Identity and Difference
Identity has been found problematic; difference, not so much. Identity has been judged a pseudo relation, but no one doubts that difference is a genuine relation between things. We can observe that one thing is different from another, but can we observe that a thing is identical to itself? Difference is essential to counting, but identity never gets us beyond a single entity. Do we really need the concept of identity? As a matter of definition, identity is said to be the relation a thing has to itself and to no other thing; difference is the relation a thing has to everything apart from itself. Everything is either identical to a given thing or different from it. Each thing is identical to itself and different from everything else. What exactly is the logical relationship between the concept of identity and the concept of difference? It tends to be assumed that identity is basic and difference is derivative—difference is simply non-identity or lack of identity—but what about considering it the other way round? What if we take difference as basic?
If difference is basic, then identity is simply non-difference or lack of difference. We can express the sentence “a is identical to b” by the sentence “a is not different from b”. We start out with the concept of difference, tied to our perception of distinct objects, and then we define identity as simply the opposite of difference: it is the relation a thing x has to a thing y when x and y are not different things. We might initially suppose that everything in the world is different from everything else, but then we make a conceptual discovery and realize that objects are not different from themselves—there is another relation apart from difference, namely identity. So identity is really the absence of difference. We thought that Hesperus was different from Phosphorus, but it turns out that the two are not different—this was an illusory difference. We could express our discovery by saying, “Hesperus and Phosphorus are not different”, but we choose to introduce a shorter form of words and say, “Hesperus and Phosphorus are identical”. We don’t thereby expand expressive power; we had already said what needed to be said by saying that the two are not different.
Thus we might offer to analyze identity in terms of difference plus negation: “identical” means “not different”. Why is this any less correct than analyzing “different” as “not identical”? In fact, it looks as if difference is more primitive in our system of concepts. Animals and young children surely make judgments of difference, but do they make judgments of identity? Someone could in principle have the concept of difference and never hit on the concept of identity, which would require conjoining difference with negation. But how could someone have the concept of identity and not have the concept of difference? Identity is the absence of difference: it is the relation a thing has to what it is not different from. I realized I was different from everyone else at an early age; it was only later that it dawned on me that I was identical to myself (funny thought). Difference is a given, a datum; but identity is more of a construction or abstraction. Identity is a sophisticated concept; difference is as plain as the nose on your face. A farmer counting his chickens needs the concept of difference; the philosopher explores the concept of identity. To reach the concept of identity you need to combine difference with negation—not a trivial operation.
The case might be compared to truth and falsehood. Philosophers tend to concentrate on truth, leaving falsehood to take care of itself, but a good case can be made that truth is definable in terms of falsehood and negation.  Thus for a proposition to be true is for it not to be false. This defines truth in terms of falsehood and negation. Similarly, we can define identity as difference plus negation—as not being different. This is a genuine definition and it provides necessary and sufficient conditions. First we master the concept of difference; later we form the concept of identity by combining difference with negation. Identity is what holds when difference doesn’t.
Nothing in the standard logic of identity will be sacrificed by adopting this position. We will still have reflexivity, symmetry, and transitivity. We can still distinguish numerical and qualitative identity: some things are numerically different without being qualitatively different, or not different qualitatively while different numerically. Leibniz’ Law can simply be reformulated to read, “If a is not different from b, then a and b have all their properties in common”. It might be a good idea to reform the symbolism we use to express claims of identity and difference, because the standard symbolism makes identity out to be basic with difference coming out as the negation of identity. Thus we now have “=” and that symbol with a slash through it, or modified by “not”. Instead we could have a new sign for difference (say “^”), taken as primitive, and then introduce identity by means of negation. We will then write “Hesperus not-^Phosphorus” to express the fact that Hesperus is identical to Phosphorus, i.e. not different from Phosphorus. We can then add this sign to the usual symbols of the predicate calculus instead of “=”; formulas will then include “a ^ b” and the like. Obviously, this will be equivalent to taking “=” as primitive and defining difference by means of it and negation. But the new formulation is conceptually more perspicuous in the light of the proper order of definition.
Understanding difference is fundamental to every cognitive relationship to the world and is present in every perception. We see difference everywhere we look. The world is packed with difference. Identity is the exception to this universal rule—not everything is distinct from everything. There is the odd case of things in relation to themselves: here there is not difference. Things assert their difference from other things, but not from themselves. This lack of self-difference is what we name “identity”.