Not so long ago, I came up with a rule that Every rule has an exception. At first glance, the rule is a contradiction in itself as there seems to be no exception to this rule. I gave up after few failed attempts at coming up with a consistent definition for the rule.
By the self-referential nature of the examples given in the said answer, it reminded me of the rule again.
To proceed further, let’s make some assumptions.
- A rule is a set of statement(s) about rules or some general phenomenon.
- An exception to a rule is any statement that contradicts with one or more statements of the rule.
- A set such that all the rules inside this set have one or more exceptions.
- Rule which says that - Every rule in set S has one or more exception.
Clearly if then there is no problem with the rule . For the case when the set is empty rule will be vacuously correct.
Things get interesting when rule is also made an element of .
Is Rule R correct?
No? Because, apparently, there are no exceptions to . Since , for to be true, all rules in including must have one or more exception.
Wait a minute. All rules in have exceptions and rule is the only one without exception! The rule is the only exceptional rule without exception!
Having no exceptions made an exception for rule . So the rule is correct!
Well, it does not stop here.
Because if rule has an exception then all the rules in the set including have exceptions. Thus rule is correct without any exception!
What I just said? Rule has no exception? But it has an exception?
The absence of exception was an exception of which in turn made true for all elements in and which in turn again contradicted with because must have an exception.
Or more clearly, for to be correct, it means:
- has no exception.
- Every rule in has one or more exception.
Thus they can’t both be true at the same time!
Let’s try to update the rule:
Every rule has an exception, except this rule.
Is it correct?
The updated rule states that apart from rule , every rule in has an exception. Thus there must be no exception to rule .
In the previous case, Because the absence of exception became an exception which made true but in this case, it makes false in the first check! Because in this case must not have any exception by its definition!
The updated part - except this rule - of can be interpreted in two ways:
- Every rule except inside the set has an exception but is without exception.
- Every rule except inside the set has an exception and can or can not have an exception. It does not say anything for itself.
The second interpretation is almost as good as saying that . Thus removing from seems to be the only way which can make this rule correct!