Curry's paradox is a paradox that occurs in naive set theory or naive logics, and allows the derivation of an arbitrary sentence from a self-referring sentence and some apparently innocuous logical deduction rules. It is named after the logician Haskell Curry.

It has also been named Löb's paradox after Martin Hugo Löb.

In natural language

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A natural language version of Curry's paradox might be a box which contains only a slip of paper stating the following:

{| border=1 cellpadding=4 If everything in this box is true, then Santa Claus exists.

Suppose everything in the box is true. The slip says that if everything in the box is true, then Santa Claus exists. We are supposing that everything in the box is true, therefore we can conclude that Santa Claus exists.

We have found, then, that if we assume that everything in the box is true, then we can prove the existence of Santa Claus. But this is precisely what the slip was telling us, so the slip was right after all. Since it's the only thing in the box, that means everything in the box is, in fact, true. It follows therefore that Santa Claus exists.

There was nothing special about the clause "Santa Claus exists"; by this means, any proposition, whether true or not, may be proved.

In mathematical logic

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Let us denote by Y the proposition to prove, in this case "Santa Claus exists". Then, let X denote the statement in the box, which asserts that Y follows from the truth of X. Mathematically, this can be written as X = (X → Y), and we see that X is defined in terms of itself. The proof proceeds:

1. X → X

identity

2. X → (X → Y)

substitute right side of 1, since X = X → Y

3. X → Y

from 2 by contraction

4. X

substitute 3, since X = X → Y

5. Y

from 4 and 3 by modus ponens

A particular case of this paradox is when Y is in fact a contradiction of the form Z∧¬Z. Then X becomes X = (X → (Z∧¬Z)). If the law of non-contradiction is accepted, then, from (X → (Z∧¬Z)), it follows that ¬X. Conversely, if the principle of explosion is accepted, and if ¬X, then from X anything follows, and in particular X → Z∧¬Z. Therefore X → Z∧¬Z is equivalent to ¬X. So X = ¬X, which is exactly the liar paradox.

In naive set theory

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Even if the underlying mathematical logic does not admit any self-referential sentence, in set theories which allow unrestricted comprehension, we can nevertheless prove any logical statement Y from the set

$X\; \backslash equiv\; \backslash left\backslash \{\; x\; |\; (\; x\; \backslash in\; x\; )\; \backslash to\; Y\; \backslash right\backslash \}.$

The proof proceeds:

$\backslash begin\{matrix\}$

\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, modus ponens} \\ \mbox{6.} & Y & \mbox{from 3 and 5, modus ponens}

\end{matrix}

Again a particular case of this paradox is when Y is in fact a contradiction. Then X becomes $\backslash left\backslash \{\; x\; |\; (\; x\; \backslash in\; x\; )\; \backslash to\; (\; Z\; \backslash wedge\; \backslash neg\; Z\; )\; \backslash right\backslash \}$, which, similar to above, is equivalent to $\backslash left\backslash \{\; x\; |\; (\; x\; \backslash notin\; x\; )\; \backslash right\backslash \}$, the set of all sets which do not contain themselves. This is exactly Russell's paradox.

Discussion

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Curry's paradox can be formulated in any language meeting certain conditions:

1. The language must contain an apparatus which lets it refer to, and talk about, its own sentences (such as quotation marks, names, or expressions like "this sentence");
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;
3. The language must admit the rule of contraction, which roughly speaking means that a relevant hypothesis may be reused as many times as necessary; and
4. The language must of course admit the rules of identity ("if A, then A") and modus ponens (from "A", and "if A then B", conclude "B").

Various other sets of conditions are also possible. Natural languages nearly always contain all these features. Mathematical logic, on the other hand, generally does not countenance explicit reference to its own sentences, although the heart of Gödel's incompleteness theorem is the observation that usually this can be done anyway. The truth-predicate is generally not available, but in naïve set theory, this is arrived at through the unrestricted rule of comprehension. The rule of contraction is generally accepted, although linear logic (more precisely, linear logic without the exponential operators) does not admit the reasoning required for this paradox.

Note that unlike the liar paradox or Russell's paradox, this paradox does not depend on what model of negation is used, as it is completely negation-free. Thus paraconsistent logics can still be vulnerable to this, even if they are immune to the liar paradox.

The resolution of Curry's paradox is a contentious issue because nontrivial resolutions (such as disallowing X directly) are difficult and not intuitive. Logicians are undecided whether such sentences are somehow impermissible (and if so, how to banish them), or meaningless, or whether they are correct 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.

External links

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• 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.

Paradoxes | mathematical logic

Paradoja de Curry | Curry paradoxonja | Paradoks Curry'ego | Curryn paradoksi