Concerning “the exception that proves the rule”, I think it’s telling that the Wiki article on the expression has a section for “jocular nonsense” and another one for “serious nonsense”.
Another one that should probably just be nuked from orbit, IMO.
“The exception proves the rule” means that the existence of an exception to a rule proves the existence of the rule. The Wikipedia article cited by Martian Bigfoot explains this with an example:
This example (like most of the article) was lifted directly from Fowler’s “Modern English Usage.” The Wikipedia article also gives the “test” explanation, and uses the book “The Etymologicon” by Mark Forsyth as a reference. I trust Fowler on this one, for a couple of reasons. Fowler links the origin of the phrase back to an earlier usage. Also, Fowler’s explanation seems more logical to me-I can see why people would use a aphorism to remember the idea behind it-while the meaning behind the “test” explanation is so obvious it’s almost a truism. Why would people use an aphorism whose meaning is so much less obvious than the idea it’s supposed to express?
“The exception proves the rule,” is a fragment. The whole quote (from Cicero) is “The exception proves that the rule is in force on all other occasions.” What it means is that if a written rule or law contains an explicit exception, you can assume there are no other exceptions, or they would be explicit as well. If a sign says “Two hour parking except on Sundays and holidays,” and you get a ticket for parking overnight on a Saturday, don’t try arguing that there’s an exception for Saturdays, because the two adjoining streets have them. The sign is explicit that the exceptions are Sundays and holidays only. If there were no stated exceptions at all, you might be able to argue implicit exceptions, but the existence of explicit ones rules out other, implicit ones.