non epistemic possibility and the lay of the land

Originally published on Language and Philosophy, May 31, 2012

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:

Classical

actual=>possible (simpliciter) [because the actual is one instance (though merely one)]

necessary=>possible (simpliciter)

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.

Non classical

actual=>necessary=>not contingent

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:

necessary=>tautological

necessarily not=>contradictory

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.

possible, but not necessary

Originally published on Language and Philosophy, May 30, 2012

[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.

possible or not

Originally published on Language and Philosophy, May 10,2012

Is it possible to swim the Atlantic?
An ex neighbor points out that if “possibly” doesn’t imply also “possibly not” then how is “possibly” different from necessity? Doesn’t “Life on Mars is possible” mean “It’s also possible that there’s no life on Mars”? And doesn’t it also mean “Life on Mars isn’t necessary”?
Grice gave an answer to this question, and I’ve written about it elsewhere in this blog, but I think there’s more to be said and I want to try to sort all of them out.
Suppose Goldbach’s postulate is possibly true. Suppose someone proves it. Now it’s necessarily true. Is it no longer possibly true?
My neighbor says no. I think Lukasciewicz agreed with him.
But suppose we have a set of conditions that fulfill the capacity to swim the Atlantic, and some person satisfies that set. Now it’s possible for someone to swim the Atlantic, and the question of whether it has been done is irrelevant.
One way to cash out the notion of truth is a saying of what is, and the notion of false, a saying of what is not. In this dichotomy, where does possibility lie?
One place is knowledge. There’s what we know exists, what we know doesn’t, and what we don’t know about is the possible.
But isn’t this Lukasiewicz trichotomy conflating knowledge with fact?
A lot depends on whether determinism holds — if every actuality is necessary, then everything that happens is necessary. But whether determinism is true is itself uncertain.
It seems to me that there are many meanings of “possible” besides “unsure.” There’s the abstract conditions of conceivability, “if someone were strong enough, someone could swim the Atlantic;” the actually satisfiable conditions of possibility, “there is someone strong enough, so it is possible,” and the actual itself, “she swam the Atlantic, proving that it is possible.”
If you choose a Lukasiewicz trichotomy, is there space for all these? It seems well suited to determinism, but not to modality. The classic Aristotelian four-way modality (possible+negation yielding possible, not possible, possibly not, not possibly not) along with the conjunction of “possible & possibly not” accommodates a distinction between actuality and necessity, where we can be agnostic about determinism and entertain possible worlds that will never be.

Lukasiewicz, bivalence and the future

Originally published on Language and Philosophy, April 7, 2011

Just now looking at David Foster Wallace’s Fate, Time and Language, I’m puzzled by Lukasiewicz’ argument, quoted in his text, that statements about the future cannot be true or false at the moment when they are stated. It seems obvious to me that any statement about the future must be true or false, it’s just that we don’t know their truth value at the moment (except for necessary truths and inconsistent statements which may be deemed false and if contradictory, plainly false).
~K(p) does not imply (p) or (~p).
Not knowing the truth value of a statement means that the epistemic certainty of it has a degree of probability <1. But that doesn’t imply that the proposition itself has a certainty <1. The proposition itself has a probability of either 1 or 0. Why would anyone conflate the epistemic with the realis assertive?

Am I missing something? The probability of a belief for a determinist depends on the known circumstances. Those known circumstances often do not suffice for certainty.

The issue for Lukaseiwicz lies in the way we speak about possibility. If I say, “I will be at your place tonight,” even I can’t say for sure that I really will get there — I could get run over, I could get distracted by a friend. So we venture to say that it’s possible I’ll get there, and, likewise, it’s possible that I won’t. Using P for “possible” and T for “I’ll get there tonight”

P(T) & P~(T)

When the future arrives, we’ll know which of the conjuncts is true. If we’re not determinists, there’s no problem. But if we’re determinists, then one of these conjuncts is necessarily true, and the other necessarily false: necessity is interchangeable with “not possibly not,” and “necessarily not” is interchangeable with “not possibly”:

N(T) = ~P~(T)

N~(T) = ~P(T)

but if T is true, then the statement before the future arrived, added to our knowledge of necessity now in the future, yields a contradiction

N(T) & P~(T) =

~P~(T) & P~(T) =

N(T) & ~N(T)

and if T turns out to be false

N~(T) & P(T) =

N~(T) & ~N~(T) =

~P(T) & P(T)

Now, if we are not determinists, there’s no problem: the future isn’t necessary, so the truth value at the future doesn’t contradict any assertion in the past. So non determinists can assert that propositions about the future have distinct possibilities. But if we buy into determinism, we can’t assign probabilities to propositions about the future. So Lukawiewicz offered to abandon bivalence: statements about the future are neither true nor false, but somewhere in between.

But all that’s ignoring the epistemic context of our assertions of possibility. The correct formulation of our assertions, if we are determinists-in-ignorance is:

B(P(T) & P~(T))

“I believe that possibly T and possibly not T” or alternatively

B(P(T)) & B(P~(T))

“I believe possibly T and I believe possibly not T”

Believing possibly T or possibly not T is in no way inconsistent with T or ~T or N(T) or N~(T).

B(P(T) & P~(T)) & N(T)

is consistent, as is

B(P(T) & P~(T)) & N~(T)

A simpler formulation uses the anepistemic mode

~K(T)

“I don’t know T for sure” which itself implies

~K~(T)

“I don’t know ~T for sure” and therefore

~K(T) & ~K(~T)

(I’m leaving out for the moment the possibility that ~K(T) can mean “I don’t know of T,” which allows for three possibilities: I don’t know that T is true, I don’t know that T is false, I don’t know of T at all)

These are also consistent with either of T or ~T or their modal necessary versions. There are no contradictions here:

~K(T) & ~K(~T) & N(T)

~K(T) & ~K(~T) & N~(T)

The implication is that “I might not be there tonight” means both that I don’t know whether I’ll be there or not — it means the exact same as “I might be there tonight.”
Elsewhere I’ve given the evidence of the equivalence:
???I might go but I will
???I might go and I will
???I might go but I won’t
???I might go and I won’t
???I might not go but I will
???I might not go and I will
???I might not go but I won’t
???I might not go and I won’t
Unless the speaker has had a change of mind mid-utterance, these sentences are semantically incoherent. It is uncontroversial that the consequent conjuncts assert certainty over intention, so, presumably the incoherence lies in the uncertainty of the antecedent conjunct. Since the same certainties clash with the negation or without, the implication is that “might” and “might not” bear the same uncertainty: “I might not go” implies “I might go” and both can be cashed out as the anepistemic

~K(G)

~K(~G)

~K(G) & ~K(~G)

but not  ~K(G & ~G) unless we’re very contrary, since we all know

K~(G&~G)

and we know that we know it, too.