Rules and Gödel

If we were trying to interpreting a language, we would test interpretation hypotheses to see whether they led to contradictions.

A contradiction should result in rejection of the interpretation hypothesis which entailed it.

Part of this hypothesis, in the context of a logical proof, is about the meanings of the rules.  Perhaps, in logic, there is nothing apart from the meanings of the rules.

Will there always be an interpretation which avoids contradiction?  Yes, but possibly not always a constructive interpretation - an interpretation which allows novel constructions from the same elements.  But of course constructive interpretations must include interpretable rules of construction ... and about what count as 'elements' and 'composites' etc.

Do Gödel's proofs show us that all 'closed' rule based interpretations must be incomplete or contradictory?  He and Turing have ruled out computational approaches to certain questions.

But the contradictions just throw us back to the incompleteness hypothesis - that our interpretations cannot protect us from future re-interpretation.  Any definite interpretation must leave open the question of its own reliability.

Here is a rule of interpretation:

To you words
understand must vertically,
this read not
rule, the horizontally.

On the basis of what further rule do we demonstrate that we have a correct interpretation of this one?  Only that we have avoided contradiction?  What rules demonstrate the contradiction?

Phenomenologically, we can have a Wittgensteinian 'Aha!' moment, and know how to 'go on' ... but this is epistemologically irrelevant.  It just tells us what a competent language user might feel on such an occasion.  It doesn't tell us anything about justification.  To try to say, as W seems to, that it's a mistake to think we need justification here is just to introduce a novel kind of justification strategy (a strategy which I, for one, don't have any 'Aha'  feelings about ...).

He might say, instead, 'to ask for more is to ask for a justification of justification' - and this is right as a global position, but is not right with respect to particular justificatory strategies, which always seem to be revisable (Quine).  What we need is a demonstration that unless this strategy works then no strategy would work - RAA.  This seems more elusive ...