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 $S$ such that all the rules inside this set have one or more exceptions.
- Rule $R$ which says that - Every rule in set S has one or more exception.
Clearly if $R \notin S$ then there is no problem with the rule $R$. For the case when the set is empty rule $R$ will be vacuously correct.
Things get interesting when rule $R$ is also made an element of $S$.
- $R \in S$.
Is Rule R correct?
No? Because, apparently, there are no exceptions to $R$. Since $R \in S$, for $R$ to be true, all rules in $S$ including $R$ must have one or more exception.
Wait a minute. All rules in $S$ have exceptions and rule $R$ is the only one without exception! The rule $R$ is the only exceptional rule without exception!
Having no exceptions made an exception for rule $R$. So the rule is correct!
Well, it does not stop here.
Because if rule $R$ has an exception then all the rules in the set $S$ including $R$ have exceptions. Thus rule $R$ is correct without any exception!
What I just said? Rule $R$ has no exception? But it has an exception?
The absence of exception was an exception of $R$ which in turn made $R$ true for all elements in $S$ and which in turn again contradicted with $R$ because $R$ must have an exception.
Or more clearly, for $R$ to be correct, it means:
- $R$ has no exception.
- Every rule in $S$ 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 $R$, every rule in $S$ has an exception. Thus there must be no exception to rule $R$.
In the previous case, Because the absence of exception became an exception which made $R$ true but in this case, it makes $R$ false in the first check! Because in this case $R$ must not have any exception by its definition!
The updated part - except this rule - of $R$ can be interpreted in two ways:
- Every rule except $R$ inside the set $S$ has an exception but $R$ is without exception.
- Every rule except $R$ inside the set $S$ has an exception and $R$ can or can not have an exception. It does not say anything for itself.
The second interpretation is almost as good as saying that $R \notin S$. Thus removing $R$ from $S$ seems to be the only way which can make this rule correct!