Friday, May 15, 2009

König's paradox and the modal view of plural quantification

In the last chapter of this thing, I defended the view according to which plural quantification:
For some a, .....
(where a can be singular, empty, or general) can (roughly speaking) be read nominalistically as:
It is possible to introduce a (singular, empty or general) name-token, such that...
One of the prima facie reasons to reject the substitutional interpretation of plural quantification was that we run out of tokens (finite sequences over a finite alphabet), if the domain is large enough.

My solution was to distinguish between different possible worlds where possible tokens are introduced, so that (assume we believe in Real Numbers):
For every real number, it is possible that it has a name.
comes out true, whereas:
It is possible that every real number has a name.
comes out false. So the basic idea is that even if in every possible world, there are only countably many names, the union of names in all accessible possible worlds doesn't have to be countable.

Now, I've been thinking about König's paradox (this is G. Priest's formulation from his essay Paraconsistency and Dialetheism, in Handbook of the History of Logic, vol 8):
There is an uncountable infinitude of ordinal numbers, but there is only a countable number of descriptions in English. Hence, there are many more ordinal numbers than can have names. In particular to turn the screw, since the ordinal numbers are well-ordered, there is a least ordinal number that has no description. But we have just described it.
Now, in the framework I like to think in (the modal framework mentioned above), the problem doesn't seem to arise (even given the assumption that ordinal numbers actually exist, which I'm not inclined to accept but that's a different story). Why?

Well, even if in any possible situation the number of English descriptions is countable, it doesn't mean that there is an unnameable ordinal number. So say, we are at certain time t. There is a set of actually existing English tokens which describe ordinals. Then at t+1 we produce the token:
The least ordinal number that has no description.
Now, as I see it, there are two different things that can be meant here.
  1. The least number for which no description actually existed at t.
  2. The least number for which no English description can be introduced, i.e. for which there is no possible description.
If 1. is meant, then no problem arises, because the newly introduced description exists at t+1 but not at t. If 2. is meant, then the description introduced doesn't pick an ordinal, because there is no ordinal for which no English description can be introduced, and empty sets don't have least members. Basically, from the fact that necessarily, the number of English descriptions is countable, it doesn't follow that there are ordinals that can't be described.

Of course, they can't be simultaneously described in one possible world. Say one points this out and tries the description:
The least ordinal that can't be possibly described in a simultaneous description of (some) ordinals.
But here, again, from the uncountability of ordinals it doesn't follow that there is such a number, because there are many ways we can go about naming things, and an ordinal not named in one scenario might be named in another one.




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