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"There are none so blind as those who will not see." --

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Saturday, January 31, 2009

A new variant of the liar paradox?

A few weeks ago I thought of the following variant of the liar paradox. I won't pretend that this post advances our understanding of the liar in any way, I'm just wondering if anyone else has thought of it before.

Consider the following statement:

(1) If there are elephants, then (1) is false.

If there were no elephants this statement would not be problematic. But there are. So let's assume that (1) is true. Given that there are elephants and that (1) is true, it follows by Modus Ponens that the consequent of (1) is true, and hence that (1) is false. So if (1) is true, and there are elephants, then (1) is false.

Let's now assume that (1) is false, granting once more that there are elephants. If (1) is false, the falsity of the consequent of (1), and hence (1) itself, cannot follow from the fact that there elephants. On this supposition (1) and its consequent are false, to be sure, but their falsity must not be entailed by the fact that there are elephants. But if the fact that there are elephants does not entail the falsity of the consequent of (1), and hence (1) itself, it must be possible for there to be elephants while they are both true. But we have already seen that if (1) is true, and there are elephants, then (1) is false. Thus it is not possible for there to be elephants while (1) and the consequent of (1) are both true. So the fact that there are elephants does entail the falsity of (1) and its consequent. But then what (1) says is true: If there are elephants, it must be false! Hence if (1) is false, it is also true.

Consequently, if there are elephants and (1) is true then (1) is false, and if there are elephants and (1) is false then (1) is true. Unless we're prepared to deny the existence of elephants, we have a paradox.