Todd (ch.6) – Part 2: Probability and the Open Future

This is part 7 of my ongoing series on Patrick Todd’s recently published book The Open Future: Why Future Contingents are All False (Oxford, 2021). (Previous installments: part 1, part 2, part 3, part 4, part 5, and part 6.)

This post continues my discussion of Ch.6, focusing on pp. 129–147, which is concerned with arguments to the effect that open futurists can’t make sense of probabilities about the future, specifically the ways we normally think about chance (i.e., single-case objective probability) and credence (i.e., subjective probability). This was a difficult post to write because there is a lot of subtle nuance to work through, not only in Todd’s chapter but with respect to the topic itself.

I. What is credence?

Before diving in to the relevant sections of Todd’s book, I’d like to provide a little background on credence. Credence or subjective probability is a measure of one’s degree of confidence in a proposition. It’s one’s personal estimate of its likelihood of being true. For example, most people who have considered the matters have maximal credence (credence = 1) that 2+2=4, a middling level of credence (0 < credence < 1) that a Republican wins the 2024 U.S. Presidential election, and zero or near-zero credence that consuming large quantities of arsenic is good for one’s health.

Todd doesn’t reference David Lewis’s so-called “principal principle”, but it’s worth mentioning because it’s a highly plausible way of relating credence and chance:

principal principle (PP): One should always match one’s credence that p to one’s estimate of the objective chance that p (i.e., Cr(p) = Ch_est(p)).

In other words, if you think the chance of p is X (e.g., the chance of a coin toss landing heads is 1/2), then your subjective confidence that p (i.e., that the coin toss lands heads) should also be X. Otherwise, you would have a kind of subjective inconsistency of the sort that could potentially be exploited by a “Dutch Book” argument—for example, if you are more confident than your chance estimations warrant, then you would be psychologically inclined to accept wagers that your own estimations indicate are too risky.

Another issue worth mentioning that Todd skips over is the idea that to believe p is to believe p is true.

believing is believing true (BBT): To believe p is to believe that p is true.

BBT is another highly plausible principle. It’s the idea that to believe p is to believe that reality is as p describes and thus to believe that p is true. If we combine BBT with PP and think of credences as degrees of belief (this step is somewhat controversial), then we can say that if one estimates the chance that p as E then one should (by PP) have a credence of E that p or, alternatively, that one should believe p to degree E and therefore (by BBT) believe to degree E that p is true.

II. The credence problem for open futurism

Suppose that a fair coin is about to be flipped and that how it lands is now perfectly indeterministic. To be extra careful, let’s stipulate that nothing can prevent its being flipped and that nothing can prevent its landing once it has been flipped. Under those assumptions, one would naturally estimate the chance of the coin’s landing heads at 0.5 and should therefore (by PP) have a credence of 0.5 that the coin lands heads.

Let’s also define open futurism since that’s the context for the credence problem:

open futurism (OF): Necessarily, for all p such that p represents a future contingent, neither Fp nor F~p is true.

From an open futurist perspective, neither F(The coin lands heads) nor F~(The coin lands heads) are true given our indeterministic coin-flipping scenario. Indeed, given OF it is impossible that either proposition be true so long as the outcome remains both indeterministic and future. It may be that one or the other becomes true if the outcome becomes determined a split-second before the coin fully lands, but it cannot be true before the outcome becomes determined. By the open futurist’s lights, therefore, the chance that either F(The coin lands heads) or F~(The coin lands heads) is true is zero, and so by PP he should have a credence of zero that either proposition is true. Of course, being reasonable, the open futurist believes that the chance is 0.5 that the coin lands heads and has a credence of 0.5 that the coin lands heads. So clearly the open futurist must distinguish between credence that p = (The coin lands heads) and credence that Fp = (The coin will land heads). Likewise, the open futurist must distinguish between the chance that the coin lands heads (=0.5) and the chance that F(The coin lands heads) is true (=0). (Todd makes roughly these same distinctions on p. 134.)

