Non-Contradiction

Maybe we should think of the rule against contradiction as a translation rule rather than as a speaking rule.  It should be a test of a translation schema that it does not render the native speaker as uttering contradictions.

This becomes a 'speaking rule' in the sense that we may say to someone:  'I do not understand you because you seem to be contradicting yourself'.  The 'seem to be' is not eliminable:  we may have made a translation error.  This can never be ruled out on purely formal, or empirical, grounds.

This renders the reductio ad absurdum correctly:  it required prior agreement on the issues which generate the contradiction, if it is to be valid.  It has this form:  I cannot make sense of what you are saying, because however I interpret ('translate' it) it produces a contradiction.

So the RAA is always vulnerable to a 'translation error' defence.

This is fine, though:  mathematics depends on agreement about how to speak, and not vice versa.