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Liar's Paradox
By Brad Rodgers (P1930) on Wednesday, September 13, 2000 - 12:45 am:

In the statement:

"This statement is false"

Can you conclude that the statement is either true or false. I think that you can conclude that it has no true false-value (as there is a contradiction if it is true and if it is false), but this apparently is not the case (or so says my maths teacher).

Thanks

Brad

By Dan Goodman (Dfmg2) on Wednesday, September 13, 2000 - 06:40 pm:

I think that you can only conclude that human intuition is inconsistent, which should come as no surprise to anyone. Another thing you can conclude is that natural language is very woolly, for instance "colourless green ideas" is a perfectly grammatical sentence which has no meaning whatsoever, similarly "This statement is false". Hopefully, someone else will give you a more interesting response, but that's the best I can come up with. If you were to talk about formal systems and provability that would be another matter.

By Michael Doré (P904) on Wednesday, September 13, 2000 - 07:06 pm:

I find it hard to see how the teacher can reject Brad's answer... I think it summarises the situation perfectly!

By Dave Sheridan (Dms22) on Tuesday, September 19, 2000 - 02:57 pm:

I'd be interested to hear what Brad's teacher thinks the answer is. The way I think about such things is that mathematical True and False only apply to mathematically constructed sentences - something you can write in a formal language obeying its natural rules. This clearly can't be done to the sentence above, so the concept of Truth or Falsity can't be applied to the sentence. Of course, you could try to construct a formal language which admits this sentence, but I doubt you'd get anything useful from it...

-Dave

By Brad Rodgers (P1930) on Tuesday, September 19, 2000 - 11:17 pm:

After bringing in several documents on this subject, I was able to make her concede that it had no true-false value. She, however, may have known all along, and just wanted the rest of the class to figure it out on their own.

Brad

By Marcus Hill (T3280) on Tuesday, September 26, 2000 - 07:07 pm:

Martin Gardner's "aha! Gotcha" contains many entertaining paradoxes such as the Liar paradox.

By Robert Bartlett (M723) on Wednesday, September 27, 2000 - 10:26 am:

The sentence in question is non-specific. Is the phrase "This statement" refering to some external statement; if so the true/false condition of the sentance is defined by the true/false condition of that outside statement. If the phrase "This statement" is in reference to itself then the sentence is of no true meaning because of the paradox. It could be argued that the sentence must be false due to its lack of fault, but if the phrase "This statement" is in reference to itself then the sentence is a logistical loop with no comparison to gauge truth or falsehood; a paradox. Basically the question revolves around what it is that is supposed to be false, and that, as I see it, is not clearly defined.

By Hal 2001 (P3046) on Monday, November 13, 2000 - 08:16 am:

"This statemant is false"

Therefore I think that the above is not a statement. This is what an AI program would say!

By Dan Coomber (P3651) on Thursday, January 18, 2001 - 11:05 pm:

I would agree that the statement is not actually a statement at all. This is very similar to the 'un-self-contained sets paradox', which is the set of all sets which do not contain themselves - the paradox being whether or not it contains itself. The general consensus here (I believe) is that this is overlooked as a set altogether.

By Anonymous on Thursday, January 18, 2001 - 11:14 pm:

The paradox arises because you break the standard linear form of mathematical argument.

Definitions -> Propositions -> Proofs

etc

and make a self referential 'definition'.