Philosophical logicians usually distinguish between qualitative and numerical identity. The former can hold between one object and another, meaning exact similarity (we can also define a notion of partial qualitative identity). Numerical identity (which from now on I will simply call identity) is supposed to relate objects only to themselves: nothing can be identical, in this sense, to an object that is not it. It is supposed that every object stands in this relation to itself, using “object” in the most capacious sense to include numbers, properties, functions, processes, etc. Identity appears to hold even between fictional objects and themselves—Sherlock Holmes is identical to himself. So the relation of identity is absolutely universal; moreover, it is necessary—everything is necessarily identical to itself. This is not true of qualitative identity, since it can be contingent that two objects are exactly similar. It is commonly accepted that the identity relation holds trivially of everything: just by being something an object is self-identical. For this reason some people have felt that identity is a pseudo relation—that there is something suspicious about it. It does seem exceptionally uninformative to be told that an object is identical to itself (tell me something I don’t know!). Anyway it is supposed that we know what we are talking about: we know what “identical” means in this special sense—to the point that we can recognize the concept as fishy in some way. But do we really know what identity is in the intended sense? Do we have a genuine concept of identity? Can we articulate what we mean by the word?
It might be thought that we have a number of possible avenues of explication available: Leibniz’s law, a famous dictum of Frege’s, and the involvement of sortal concepts in identity.  Taking the last first, it is sometimes said that identity statements are incomplete without the specification of a sortal, as in “Hesperus is the same planetas Phosphorous”; accordingly, identity is sortal relative, or at least sortal dependent. Thus we can explicate the nature of identity by saying that it essentially involves the kinds of objects—same planet, same animal, same number. It is not just an elusively bare abstract relation that holds indifferently of everything thing there is (and is not); it has specific concrete substance (in the Aristotelian sense of “substance” as well). But this doctrine cannot be right: it confuses identity statements and identity facts. Maybe statements of identity need sortal supplementation (or maybe not ), but the nonlinguistic fact of identity surely does not. How can an object’s identity with itself depend on its kind? What does that even mean? Does it mean that there is no relation of identity except one that incorporates a sortal kind—as with planet-identity, animal-identity, and number-identity? But it is hard to see what this is supposed to mean: aren’t these all instances of identity tout court? Isn’t there an overarching relation of simple numerical identity? Nor is it clear how much elucidation this doctrine affords: we are still left wondering what the import and point of identity is supposed to be. I am the same human being as myself: big deal, what’s the point of saying that? We have a bunch of sortal-relative identity relations, but we still don’t know what they are exactly—and why objects bother to instantiate them. What does it mean to say that x is the same F as y? What is this sameness with oneself?
Here we reach for Frege’s famous dictum: identity is that relation a thing has to itself and to no other thing. This is ritually intoned, as if it contains self-evident wisdom, but it is not critically examined. The thought is that other equivalence relations don’t satisfy the definition because they can relate an object to other objects: for example, I am the same height as myself, but also the same height as other people—whereas I am identical to myself, but not identical to anyone else. The identity relation can only relate an object to itself, but other equivalence relations can hold between an object and itself and other objects. There are three problems here. First, what about a universe empty save for one object? In such a universe sameness of height does not relate me to other objects, since there are none—so it would count as the identity relation according to Frege’s dictum. Intuitively, what have other objects got to do with my self-identity? Not relating me to other objects can hardly count as essential to my identity with myself. Second, the dictum is circular if offered as a definition: for we need to understand what is meant by “other things”. Surely this phrase means “things not identical to the given thing”, but then the concept of identity is being presupposed: you already have to grasp what identity is before you can understand Frege’s dictum. Third, and most telling, the dictum doesn’t single the identity relation out even in the actual world; other relations satisfy Frege’s condition. Take the part-whole relation: certain objects stand in this relation to me, but they don’t stand in this relation to anyone else. My right arm is part of me but not part of anyone else—so the part-whole relation holds between me and parts of me but not between me and parts of other objects. You might object that the part-whole relation doesn’t relate me to myself but only to parts of me (unlike the same-height relation), but consider the relation of improper part: that does relate me to myself but not to anyone else—I am not a part, proper or improper, of anyone else. Yet part-hood and identity are not the same relation. This could have been expected on intuitive grounds, because Frege’s dictum is very general—identity is being said to be anyrelation that relates an object to itself but not to anything else. That is unlikely to single identity out uniquely, save per accidens (this why it fails for the single-object universe). The dictum fails to capture the specific idea of numerical identity. Not that Frege meant it as a strict definition (he was too circumspect for that) but more as a useful heuristic; in any case, it doesn’t help with the task of giving the notion of identity clear content. We can’t complacently cite the dictum as explaining what that concept consists in.
