First of all I disagree with the premises that swinging a sword through an illusion constitutes as proof of its non existence; however, certain illusion spells address this directly.
Major image dictates that the illusion vanishes upon being struck unless the caster causes the illusion to react appropriately. Persistent image reads like you can program it to react appropriately upon being struck.
From what I've been reading up on it it doesn't look like you even get to roll against a figment based illusion unless you interact with it. i.e. Noticing that the goblin created by silent image doesn't make sound. Swinging a sword through a major image.
From what I see swinging a sword through an illusion would only make some of the not concentrated on early illusions fail. And a figments ac is 10+size modifier.
I figure the best way to run an illusionist is combine everything. Drop a few walls, a couple of pits, set up an illusionary terrain, which comes with tactile sensation, and drop down a couple of shadow based summon spells set up a doppelganger and drop more spells while invisible. I've changed my position on never attacking a abjuration mage in his home to never attack an illusionist in his.
What? No it doesn't. That doesn't even make sense.
It provides evidence and allows a will save. Its right there in the spell rules. It even goes so far as to state that if the illusion doesn't react in a realistic way it vanishes.
Proof and evidence are different things. The will save is allowing you to pay more attention to the fact that the sword had no friction than what your other senses are telling you. Even if it is attended, and it moves accordingly, you still get a roll since you interacted with it to notice that it didn't move naturally. Heck if you touch the illusion it constitutes as automatic proof.
Everyone seems to be ignoring that "proof = no save required" clause.
And that's because this clause means "if your DM wants you to disbelieve the illusion without save"
By the way to me proof is something like that:
Illusionary wall in front of you, you are being bull rushed into it, and suddenly you are on the other side, now you disbelieve it without save and you can see through the illusionary wall and fire arrows or something to the one who bull rushed you.
You guys have weird definitions of proof.
Wikipedia mathematical proof. That is the one I've been using. In order for it to be proof it has to be conclusive that the subject is invariably true.
Swinging a sword without an appropriate reaction from the illusionist constitutes as proof simulated by the illusion vanishing.
Swinging a sword with an appropriate reaction and failing a will save constitutes as evidence because we do not yet know it to be a proof without further testing.
Swinging a sword with an appropriate reaction and succeeding a will save constitutes as proof because we know it to be true.
Which definition are you using? Perhaps that is the trouble.
The non mathematical one reads
"A proof is sufficient evidence or argument for the truth of a proposition"
Key word reading sufficient. It has to prove the truth of the proposition.
Swinging a sword and noticing that the reaction was weird and that there was no friction on the sword proves that there is something strange going on hence the will save.