Probability and Open Future Views
December 8, 2008 — 11:30

Author: Alexander Pruss  Category: Open Theism  Comments: 19

This is an attempt to make precise something I said in the comments to my previous post, and analyze it. Let T(p) be the claim that p is true. Work with epistemic probabilities. Assume the following three principles:

  1. p entails T(p)
  2. If p entails q, then P(p) is no greater than P(q)
  3. P(not-p)=1-P(p)

Let OF be the claim that the future is open, so that no proposition about a future contingent is true (i.e., it either lacks truth value, or it is false, depending on the version of OF). Let q be the claim that tomorrow I will toss a quarter a thousand times and each time it will land tails. Let p be not-q. Intuitively, P(p) is more than 1-2-1000 (more, because the chance that I will both to do the tossing is very small). Now OF entails not-T(p) (by definition of OF). Hence, P(OF) is no bigger than P(not-T(p)) = 1-P(T(p)) by (3). But P(p) is no bigger than P(T(p)) (by (1) and (2)) and hence 1-P(T(p)) is no bigger than 1-P(p). Thus, P(OF) is no bigger than 1-P(p). But P(p) is more than 1-2-1000. Hence, P(OF) is less than 2-1000. Hence, OF is not worthy of belief.

But in writing this argument out rigorously, it became clear that there is a way in which it begs the question against those defenders of OF who deny excluded middle. For (3) implies a probabilified version of the axiom of double negation (the axiom of double negation is: not-not-p entails p), and double negation has to be denied by those who deny excluded middle. So the defender of OF can get out of the argument if she denies the axiom of double negation, and gives a propositional probability theory compatible with that denial. But to deny double negation is to go pretty far down the road of implausibility–all reductio arguments go down the drain at that point. Of course one might maintain double negation for non-future-tensed claims, but that’s ad hoc.

Comments:
  • A minor screwup in my argument. So as not to beg the question against Rhoda, I shouldn’t define q, and then define p as not-q. Instead, I should directly define p as: It will be the case that I don’t both toss a coin a thousand times and get heads each time.

    December 8, 2008 — 13:15
  • Mike Almeida

    Alex,
    Very cool! Small point. Don’t you get that P(Tp) = P(p), since Tp and p entail each other. And so 1-P(T(p)) will equal 1-P(p). Thus, P(OF) equals 1-P(p).

    December 8, 2008 — 13:21
  • Yes, you do get P(Tp) = P(p). But the conclusion to be drawn is that P(OF) is less than or equal to 1-P(p), not that P(OF) = 1-P(p).

    December 8, 2008 — 13:25
  • Alan Rhoda

    Hi Alex,
    Your argument, in a nutshell, seems to be that OF entails that some propositions having high epistemic probability are nonetheless not-true, which counts strongly against OF.
    The argument is unsound, however, because it conflates two possible readings of the future tense operator “will”.
    The claim of OF, that there are no true “will” or “will not” propositions about future contingents presupposes a Peircean semantics according to which for it to be “true” that event E “will” happen is for it to be now-inevitable that E. Let E be the event of your tossing a coin 1000 times tomorrow and its landing tails each time. Given that reading, q is true iff E is now-inevitable. Hence, p is true iff it is not the case that E is now-inevitable.
    Now, if E is a future contingent, then E is not now-inevitable, in which case P(p) = 1. In this case, OF does not entail that it is not true that p. Quite the contrary, it entails (correctly) that p is true. Conversely, if E is now-inevitable, then P(p) = 0. In this case, OF entails (correctly) that it is not true that p.
    Now suppose we adopt a non-Peircean semantics according to which it can be “true” that E “will” happen even if E is not now-inevitable. If p is understood in such terms, then a proponent of OF need not agree that OF entails not-T(p). Instead, she will think that “will” and/or “true” are being used loosely, in a manner that does not conflict with what she means to deny, namely, that there is (and always has been) a fixed and complete set of truths describing a unique “actual” future (as of time t).

    December 8, 2008 — 15:04
  • Alan:
    I am afraid I don’t understand what you mean by “will” and/or “true” being used loosely. I doubt that there is any loose sense of “will” and “true” that lets one say in this context that P(p) is more 1-2^(-1000).
    Suppose “will”, speaking loosely, means a high degree of probability. Then P(p)=1. For it is not merely probable but certain that the event described by p (my not getting a thousand tosses of heads tomorrow, say) has high probability.

