There are certain axioms and proverbs that amuse and/or fascinate me. The above is one of them. It seems on the surface to be either nonsensical or paradoxical or perhaps both. This then becomes a great reason to take this axiom and actually ponder and contemplate upon it. What does this really mean?
Not too surprisingly we get a variety of interpretations. But in all fairness, we want to hold onto the interpretation(s) that appears to be the most correct and most in keeping with this proverb. For instance, you could break the saying down a little further to understand it to mean, that if something is postulated as being an exception, than, by definition, there must be a rule. In other words, you can't have an exception, without a rule. This doesn't yet prove the rule but it does demonstrate that the rule exists.
Additionally, let us take, for instance, the traffic example of "no right turns on a red light between the hours of 7AM-9AM and 5PM-7PM, Monday thru Friday." This would be the exception. The rule would be that right turns on a red light are therefore permitted at all other times. So that this would be the exception that would indeed prove the rule and would make this proverb both sensible and correct.
Sometimes when a rule is made there are exceptions put into it as a matter of course. For instance, in Major League Baseball, the pitcher has to take his turn at bat, but this was later replaced by the Designated Hitter in 1973 but only in the American League. Then when interleague play began, the rule was further modified so that the DH is only used at American League stadiums in which it is utilized by both teams, whereas in National League stadiums the use of the DH is prohibited by both teams. So once again these are exceptions that uphold the rule.
Do we need exceptions to prove the rule? Of course not. For instance, as Descartes stated: "I think therefore I am," cannot logically be refuted, because if you can think that thought you must exist. To take this a little bit further, you will by definition never be able to say to yourself, that you are dead, you may be able to say that you are dying, but the fact that you can still think, that you can still contemplate, justifies that you exist on some plane or on some level, perhaps in a yet undiscovered country and that you are indeed alive.
Yet, despite this straightforward maxim, there have been serious doubters such as Bertrand Russell and Friedrich Nietzsche. Russell stated"…Here the word ‘I’ is really illegitimate; He ought to state his ultimate premises in the form ‘there are thoughts’", and Nietzsche stated: ” …a thought comes when “it” wishes, not when “I” wish, so that it is a falsification of the facts of the case to say that the subject “I” is the condition of the predicate “thinks.”
It seems like spurious philosophy. But perhaps, this is yet again the exception that proves the rule, because I definitely exist.