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

Roger Bishop Jones rbj at rbjones.com
Fri Oct 30 17:47:33 EDT 2009


On Friday 30 October 2009 19:44:23 Danny Frederick wrote:

> Incidentally, the reason I put 'proof' in quotes all the time is that we
> can never be sure that a proposed proof is really a proof. We can give a
> 'proof' in standard logic, say; but the 'proof' is questionable because
> standard logic is questionable. The same applies to every other logical
> system. Whether or not something is a proof is something we can only guess
> at.

If the "standard logic" you speak of here is first order predicate logic then 
there are straightforward metamathematical proofs of its soundness.
These make the possibility that this logic is unsound not much more doubtful 
than the truths of elementary arithmetic (say, the associativity of 
multiplication), and much less doubtful than generally accepted (and 
themselves relativity straightforward by the standards of professional 
mathematicians) results such as Godel's two incompleteness theorems.

Skeptical doubts know no bounds, but it is useful to know whether a doubt is 
serious or academic, or better, to have a sense of the ranking of doubtfulness
of propositions.

Doubt about the applicability of this logic to any argument in a natural 
language are of an entirely different order.
For example, in the example discussed in this thread, the supposition that the 
claim:

if it rained today, it did not rain heavily

can properly be understood as a material implication, is highly doubtful,
and hence the value of a naive translation into "standard logic" is moot.

Roger Jones

-- 
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