More precisely, proving a negative requires quantifying over the entire underlying model and applying the law of excluded middle or double negation elimination. Proofs of a positive are generally constructive, and valid in intuitionistic logic which rejects the law of excluded middle as an axiom.
So you can't prove a negative if:
* It is infeasible to quantify over the entire underlying model
* You're working within a logic that doesn't admit it as an axiom, for practical reasons.
It's easy for me to argue that there are no pink ducks on my desk. It's very difficult to argue that there are no pink ducks.
People usually use "pink duck" as "example of thing that is unlikely to exist". If you choose "proposition guaranteeing that the thing doesn't exist", I think you get this argument:
Claim: There are no non-existent objects.
Proof. Let x be a non-existent object. x is a witness for its own proof, so x exists. Contradiction.
That one works in a couple logics I looked up. People boldly claiming that "For any predicate P, forall x. ~P(x) is unprovable" would have to reconcile this example. Also, the empty model satisfies that.
Of course, he may have literally meant "You can't prove a negative", as in "lumberjack on Hacker News is unable to produce a proof of any formula in the format given above". That depends on how eagerly you evaluate "You".
So you can't prove a negative if:
* It is infeasible to quantify over the entire underlying model
* You're working within a logic that doesn't admit it as an axiom, for practical reasons.
It's easy for me to argue that there are no pink ducks on my desk. It's very difficult to argue that there are no pink ducks.
People usually use "pink duck" as "example of thing that is unlikely to exist". If you choose "proposition guaranteeing that the thing doesn't exist", I think you get this argument:
Claim: There are no non-existent objects.
Proof. Let x be a non-existent object. x is a witness for its own proof, so x exists. Contradiction.
That one works in a couple logics I looked up. People boldly claiming that "For any predicate P, forall x. ~P(x) is unprovable" would have to reconcile this example. Also, the empty model satisfies that.
Of course, he may have literally meant "You can't prove a negative", as in "lumberjack on Hacker News is unable to produce a proof of any formula in the format given above". That depends on how eagerly you evaluate "You".