Allen claimed that both
P can be proven
and
~P can not be disproven
are true. This is an obvious contradiction, any proof for P is necessarily a disproof of ~P.
No, that's not how things were framed. Logic lesson part 2:
Let P(x) = "x is participating in a covert operation". Then:
a) ∃xP(x) is an existential (∃) statement. Asserting this means you can show a proof, otherwise the statement is as empty as "invisible pink unicorns exist".
b) The negation is ¬∃xP(x) ≡ ∀x¬P(x). This is an universal (∀) statement. This can be both true and unprovable at the same time.
Take for example Goldbach's conjecture. This is an universal (∀) statement: for all natural numbers greater than 5 there's a decomposition of each into the sum of three primes. The opposite is an existential (∃) statement: A number greater than 5 exists such that it's not the sum of three primes.
Now, whoever asserts that Goldbach's conjecture is false can just show a counterexample and settle the question. But the assertion that it's true may be unprovable. I learned that here:
https://www.physicsforums.com/threads/goedel-vs-goldbach.476633/ (unfortunately the reference link in matforum.org that explained the above is now dead). As indicated there, a delicious corollary is that if Goldbach's conjecture is unprovable, then it's true.
Allen went a bit far saying that you can't prove a negative, but I understood it as "you can't always prove a negation of an existential statement" and it was unreasonable of you to ask for a proof of such a statement because it's clearly impossible in this case to prove it.
Now, we can say with a very high degree of confidence that Goldbach's conjecture is probably true. The probability that Goldbach's conjecture is false is not 50%. There are heuristic arguments supporting it, and several conjectures which were supported by heuristic arguments turned out to be true and are now theorems, like the four colour map theorem or Fermat's last theorem.
Likewise, we can say with a very high degree of confidence that there were no covert operations in 9/11, and there are heuristic arguments supporting it. I gave you one, that there's no precedent in history for such a mass killing of own people.