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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.

2 comments:

Colin said...

Jason, I think you are mistaken. My assessment is that (*) is false. The reason is very simple: (*) entails (*). Contrary to your argument, I say that this does not collapse into a contradiction. A crucial step in your argument is the following conditional.

"...if something entails that (*) is true, then (*) is true."

But this is just wrong. It is possible that A entails B where B is false. Just consider that by standard logic "The king of France is bald and is not bald" entails "Paris is the capital of England". In this case, the conclusion of the inference is false, but that doesn't undermine the fact that it is entailed by the premise(s). So your conditional is wrong.

Jason Zarri said...

Hi Colin,
You're absolutely right, I posted late at night and I should have given it more thought. Entailment is is certainly not the right concept. But consider the following attempted repair:

Define “P necessitates Q” as “P is true and P entails Q”. It is thus impossible that P necessitates Q and Q is false.
Now we have:
(*’): Nothing necessitates that (*’) is true.
Suppose (*’) is false. In that case, it is false that nothing necessitates that (*’) is true. So something necessitates that (*’) is true. But if something necessitates that (*’) is true, then (*’) is true, because by the definition of ‘necessitates’ some true statement entails (*’). But then what (*’) says must be the case, and hence it follows that nothing necessitates that (*’) is true. So if (*’) is false, it is true both that something necessitates that (*’) is true and that nothing necessitates that (*’) is true, which is a contradiction. (*’) must, in consequence, be true. So it is true that nothing necessitates 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 necessitated by every true statement, by virtue of the fact that every statement entails (*’) and the definition of ‘necessitates’. So if (*’) is true, it is true both that nothing necessitates that (*’) is true and that every true statement necessitates that (*’) is true. This too is a contradiction. So it seems that this modified version of my example is a genuine paradox.