Most philosophers are familiar with the Liar Paradox:
L: This sentence is not true.
If L is true, then things are the way it reports, namely, L is not true.
If L is not true, then thing are not the way it reports, namely, L is true.
So, we seem to be able to conclude that L is true iff L is not true. Paradox.
But let’s consider some related sentences:
P: I promise not to fulfill this promise.
C: Do not comply with this command.
Q1: What is an incorrect answer to this question?
Q2: What is not an answer to this question?
Now, P, C, Q1 and Q2 seem to admit of similarly paradoxical results.
An action fulfills the promise made by P just in case it does not fulfill the promise.
An action complies with the command issued by C just in case it does not comply with the command.
Something is the correct answer to Q1 just in case it is not the correct answer to Q1.
Something is an answer to Q2 just in case it is not an answer to Q2.
It is worth mentioning that, like L, each of the above can be reformulated in non-directly self-referential terms (replacing name for the sentence with “the the Nth labeled sentence in such-and-such blog post” and rephrasing slightly).
One thing that I want to note is that, if these are genuinely Liar-like paradoxes, they might be taken to suggest that focusing on “truth” in the Liar paradox is something of a red-herring. Prima facie, none of these invokes the truth-predicate, but seem to be of a piece with the Liar paradox.
One thing I want to ask is whether these paradoxical sentences have been discussed in the literature. I’ve read a fair amount about the Liar and the truth predicate, but haven’t come across any discussion of these sorts of sentences. The closest thing I know of is Markosian’s paradox of the question (“What is the pair <Q, A> such that Q is the most useful question, and A is its correct answer?”).