The expression is referring to an implicit or unstated rule. Defining it is hard but people know when it has been broken. Hooters is an exception, the rule is, don't be like Hooters.

I see from Wiktionary that it was originally a legal concept, expressed in medieval Latin.

https://en.wiktionary.org/wiki/exception_that_proves_the_rul...

Just as you say, the point is that a rule is implied by a specific exception, as in the example "free entry on Sundays", which implies the unstated rule "pay for entry on other days".

The exception weakens the rule, it's true, but may also reveal the rule.

Thank you! I never understood the expression, but this explanation was immediately clarifying for me.

It’s a folkism, but consider this: If a rule doesn’t have any exceptions, is it really a rule? If a rule doesn’t exist how could there be any exceptions?

‘Prove’ was historically used as a synonym for ‘test’, which gives the phrase quite a different meaning. Like how ‘result’ is now sometimes used to mean ‘positive outcome’, as in a football fan saying ‘we got a result’

This phrase only made sense when it was explained to me that it’d be better phrased as “the exception that proves the existence of the rule”.

(if there were no rule, there wouldn’t be any exceptions to it, or nothing would seem exceptional with respect to the (previously unstated) rule)