paradox as self-referential.[6]
Explanation and variants
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The problem of the liar paradox is that it seems to show that common beliefs about truth and falsity actually lead to a contradiction. Sentences can be constructed that cannot consistently be assigned a truth value even though they are completely in accord with grammar and semantic rules.
The simplest version of the paradox is the sentence:
A: This statement (A) is false.
If (A) is true, then "This statement is false" is true. Therefore, (A) must be false. The hypothesis that (A) is true leads to the conclusion that (A) is false, a contradiction.
If (A) is false, then "This statement is false" is false. Therefore, (A) must be true. The hypothesis that (A) is false leads to the conclusion that (A) is true, another contradiction. Either way, (A) is both true and false, which is a paradox.
However, that the liar sentence can be shown to be true if it is false and false if it is true has led some to conclude that it is "neither true nor false".[7] This response to the paradox is, in effect, the rejection of the claim that every statement has to be either true or false, also known as the principle of bivalence, a concept related to the law of the excluded middle.
The proposal that the statement is neither true nor false has given rise to the following, strengthened version of the paradox:
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