Logic and Interpretation

Here are two compound expressions:

~(P&~P)

    P&~P

Conventionally, the first is a tautology and the second a contradiction.  However, we might introduce a 'rule of interpretation' here:

When an expression appears on the left of an ampersand, it means the contrary of what it would mean if it appeared on the right.

Applying this rule, we make the first expression a contradiction, and the second a tautology.

Since it is always possible to distinguish expressions that are sentences, and to distinguish whether they are on one side of an ampersand or the other, it seem likely that this rule would not render the notation inconsistent.  By putting '~( ... )' round all sentential expressions on one side of an ampersand we could 'encode' and 'decode' our normal propositional logic.

If this was a rule that a reader might mistakenly apply when doing logical calculations, we would have to warn against it.  On the other hand, such a warning would sound mad to a reader who had never thought of such a rule.  It would say 'Remember to treat an expression on one side of the ampersand as meaning the same thing as when it appears on the other side'.  But this, of course, is another rule of interpretation, just like the first one.

Neither rule is more fundamental than the other.  Each resolves an ambiguity which, if we did not see it, would not need resolving.

Not only is it absurd to consider ourselves to be applying a myriad of interpretational rules in order to eliminate unimagined ambiguities, it is also incoherent.  In order to say 'I am applying rule X', we need to articulate rule X, and so also interpret it.  Not only would every rule, itself, require interpretation; but there would be rules whose correct interpretation would depend upon their own application.