Todd (ch.4) – Against Conditional Excluded Middle

By | May 4, 2022

This is part 4 of my ongoing series on Patrick Todd’s recently published book The Open Future: Why Future Contingents are All False (Oxford, 2021). You can find part 1 here, part 2 here, and part 3 here.

Ch. 3 dealt with will excluded middle (WEM), the thesis that Fp ∨ F~p (i.e., that for any possible future state, p, either p will obtain or p will not obtain). There Todd argued that WEM is strictly false but can often seem true (and perhaps be true in practice) if one assumes a certain metaphysical model (like a unique actual future, or UAF) that rules out cases where ~Fp is true and F~p is false.

Ch. 4 proceeds in similar fashion, except here the target is conditional excluded middle (CEM), the thesis that (A > C) ∨ (A > ~C). In other words, for any given antecedent (A) to a subjunctive conditional and for any given consequent (C), either C would be true given A, or C would not be true given A. Following Todd, I just used ‘>’ to indicate a subjunctive would conditional, but one major downside to this notation is that it obscures the distinction between would and might conditionals. Todd doesn’t give the would/might distinction much play, but since I regard it as important for assessing CEM, I prefer to represent CEM with that distinction in mind. If we use ☐→ to represent a would-conditional and ◇→  to represent a might-conditional, then CEM takes the following form:

CEM: (A ☐→ C) ∨ (A ☐→ ~C)

As with WEM, Todd argues that CEM is strictly false but can often seem true (and perhaps be true in practice) if one assumes a certain metaphysical model, in this case, that there eternally exists something like a “book” of “brutely true, ungrounded counterfactuals” (p. 94).

Throughout Ch. 4 Todd repeatedly draws parallels between WEM and CEM. His stated goal is not so much to refute WEM and CEM, but to explain how both can be reasonably denied by rejecting their metaphysical underpinnings. What follows are my comments on the main themes of Ch. 4.

I. CEM is not a logical truth

To argue against CEM, Todd draws on the work of distinguished philosopher Timothy Williamson. Todd and Williamson ask us to consider cases where the antecedent is neutral with respect to two or more different outcomes. For example (cf. p. 85):

  1. If I had flipped this fair coin, it would have landed heads.
  2. If I had flipped this fair coin, it would have landed tails.

If we stipulate that “tails” = any result that is “not heads”, then (b) can be rewritten as

  1. If I had flipped this fair coin, it would not have landed heads.

When joined together in a disjunction, (a) and (c) give us an instance of CEM in which neither disjunct seems true. This example therefore suggests that CEM is false as a principle of logic. Not only does neither disjunct seem true, but it seems “metaphysically arbitrary” to suppose that reality somehow decides in favor of (a) rather than (c), or vice-versa (p. 87).

To the extent that CEM seems to be a logical truth, this may be due to a conflation of (A ☐→ C) ∨ (A ☐→ ~C) with A ☐→ (C ∨ ~C). In the latter case, the consequent (C ∨ ~C) is an instance of the law of excluded middle and is therefore necessarily true. Given a necessarily true consequent, it’s hard to see how A ☐→ (C ∨ ~C) could be false (except, perhaps, for reasons of relevance, but let’s set that aside). But (A ☐→ C) ∨ (A ☐→ ~C) is not an instance of excluded middle or any other generally accepted logical truism. It says that the implications of any hypothetical scenario (A) are always fully determinate, such that for any consequent C it is either determinately the case that C (given A) or determinately the case that ~C (given A). That’s not something logic or semantics alone can plausibly guarantee.

II. CEM is based on metaphysical assumptions

How then might one uphold CEM as true—not as a truth of logic, but as true in practice? As Todd points out (pp. 91–94), the situation is parallel to that with WEM. While not will (~Fp) and will not (F~p) are semantically distinct, the difference between them can be “masked” or “suppressed” if one adopts a metaphysical commitment to a unique actual future (UAF). Likewise, while not would (~(A ☐→ C)) and would not (A ☐→ ~C) are semantically distinct, the difference between them can be masked or suppressed if one adopts a metaphysical commitment to something like “a book of primitive counterfactuals” (p. 91), one that specifies for every hypothetical A–C pair whether C or its negation (~C) would obtain given A.

Because WEM and CEM depend on such metaphysical commitments, one obviously cannot use WEM and CEM to justify those commitments. This observation is surely inconvenient for those, like Ockhamists and Molinists, would who would like to be able to ground their metaphysical commitments in WEM and CEM, respectively.

III. Do we “ordinarily” presuppose that there are primitive counterfactuals?

While Todd rejects the idea that there is any like a book of primitive counterfactuals, he nevertheless regards that assumption as “perfectly reasonable” (p. 94) because he thinks that

in ordinary life, we naturally tend to prescind from such inconvenient facts as that there is no fact of the matter about what would have happened in indeterministic scenarios that never occur. (p. 92)

I think Todd is wrong here. We don’t normally “prescind” from the idea that there is no “fact of the matter” and thereby presuppose that there is such a fact of the matter. Rather, epistemic modesty generally encourages a position of neutrality—maybe there is such a fact and maybe there isn’t—we often aren’t in a good position to say either way. In cases where the antecedent seems clearly neutral with respect to a given consequent and its denial, most people are not only hesitant to endorse either the would or the would not claim but also hesitant to admit—apart from antecedent commitment to a theory like determinism or Molinism—that there already is a fact of the matter as to which consequent would result.