    December 8, 2008 — 15:12
  • john a

    Alexander
    I am confused on why you think there is a problem for OT. The problem I am having is understanding exactly how you are using the word ‘proposition.’ If a proposition has to affirm or deny that something is the case, then what exactly what is being affirmed by your premises that a defender of OT must worry about? If p (“I will freely eat dinner tomorrow”) is true as you originally stated, it must also mean that there is a possibility, albeit a small one, that I will freely not eat dinner tomorrow being part of the truth condition of that proposition (if it is a proposition). This means that I do not know what I am going to do tomorrow regarding eating dinner if part of what it means to ‘know that p’ is that ‘p is true’ where ‘true’ refers to the way things are (where ‘will means ‘must’). OT is concerned with defending the idea that God does not know what I am actually going to do tomorrow where ‘will’ means ‘must,’ but is not concerned with what I probably will do tomorrow where ‘will’ means ‘may,’ To say that ‘I will probably freely eat dinner tomorrow night’ is true entails that ‘It is possible that I will freely not eat dinner tomorrow night’ is also true. So even if God knows p as you stated it, He still does not know in a non-probabilistic sense that I will eat dinner tomorrow, anymore then I do. I can even be certain, in Moore’s sense of being psychologically certain (e.g. I do not doubt that I will freely eat dinner tomorrow), that I will freely eat dinner tomorrow, but this does not count against OT. I think you need to demonstrate that statements about the future are propositions that actually affirm or deny anything in a non-probabilistic sense of affirming or denying before your argument works.

    December 9, 2008 — 8:40
  • Alan Rhoda

    Alex:
    Set aside what I said about speaking “loosely” and such. I think I can come at my point more clearly as follows:
    Let’s distinguish between two types of semantic theories concerning the relation of modality and tense, particularly, the future tense.
    On the first type of theory, the future tense is intrinsically “probabilified” – to say that E “will” happen is always elliptical for something like “will probably (to a degree ≥ x)”. In other words, it is to say that the world is, at the very least, strongly tending toward E.” The value of x might vary from sentence token to sentence token, but it is never less than 0.5. On any such semantic account, of which the Peircean proposal is one variant, the epistemic probability of “ it will be the case that tomorrow you toss a coin 1000 times and it lands tails each time” is determined by the putative relation between the probability of your tossing a coin 1000 times tomorrow and its landing heads each time and the assumed value of x.
    On the second type of theory, the future tense has no intrinsic modality – to say that E “will” happen is just to say that E subsequently “does” happen, nothing more. It’s probability may lie anywhere in the half-open interval (0,1], but what matters for the truth of “E will happen” is not the probability of E, but the mere fact that E happens in the course of events. This is the Ockhamist’s way of construing the future tense.
    My chief objection against Ockhamism (not my only one) is that it cannot account for the assertibility of statements about the future. For any claim to the effect that E will happen to be assertibility for me, I must believe that present trends and tendencies are sufficiently strong to render E’s happening at least probable. When I assert “E will happen” I express just such a belief. The proposition I assert doesn’t merely represent E’s happening subsequently, but it’s doing so as a consequence of present trends and tendencies. Hence, my use of the future tense is probabilistically loaded.

    December 9, 2008 — 12:06
  • Alan:
    So, on the first type of theory, P(It will be the case that tomorrow you toss a coin 1000 times and it lands tails each time)=0. For, it is presently epistemically certain that not(the world is tending towards 1000 tail tosses with strength x), at least as long as is at least 0.5.
    On the second type of theory, P(It will be the case that tomorrow you toss a coin 1000 times and it lands tails each time) has a value between 0 and 1.
    I am afraid I do not fully understand your objection to the Ockhamist. One way to put my worry is that the argument you give would work just as well in a lot of contexts where, surely, we wouldn’t want it to work. For instance, by parallel, for a claim to the effect that E did happen to be assertible for me, I must believe that present events are such as to have been unlikely if E didn’t happen. But the claim that E happened is not a claim about present events and what circumstances they would have been likely under, but it is a claim about the past. Similarly, for the claim that there are aliens in a distant galaxy to be assertible for me, I must hold myself to be somehow causally connected to the aliens (where “causal connection” is the transitive closure of the relation “is causally prior or posterior to”). But the assertion that there are aliens in a distant galaxy does not represent the aliens as causally connected to me.

    December 9, 2008 — 12:53
  • Alan Rhoda

    Alex:
    My anti-Ockhamist argument was only concerned with the semantics of the future tense. And if Ockhamist fails in that case alone, I would take that as strong reason to reject it. Of course, if I can extend my argument beyond the future tense, that’s all to the better for me. Application to the past tense would require some modification given the asymmetry of time, but I think a plausible account could go something like this:
    For the claim “E happened” to be assertible for me, I must believe that the traces (e.g., memories, testimony of eyewitnesses, etc.) of E of which I am presently cognizant are reliable, i.e., that those traces probably could not have been produced if E had not happened. When I assert “E happened” I express just such a belief. The proposition I assert doesn’t merely represent E’s having happened, but E’s having initiated a series of events that has left its traces in the present.
    Similarly, I think it plausible to argue that “There are aliens in a distant galaxy” is assertible for me only if I believe that some state of affairs of which I am presently cognizant is such that either it probably would have caused their to be such aliens or probably would have been caused by such aliens. Etc.
    I’ll grant that I’m not as confident of the above two accounts as I am of the future tense case (Why the difference? I’m not sure).

