[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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