How can we say it is well-formed when natural language is informal, and hence does not have a notion of well-formed statement? Do you just mean it is grammatically correct? If so, that doesn't tell us much as the statement 'The cheese of five is sad' is also grammatically correct.D is a well formed definition in the natural language. — Meta
Depending on what language we are using, this may be possible. For instance, the sentence:Same goes for sentences. We can't have a statement A where the definition of A mentions A. — Meta
This argument fails when applied to a natural language, because there is no precise definition for <is a well-formed formula>. Only our intuition can tell us what do we consider a WFF. There isn't a fixed set of relation symbols either. — Meta
Any anyways we are not talking about real self-reference just some kind of reflection. Let's modify the statement of the Berry paradox:
"The definition with the least Godel number not definable in fewer than 20 words."
This is also paradoxical. — Meta
Or "The statement with the least Godel number that does not contain the first letter of the alphabet."
This can be a paradox. — Meta
As I mentioned in a private message, I'm not entirely sure this step holds. What is the reference of "this" above? If it is "this statement is false" (taking it to have small scope), then your proposed rule would take us from "All statements are false" to ""This statement is false" is false". But the latter one is false, not truth-valueless. So the whole conjunction is false. — Nagase
We are dealing just in the scope of classical (two-valued) logic and there a statement like "This statement is false" is just without a truth-value as far as I know. — Pippen
So not having a truth value is the third option. If we have the conjunction p ∧ q and if p is false and q doesn't have a truth value then the conjunction as a whole is false. — Michael
Not in classical logic. Not having a truth value is not a third option there. I think 2. of my proof is dubious, maybe nagase will check that out. — Pippen
Get involved in philosophical discussions about knowledge, truth, language, consciousness, science, politics, religion, logic and mathematics, art, history, and lots more. No ads, no clutter, and very little agreement — just fascinating conversations.