RETURN TO THOUGHTS
The Exception Rule
Here is an interaction you’ve probably had or seen:
Mr Jones: Everything A is also a B.
Ms Smith: What about C? C is an A, but not a B.
Mr Jones: Well, C is the exception that proves the rule!
What exactly is going on here? Why does Mr Jones think that this sort of retort preserves the truth of his generalisation? Surely given that Ms Smith has presented a counter-example to the statement Mr Jones made, and Mr Jones accepts this is a counter-example, he should proceed to give up his initial generalisation.
Here is one way of understanding what Jones has done. Jones is invoking a rule*, let us call it the Exception Rule, which states:
Exception Rule: For every rule, there is an exception.
The next question of course should be: is the Exception Rule true? It might seem obvious that it is not. Surely there are exceptionless rules: "All bachelors are unmarried", "Every even number is divisible by 2". These both seem like candidates. So surely, the Exception Rule is false.
Not so fast. Certainly, these exceptionless rules are exceptions to the rule, but that should pose no threat to the truth of the Exception Rule, because the existence of such exceptions is exactly what the Exception Rule predicts. Therefore, the Exception Rule is true. Naturally, it has exceptions because, as it states, all rules have exceptions.
Now, suppose, putting aside the Exception Rule itself, there were no exceptionless rules like the rules I have given as examples. Would the Exception Rule still be true in this case? It seems like given what I have said, it wouldn’t be true. The Exception Rule predicts that there is an exception to itself, i.e. an exceptionless rule, but by hypothesis, there is no such rule. So even though every rule (excluding the Exception Rule itself) has an exception, still the Exception Rule is not true.
So, the Exception Rule itself becomes the exception to the Exception Rule, being the one exceptionless rule, therefore still there is an exception to every rule, so the Exception Rule is proved. But now, every rule, including the Exception Rule, has an exception, so there can no longer be an exception to the Exception Rule.
This is a muddle. What is going on now? Now I will try to reveal the trick. The Exception Rule is in fact false, and it has an exception. Now I will try to prove this. First, I will prove that the Exception Rule has an exception:
First Proof: Suppose there is no exception to the Exception Rule, which is to say, "For every rule, there is an exception". Then, given that the Exception Rule is itself a rule, there is an exception to the Exception Rule. But this directly contradicts our supposition. Therefore, it is false that "there is no exception to the Exception Rule".
Now, I will prove that the Exception Rule is false:
Second Proof: Notice that in any other circumstance, we would accept that:
There is no exception to the Exception Rule if and only if the Exception Rule is true.
This is simply what we understand the truth-conditions of generalisations to be. Generalisations (as in "All Cs are Ds") are true when (and only when) there are no exceptions to them (as in, a C that isn’t a D). The only reason we would deny this equivalence is if we already accepted the Exception Rule as true. But to do so, at this stage, would be circular. Therefore, we should accept the equivalence. Once we do this, it is clear that because "there is no exception to the Exception Rule" is false (see the first proof), then "the Exception Rule is true" is false, because of the equivalence. Therefore, the Exception Rule is false.
Where did I go wrong before? The faulty reasoning is as follows: when talking about the existence of an exceptionless rule I said it "should pose no threat to the truth of the Exception Rule, because the existence of such exceptions is exactly what the Exception Rule predicts. Therefore, the Exception Rule is true."
This inference is invalid. It would only "pose no threat" to the Exception Rule, if we already accept the truth of the Exception Rule (and therefore deny that equivalence I talked about in the second proof). But there is no reason to do that at that stage. So, there is no proof of the Exception Rule.
Therefore, Mr Jones is quite wrong in his reasoning, because he is relying on a false rule, the Exception Rule.
* There is a different usage of this saying, which is much more reasonable, although I don’t think I’ve ever heard someone use it this way. According to this, the existence of an exception grants the inference that there is a rule that is being excepted. "Rule" here is not understood as a true generalisation/regularity, but as something moral or legal. The rule underlying this usage is as follows:
Exception Rule*: For every exception, there is a rule.
I have nothing more to say about this usage.