Third, we have Leibniz’s law of the indiscernibility of identicals: does this tell us what we are talking about when we talk of identity? The law has the advantage of being true, indeed necessarily true, but it is limited as a method for explaining the concept of identity. It offers only a necessary condition to begin with, and the converse principle is far from self-evident on a natural understanding of it (i.e. the identity of indiscernibles). But the main problem is one of triviality: what precisely does this law assert? It is awkward to state because we have to say something like, “If two objects are (numerically) identical, then they must share all their properties”. Two objects? We blushingly shift to using variables: “If x and y are identical, then they must share all their properties”. But this is scarcely any better—x AND y? What is really meant is just that an object is always exactly similar to itself: an object is always qualitatively identical to itself. Where there is numerical identity there is qualitative identity. True enough, but does it help with understanding the concept of (numerical) identity? We are being told that objects always have the properties they have and no others—again, that is hardly news. But worse, it uses qualitative identity to explain numerical identity: it derives the latter concept from the former. Construed as an effort to get a handle on identity proper, it invokes qualitative identity, stating weakly that objects are always exactly like themselves. Surely we are entitled to expect something better—something meatier, more apropos (but see below). So we are still lacking any decent account of what this alleged special relation of numerical identity comes to—some kind of elucidation, analysis, insight. Instead we just have the identity relation staring blankly and inarticulately back at us, hoping we will somehow get the hang of it. It seems unnervingly self-effacing.
At this point the disturbing figure of the skeptic enters the conversation: what if there is no such relation as identity? It is proving so elusive because it isn’t really there. What do we mean when we say an object is identical to itself—what are we thinking? Nothing, according to the skeptic: our wheels are spinning, our thought process deceiving us. It might be contended, skeptically, that the concept has its origin in certain epistemic and linguistic practices but that it has no reference in objective reality; it is a kind of illicit projection, a phantasm of the intellect. Things are not round and heavy and red and self-identical: that last is just not a real property of things, but a reification of our epistemic and linguistic practices. We often don’t know that we are dealing with a single object and can therefore discover the truth of a statement of the form “x is identical to y”, but that doesn’t imply that the real world contains objective identity relations. The concept of identity is useful to us in recording our epistemic dealings with the world, but it shouldn’t be taken to denote a genuine constituent of objective reality—for what kind of constituent is it? Do we see or touch it, or need it in our scientific theories, or feel it in ourselves? Why not just admit that it isn’t part of a truly objective conception of things (perhaps rather like the commonsense concept of an object). There are objective similarities between things to be sure, and we can speak of things as indistinguishable, but the idea of numerical identity is a chimera. So says the skeptic, and he is not without rational grounds for his opinion. However, the position is extreme and I am inclined to suggest something weaker, though in the spirit of the skeptical position. This is that talk of numerical identity is best interpreted as an extension of the concept of qualitative identity, which is perfectly meaningful: to say that an object is identical to itself is just to say that it is exactly similar to itself. Two distinct objects can be exactly similar, thus warranting talk of identity between them (“these two balls are identical”), and a single object can be exactly similar to itself too. So there is not an extra primitive relation in the world called “numerical identity”; this talk is really just the application of the concept of qualitative identity to solitary objects. Every object is qualitatively identical to itself—that is, every object is self-identical in just the sense that two objects can be said to be identical. There is really just the concept of qualitative identity, and it can hold between distinct things or one thing and itself. Of course, statements of qualitative identity between an object and itself are trivially true, but then