We can now state the credence problem. Todd uses MacFarlane’s statement of the problem as his launching pad (pp. 129–130). I’ve liberally modified it here to stick with the coin-flipping example instead of MacFarlane’s sea battle scenario:

The open futurist who believes that a given coin flip is indeterministic should not believe either F(The coin lands heads) or F~(The coin lands heads). But what credence should he have in F(The coin lands heads)? Here competing considerations seem to point in different directions. On the one hand, he knows that F(The coin lands heads) is not true. Normally we give a very low credence to things that we are certain are untrue. That suggests that he should have a very low (perhaps 0) credence in both F(The coin lands heads) and F~(The coin lands heads). On the other hand, when an agent has a credence of 0 that p, we generally take it to be irrational for her to accept a bet on p at any odds. So the open futurist who believes that both Heads and Tails are objectively possible outcomes of a coin flip should not accept a bet at any odds on the outcome. And that is surely wrong.

The problem that MacFarlane poses is that if the open futurist follows PP in assigning very low (indeed zero) credence to future contingent propositions that he thinks have zero chance of being true, then he allegedly cannot make sense of rational action (e.g., placing bets) based on those zero credence propositions. The zero credence the open futurist assigns to F(Future contingent E occurs) is supposedly in tension with what we would normally take to be reasonable bets on such propositions, such as betting on heads where the expected winnings are 100-to-1 in your favor.

In response to MacFarlane, Todd’s first point is to concede that, when it comes to probabilities, open futurists must say things that seem “revisionary” or “unfamiliar”, but that this isn’t particularly worrisome. As long as the underlying metaphysical picture is well-motivated and coherent, open futurism stands in good company with other philosophical positions that require making distinctions or talking in ways that do not seem “natural” (pp. 130–131). For example, in Ch.8 Todd compares open futurism to the metaphysical position of “eliminativism”, more commonly known as “mereological nihilism“. It’s the metaphysical view that compound objects—objects with separable or “proper” parts—don’t exist; only non-compound “simples” do. On this view, compound objects like chairs don’t exist. What exist, rather, are simples-arranged-chairwise, which for convenience we commonly call “chairs”. Now this is obviously a revisionary way of thinking about what most people regard as compound objects, but Todd’s point is that if the arguments for mereological nihilism are good then that’s an acceptable price to pay. You can’t refute mereological nihilism simply by pointing to our common practice of talking as if there are compound objects. Similarly, you can’t refute open futurism simply by noting that it leads to some seemingly awkward departures from normal discourse.

I like the comparison with mereological nihilism and the insistence that merely linguistic considerations are not compelling defeaters for well-motivated philosophical theories, but I’m not persuaded that open futurism is committed to as much revisionism as Todd seems to think. At any rate, it seems dialectically out of place to lead with this kind of concession. Let’s first convince ourselves that a problem is legit before deciding what concessions to make in dealing with it.

Todd’s substantive reply to MacFarlane begins by dividing the credence problem into two parts: (1) the zero credence problem and (2) the linguistic problem. Following Todd, we’ll tackle these separately.

III. The zero credence problem

If one has zero credence that F(The coin lands heads) is it irrational to accept a bet on heads at any odds? MacFarlane seems to think so, but Todd answers negatively (p. 131). More exactly, he thinks that irrationality follows only on Ockhamist assumptions, specifically will excluded middle (WEM) and bivalence (BV). If my credence that F(The coin lands heads) is zero, then I am certain that F(The coin lands heads) is false and therefore, by WEM and BV, certain that F~(The coin lands heads) is true. Hence, I’m certain that the coin will not land heads. But if I’m certain about that then I’m not going to bet on heads at any odds. So on Ockhamist assumptions, betting on heads with zero credence that F(The coin lands heads) is indeed irrational.

But open futurists aren’t Ockhamists. They reject either BV or WEM. If they affirm BV and reject WEM (as Todd does), then they will deny that zero credence that F(The coin lands heads) implies positive credence that F~(The coin lands heads). So one can continue to have zero credence in both F(The coin lands heads) and F~(The coin lands heads) while reasonably maintaining positive credence (0.5) that the coin lands heads.

If the open futurist affirms WEM and denies BV, then I think he will have to deny BBT and say that credence that p is not credence that p is true where p is a future contingent. This way he can deny the move from zero credence in F(The coin lands heads) to certainty that F~(The coin lands heads) is true. On this version of open futurism, both F(The coin lands heads) and F~(The coin lands heads) are instead neither true nor false.

