Saturday, February 12, 2005

Rules and Exceptions

One often hears, 'The exception proves the rule.' But what could that mean?

Suppose that the rule is of the form All Fs are Gs. A rule of this form is not proven, but definitively refuted, by an exception, an F that is not a G. A rule of the form, Some/Most Fs are Gs, is not refuted by an F that is not a G, but it is not proven by it either.

What we should conclude is that an exception ‘proves’ the rule only in the sense that it makes the rule stand out by contrast; the former throws the latter into relief. That is an interesting use of ‘prove’: an exception can literally contradict the rule and yet one says that it proves the rule!

One lesson to be drawn from this is that the meaning of a word often depends on its context, in this case the sentence in which it is embedded.

Now given that many rules have exceptions, some will be tempted to affirm

R. Every rule has an exception.

But (R) is itself a rule, whence it follows that (R) has an exception. Now an exception to (R) would be a rule that has no exception. Therefore, (R) entails the negation of (R), namely,

~R. Some rule has no exception.

Now either (R) or (~R). Since (R) entails( ~R) and (~R) also entails not (~R), it follows that (R) is not merely false, but necessarily false. Therefore, ‘Some rule has no exception’ is necessarily true. An example of such a rule would be (~R) itself.

It follows that there is no possible world in which the rule that some rule has no exception can be 'proven' by an exception.

Ah, the pleasures of analysis!