Logic

Is it possible to construct a statement P (in English) such that both P and NOT P are true?

This statement is false.

No.

Yes, it isn't.

This statement isn't false.

>Let R be the set of all sets that are not members of themselves.
It's called a paradox and because it's paradoxical, it's useless in any formal system.

No, because there are no contradictions in reality.

You can, constructively, have such a statement, but with the little thing that it's not rue or false. The king of the USA is bald. If it was true, then there would be a king, and similarly for the case in which it is false. Since such a king doesn't exist, you have a statement neither true or false.

But this requires non-classical logic.

Let's say 'this statement is false' is P

Since it refers to P, P is true(1) if P is false(-1):

P = -P

Since this equation is P, the solution is very simple. Let False be that which is not True. Since P = -P is not true, it is False. The mistake is not necessarily on the value it assigns to P. You could say: The Bird On This Table has no wings. If there is no bird on this table, the affirmation is false. Similarly, every time P is called, there is a parameter missing, so saying A statement which doesn't exist is either True or False, is False. Since there is a parameter missing, and since you are defining False as anything that is not a True Statement, even Non-statements can be assigned False, so P is false and it remains false because it's not a Statement. Example of similar sequence of words: Fax Phone Is Is, Evaluate()=True.

The king of the USA is bald is false, my friend. This is basic classical logic.