On counterexamples and exceptions
Given a general statement like a definition or rule, counterexamples either don’t fit the criteria under consideration and our intuition is that they should be included, or they do fit the criteria, but our intuition is that it should not be included. In short: counterexamples disprove general statements. Logically, the argument is:
P1: All As have properties B.
P2: D has properties B but is not an A.
Therefore,
C1: P1 is false as stated.
P1 is the general statement and D is the counterexample. For example: all swans are white… but here’s a black one… nevermind, not all swans are white.
In the face of a counterexample, one is rationally obligated to change something. Given P1 and P2, there are a few possibilities:
S1: P2 is wrong: D is an A.
S2: P1 stays intact but D is allowed as an exception.
S3: P1 is revised to include cases such as D.
Note that in all cases, D ceases to be a counterexample. Counterexamples are (ideally) temporary conditions that force a logical reconciliation and it is irrational to persist for too long without accepting one of these possibilities.
Consider an example for illustration: imagine you are the judge of a prize competition and the criteria are that submissions must be written works of non-fiction less than 10,000 words. You receive an outstanding submission that is 20,000 words and the author refuses to change it. If you are going to award it the prize, then you have two options: you can leave the criteria as they are and make an exception in this case or you must change the criteria to include cases like this.
But keeping the criteria as they are and granting an exception is in effect to change the criteria. There is no meaningful difference. The pragmatist would say: for the sake of parsimony, we needn’t introduce “special” statuses to save the sanctity of the rule in P1. The rule wasn’t effectively absolute to begin with. The criteria are effectively: written works of non-fiction less than 10,000 words…or they have been granted an exception (exceptional). In logical terms, an additional and distinct sufficient criterion (a disjunct) is added: A or B.