David Wyde

Negating Self-Referential Sentences


What is the opposite of a self-referential sentence?

Two Ways to Negate Self-Referential Sentences, 2 pages.

One form of the liar paradox is “this sentence is false.” Trying to evaluate this paradoxical statement leads to a type of infinite loop: if it is true it must be false, and if it is false it must be true.

Kurt Gödel’s incompleteness theorems are based on an undecidable sentence, roughly “this statement is not provable.” As in the liar paradox, neither that sentence nor its negation is provable.

One way to bypass the liar paradox and Gödel’s incompleteness theorems is to have the negation of a self-referential sentence refer to the new sentence (itself) rather than to the original statement.

With this approach, the negation of the liar sentence is “this sentence is true,” which resolves the paradox. A similar analysis applies to Gödel’s undecidable sentence.