Related: The map is not the territory, Unresolved questions in philosophy part 1: The Liar paradox

A well-known brainteaser asks about the truth of the statement "this statement is false". If the statement is true, then the sentence must be false, but if it false then the sentence must be true. This paradox, far from being just a game, illustrates a question fundamental to understanding the nature of truth itself.

A number of different solutions have been proposed to this paradox (and the closely related Epimenides paradox, Pinocchio paradox). One approach is to reject the principal of bivalence - that every proposition must be true or false - and argue that this statement is neither true nor false. Unfortunately, this approach fails to resolve the truth of "this statement is not true". A second approach called Dialetheism is to argue that it should be both true and false, but this fails on "this statement is only false".

Arthur Prior's resolution it to claim that each statement implicitly asserts its own truth, so that "this statement is false" becomes "this statement is false and this statement is true". This later statement is clearly false. There do appear to be some advantages to constructing a system where each statement asserts its own truth, but the normative claim that truth should always be constructed in this manner seems to be hard to justify.

Another solution (non-cognitivism) is to deny that these statement have any truth content at all, similar to meaningless statements ("Are you a?") or non-propositional statements like commands ("Get me some milk?"). If we take this approach, then a natural question is "Which statements are meaningless?" One answer is to exclude all statements that are self referential. However, there are a few paradoxes that complicate this. One is the Card paradox where the front says that the sentence on the back is true and the back says that the sentence on the front is false. Another is Quine's paradox - ""Yields falsehood when preceded by its quotation" yields falsehood when preceded by its quotation". One other common example is: "The statement on the blackboard in Carslaw Room 201 is false". The Card paradox and blackboard paradox are interesting in that if we declare the Liar paradox to be meaningless, these paradoxes are meaningless or meaningful depending on the state of the world.

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