 where the formula \(\phi(x)\) is like the formula \(\phi(y)\) except for having occurrences of \(x\) at some or all of the places \(\phi(y)\) has occurrences of \(y\) (see Enderton 2000, for a precise definition). Ref is the principle of the reflexivity of identity and LL (Leibniz’ Law) is the principle of the indiscernibility of identicals. It says in effect that identical objects cannot differ in any respect. The other characteristic properties of identity, symmetry \((x = y \rightarrow y = x)\), and transitivity \((x = y \amp y = z \rightarrow x = z)\), may be deduced from Ref and LL. Any relation that is reflexive, transitive, and symmetric is called an ‘equivalence relation’. Thus, identity is an equivalence relation satisfying LL. But not all equivalence relations satisfy LL. For example, the relation x and y are the same size is an equivalence relation that does not satisfy LL (with respect to a rich language such as English).