## Vicious circles and self-reference

Most of what I write on this blog consists of rough thoughts, and here again this contribution is ‘rough’. Suppose a sentence reads “this sentence is false” or for that matter reads “this sentence is true” (or “this sentence is neither true nor false” or “this sentence is both true and false” or “this sentence is more false than it is true”) – is it possible to interpret this sentence as a proposition? On the assumptions adopted in modern logic, one may be inclined to interpret the second as a proposition precisely because it can logically possibly have a truth value, whereas the former cannot have a fixed truth value. However, if we construe propositions as proposals about the way the world is or could be (allowing room for analytic propositions) then we should be inclined to think that neither of these sentences are propositions since neither one proposes anything about the way the world is or could be.

Now, I used to think that some sentences could refer meaningfully to themselves, such as the sentence “this sentence has thirty-two letters.” Since it was predicating about the sentence rather than the proposition, I thought this was surely a case of self-reference. However, a friend recently pointed out to me that one could interpret this proposition as follows: “this sentence token has thirty-two letters.” If we can make a distinction not only between a proposition and a sentence, but also between a sentence and it’s token, then we can predicate without self-reference here. However, one might think that, if there’s a difference between a sentence and it’s token, then the sentence “This sentence token has thirty-two letters” is false, and that, instead, “This sentence token has thirty-six letters” would be correct. One might begin to suspect that the question hasn’t been answered, for what of a sentence, instead of a sentence token, which refers to itself? After all, the sentence-token seems to refer to itself (though perhaps not).

However, if the sentence has any ‘sense’ or meaning at all, and it isn’t a proposition, then maybe the only way to interpret it is as referring to it’s token. We should just interpret “This sentence has thirty-two letters” to mean that the sentence token has thirty-two English letters. Construed this way, the sentence is a proposition with a truth value. Clearly, then, the proposition is not about itself per se, but about the sentence token. Other sentences may seem more problematic, such as the sentence “this sentence is in the indicative mood” and I’m still not entirely sure what to do about such cases of self-reference, since I don’t think that refers to the sentence token.

We’re all aware of viciously circular explanations and how worthless they are as explanations, but I think there’s an analogy from vicious circularity to the problem of self-reference. In a proposition, a predicate and subject cannot be identical (unless the predication is done by analogy, for instance as in the case of saying “God exists”). So also in an explanation, the thing explained (which doesn’t explain itself) cannot be identical to the thing explaining it. if P explains Q and Q explains P, then ultimately Q explains Q, but Q is not self-explanatory (or else it couldn’t possibly be explained by P), ergo etc. In the case of the vicious circle, the explanandum and the explanans are identical and not self-explanatory, and in the case of self reference the sentence is about itself. The problems are analogous.

Thus, to say “There is a set of all sets which are not members of themselves,” which was Russell’s paradox, one has to say that this is a more subtle but just as pernicious case of self-reference, and by reason of that alone is evacuated of propositional content, and evacuated of ‘sense’ or ‘meaning’ in the wider Fregean sense (no pun intended). I’m not sure whether “there is a set of all sets which are not members of themselves” is a sentence at all (if a sentence must have a ‘sense’ or have ‘meaning’ as sentences in the imperative-mood clearly do), but I’m quite sure that it isn’t a proposition at all. Thus, by eliminating some vicious cases of self-reference we can see our way through paradoxes like Russell’s set-theoretic paradox.