
Curry's paradox: "If this sentence is true, then Santa Claus exists." - byrneseyeview
http://en.wikipedia.org/wiki/Curry%27s_paradox
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hobbs
The article has one point wrong. The statement "If this sentence is true, then
Santa Claus exists." is not equivalent to "If this sentence is false, then
Santa Claus doesn't exist." It _is_ equivalent to "If Santa Claus doesn't
exist, then this sentence is false".

To break it down into standard boolean algebra, the statement "if A then B" is
logically equivalent to saying "not A or B". !A|B = B|!A = "If not B then not
A".

Now back to the paradox, using boolean terms. The statement can be rephrased
as !A|B: "This sentence is false or Santa Claus exists." Since we know that
Santa doesn't exist, the sentence must be false: !(!A|B). That is equivalent
to A&!B: "This sentence is true and Santa Claus doesn't exist." Here is where
we have the paradox. We just said that the sentence must be false, but that
then implies that the sentence is true.

The main mind-bender here is to realize that the negation of "If this sentence
is true, then Santa Claus exists." is "This sentence is true and Santa Claus
doesn't exist." We are not able to get rid of the statement "this sentence is
true" no matter which way we toggle it.

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mynameishere
_The article has one point wrong._

See the [edit] link?

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hobbs
I can't now, since _my_ statement would then be false. :)

Seriously though, I thought about it, but then I realized that I would
probably have to replace the entire paragraph, since it seems to be a key step
in the original author's explanation.

Maybe I will replace it anyway, since it can't be a very good explanation if
it relies on faulty logic.

-

Update: Did it. Now I'm contradicting myself. Gotta' love these logic games.

~~~
Retric
If this sentence is true, then Santa Claus exists.

Basically assume A is true then the non existence of Santa Clause is a
contradiction so A is false.

If (A then B) says nothing if A is false so your done.

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byrneseyeview
Another explanation: Penguins rule the universe
(<http://xeny.net/PenguinsRuleTheUniverse>).

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dusklight
Sorry I am a bit lost here. How is this a paradox?

Now we can conclude that if this sentence is true, then Santa Claus exists,
then necessarily,

if Santa Claus does not exist, this sentence is false, but

if this sentence is false, we cannot conclusively make any statement about the
existence of Santa Claus.

Even if you modified the above statement to

iff this sentence is true, then Santa Claus exists,

the above would still apply. If the sentence is false, then the iff condition
is also false, so if the sentence is false, Santa Claus may or may not exist.

~~~
bayareaguy
Sometimes the term paradox is used for situations that are merely surprising.

~~~
derefr
I take a more practical definition of "paradox": anything that throws a Prolog
interpreter into an infinite loop ;)

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redorb
Someone (perhaps the submitee) please tell us why, this is relevant, dealing
with infinite loops? or something? -

~~~
byrneseyeview
If you had to reduce it to a sentence, it would be: this is something that
gratifies my intellectual curiosity.

