[hist-analytic] The status and relevance of "standard logic"

Roger Bishop Jones rbj at rbjones.com
Sat Oct 31 16:48:47 EDT 2009

On Saturday 31 October 2009 16:44:05 Danny Frederick wrote:

> The metamathematical proofs to which you refer have hidden axioms that are
> far from obvious.

There are no "hidden axioms" involved.

> In order to avoid paradoxes which refute the whole
> system, they typically have a theory of logical or linguistic types which
> is built into the formation rules for formulae of the system. Such
> type-theories are a lot more doubtful than the truths of elementary
> arithmetic.

All the results I mentioned (including Godel's incompleteness results) are 
provable in logical systems weaker than PA (first order arithmetic).
A type theory is not necessary.

> Besides, the question of how doubtful something appears does not bear upon
> its truth.

How can this be?

> Frege was one of the greatest mathematicians. His Basic Law V
> seemed self-evident to him. Yet it was shown by Russell to be logically
> false. As Russell stated in PM (second edition, p.59) we have no guarantee
> that another paradox will not surface however happy we may be with the
> latest set of axioms (open and hidden).

The problem you cite concerns reliability of our intuitions about the truth of 
basic axioms.

I made no appeal to intuitions about truth but cited results about the 
strength (and hence riskiness) of the logical contexts in which these results
(about soundness, which includes the truth of the axioms) are formally 

As I stated in my message, nothing is beyond doubt and we have no absolute 
warrants of truth, which is why I gave evidence about relative dubitability. 
suggesting that doubting the soundness of classical logic is equivalent to 
doubting the truth of quite elementary parts of established mathematics.

Roger Jones
rbjones.com                        PGP public key at: rbjones.com/rbj.asc

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