For my part, I think the first way out (rejecting WEM and not BV) is significantly better. We shouldn’t reject either BBT or BV if we don’t have to—and we don’t have to, at least not for open futurist reasons. There are, of course, ways to motivate denials of both BV and BBT, but they don’t have any essential connection to future contingency, which is what open futurism is concerned with. Thus, it may be that considerations of vagueness give us some reason to deny BV, but future contingency isn’t about vagueness. And it may be that lottery cases provide some reason to deny BBT—e.g., a person might say “I believe I will lose the lottery” but balk at saying “I believe it’s true that I will lose the lottery” on the grounds that it’s only probable—but then we could defend BBT by saying that what the person really believes, even if he didn’t state it that way, is that he will probably lose the lottery, not that he unqualifiedly will lose it.

In any case, after pointing out that the zero credence problem only really gets purchase on Ockhamist assumptions, Todd elaborates with a helpful distinction between “tendency facts” and “facts of resolution” (underlined text is my emphasis):

[R]econsider the case of a simple wager of £5 on rain tomorrow. What should help to determine whether I should make this bet? Not, I have suggested, my beliefs about the chance that a rain future is our “actual” future. But then what? More particularly, I claim, there is no reason, in this scenario, to regard it as irrational for me to stake a claim to the rain futures, if I think that current reality is tending towards the realization of one of those futures, [and] even if there is no fact of the matter concerning how those tendencies will be resolved. … [I]t is fundamental to the open futurist’s picture of reality that the world could be strongly tending in a certain direction, without this implying anything about a likelihood of a current fact about the resolution of those tendencies. There are the tendency facts, but no further facts about the resolution of those tendencies. Since there are no such facts, claims purporting to report such facts—claims, for example, to the effect that there will be rain tomorrow—are claims in which I will … accordingly have credence 0. (pp. 132–133)

Todd then contrasts this open futurist “picture of reality” with the “common-sensical” Ockhamist picture according to which in addition to the tendency facts there are also facts of resolution (albeit not directly knowable by us). On the Ockhamist account, we use our knowledge of the tendency facts to make an educated guess about what the facts of resolution will be (p. 133) and bet accordingly. But Todd stresses that this further step of judging what the facts of resolution are going to be is completely unnecessary to underwrite the rationality of betting since one can ground one’s bets directly on the tendency facts instead (p. 135).

I think Todd is exactly right about this. The open futurist has a perfectly coherent picture of reality that underwrites our betting practices by grounding our estimates of the chances of future contingent events in the tendency facts without any need to posit further facts of resolution. My only qualm is Todd’s concessive description of the Ockhamist alternative as “common-sensical”. He may be right to use that description in light of the linguistic problem, which we are about to look at, but not in light of the zero credence problem, for which the open futurist picture is not only pragmatically equivalent to Ockhamism but more ontologically parsimonious to boot. Here again, I think Todd gets ahead of himself by making concessions for which no clear reasons have yet been given.

IV. The linguistic problem

We finally come to what Todd characterizes as a “difficult” objection to open futurism and, indeed, a “cause for despair” (p. 136). The objection is that the open futurist—particularly one who holds the all falsist version that Todd endorses—is committed to accepting some linguistically awkward or “infelicitous” things, such as

1. It is false that it will rain tomorrow, but it is probable that it will rain tomorrow.
2. It is false that it will rain tomorrow, but it will probably rain tomorrow.

Here I think Todd makes a serious misstep by conceding the charge and playing damage control rather than by flipping things around and considering whether Ockhamists aren’t also committed to saying similarly awkward things.

In the first place, (1) and (2) are not equivalent. The open futurist should reject (1) and accept (2). He should reject (1) because the second half of (1) reflects an Ockhamist way of thinking about future contingents by affirming that “It will rain tomorrow” is probable. For the open futurist, it is rain tomorrow that is probable, not F(It rains tomorrow). It is the Ockhamist, not the open futurist, who thinks that the probability of rain tomorrow entails the probable truth of F(It rains tomorrow). So one reason why (1) sounds so bad is because it incoherently combines an open futurist perspective (“It is false that it will rain”) with an Ockhamist perspective (“it is probable that it will rain”). (2) does not suffer from this problem and so should be endorsed by the open futurist (so long as its raining tomorrow remains a future contingent that is more likely to happen than not). And (2) is noticeably less awkward than (1), for to say that it will probably rain tomorrow does not obviously entail either the truth or the probable truth of F(It rains tomorrow).

Second, whatever residual infelicity there may be in (2) is wholly counterbalanced by the kinds of infelicity to which Ockhamists are committed. For suppose that rain tomorrow is a future contingent that is much more likely to happen than not. And suppose tomorrow comes and the now-probable rain fails to eventuate. In such cases—ones where either the improbable occurs or the probable fails to occur—the Ockhamist is committed to accepting things like this:

3. It is false that it will rain tomorrow, but it is probable that it will rain tomorrow.
4. It is true that it will rain tomorrow, but it is improbable that it will rain tomorrow.

