Friday, June 03, 2011

Aristotle is a non-vegetable, and someone else is not.

Suppose Max, a Cretan, says, “All Cretans are liars.” Assuming that the quantifier ‘all’ in conjunction with the term ‘Cretans’ is to be understood divisively (i.e., denoting each and every Cretan), we seem to have a paradox on our hands. For…

Suppose it’s true that “all Cretans are liars”, then it’s false, for Max, being a Cretan, would be telling the truth, in which case it’s false that “all Cretans are liars.” Alas, it seems to be the case that at the moment when Max says, “All Cretans are liars”, his statement becomes both true *and* false at the same time.

Assuming truth and falsity are contraries (viz.—that it’s a contradiction to predicate truth and falsity of the same subject at the same time), Aristotle was apparently not only begging the question when he argued in Metaphysics Gamma that the principle of non-contradiction is necessarily true (a pseudo-charge, given that Aristotle readily admits that the defense of any genuine first principle must, being a first principle, be employed in its own defense), Aristotle was a vegetable—that is, he wasn’t being rational—for in fact there are “true” contradictions!

Well, the Cretan “paradox” above is no reason to think Aristotle was a vegetable.

On the Contrary: let’s suppose that prior to the moment before Max says “all Cretans are liars” it’s true that

(1) All Cretans are liars.

Well, by the “question begging dogma” of the necessity of logical entailment, if p is true at time t, then it follows necessarily that ~p is false. Symbolically

LE: (p/t ⊃~~p/t) [Necessarily, if it’s true that p at t then it’s true ~~p at t]
equivalently: [Necessarily, if it’s true that p at t then it’s false that ~p at t]
Well, let p = (1). By LE we get
(2) ((1)/t ⊃ ~~(1)/t) [Necessarily, if it’s true that “All Cretans are liars” at t then it’s false that “All Cretan are not liars” at t].
Now, as we’ve already supposed, let the antecedent of (2) be true, i.e.,
(3) (1)/t
by modus ponens
(4) ~~(1)/t.
But, now suppose that while (4) is true, Max says “all Cretans are liars”. But this would entail a contradiction, for at the moment Max says, “all Cretans are liars”, he’s telling the truth since he’s lying. But if such were to happen, the following would be true
(5) (~~(1)/t ^ ~(1)/t)
which is equivalent to
(5’) (~(1)/t ^ (1)/t)
Which is a contradiction and thereby impossible simpliciter.
All this to say, if it’s true that
(1) All Cretans are liars
then it would be logically impossible for any Cretan, such as our Max, to say so. And if it’s impossible for any Cretan to say that “all Cretans are liars”, then there’s no paradox, for no proposition would be both true and false.

Ergo, Aristoteles non vegetabili et Sacerdos est.


Blogger Jonathan Charles Wright said...

I think you're right to conclude that, given this setup, we are to reject (1) as false. If it were true we would be forced to conclude the obvious falsehood that it's logically impossible for Max to utter some string of phoneme's because he was born in some place. So I agree with what you've got here.
This seems parallel to the barber paradox: we should conclude that no such barber exists.

But this doesn't touch the more austere Liar paradox which we've argued over before:

(L). (L) is not true (is false, is non-truth-evaluable, etc.)

Ditto for Russell's paradox (which stands as an analogue to the barber paradox as the formal Liar paradox is an analogue to the Cretan paradox).

4:49 PM  

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