Curry's Paradox

[FleshLight]

http://en.wikipedia.org/wiki/Curry%27s_paradox

In logic, specifically mathematical logic, Curry's paradoxes are a family of logical paradoxes that occur in naive set theory or naive logics. They are named after the logician Haskell Curry.

An informal version runs as follows:

Abelard: "If I'm not mistaken, then Santa Claus exists."
Eloise: "I agree: if you are not mistaken then Santa Claus exists."
Abelard: "You agree: what I said was correct."
Eloise: "Yes."
Abelard: "Then I am not mistaken."
Eloise: "True."
Abelard: "If I am not mistaken, then Santa Claus exists. I am not mistaken. Therefore, Santa Claus exists."
By this means, any proposition, whether true or not, may be proved.

Curry's paradox is: "If I'm not mistaken, Y is true", where Y can be any statement at all. ("black is white", "2 = 1", "Gödel exists", "the world will end in a week").

 

[CrackRabbit]

So if you can get two or more people to agree on something then it is true?

 

[KLin]

If i'm not mistaken, you're a tool.

 

[Stumps]

true

 

[CrackRabbit]

Originally posted by: KLin
If i'm not mistaken, you're a tool.

I agree!

 

[Howard]

Is that supposed to be confusing? The conclusion is valid only if all the premises are true, and his not being mistaken is false (he is mistaken).

 

[EULA]

Liar paradox

This statement is false.

The following sentence is true.
The preceding sentence is false.

 

[Stumps]

ummm...confused

 

[FleshLight]

Originally posted by: Stumps
ummm...confused

Something a little more simplistic:

Consider the following list of sentences, named ?The List?:

1. Tasmanian devils have strong jaws.
2. The second sentence on The List is circular.
3. If the third sentence on The List is true, then every sentence is true.
4. The List comprises exactly four sentences.

Although The List itself is not paradoxical, the third sentence (a conditional) is. Is it true? Well, suppose, for conditional proof, that its antecedent is true. Then the third sentence of The List is true is true. By substitution, it follows that If the third sentence of The List is true, then every sentence is true is true. But, then, Modus Ponens on the above two sentences yields that every sentence is true is true. So, by conditional proof, we conclude that If the third sentence of The List is true, then every sentence is true
is true. By substitution, it follows that the third sentence of The List is true is true. But, now, by Modus Ponens on the above two sentences we get that every sentence is true is true. By naive truth theory we disquote (or, in this case, dis-display, as it were) to conclude: Every sentence is true! So goes (one version of) Curry's paradox.

http://plato.stanford.edu/entries/curry-paradox/

 

[Stumps]

I was being sarcastic!

 

[puffff]

interesting stuff