Nicholas Rescher cites this example from Buridan. The proposition is false, but not self-refuting. If every proposition is affirmative, then of course *Every proposition is affirmative* is affirmative. The self-reference seems innocuous, a case of self-instantiation. But *Every proposition is affirmative* has as a logical consequence *No proposition is negative.* This follows by Obversion, assuming that a proposition is negative if and only if it is not affirmative.
Paradoxically, however, the negative proposition, unlike its obverse, is self-refuting. For if no proposition is negative then *No proposition is negative* is not negative. So if it is, it isn't. Plainly it is. Ergo, it isn't.
Rescher leaves the matter here, and I'm not sure I have anything useful to add.
It is strange, though, that here we have two logically equivalent propositions one of which is self-refuting and the other of which is not. The second is necessarily false. If true, then false; if false, then false; ergo, necessarily false. But then the first must also be necessarily false. After all, they are logically equivalent: each entails the other across all logically possible worlds.
What is curious, though, is that the ground of the logical necessity seems different in the two cases. In the second case, the necessity is grounded in logical self-contradiction. In the first case, there does not appear to be any self-contradiction.
It is impossible that every proposition be affirmative. And it is impossible that no proposition be negative. But whereas the impossibility of the second is the impossibility of self-referential inconsistency, the impossibility of the first is not. (That is the 'of' of apposition.)
Can I make an aporetic polyad out of this? Why not?
1. Logically equivalent logically impossible propositions have the same ground of their logical impossibility.
2. The ground of the logical impossibility of *Every proposition is affirmative* is not in self-reference.
3. The ground of the logical impossibility of *No proposition is negative* is in self-reference.
The limbs of this antilogism are individually plausible but collectively inconsistent.
REFERENCES
Nicholas Rescher, Paradoxes: Their Roots, Range, and Resolution, Open Court, 2001, pp. 21-22.
G. E. Hughes, John Buridan on Self-Reference, Cambidge UP, 1982, p. 34. Cited by Rescher.
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