 Numerical identity can be characterised, as just done, as the relation everything has to itself and to nothing else. But this is circular, since “nothing else” just means “no numerically non-identical thing”. It can be defined, equally circularly (because quantifying over all equivalence relations including itself), as the smallest equivalence relation (an equivalence relation being one which is reflexive, symmetric and transitive, for example, having the same shape). Other circular definitions are available. Usually it is defined as the equivalence relation (or: the reflexive relation) satisfying Leibniz’s Law, the principle of the indiscernibility of identicals, that if x is identical with y then everything true of x is true of y. Intuitively this is right, but only picks out identity uniquely if “what is true of x” is understood to include “being identical with x”; otherwise it is too weak. Circularity is thus not avoided. Nevertheless, Leibniz’s Law appears to be crucial to our understanding of identity, and, more particularly, to our understanding of distinctness: we exhibit our commitment to it whenever we infer from “Fa” and “Not-Fb” that a is not identical with b. Strictly, what is being employed in such inferences is the contrapositive of Leibniz’s Law (if something true of a is false of b, a is not identical with b), which some (in the context of the discussion of vague identity) have questioned, but it appears as indispensable to our grip on the concept of identity as Leibniz’s Law itself.