As to whether multiple negations (in language, as opposed to algebra) really
DO strengthen or reinforce each other in the relevant dialects/languages, my
sense is that the phenomenon is basically akin to other sorts of concord. We
wouldn't expect singular (or plural) agreement on a verb to strengthen or
reinforce the singularity (or plurality) of the subject, nor the
masculine/feminine/neuter form of an article to reinforce the gender of the
head noun with which they agree, and the same is generally true of negative
concord. This is not to say that negative sentences may not (!) have other
devices to strengthen the negative force; Bolinger has written insightfully on
the differences between "not...any" and "no" in the standard dialect (I don't
want any of that vs. I want none of that), but this distinction is independent
of the negative concord question.
(P.S. Incidentally, the postings here on langauge vs. logic seem to assume
there's just one monolithic "logic", with which language may or may not agree.
But there are as many logics as there languages, and their properties differ
from each other in interesting ways. The issue isn't really language vs.
logic, but what the mapping rules are between grammar (morphosyntax) and