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:

- Its negation is true.
- 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.