David Wyde

Curry's Paradox

2021-02-18

Curry’s paradox arises from statements of the form “If this sentence is true, then [a false proposition].” Wikipedia gives the example of “If this sentence is true, then Germany borders China.” Unlike the liar paradox, Curry’s paradox does not directly use negation (SEP).

I have two observations about Curry’s paradox:

  1. Its negation is true.
  2. It is equivalent to the liar paradox.

Both points require that the negation of a self-referential sentence refers to the new sentence (itself) rather than to the original one, which I have previously discussed for the liar paradox and Gödel’s incompleteness theorems.

Negating Curry’s Paradox

Curry’s paradox features implications of the form p → q, where p is “This sentence is true” and q is a false proposition. The negation of such an implication is p ∧ ¬q. Taking the example from earlier, we have “This sentence is true and Germany does not border China,” which is true.

Equivalence to the Liar Paradox

p → q is logically equivalent to ¬p ∨ q. Curry’s paradox uses a false q, so ¬p ∨ q reduces to ¬p. The negation of “This sentence is true” is “This sentence is false,” which is the liar paradox.

Validity Curry

In their paper “Two flavors of Curry’s paradox,” Beall and Murzi give a variant of Curry’s paradox called v-Curry, where “v” stands for “validity”. An example version is “The argument from me to absurdity is valid.” Its negation, “The argument from me to absurdity is not valid,” is true.

Conclusion

A different way of negating self-referential sentences resolves two forms of Curry’s paradox.