[FOM] origins of completeness in modal logic

[Max Weiss]

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