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Could there be more than a countably infinite number of propositions?
October 3, 2009
If the term 'proposition' is used to mean a sentence -- a string of symbols, be they spoken, written, gesticulated or whatever -- then I suppose there could be uncountably many propositions if we allow there to be propositions of infinite length. In that case, one ought to be able to diagonalise on them just as one does with the infinite decimal expansions of the real numbers. But I think it's reasonable to stipulate that we're only going to countenance finitely long sentences. After all, they wouldn't be much use in communication, if you could literally never get a sentence out.
Alternatively, if the set of symbols is itself uncountable, then that will certainly lead to an uncountable infinity of strings of such symbols. But it seems reasonable to stipulate against that case too. Communication would once again be thwarted, because we don't seem to have the perceptual capacity to discriminate between uncountably many different symbols -- indeed, our discriminatory abilities probably only extend to a finite (though no doubt very large) number.
But 'proposition' has been used in other ways over the years. So what if we take it to mean not the string of symbols itself, but rather the non-linguistic thing that such a sentence is expressing: a thought, or something of that kind? Well, then maybe there could be uncountably many propositions: we'd need to have a great deal more spelt out about the details of this notion before we'd be in a position to assess that. But (leaving aside any eccentric conceptions of sentences and symbols like those just discussed) there could no longer be a one-to-one correspondence between propositions and sentences. For there to be uncountably many propositions, while there are only countably many sentences, either there would need to be propositions that couldn't be expressed by sentences, or else a single sentence would need to correspond to multiple propositions at once. Either way, it would mean that sentences couldn't do a very good job of expressing propositions any more -- but that was surely the very thing that sentences were invented for in the first place.
If I remember correctly, and I may well not, David Lewis explicitly argues that there are uncountably many propositions in Plurality of Worlds and uses this as an argument against any view that would try to reduce propositions to sentences. At the very least, he does consider this issue. So here's an argument that I think I remember from that book that we can consider, anyway. It is based upon the claim that, for any real number x, there ought to be a proposition---a possible content of thought---that I am shorter than x inches tall. Indeed, each such proposition could be expressed by a sentence. All we have to do is give the real number x a name, say, "Fred", and then the proposition will be expressed by the sentence "I am shorter than Fred inches tall". But if so, then there are at least as many propositions as there are reals.
The key to this argument, note, is the observation that the claim "For every proposition p, there could be a sentence S that expressed it" is much weaker than "For every proposition p, there is a sentence S that expresses it". It's the latter that the reductionist needs. But the argument needs only: If there could be a sentence S that meant that p, then there is a proposition that p---assuming that other, uncontroversial conditions required for the proposition to exist are met. E.g., there might have been propositions other than the ones there are, because there might have been people other than the ones there are, and then there would have been propositions about them that there in fact aren't. But that's not what's happening in this case.
That there should be uncountably many propositions for this kind of reason does not, so far as I can see, imply anything bad about the possibility of communicating. Each actual (meaningful) sentence corresponds to---that is, expresses---just one proposition (modulo worries about context). But there are lots of propositions that are not, in fact, expressed by any sentence.
My own preference, for what it is worth, is to refuse to talk of propositions, period. But that's a different matter. If we are going to talk of propositions, then it actually seems quite hard, for the reasons just given, to keep them countable.
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