us in this paper. Its definition can be seen as a relativization to a first-order language of the essentially second-order Leibniz principle of the identity of the indiscernibles. Namely, if A is a structure for L and a and b are two members of its domain, a and b are related if the condition