[This post has been updated for clarity.] Łukasiewicz supposed that the actual is necessary (if I have no coins in my pocket, then it is not possible that I do have a coin there) and that possible implies possibly not. I want to contest both of these. There is good reason to distinguish the actual from the necessary — the earth revolves around the sun is an actual fact, but that the sun is the center of the solar system is necessary (on the grounds that “solar” means “sun”). But if the earth does revolve around the sun, and it’s not possible therefore that it doesn’t revolve around the sun, then isn’t the earth’s revolution around the sun necessary (Łukasiewicz’ not-possibly-not)? So hasn’t he leveled a useful distinction?
Consider actuality from the perspective of possible worlds. “The earth revolves around the sun” is actually true now. But at another time, maybe not. So to identify the world in which it is actual, the statement of it must include a temporal index. In English it’s the present tense. In the statement “The earth revolves around the sun now,” then it must hold at one world, the now actual world, if the statement is true.
Now suppose it is so (and it is, of course). The earth revolves around the sun in the actual world. Does that mean that it is necessary? Well, it is not possible for it not to revolve in this now-world. But is the fact of the matter in one world sufficient to ascribe “necessity” to it? If there is only one accessible world to evaluate, is “necessary” meaningful for that one world? Why not require that necessity be evaluated at all possible worlds irrespective of the temporal index now?
When reading Łukasiewicz describing the necessary, I feel as if the whole meaning of “necessity” is being lost. He doesn’t seem to entertain the notion of the irreal — the realm of possibilities contrary to fact. The modal notions, to me, are beyond the real. They are the realm of speculation and conditions. If a statement is necessary, it means not that it is merely factual, but that it must hold regardless of conditions. That’s the whole notion of modality: considering the actual from a non actual perspective, what might have been and what might not have been.
The second puzzle — if the possible is divorced from the possibly-not, isn’t it the same as the necessary? — doesn’t seem puzzling at all. Start with the relationship of the actual and the possible. For the actual to be not possible is a contradiction: the actual would then be impossible and couldn’t be actual. So assume the actual is possible. So far, no problem.
Here’s the trouble: if the possible entails the possibly not, and the actual is possible, then the actual would also have to be possibly not.
There are two ways of dealing with this. In one way, the statement “the earth revolves around the sun now” is possibly not true in the sense that there is some possible world (not now) where the earth doesn’t revolve around the sun. That’s on all fours with treating necessity above as evaluating all possible worlds irrespective of the temporal index.
On the other way of dealing with it, “the earth revolves around the sun” cannot be possibly false now (possibly-not), and therefore the possible cannot entail the possibly not. More plainly: “the earth revolves around the sun now and the earth may not revolve around the sun now” can’t be accepted. So the premise must be false: the possible cannot entail the possibly-not.
Either way of dealing seems acceptable to me for different purposes.
The second way answers my ex-neighbor’s objection that if the possible doesn’t also mean possibly not, then doesn’t possible simpliciter mean the same as necessary or actual? The answer is, yes, exactly. If possible is not conjoined with possibly not, then possible is just a corollary of necessary or actual: if the earth revolves around the sun, then it is possible that it so revolves. Necessity => possibility; possible + possibly-not => non-necessary. If it is possible that there is a ninth planet revolving around the sun, and it is possible that there is no ninth planet revolving the sun, then “there is a ninth planet revolving the sun” can’t be evaluated as true necessarily.
What I want to say is this: the second way should apply to necessity, the first way to actuality.
Going back to the notion that necessity can’t be evaluated at the actual alone, but is always relative to all possible worlds (modulo sets of accessibility relations that facilitate and distinguish actuality and beyond), and possibly-not must be evaluated not just at the actual but at all the other possible worlds, then why, you might ask, is the possible not also independent of the actual too? In other words, why should the actual entail the possible; why can’t it be that the earth revolves around the sun now, yet not be possible? After all, I am rejecting the actual world as necessary, and I am rejecting that the actual is not possibly-not (I’m allowing that the actual can be possibly other than it is — at some other world), then why insist, you might ask, that the actual world entail anything about possibility?
Here the parallel with quantification is exact. The possible simply means that there is at least one instantiation of the proposition, and the actual is one such instantiation. The possible has instantiational weight, just as the ‘existential’ (instantiating) quantifier some, has instantiational weight. And the actual has existential, i.e., instantiational, weight. It’s an instance.
I don’t mean to imply that the Łukasiewicz modalities are not useful. But I don’t find them required for the understanding of natural language notions of ‘can’ and ‘must’ and ‘true’ and ‘false’. His modality of probability also has its usefulness. But maybe it’s only the probability that is useful, not the trivalence.