Sunday, 18 January 2015

Logical paradoxes

I'm talking about logical paradoxes (This lecture will surprise you: when logic is illogical) at Gresham College on Monday 19th January, which is tomorrow as I write this on Sunday afternoon.  I've been fascinated by these for years, thanks to writers like Martin Gardner, Raymond Smullyan, Douglas Hofstadter.

It's nice to prove things: suppose I want to prove something which is slightly doubtful (like "Arsenal will beat Manchester City this afternoon" - a very unlikely proposition).  Here's a proof from Martin Gardner.  Consider these two statements:

A: Both these statements are false
B: Arsenal will beat Manchester City this afternoon.

Clearly A cannot be true, since if it were it would contradict itself.  So A is false, and if B were also false, then A would be true,  So B must be true.

One proof isn't always enough,  So here's another - this one is Curry's Paradox.  Consider the statement:

If this statement is true, then Arsenal will beat Manchester City this afternoon.

Is this statement true?  It's of the form "If A, then B", and we test that by seeing what happens when A is true.  So assume that the first part of the statement above is true - which means that the whole statement is true, because that is what that clause asserts.  And if that whole statement is true, and the first part is true, then the second part is true. So we have established the truth of the statement above, And if it is true, then Arsenal will win.

So I've proved in two different ways that Arsenal will win, despite almost all the pundits and 76% of the BBC poll thinking the opposite.

ADDED AT 6pm: Arsenal did win.  Which proves the power of mathematical logic.


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  2. Good evening!!
    I just watched your lecture on you tube,
    I have a bit of a question, please help me if you have a moment?!?

    You said:

    "A: Both these statements are False
    "B: I am the greatest mathematician in the world".

    "If B is false, then A would be true, yet A states in itself that it is False, so we have a contradiction."
    "But, If B is true, then A would be false, just as A says it is false, so we have no contradiction if B is True. Thus: B must be true."

    Now, I don't see how this escapes the paradox, nor does it avoid contradiction....


    If B is FALSE, then A would be true, and if A is true, A then becomes false. If A is then decided to be false, it is actually true, and on and on.....

    If B is TRUE, then A would actually be false, which is exactly what A says it is, so therefore A would be true, but if A is true, then it is false, and when A becomes false, we know then that it is true...and on and on and on again....

    This doesn't seem to escape the contradiction. This is really just a pesky liar paradox hanging around and imposing itself upon another statement.

    A paradoxical statement, which is both self-referential and referential to another non-self-referential statement, one who's truth or falsehood depends on the same of the paradoxical latter, only leaves us with the inability to determine anything about either statement. We remain (by either begining or ending) in a cyclical loop of uncertainty within the first sentence, thus not being able to decide anything about the validity of the second.

    You claim this example escapes the contradiction, with the only logical answer being that "B is true". I do see how you got here, but I don't see how it can stop here.

    At the same point that we decide that you ARE actually the greatest mathematician in the world, the first statement instantly becomes false, which then, by it's own self-proclamation of it's falsity, makes it true, so then in turn it makes us realize that you are no longer the greatest mathematician in the world....(no worries, keep going and you'll be back to the greatest again, in @ half a second.)

    Please correct me here, or explain if I am wrong?

    What I think is that logic itself is self-referential.
    You want to be the greatest mathmatician in the world, (maybe you are? Who knows!) but because this is your identity, this is where reasoning ends.

    Maybe no self-referential system, which makes claims of it's own truth or falsehood, should be trusted to determine anything for certain about elemets of either itself or outside of it?

    The Bible, Science, and reason itself, all seem to do this.
    Oh dear, now what?

    Certain uncertainty!!

    I've never studied logic, but this sure is fun!

    1. Thanks for your comment. First of all, I am certainly not the greatest mathematician in the world (do I want to be? I'm not sure!)
      I think the argument is that if we have two statements, A which says "A and B are both false" and B which can be anything you like, then the only consistent arrangement is that B is true and A is false. Then A is indeed false, because it is not true that both statements are false, since B is not false. But any other arrangement leads to a contradiction. Thus the argument appears to prove that B must be true (even if B is complete nonsense). Which is a little worrying.

  3. nice blog.