It's a "Catch-22" paradox!
The Catch-22 paradox is that behaving a certain way because of some desire is a sure way to not get what you desire. If you want to be declared insane in order to be discharged, you can't act insane as only sane men can act insane. But if you don't exhibit signs of being insane, you will not be discharged. The same analysis works for the witch paradox.
One point of Catch-22 is to illuminate the dysfunction of the US Army and bureaucracy in general. The continual use of paradox accomplished that goal in my opinion.
The point isn't to prove the original statement, but to demonstrate a particular type of absurdity. Another example that I've seen professionally is that in order to get a job that requires a security clearance, you need to already have clearance, but you can't get clearance until you have a job that requires it. In the entertainment industry, blockbuster movies require a lead actor who is a "name", but the only way to become a "name" is to star in a blockbuster movie. The usual solution to these conundrums is to fudge the rules a bit: interim security clearances are granted to people who start jobs that require clearances and actors can star in blockbusters if they are particularly well received in smaller roles.