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In a piece of paper, it was written:

The statement written in the paper is false.

Is that statement actually true or false?

  • There is no duplicity: the issue here is about solutions in general, not only involving formal logic, but in formal languages​​, semantics, etc. – Annotations Apr 16 '13 at 15:54
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    I can see an argument for this not being a duplicate, but then such a formulation borders being non-constructive. I'm inclined to let the community decide at this point. – stoicfury Apr 17 '13 at 19:46
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    It's false! No, it's true! No, it's false! No, it's … "this question will likely solicit debate, arguments, polling, or extended discussion" inside your head :) Seriously, as it stands the question is non-constructive. Vote to close. Could @sidarth explain what kind of answers he expects? E.g. solutions where the statement has a definite truth value (like Prior's)? – DBK Apr 17 '13 at 22:38
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I like the solution of Tarski:

Alfred Tarski diagnosed the paradox as arising only in languages that are "semantically closed", by which he meant a language in which it is possible for one sentence to predicate truth (or falsehood) of another sentence in the same language (or even of itself). To avoid self-contradiction, it is necessary when discussing truth values to envision levels of languages, each of which can predicate truth (or falsehood) only of languages at a lower level. So, when one sentence refers to the truth-value of another, it is semantically higher. The sentence referred to is part of the "object language", while the referring sentence is considered to be a part of a "meta-language" with respect to the object language. It is legitimate for sentences in "languages" higher on the semantic hierarchy to refer to sentences lower in the "language" hierarchy, but not the other way around. This prevents a system from becoming self-referential.

And Prior too:

Arthur Prior asserts that there is nothing paradoxical about the liar paradox. His claim (which he attributes to Charles Sanders Peirce and John Buridan) is that every statement includes an implicit assertion of its own truth. Thus, for example, the statement, "It is true that two plus two equals four", contains no more information than the statement "two plus two equals four", because the phrase "it is true that..." is always implicitly there. And in the self-referential spirit of the Liar Paradox, the phrase "it is true that..." is equivalent to "this whole statement is true and ...". Thus the following two statements are equivalent:

This statement is false.
This statement is true and this statement is false.

The latter is a simple contradiction of the form "A and not A", and hence is false. There is therefore no paradox because the claim that this two-conjunct Liar is false does not lead to a contradiction.

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    Tarskis idea sounds similar to truth stratification in Russells theory of types - except that was motivated by his own paradox. – Mozibur Ullah Apr 16 '13 at 17:15

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