Curry’s Paradox

Today at lunch, we ended up spending a fair amount of time pondering and debating Curry’s Paradox. None of us were able to explain it very well even by the end of lunch, and some didn’t think it was meaningful. After reading a couple of web sites, I now understand it, so I’m going to try to explain it in logical symbols.

Here’s the paradox:
“If this sentence is true, then Santa Claus exists”

Obviously the Santa part can be replaced with any statement. I’ll borrow some notation from the wikipedia entry to show the proof.

Let X be the sentence in quotes above (in other words, if X is true, then the sentence is true, if not then the sentence is false).
Let Y be the existence of Santa Claus.

The sentence can then be written:
1. X → Y
since this means that X implies Y, i.e. if X is true, then Y is true.

thus this is true:
2. X = X → Y

and the identity
3. X → X
is always true. (This means “if this sentence is true, then this sentence is true”)

Since we know what X is, we can perform substitution from #2 and get another true statement:
4. X → (X → Y)
It’s hard to translate this part into a sentence, which is why I think it’s hard to describe this without logic symbols. The best I can do is “If X is true, then the statement ‘if X is true then Y is true’ is correct”.

Now, simplifying this to X → Y seems like it shouldn’t be too hard, based upon the English translation of #4. Contradiction does this: Let’s say that X → Y is false (HTML doesn’t have a symbol for → with a slash through it). If X → Y is false, then there must be some case in which X is true and Y is false, thus breaking the implication. However, if X is true, then the previous statement (X → (X → Y)) tells us that Y must be true, thus a contradiction ∴
5. X → Y.

Looking at the definition in #2 and the statement in #5, we can now say:
6. X
.. in other words, the sentence X is true.

Using #5 and #6, we can also say:
7. Y
.. and thus we have proved Y to be true. If we set Y to be a contradiction (such as (Z ∧ ¬Z)), the paradox is created. QED

OK, that’s enough of that for now.

HTML sometimes makes it hard to write stuff like this, but I do like how ∴ is written ∴.