The peculiarity of classical notion of possibility is that it has a relation to the actual world as well as a relation to the irreal world of conditions counter to the actual and the epistemic world of certainty and uncertainty. Lukasiewicz’ notion of possibility seems to apply only to uncertainty — it seems essentially epistemic.
So here’s the lay of the land, as I see these two modal programs:
actual=>possible (simpliciter) [because the actual is one instance (though merely one)]
contingent (possible & possible-not)=>not necessary
contingent (possible & possible-not)<=>not necessary & not necessarily not
actual & not necessary=>contingent [possible-not in some plausible world beyond the actual world]
The actual and its entailments are non epistemic; contingency and necessity are epistemic.
necessarily not=>actually not (i.e., determinate falsehoods)=>not contingent
possible<=>possibly-not<=>contingent<=>not necessarily & not necessarily not
The actual again is nonepistemic; the contingent/possible are epistemic; by implication, necessity is also nonepistemic, but perhaps not always or only epistemic.
Alternatively, the many-valued logic could abandon actuality altogether and treat the entire realm of assertions as epistemic except the logical truths. This makes a lot of sense to me, since I don’t believe that there are any certainties beyond maybe analytical truths or truths by definition. On this model the tautologies have a probability of 1, contradictions 0 and everything else is somewhere on the scale in between. I would dispense with the notion of “actual” entirely, since, on the one hand, tautologies do not imply actual instances of anything in the real world besides their tautological truth — all unicorns are unicorms, but there’s nothing in the actual world that bears on this truth or that this truth bears on actuality; and on the other hand, the empirical world of phenomena are not entirely certain — why fool ourselves with a pretense of knowledge? So:
non necessary & not necessarily not<=>degrees of probability <1 and >0
Here necessity is analytical; everything else is epistemic. This model seems completely consistent with the classical model, where the actual is not necessary.
It has a philosophical consequence: on this view of modality, actual empirical events are never completely certain. So there is no “actual world” among possible worlds. Instead, there are apparent experiential worlds — phenomenal worlds in a more or less Kantian sense; subjective worlds of relative conviction. Along with the experiential worlds there are possible worlds of conjecture. Necessary truths will be true in all of them, etc., etc.
This seems much closer to the realm in which my mind lives where nothing is absolutely certain but tautologies and contradictions.