[FOM] origins of completeness in modal logic
Max Weiss
30f0fn at gmail.com
Mon Mar 2 13:07:44 EST 2009
For first-order logic, there is some intuitive notion of (classical)
validity that is sufficiently stable to allow axiomatization and even
the proof of a completeness theorem before the relevant semantical
notions were completely formalized. However, for modal logic this
seems not clearly to be the case: the question of an arbitrary modal
formula "is it valid?" tout court, seems simply ill-posed.
But moreover, it is not just any mathematically precise method of
"interpretation" that delivers a notion of validity nor hence of
completeness. Consider, for example, the result of McKinsey and
Tarski (1944) that a formula is a theorem of S4 iff it is always
assigned the top element in all closure algebras. My impression is
that this would not be considered a completeness theorem, since
"always denotes the top element" is not an intuitively plausible
notion of validity. (For whatever reasons, the authors don't call it
a completeness theorem.)
Roughly speaking, what I'm wondering is what sort of basis there might
be for considering, say, Kripke (1959), as opposed to such earlier
work, indeed to contain "a completeness theorem in modal logic". The
motivation is not to try to establish relationships of historical
priority, but rather to try to understand how the notion of
completeness in modal logic arose in the first place.
Perhaps understanding the development of the relevant notion of
completeness would help to explain why Jonsson and Tarski don't, in
their (1950), draw the retrospectively obvious connections to logic
for their representation theorem.
Any references, hypotheses, critical remarks, etc. would be much
appreciated.
Thanks!
Max Weiss
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