so is the proposition that every object is self-identical. An advantage of this way of seeing things is that we need not recognize any ambiguity in the word “identical”: it always means so-called qualitative identity. And there is little intuitive plausibility in the view that “identical” varies in meaning as between a numerical and a qualitative sense. If this position is correct, there is no identity relation such as philosophical logicians have supposed—no separate kind of identity; there is just a single relation of similarity—but objects can stand in this relation to themselves. To be self-identical is to be self-similar. I am completely similar to myself, hence “self-identical”. We can easily specify what this identity relation consists in: the sharing of properties. We know what properties are and we know what sharing them is—well, that is what identity is all about. This relation can hold between several objects and it can hold between a single object and itself. If I say that I am identical to myself, I am saying that I am exactly similar to myself—just as I can be similar to other people (perhaps exactly similar). The statement is no doubt peculiar, because hardly disputable, but it is the interpretation that makes the most sense of identity talk; it’s either that or skepticism about the whole concept. We could try maintaining, feebly, that the concept of numerical identity is primitive and inexplicable—simply not capable of any articulation—but that seems unattractive in the light of the alternatives. It is preferable to hold that so-called numerical identity is analyzable as reflexive qualitative identity. After all, that relation clearly exists and has a clear content—why introduce anything further?
What are the consequences of this revision in the way we think of identity? All those puzzles of identity must now be recast in terms of self-similarity, as must the idea of a criterion of identity. This may not (should not) make a difference to the substantive issues, but to be clear in our mind we should think in the recommended terms. There is nothing real to identity over and above self-similarity. And since philosophy is very largely concerned with questions of identity, particularly the identity of concepts and properties, the revision must have an impact on how we understand philosophical questions. Concepts (meanings, intensions, properties) can be exactly similar to themselves, this being what concept identity comes down to. If the concept of knowledge, say, has a property not possessed by the concept of true justified belief, then the two concepts cannot be identical; for then the concept of knowledge would not be qualitatively identical to the concept of true justified belief. Identity is always qualitative identity, so concepts can’t be identical unless they share the same qualities (this is Leibniz’s law in another form). In a way the concept of identity already contains Leibniz’s law, because what it means to say that x is identical to yis defined as sharing the same properties. It is not some further tacked on thesis that identical objects are always exactly alike: self-identity simply is sharing the same properties—x being identical to x just is x being qualitatively identical to x. This is why Leibniz’s law is so self-evident: it is really a kind of tautology. This is as it should be. 
 I have consigned to a footnote another familiar attempt to explicate identity because the attempt barely gets off the ground and is lamentably confused, namely that identity is a relation between signs. That is, for objects to be identical is for them to be the single denotation of two terms. The trouble, obviously, is that object identity can’t depend on language. Still, the suggestion is helpful in illustrating what a substantive account of identity might look like: at least we are given a nontrivial analysis of the concept (just a wrong one). Compare: identity is the co-reference of ideas in God’s mind—substantive enough but none too plausible.
 The sortals go into the fixation of reference not the type of identity relation involved, as in “that elephant is identical to that elephant” said while pointing to different parts of the same elephant. But objects cannot be incomplete in the way bare demonstratives are.
 Old hands will see the imprint of several philosophical logicians on what I write here (Geach, Wiggins, Kripke, and others). It should be evident that what I say is radical to the point of heretical—I myself have always assumed that numerical identity is transparently a concept in good standing. I am as shocked as anyone by the skeptical reflections herein sketched (contrast my Logical Properties (2000), chapter 1).