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Thursday, December 24, 2009

Another new variant of the liar paradox?

I've thought of another variant of the liar to add to my collection--I'll leave it to my readers to tell me if someone has already thought of it.

Consider the following statement:

(*): Nothing entails that (*) is true.

Suppose (*) is false. In that case, it is false that nothing entails that (*) is true. So something entails that (*) is true. But if something entails that (*) is true, then (*) is true. But then what (*) says must be the case, and hence it follows that nothing entails that (*) is true. So if (*) is false, it is true both that something entails that (*) is true and that nothing entails that (*) is true, which is a contradiction. (*) must, in consequence, be true. So it is true that noting entails that (*) is true. (*), however, is not only true, it is necessarily true, for its falsity would entail a contradiction. However, if (*) is necessarily true, its truth is entailed by every statement whatever. So if (*) is true, it is true both that nothing entails that (*) is true and that everything entails that (*) is true. This too is a contradiction. So no matter whether (*) is true or false, it must be both true and false.