# Curry's paradox (a.k.a. omigod, this is so weird. I just proved that 1+1=3 w/ LOGIC)

1. Jul 16, 2009

### lolgarithms

Curry's paradox can be used to (dis)prove the riemann hypothesis and string theory, and even prove the (non)existence of God... no, actually, Curry's paradox IS God.

Just kidding... I am now (speaking somewhat hyperbolically) freaked out. Does Curry's paradox go like this (try "1 = 0" or anything you like for P): I don't think you need the whole contraction (A->(A->B) = A->B) thing for this paradox to appear. Contraction is just substituted by properties of OR, and I use the definition of the material conditional.

1. Let S := "If (material conditional) S is true, then P" = "S -> P"
2. Which also means: "P, or S is false" = "P or not-S"
3. S implies itself: S -> S
4. substitute: "If S is true, then (P, or S is false)" = S -> (P or not-S)
5. the material conditional means: (P or not-S) or not-S
6. OR (logical disjunction) is associative: P or (not-S or not-S)
7. x OR x = x: P or not-S

8. Hey, that's... S. Both the conditional and the condition (S is true) are now proven.
9. Therefore, P.

$$\mathcal{Q.E.D.}$$
" *evil laugh* now all your common sense is (not) destroyed... muhahahaha.... "

Contraction? Some logics explicitly allow it. but I think it only depends on the definition of material conditional, associativity and idempotence (X or X = X) of OR, and S->S, which are more fundamental in ways.

P.S. Simpler presentation for laymen:
consider the sentence: "If this sentence is true, then the Flying Spaghetti Monster exists"

Alright. suppose the sentence is true. then:
*If the sentence is true, then the Flying Spaghetti Monster exists.
*the sentence is true.
*Therefore, the Flying Spaghetti Monster exists.
So if the sentence is true, then the Flying Spaghetti Monster exists.

But that's what the sentence says, so the sentence is true.
Therefore, the Flying Spaghetti Monster exists.

Last edited: Jul 16, 2009
2. Jul 16, 2009

### CompuChip

Re: Curry's paradox (a.k.a. omigod, this is so weird. I just proved that 1+1=3 w/ LOG

I don't really see the problem. In statement 1 you have S. In statement 2 you have S. In statement 7 you derive S. You have derived S from S. How does that imply P?

3. Jul 16, 2009

### lolgarithms

Re: Curry's paradox (a.k.a. omigod, this is so weird. I just proved that 1+1=3 w/ LOG

I didn't *assert* S in 1 and 2. I simply *defined* S to be P or not-S, not assert its truth.
Then I substituted P or not S for S in "S or not-S", which is a true statement in classical logic: (P or not-S) or not-S). But then OR is associative, and idempotent: (P or not-S) or not-S = P or (not-S or not-S) = P or not-S. But that's what S says. We have both the premise and the conditional, so by modus ponens we have derived P.

Last edited: Jul 17, 2009
4. Jul 17, 2009

### CompuChip

Re: Curry's paradox (a.k.a. omigod, this is so weird. I just proved that 1+1=3 w/ LOG

Ah I see it now. Basically, it's saying that certain self-referential statements cannot be assigned a truth value without making any arbitrary statement tautological.

5. Jul 17, 2009

### lolgarithms

Re: Curry's paradox (a.k.a. omigod, this is so weird. I just proved that 1+1=3 w/ LOG

But the proof of Curry's paradox tells you that you don't HAVE to assign any truth value to statements like "If this statement is true, then P" to begin with - the mere existence of such statements automatically proves them in the logic we normally use. So is there a way to somehow make them nonexistent or "grammatically incorrect"?

Last edited: Jul 17, 2009
6. Jul 18, 2009

### Gil Grissom

Re: Curry's paradox (a.k.a. omigod, this is so weird. I just proved that 1+1=3 w/ LOG

In formal logic, they are non-existent, because in e.g. propositional calculus you can't construct self-referential statements.

In natural language, I don't think you can do anything about it.