A generalization is a claim of the form:
(1) All A’s are B’s.
A generalization about generalizations is thus a claim of the form:
(2) All generalizations are B.
Some generalizations about generalizations are true. For example:
(3) All generalizations are generalizations.
and some generalizations about generalizations are false. For example:
(4) All generalizations are false.
In order to see that (4) is false, we could just note that (3) is a counterexample to (4). The following argument is a bit more interesting, however.
Proof: Assume that sentence (4) is true. Then, given what (4) says, all generalizations are false. But (4) is a generalization, so (4) must be false, making (4) both true and false. Contradiction. So (4) can’t be true, and hence must be false.
(4) has an interesting property, however. Although, as we have seen, not all generalizations are false, and hence (4) fails to be true, it is itself a false generalization, and hence is a supporting instance of (4). In other words, since (4) is false, the existence of (4) provides some small amount of positive evidence in favor of the truth of (4), even if, in the end, our proof of (4)’s falsity trumps this small bit of defeasible evidence.
Thus, we can introduce the following terminology. A generalization about generalizations of the form:
(5) All generalizations are B.
is a self-supporting generalization if it in fact has property B – that is, if it has the property that it ascribes to all generalizations (regardless of whether all other generalizations have this property, and thus regardless of whether the generalization in question is itself true or false). In short, a generalization about generalizations is self-supporting if it has the property that it says all generalizations have.
It is easy to show that any true generalization about generalizations will be self-supporting (proof left to the reader). But false generalizations might be self-supporting, like (4) above, or they might not. For example:
(6) All generalizations are true.
is false, since (4) is false, and hence a counterexample to (6). But it is not self-supporting, since it would have to be true to be self-supporting.
To obtain the paradox promised in the title of this post, we need only consider:
(7) All generalizations are not self-supporting.
Note that we could express this a bit more colloquially as “No generalizations are self-supporting.”
Now, (7) is clearly false, since we have already seen an instance of a self-supporting generalization. But is (7) self-supporting? As the reader has no doubt guessed, there is no coherent answer to this question:
Proof of Contradiction: (7) is either self-supporting, or it is not self-supporting.
Case 1: Assume that (7) is self-supporting. If (7) is self-supporting, then it has the property that (7) says all generalizations have. (7) says that all generalizations are not self-supporting. So (7) is not self-supporting after all. Contradiction.
Case 2: Assume that (7) is not self-supporting. But (7) says that all generalizations are not self-supporting. So (7) has the property that (7) says all propositions have. So (7) is self-supporting after all. Contradiction.
Note that this paradox, unlike the Liar paradox (“This sentence is false”) does not involve any problems with regard to determining the truth-value of (7). As we have already noted, we can straightforwardly observe that (7) is false. The paradox arises instead with regard to whether (7) has the rather more esoteric property of being self-supporting. It turns out that (7) has this property if and only if it doesn’t.
Note that this is exactly isomorphic to the Grelling–Nelson paradox
[…] the self-referential sentences discussed here, here, and here is that determining whether a particular self-referential sentence is true or false […]
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