Wednesday, November 25, 2009

Crispin Wright and Rule Following

In the paper on Rule Following here, Wright suggests that good reasoning proceeds in obedience to with concrete rules, rules which we can in principle give at least a rule-circular justification for. I claim that Wright's view is only tenable, IF humanlike reasoners count as `obeying' infinitely many rules in this sense.

For, suppose a good reasoner only obeys finitely many such inference rules. And suppose, as Wright wants to claim that reasoner can come (by reasoning) to provide a rule circular justification for each such rule, (i.e. show that applying this rule cannot lead from truth to falsehood). But then our reasoner can combine all these finitely many justifications, to arrive at the conclusion that anything arrived at by applying some combination of these rules must be true. Hence, he can derive that the combination of these rules doesn't allow one to prove "0=1".

But remember that the rules are supposed to be concretely described. So, our reasoner can syntactically characterize the system which combines all these rules (the one which, unbenounced to him capthers all his good reasoning), and state Con(some formal system which allows exactly the inferences allowed by these rules). But he knows the combination of rules is consistent, so he can derive the con sentence for this set of rules. But, by incompleteness II (on the assumption that good reasoner's reasoning extends Robinson's Q, so that the theorem applies) this is impossible.

Hence, if Wright's theory about obedience to rules is correct, any good reasoner who accepts principles of reasoning that include Q (like us) must be obeying infinitely many rules.

[This may be problematic if one wants the notion of obedience to a rule to have some kind of psychological reality]

[Note that this doesn't mean the good reasoner's behavior won't be describable in some more efficient way by some finite collection of rules, just that the reasoner doesn't have access to these rules, in Wright's sense of being able to prove that they are truth preserving]

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