That is accurate both for propositional logic and programming, but they still deal with positive claims in one way or another. You positively state something that is provable - ¬P falls into that category.

I agree in the sense that the validity of a proof relies on the relationship between

*true* premises. In that regard we're always talking about true assertions and not false ones. But the premises may contain negations, and negations are logically equivalent to 'false' in propositional logic.

Maybe I'm still misunderstanding you, but I'm saying that 'Not P' and 'P is false' are logically equivalent. One can assert that P is false by asserting 'Not P' with no problems. It seems like you're saying that one cannot assert that something is false in propositional logic.

I think this 'negative/positive' label is what's causing the confusion. I dunno what it means to positively or negatively state something, and I think that the distinction is entirely superfluous. Assertions are assertions. They may contain negations.

However, in non-mathematical arguments, it is fairly easy to prove ∃ (you show that something exists, bam, done), and essentially impossible to prove ¬∃, because you'd then have to somehow exhaust the domain of the debate (which is often impossible to even define, and even more often simply inaccessible to the parties discussing). It is exactly because of the vagueness and inaccessibility of the domain that this sort of logic falls short. Can you prove that there exist no handkerchiefs in my pockets? You can't, and it would be unfair for me to request that. You can't access my pockets, you don't even know if I'm wearing clothes right now, so you can't possibly exhaust the domain of "all the things in my pockets".

If the domain is truly inaccessible or inexhaustible, then I agree with you. But that's a question of soundness, not validity. I totally agree that there is much room for debate about the truth of the premises in any of these discussions. But those debates can be resolved, and those domains can be restricted.

That I personally cannot verify the contents of your pocket does not mean that it's impossible to prove that it contains no handkerchiefs.

P1. If PP's pocket contains a kerchief, then I will find a kerchief when I reach my hand into PP's pocket.

P2. I find no kerchief when I reach my hand into PP's pocket.

C: PP's pocket does not contain a kerchief.

I agree that I cannot resolve the truth of the premises from my current location, and we could probably debate/modify them to make them more accurate/specific/whatever. But the truth value of the conclusion is logically provable.

Yes, it sometimes is possible to prove a negative, usually by proving another claim that implies said negative (e.g. if I can prove that my name is Frank, that implies that my name is not John, or, in propositional logic, any invocation of *modus tollens*), but it is often impossible when no such implication can be made. As people have already pointed out in this thread, absence of evidence is not evidence of absence, so if I do not provide you with any information about the contents of my pockets, my handkerchief hypothesis is unfalsifiable. Of course, that causes problems of its own, but you certainly can't *disprove* it.

Same as above, basically. It might be difficult or impossible to resolve the truth of a particular premise, but that doesn't make the conclusion unprovable. It's often simply a matter of modifying a premise or formulating the proof in a different way.