Remember the Hanging Judge entry, which uses the Liar Paradox concept? Liar Paradox is generally credited to the Greek philosopher Epimenides. It asks us to consider the man who says, "What I say is false.", or "All men are liars" etc. I happened to stumble upon the solution to this paradox in a book I'm reading and I'll share it with all of ya. It's a bit chim though, but I'll put it as simply as possible.
(After writing the explanation, I'll rate this explanation as VERY chim, since I took 2 days to fully absorb it. You can take it as a challenge to see whether you can understand the explaination =)
The problem is solved by Jon Barwise and John Etchemendy who applied techniques of situation theory in 1986. (mai siao siao) The real root of the problem is an unacknowledged context. It's similar to the conflict between a American who thinks June is a summer month and a Australian who think June is a winter month. It all depends on context. Let me elaborate using algebra.
Let the what the man say be P (for proposition),
P = [What I say is false]
therefore, P = [P is false] - eqn (1)
Now, to when we state whether a proposition is true (or not), we must take into account the context.
Assuming P is true in context C (which is the context in which the statement is uttered),
substitute eqn (1), we get
[P is false] is also true in context C
This contradict our initial assumption!
Since there is a contradiction when we assume P is true in context C , the ONLY logical explaination is that the statement is false. That's what get us into a dilemma. But wait a minute. What's the context for this statement?? Let's see what happen if we assume P is false in context C.
Assuming P is false in context C ,
substitute eqn (1), we get
[P is false] is false in context C
ie. P is true in context C
This also contradict our initial assumption!
From the 1st assumption (of P is true in context C), we conclude that P MUST be false since the 1st assumption leads to contradiction. However we do not know the context in which P is false. So we assume (P is false in context C). However this 2nd assumption is also wrong since it also leads to contradiction. So the only conclusion is that P is false, but NOT in context C.
The above is the gist of the whole problem, if you cannot understand, maybe this will be of some help:
If a person in country X says truthfully that June is a winter month, we know X is Australia. X is the context of the true statement. If the statement is false, X CANNOT also be Australia.
To put it 'simply', in saying "What I say is false", the person is saying making a claim that refers (implicitly) to a particular context C, the context in which this sentence is uttered. If the claim is true, then it's true in context C, but it leads to a contradiction. So it must be false! But the context for making the observation that the claim is false cannot be C, since if it was, then that too leads to a contradiction. That means the person is making a false statement. However, the fact that the statement is false cannot be asserted in the same context C.
Frankly speaking, I was very confused myself when I wrote the book's explanation. Now I finally (think I) understood. Actually what it's trying to say is that:
This damn stupid statement is neither true nor false in the context in which it was uttered. However, the statement is false when taken out of context (whatever rubbish that means). Pay attention to the context!!! Problem solved.
Hope you all can understand the explanation. Any disagreement or anything you don't understand please voice up.
Solution from Goodbye Descartes by Keith Devlin
PS. Finally the Liar Paradox is put to rest and we've no more rubbish about Liar Paradox in JF's blog again. *Heaving a sigh of relieve* Haha, you must be thinking this, but you're wrong...
