Logic

Does the relation of self-similarity exist? It seems obvious that it does, since nothing is self-dissimilar. But if it does then it, as a relation, must be self-similar, and this second relation of self-similarity must be self-similar, and so on ad infinitum. And surely the Universe is not crammed with an infinity of relations of self-similarity. But does that mean that nothing is self-similar?

Comments

I lost you right here: "this second relation of self-similarity must be self-similar". What is the "second" relation? I thought it was just the relation of self-similarity, which, as you say, the relation of self-similarity presumably has to itself, since everything is self-similar.

It's perhaps worth noting that we can construct an analogous set of questions using identity rather than self-similarity: Everything is identical with itself; so the relation of identity is, as a relation, also identical with itself. But again, the relation identity has to itself is just identity: The relation of identity bears itself to itself.

So it doesn't seem to me that this line of argument requires "the Universe [to be] crammed with an infinity of relations of self-similarity", and I'm not sure why it seems so obvious that it isn't. That said, however, the kind of language you use here—talking of a relation standing in itself to itself—can cause problems.

Some relations, as we have just seen, stand to themselves in themselves. Similarly, some properties have themselves. For example, the property is expressible in English is itself expressible in English; the property is a property is itself a property. Of course, not all properties have the properties they express. For example, the property is a lion is not itself a lion. So consider the property is a property that does not have itself. Does this property have itself?

Exercise: Construct a similar paradox using is a relation that does not stand in itself to itself.

One way out of this puzzle is to deny that it makes sense to talk of a property as having itself, or of a relation as standing to itself in itself. There are other solutions, too, but none that commands universal assent.

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