# If I ask a person if they can say "no" and they say "no", is this a paradox?

If I ask a person if they can say "no" and they say "no", is this a paradox?

If they answer "no" it means they can't say "no", but they just said it

• Not necessarily. He could just be a liar :)
– Joe
Commented Mar 1 at 22:19
• Why such a problem is important? Please provide some details and motivations.
– Amir
Commented Mar 1 at 22:25
• Or it could be something like saying "No hablo español", which means that you know just enough Spanish to inform someone that you don't speak Spanish.
– Dan
Commented Mar 1 at 22:28
• Commented Mar 1 at 22:29
• Sure you are not asking about being able to pronounce the sound or type the characters, I guess you are asking about the ability to declare a (mathematical) statement as false? Uh-Oh, big trouble, see en.wikipedia.org/wiki/Entscheidungsproblem . Commented Mar 1 at 22:37

It's not a paradox, but you're close to something which is. It might be a reasonable assumption that every statement has a truth value. That is, every statement is either true or false. We may not know which, but it certainly is one or the other. This assumption, however, is wrong.

Classically, the example is the "Liar's Paradox:"

"This sentence is false"

Think about this for a moment, and see that it has no consistent truth value. If it that statement is false, we are lead to the conclusion that it is actually true. If instead we see the statement as true, we are forced then to accept that it is false. This statement has no truth value.

There are (informal) set theoretic analogs in Russel's Paradox, which is one of causes of the rigorous axiomatization of math and set theory in the first place. However, even in this rigorous setting (or any rigorous, powerful enough setting), it is a result of Gödel that there are true statements which are unprovable.

• That assumption is not wrong if one is using proposition in the technical sense (as in formal logic as opposed to the Logic of natural language). Not per chance, Russell's Paradox has to rather be ruled out from set theory. And this is really unrelated from the issue of incompleteness: that some propositions are undecidable and/or unprovable does not change the definition of proposition. Commented Mar 2 at 14:58

The "trick" lies in the fact that the two no's don't play the same role. Indeed, the first no refers to a particular object/word inside the proposition "Able to articulate [a given word]", when the second one is a truth value assigned to this same proposition.

In a way, this is an analog of Russell's paradox, as mentioned by Malady in his answer. This kind of paradoxes show that a language cannot handle propositions about itself, but a metalanguage is needed to treat such propositions.

• It’s “her,” but I agree. Commented Mar 2 at 15:30