    December 9, 2008 — 13:29
  • Alan:
    I agree with you about the assertibility conditions for claims about the past and about distant galaxies. But it seems very implausible that it is a part of the truth-conditions of “E happened” that there are traces of E or that it is a part of the truth-conditions of “There are aliens in a distant galaxy” that something here is causally connected to something there.

    December 9, 2008 — 14:00
  • Mike Almeida

    When I assert “E happened” I express just such a belief. The proposition I assert doesn’t merely represent E’s having happened, but E’s having initiated a series of events that has left its traces in the present.
    How could that be right? If the proposition expressed by ‘E happened’ included E’s having initiated a series of events that left it’s traces in the present, then I could discover a priori, from the truth of the sentence something (E) happened, among many other things, that causation occurs between events, that events leave traces, etc. But certainly I don’t learn these things a priori.

    December 9, 2008 — 15:18
  • Alan:
    I hope you don’t mind this interrogation. The arguments here are fun.
    Let p be the proposition: It is always causally possible for God to create a purple horse, but God never creates a purple horse.
    It seems that p is logically possible. There, surely, is a possible world where God never creates a purple horse, but neither does he ever do anything that could prevent him from doing so (e.g., promise not to create a purple horse).
    But on your view, p is not logically possible. For a proposition to be logically possible, it would have to be true at some world, either timelessly or at some time. Let say it’s true at w. I take it you don’t believe in the possibility of timeless truths. (If you do, and hence believe presentism to be contingent, this argument won’t work.) Then it’s true at w at some time t that God never creates a purple horse, but neither does he ever do anything that could prevent him from doing so. But that God never will create a purple horse is a future contingent, if he never did anything that could prevent him from doing so, and hence is false.

    December 9, 2008 — 15:39
  • Alan Rhoda

    Mike,
    Thanks for your comment. I certainly don’t mean to endorse the idea that all propositions about the past include the idea of “having initiated a series of events that has left its traces in the present”.
    I do think, however, that some propositions about the past have implications like that. There is a certain group of grammatical morphemes that linguists call “resultatives” which indicate that a present state exists as a result of an action in the past (John Bybee et al., The Evolution of Grammar, p. 18). For example, “He is (still) gone”, “The door is (still) closed”, “We are (still) waiting for him to return.”

    December 18, 2008 — 22:22
  • Alan:
    But unless you endorse something like this view, doesn’t the argument from presentism to open future either fail or generalize to an argument for an open past?
    (By the way, it’s an interesting philosophical and sociological question why so few people are willing to endorse an open past.)

    December 18, 2008 — 22:38
  • Alan Rhoda

    Alex,
    I just finished grading a stack of papers, so I can jot a quick reply before I have to tackle tomorrow morning’s final exams.
    Anyway, I deny that p (=It is always causally possible for God to create a purple horse, but God never creates a purple horse.) is logically possible.
    One reason is that I think God necessarily does exhaustive decision-making ab initio. So either he has decided to create a purple horse eventually, or he decided never to do so. Either way, p is not logically possible, assuming that God can’t retract a decision.
    But, setting that aside, there is a more fundamental reason. I take presentism to be necessarily true (if true at all). And I do not think there are any timeless truths (there are, of course, temporally invariant truths). Nor do I think that “possible worlds” consist entirely of temporally invariant truths. It follows that which possible world is actual is not fixed. (If you insist that “possible worlds” include a complete, determinate history, and that the “actual world” designates one of those possible worlds, then I deny that there is any such thing as the actual world, so defined.)
    Suppose, then, that p is true in the current actual world (that is, right now). In that case God can create a purple horse and can refrain from creating a purple horse. Hence, neither “God will create a purple horse” nor “God will not create a purple horse” are true. Both are false. And so is “God never will create a purple horse”. It is impossible, therefore, for both clauses of p to be satisfied simultaneously. Hence p is not logically possible.

    December 18, 2008 — 22:51
  • And by the same token, you will say that it is impossible that a person who lives forever and daily makes libertarian significantly free choices makes the right choice each day.
    At this point, we do have a disagreement of the standard form–you think these things are consequences, I think they’re reductios. 🙂

    December 18, 2008 — 23:00
  • Alan Rhoda

    Alex:
    I’m not sure exactly what view you have in mind, but at any rate I don’t see why I have to endorse any particular semantics for propositions about the past lest the argument from presentism to open future either fail or generalize to an argument for an open past.
    Given presentism, “will”/”will not” propositions about future contingents lack truthmakers, but that’s not true for propositions about the past given that God’s memories provide truthmakers for them.
    Well, I gotta run. This will probably be my last comment until after the APA.

    December 18, 2008 — 23:03
  • I guess I took you to think God’s memories were traces of past events, and hence I was surprised by your allowing that that there might be past events that left no trace. But I suppose that was about physical traces or maybe causal traces.

    December 18, 2008 — 23:07
  • Alan Rhoda

    OK, one more.
    It can’t be true that such a person will never err, for it is always the case that they might. But, by the same token, it is not true that such a person will eventually err, for it is always true that they might not.
    Merry Christmas!

    December 18, 2008 — 23:10