The Liar Paradox is an argument that arrives at a contradiction by reasoning about a Liar Sentence. The classical Liar Sentence is the following self-referential sentence:
(1) This sentence is false.
Experts in the field of philosophical logic have never agreed on the way out of the trouble despite 2,300 years of attention. Here is the trouble--a sketch of the Liar Argument that reveals the contradiction:
If (1) is true, then (1) is false. On the other hand, assume (1) is false. Because the Liar Sentence is saying precisely that (namely that it is false), the Liar Sentence is true, so (1) is true. We've now shown that (1) is true if and only if it is false. Since (1) is one or the other, it is both.
http://www.iep.utm.edu/p/par-liar.htm
The paradox goes like this:
1. Epimenides is a Cretan.
2. Epimenides states, "All Cretans are liars."
On the face of it, this appears to be a paradox. Epimenides, being a Cretan, must either be a liar or a truth-teller. Thus his statement must be either true or false. But if it's true, then he (being a Cretan) must be a liar, so the statement can't be true. On the other hand, his statement is false, then he can't be a liar, so the statement must be true. This is a paradox.
Or so it would seem. Actually, the trouble lies in the interpretation of the statement "The statement 'All Cretans are liars' is false".
The solution goes like this:
p1. Epimenides is a Cretan.
p2. Epimenides is either a liar or a truth-teller.
p3. His statement is either true or false.
Assume that there is more than one Cretan:
p4. There is more than one Cretan.
Also assume that Epimenides is indeed a liar:
p5. Epimenides is a liar.
p6. Thus Epimenides's statement is false.
p7. Thus "All Cretans are liars" is false.
p8. Thus not all Cretans are liars.
p9. Thus some (one or more but not all) Cretans are not liars.
p10. Thus at least one (but not all) of them is a liar.
p11. Thus Epimenides, a Cretan, could be a liar.
We assumed that Epimenides was a Cretan (p1) and a liar (p5). Therefore, there is no paradox.
Another way of looking at it is to realize that, unless there is only one member of a set, then the negation of "all members of the set", i.e., "not all members of the set", is not "no members" but "some members".
If Epimenides is the only Cretan (so the set of Cretans has only one member), then there would be a paradox, since "not all Cretans" would mean "no Cretans".
...............
(1) This sentence is false.
If (1) is true, then (1) is false. On the other hand, assume (1) is false. Because the Liar Sentence is saying precisely that (namely that it is false), the Liar Sentence is true, so (1) is true. We've now shown that (1) is true if and only if it is false. Since (1) is one or the other, it is both.
http://david.tribble.com/text/liar.htm
http://www.google.com/search?hl=en&ie=UTF-8&oe=UTF-8&q=solution+to+the+liars+paradox
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