Curry's paradox

Curry's paradox is a paradox that Haskell Curry in 1942 described; allows the derivation of any evidence of a self-referential expression which presupposes its own validity by means of simple, general logical rules. He showed in this way the inconsistency of axiomatic systems with such an expression.

Verbal version

Curry's paradox can be expressed verbally and derived by the following self-referential sentence:

Currys derivation of the paradox is easily understandable if this sentence is abbreviated to S. Thus, S is in a nutshell: If S is true, then applies A. Well obviously applies: If S is true, then applies S. Substituting S here in excerpt one, the result is: If S is true, then applies "If S is true, then A ". Now you can but simply omit a repeated condition without meaning change, so that the result is: If S is true, then applies A. This is exactly the set S. This applies the premise of S and one can infer A. Thus, any statement is provable, even if one chooses it absurd.

Language requirements

Curry's paradox can be formulated in any language that meets the following conditions:

• The language allows the modus ponens: From A and " if A then B" is B includes:

• The language allows the contraction, after a repeated premise without change of meaning can be omitted:

• The language allows the tautology " if A, then A":

• The language can express a self-reference by a statement that has an equivalent formula in which occurs, so that the self-reference has the following form:

Classical logic and many non-classical logics, in particular intuitionistic logics and paraconsistent logics even meet these criteria, of course, but also the verbal language, the self-references with pronouns instead of variables expresses. Curry's paradox deliberately used no negation and no indirect evidence, which usually paradoxes are derived, but developed a more general direct derivation.

Derivation of the paradox

The special self-reference in curries paradox is a free variable for an arbitrary statement. The formal proof of this statement variable is then:

As a tautology applies:

The replacement of the right side by self-reference gives:

It follows by contraction:

The replacement with the self- respect leads to:

From (4) and (3) follows the mode finally ponens:

With this derivation, the inconsistency of the axiom system is shown because all statements are provable. It should be noted that the self-reference is an additional axiom, which is applied twice in addition to the above language requirements in the derivation! The derivation shows that if this self-reference used as an additional argument is wrong: it is not relatively consistent for Axiom systems in which are the language requirements; here that is only the formulation properties of the self-reference is required, but not its validity.

Theoretical amount of variation

In naive set theory created a variant of the paradox at the following class:

From it is based on application of the unrestricted abstraction principle the following self-reference:

For this self-reference, the statement can be proved as above, and thus prove the inconsistency of this self-reference.

As for the logic of classes and the general set theory are provided (without axiom of infinity ) consistency proofs, curries derivation leads to no contradiction here, but proves that the class is not a lot, but a so-called real class. The self-reference here follows from the naive unrestricted abstraction principle, which can not apply; only a bound, quantified abstraction for amounts is permitted.

Classic special cases

Special cases of the paradox arise in the classical logic or intuitionistic logic, if the free variable a contradiction is established, then it follows from the self-reference. Then is equivalent to by contraposition and the principle of contradiction. The paradox thus has the form of the liar paradox in a propositional formulation by negation. In the set-theoretical variant is equivalent to Russell's class, which is responsible for Russell's antinomy.

Loeb's application

Curry's paradox in 1955, applied by Hugo Martin Loeb, to show that sentences that assert their own provability, must be true. Therefore, it is sometimes referred to in the literature as Loeb's paradox.