I am irritated and confused by my complete lack of understanding what people mean when they say "the exception that proves the rule."

Exceptions, by their very definition, disprove rules. For example, consider the rule: "Every swan is white." Translated into logic, it reads: "For all X (if X is a swan, then X is white)."

This is actually a real and famous example. It used to be an expression in England, which expression communicated rarity, to call something a Black Swan. The force of the term came from the fact that, as far as they knew, no such thing actually existed. But then, a Dutch explorer actually found black swans in Australia.

At this point, philosophers like Hume and Mill who were concerned with induction, the method in which you draw general rules from a pattern of particular instances, began using it as an example. Induction, you see, is a good but logically imperfect way of figuring things out.

Just because every swan you have seen so far has been white, that does not mean that the next one will be.

Deduction, the method in which you draw particular instances from general rules, is superior. We could deduce from the formal logical proposition above, for example, that any given swan is going to be white.

But, as we mentioned, any given swan is not going to be white. Which brings me to the heart of my confusion: Since a black swan is forbidden by the rule, a single black swan overturns it. Exceptions, by their very nature, disprove rules. The fact of the existence of a black swan means that we have to deny that proposition.

So what does "the exception that proves the rule"" mean?

Is this just a failure of creativity on my part? I think I can remember using it and understanding it in the past, though I have no idea now what that understanding was. Since the company I keep, to which I might address a question about this, consists mainly of students of logic and the related subjects, nobody has been able to help me understand this so far.

Wikipedia has been similarly unhelpful. They say that it is "frequently misused", but that its real meaning is that an exception in a specific case indicates that the rule exists outside of that case.

Which is emphatically does not.

They give the example of a sign which says: "Parking prohibited on Sundays." They say that that exception proves that parking is not prohibited on the other six days of the week.

Which it does not. I suppose it indicates the likelihood of that, but it by no means tests one's imagination to conceive of a case in which parking is prohibited on Sunday and also on one or more other days.

So I am back in the fog in which I started. I don't even have a candidate answer. Help me!