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