Curry's 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").

Curry's paradox is one of a group of paradoxical sentences (which also includes the liar paradox) which can be formulated in any language meeting certain conditions. These include: (1) The language must contain apparatus which lets it refer to, and talk about, its own sentences (such as quotation marks, names, or expressions like "this sentence"), and (2) The language must contain its own truth-predicate: that is, the language, call it "L", must contain a predicate meaning "true-in-L", and the ability to ascribe this predicate to any sentences. (Various other sets of conditions are also possible.) Natural languages nearly always contain all these features.

Logicians are undecided whether such sentences are somehow impermissible (and if so, how to banish them), or meaningless, or whether they are fine and reveal problems with the concept of truth itself (and if so, whether we should reject the concept of truth, or change it), or whether they can be rendered benign by a suitable account of their meanings.

Let us denote by X the proposition "If I am not mistaken, Y is true." The paradox results from the fact that X itself asserts that if X, then Y. Because that true statement is equivalent to X, X is true. Therefore, Y is true, and Santa Claus exists.

Another way of understanding this paradox is to observe that X = (X → Y), so if Y is false then X = (X → false) or equivalently (X = not-X). In other words, if Y is false, the proposition X morphs into "this statement is false", which is a contradiction.

Note that unlike Russell's paradox, this paradox does not depend on what model of negation is used, as it is completely negation-free. Thus paraconsistent logics still need to take care.

The resolution of Curry's paradox is a contentious issue because nontrivial resolutions (such as disallowing X directly) are difficult and not intuitive.

In set theories which allow unrestricted comprehension, we can prove any logical statement Y from the set

<math>X \equiv \left\{ x | x \in x \to Y \right\}.<math>

The proof proceeds:

<math> \begin{matrix}

\mbox{1.} & X \in X \iff ( X \in X \to Y ) & \mbox{definition of X} \\ \mbox{2.} & X \in X \to ( X \in X \to Y ) & \mbox{from 1} \\ \mbox{3.} & X \in X \to Y & \mbox{from 2, contraction} \\ \mbox{4.} & (X \in X \to Y) \to X \in X & \mbox{from 1} \\ \mbox{5.} & X \in X & \mbox{from 3 and 4} \\ \mbox{6.} & Y & \mbox{from 3 and 5}

\end{matrix} <math>

External links

• http://luddite.cst.usyd.edu.au/cgi-bin/twiki/view/Jason/PenguinsRuleTheUniverse - A short discussion of Curry's paradox
• http://plato.stanford.edu/entries/curry-paradox/ - The Stanford Encyclopedia of Philosophy has an in-depth technical discussion.pl:Paradoks Curry'ego