Argument
Curry's paradox
Overview
Curry's paradox is a paradox in which an arbitrary claim F is proved from the mere existence of a sentence C that says of itself "If C, then F". The paradox requires only a few apparently innocuous logical deduction rules. Since F is arbitrary, any logic having these rules allows one to prove everything. The paradox may be expressed in natural language and in various logics, including certain forms of set theory, lambda calculus, and combinatory logic.
The paradox is named after the logician Haskell Curry, who wrote about it in 1942. It has also been called Löb's paradox after Martin Hugo Löb, due to its relationship to Löb's theorem.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Possible passages
Found by search and not yet checked by a person; they may only mention the idea in passing.
Since it is impossible for different numbers to have any of their modes of formation completely in common, it is a kind of paradox to say, that all propositions which can be made concerning numbers relate to their modes of formation from other numbers, and yet that there are propositions which are true of all numbers. But this very paradox leads to the real principle of generalization concerning the properties of…
John Stuart Mill, A System of Logic, Ratiocinative and Inductive
States what Curry's paradox is.
In this case, however, the predicate and subject are not merely reversed, but one of them is changed. Instead of [A] and [B], the terms of the new proposition are [a thing which is not B], and [A]. The original proposition, Some A _is not_ B, is first changed into a proposition æquipollent with it, Some A _is_ "a thing which is not B;" and the proposition, being now no longer a particular negative, but a particular…
John Stuart Mill, A System of Logic
States what Curry's paradox is.
This postulate is called the “principle of limitation of variety”. Mr. Keynes again finds that it is needed in attempts to establish laws by statistics; if he is right, it is needed for all our scientific knowledge outside pure mathematics.
Bertrand Russell, Philosophy
States what Curry's paradox is.
Related ideas
- Germany
- Law of identity
- Zermelo–Fraenkel set theory
- Modus ponens
- Paradox
- Set theory
- Self-reference
- Tautology (logic)
- Russell's paradox
- Logic
- Peirce's law
- Conditional sentence
- Natural deduction
- Axiom schema of specification
- China
- Gödel's incompleteness theorems
- Conditional proof
- Consistency
- Implicational propositional calculus
Place in the tradition
Part of Paraconsistent logic.