Liar Paradox
In most common philosophy the liar paradox is the statement “this sentence is false.” An attempt to assign to this statement a classical binary value of truth leads to a contradiction. If “this sentence is false” is true, then the sentence is false is a contradiction. An example of the liar paradox is the Epimenides paradox. The Epimenides paradox is suggested to be an example of the liar paradox, but isn’t as equivalent as it. The semi-mythical seer Epimenides, a Cretan, reportedly stated that "The Cretans are always liars." However, Epimenides' statement that all Cretans are liars can be resolved as false, given that he knows of at least one other Cretan who does not lie. One version of the liar paradox is attributed to the Greek philosopher Eubulides of Miletus who lived in the 4th century BC. Eubulides reportedly asked, "A man says that he is lying. Is what he says true or false?" In early Islamic tradition liar paradox was discussed for at least five centuries starting from late 9th century apparently without being influenced by any other tradition. 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:
This statement is false. (A)
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". 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.