nLab
Curry's paradox

Context

Foundations

Type theory

natural deduction metalanguage, practical foundations

  1. type formation rule
  2. term introduction rule
  3. term elimination rule
  4. computation rule

type theory (dependent, intensional, observational type theory, homotopy type theory)

syntax object language

computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory

logiccategory theorytype theory
trueterminal object/(-2)-truncated objecth-level 0-type/unit type
falseinitial objectempty type
proposition(-1)-truncated objecth-proposition, mere proposition
proofgeneralized elementprogram
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
cut elimination for implicationcounit for hom-tensor adjunctionbeta reduction
introduction rule for implicationunit for hom-tensor adjunctioneta conversion
logical conjunctionproductproduct type
disjunctioncoproduct ((-1)-truncation of)sum type (bracket type of)
implicationinternal homfunction type
negationinternal hom into initial objectfunction type into empty type
universal quantificationdependent productdependent product type
existential quantificationdependent sum ((-1)-truncation of)dependent sum type (bracket type of)
equivalencepath space objectidentity type/path type
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
completely presented setdiscrete object/0-truncated objecth-level 2-type/preset/h-set
setinternal 0-groupoidBishop set/setoid
universeobject classifiertype of types
modalityclosure operator, (idempotent) monadmodal type theory, monad (in computer science)
linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation
proof netstring diagramquantum circuit
(absence of) contraction rule(absence of) diagonalno-cloning theorem
synthetic mathematicsdomain specific embedded programming language

homotopy levels

semantics

Contents

Idea

In mathematical logic and mathematical foundations Curry’s paradox is a paradox which is a version of Russell's paradox that does not involve the use of negation.

Russell’s paradox is a proof of falsehood, rendering a certain class of formal systems (those including some kind of unlimited axiom of comprehension) “inconsistent” in the sense that they prove false, which by the law of ex falso quodlibet entails that they prove anything at all. Curry’s paradox is a direct proof of “anything at all” without the detour through falsehood. Thus, it applies more generally than Russell’s paradox, e.g. to theories lacking a notion of “false”, or to paraconsistent theories whose notion of “false” does not satisfy ex falso quodlibet.

Curry’s paradox does, however, still depend on the structural rule of contraction. Thus, it can be avoided by systems based on linear logic.

Argument

Let PP be any statement at all, and consider the set

C={x(xx)P}. C = \{ x \mid (x\in x) \Rightarrow P \}.

Then if CCC\in C, by definition we have (CC)P(C\in C) \Rightarrow P, and hence by modus ponens we have PP. Therefore, (CC)P(C\in C) \Rightarrow P. But by definition this means that CCC\in C, and therefore (as we just proved) PP.

Note that, if negation is defined as ¬P=(P)\neg P = (P \Rightarrow \bot) for some notion of falsehood \bot, then Russell’s paradox is the special case of Curry’s paradox for P=P=\bot.

References

In relation to the axiom of full comprehension: