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.