Notice that (3) is the same as (1) and that both (3) and (4) are palpably infelicitous. Now, Ockhamists can of course gloss over these in a way that makes sense from their perspective, just as an open futurist can gloss over (2). My point is that the so-called linguistic problem is at least as bad for the Ockhamist as it is for the open futurist. So why is Todd registering notes of “despair” and issuing preemptive concessions that Ockhamism is more “common-sensical” when both sides are sometimes committed to accepting as true things that sound awkward? I’ll have to leave that for Todd to answer.

V. Probability and the future tense

The last several pages of Todd’s chapter are concerned with how probability claims interact with the future tense. Todd has a lot of interesting things to say here with respect to how the future tense functions in counterfactual and fictive contexts, but I’m going to skip over most of that to focus on the central issue.

When it comes to “will probably” sentences, my contention is that we cannot semantically “separate out” the will and the probably in terms of any simple scope distinction: the probably cannot be given wide scope over the will, … nor can it be given narrow scope with respect to will—that is, it cannot be rendered as a will be probable. Will probably then, is neither of those two things, but some third thing. (p. 145; underlining reflects Todd’s emphasis)

I think this is exactly right. To say that it will probably rain tomorrow is not ipso facto to say either (5) or (6):

5. Pr[F(It rains tomorrow)] = It is probable that it will rain tomorrow.
6. F[Pr(It rains tomorrow)] = It will be probable that it rains tomorrow.

In (5) the probability operator Pr() has wide scope, and in (6) the future tense operator F() has wide scope. But clearly neither has the same entailments as

7. It will probably rain tomorrow.

Contra (5), (7) doesn’t say that the proposition F(It rains tomorrow) is probable, but rather that the event (it’s raining tomorrow) is probable. Contra (6), (7) doesn’t say that it will (in the future) be probable that it rains tomorrow, but rather that it’s probable now that it rains tomorrow.

So how do probability claims interact with the future tense? Todd throws up his hands and says he doesn’t know (p. 146). Contrary to Todd, however, I think there is a natural way to relate probability and the future tense—it’s an idea that Todd mentions and discards in a footnote on pp. 134–135. The idea is that will probably should be conceptualized as a proportion of available futures. From an open futurist perspective, the rationale behind this idea is clear because the future tense is understood in explicitly quantificational terms. Thus, on Todd’s own semantic proposal in Ch.2,

• F(p) = In all available futures, p.
• F~(p) = In no available futures, p.
• ~F(p) and ~F~(p) = In some but not all available futures, p.

It is perfectly natural on this quantificational scheme to understand will probably in terms of a proportion of available futures, ranging from F() = “will with probability 1” to F~() = “will with probability 0”.

The reason why Todd discards this idea is because he’s worried about a certain hypothetical scenario, one in which there are only two available futures—let’s call these Rain and No Rain—because everything else is determined. And let’s suppose further that Rain is more probable than No Rain because reality is more strongly tending in that direction. But now the problem is that it seems that the proportional understanding of will probably forces us to say that Rain and No Rain are equally probable (50/50) when by stipulation they aren’t.

This is a clever objection to the proportional understanding of will probably, but it fails for reasons that Todd himself should appreciate, for if there’s one key takeaway repeatedly emphasized throughout his book, it’s that semantics must not be conflated with metaphysics. As Todd has extensively argued, IF Ockhamist theses like will excluded middle and retro-closure are true, they are true for metaphysical reasons, not semantic ones. Now consider this: the idea that will probably should be understood in terms of a proportion of available futures is a semantic proposal. As such, it provides a way of conceptualizing the relation between probability and the future tense. Because of this, the two-future Rain/No Rain metaphysical scenario proposed by Todd doesn’t matter. To accommodate it all we need to do is introduce a conceptual device that allows us to multiply the number of available futures indefinitely. Here’s one: We make use of Peter van Inwagen’s well-known rollback scenario and imagine the current indeterministic Rain/No Rain situation beting replayed indefinitely and then associate the probability of Rain with the idealized (long-run) frequency of Rain futures. In other words, we can say this:

• Will-probably_k(p) = “It will be the case with probability k that p” = The proportion (i.e., the idealized frequency) of available futures featuring p is k.