The word is is multiply ambiguous. When it can be expanded to read "is the same thing as," or "is identical with," or (in numerical contexts) "is equal to," it expresses the relation of identity. The simplest identity statements contain the "is" of identity flanked by singular terms, either names or definite descriptions: "Samuel Clemens is Mark Twain"; "The U.S. president in 1996 was Bill Clinton"; "Four is the sum of two and two." A more complex identity statement might, for example, combine the "is" of identity with quantifiers: "Every even number is the sum of two primes."

Identity, on its face, is simple and unproblematic: It is that relation that everything bears to itself and to nothing else. Yet discussions of identity in contemporary philosophical logic and metaphysics are brimming with controversy. From where does this controversy arise? Some of it is not genuine, being based on...