Every rule has an exception

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.

Recently I went through an awesome explanation of Gödel’s incompleteness theorems in a Quora answer. A must read for anyone even remotely interested in the subject.

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.

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?

To summarize,

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!

Eh?

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!