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I've seen this used enough times that there's probably a formal name for it, but I don't know what it is. For example, from Catch-22:
"I think I'm going crazy."
"Only a sane person is capable of doubting their own sanity. Therefore, you're just fine."
Or, conversely:
"Of course I'm not a witch! There's no such thing as witches!"
"Only a witch would say that! Burn her!"
Is there a formal name for this type of argument?
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.
I'm not sure if this fits the pattern you're looking for, and I realize that this may a bit controversial, but there is the saying (not exactly a rule of inference) that:
The exception proves the rule.
I've always taken this very informal almost joking statement to mean that one has a general rule in mind, one sees an exception to the rule, but the circumstances of the exception are so rare that it implies that rule probably is pretty accurate in most circumstances. If forced to formalize and pedanticize this I would say it's a Bayesian confirmation by improbable data.