Todd presents an imagined dialog to support his position (p. 92):

Consider a scenario … in which a sports fan asks another, “God! Do you think we would have won in only Jones had made that catch?” Note the stark difference between the following replies:

I don’t know! On the one hand, we would have had the momentum, but on the other hand, there was plenty of time left on the clock, the game is still a chancy game, and it could still then have gone either way …

That’s a perfectly respectable, cooperative reply. But then consider:

Well, there’s really no fact of the matter concerning whether we “would” have won. After all, given the indeterminism inherent in the game, there are approximately equiprobable scenarios under which Jones makes that catch and we go on to lose, and scenarios in which Jones makes that catch and we go on to lose, and scenarios in which Jones makes that catch and we go on to win, and nothing to break the tie. But in such a circumstance, reality simply doesn’t decide …

In Todd’s view, the first response is natural and the second is “asinine” (cf. footnote 6 on p. 93), and the lesson we’re supposed to draw is that “ordinary thought and talk presupposes that there is a fact of the matter concerning who would have won the game” (p. 93).

But Todd’s reasoning here is flawed. The first reply is more reasonable because the interlocutor is appropriately uncertain in contrast to the second interlocutor, who seems inappropriately dogmatic. The first interlocutor does not presuppose that there IS a fact of the matter concerning who would have won the game. He merely leaves that as an epistemic possibility. So the example doesn’t support Todd’s claim at all. Indeed, since there is often more to reality than we are cognizant of, it’s commonsense that even if a scenario seems “chancy” to us, it may not in fact be chancy. That’s why the second response seems oddly dogmatic—how does he know that the outcome of the game was highly indeterministic with nothing to “break the tie”? For all he knows, there were factors at work that would have broken the tie and made one outcome far more likely than the other. In short, the most this example (and the one Todd gives in the footnote on p. 93) shows is that we don’t normally presuppose that there is not a determinate fact of the matter concerning what would have happened. It doesn’t show that we tend to presuppose that there is such a determinate fact. Given our epistemic limitations, the safe and modest position in most cases is to assume that, for all we know, maybe there is such a fact and maybe there isn’t.

Contrary to Todd, then, the supposition that there is a book of primitive counterfactuals is not “perfectly reasonable”. In most contexts it’s an epistemically immodest position that does nothing to allay the charge of metaphysical arbitrariness.

IV. The would / might connection

I was very much surprised how little attention (barely over 1 page) Todd gives to the would / might connection. At least since the ground-breaking work of David Lewis on counterfactuals, the relation between would-counterfactuals and might-counterfactuals has been a major issue, and it’s one that both directly threatens CEM and offers relevant support to Todd’s own position regarding the falsity of will and will not future contingent propositions.

Lewis endorses a principle that Todd calls “Duality”. This is the idea that would entails might but is opposed to might not, whereas would not entails might not and is opposed to might. More formally,

Would/Might Duality:
would (A ☐→ C) = not-might-not ~(A ◇→ ~C)
might (A ◇→ C) = not-would-not ~(A ☐→ ~C)

Given Duality, CEM is clearly false, as would and would not do not exhaust the possibilities. In contrast to CEM, Duality implies that both would and would not are false when both might and might-not are true. In other words, instead of

(A ☐→ C) ∨ (A ☐→ ~C)

we get

(A ☐→ C) ∨ (A ☐→ ~C) ∨ ((A ◇→ C) ∧ (A ◇→ ~C))

Why endorse Would/Might Duality? Just think of cases where A seems to be neutral with respect to C and ~C. If nothing in reality decides for either C or ~C, then it is reasonable to think that (A ☐→ C) and (A ☐→ ~C) are false and that what’s true instead is that if A then C might-and-might-not obtain. In other words, C is contingent relative to A.

Or think of it this way. Like willwould is a determinacy indicatorWill represents the future as determinate in some respect (e.g., it will rain tomorrow). Likewise, would represents C as a determinate consequence of A (e.g., if I were to toss this coin then it would—not just might—land heads). But what happens if the future or the hypothetical consequence are not determinate. If rain tomorrow is a future contingent then it seems like neither <It will rain tomorrow> nor <It will not rain tomorrow> are true. Arguably, both are false because what’s actually true is <It may-and-may-not rain tomorrow>. Likewise, if the coin’s landing heads upon being tossed is a hypothetical contingent, then it seems like neither <If tossed, this coin would land heads> nor <If tossed, this coin would not land heads> are true. Arguably, both are false because what’s actually true is <If tossed, this coin might-and-might-not land heads>.

The obvious parallels between future contingency and hypothetical contingency tend strongly to support Todd’s overall case against WEM and CEM. So why does he downplay the wouldmight connection? I think it’s because he worries (wrongly) that the endorsing either Would/Might Duality or the parallel claim about will (call it Will/May Duality) would (1) undercut his project of keeping metaphysical and semantic issues distinct and (2) commit him to something like a “Peircean” semantics that he regards as untenable. Thus, on pp. 94–95, Todd writes:

My complaint against CEM is metaphysical, not semantic. However, the primary arguments considered in the literature against CEM seem to me to be semantic arguments—roughly, “might” arguments from the truth of a principle sometimes called “Duality”.… These points have an important analogue in the case of future contingents. For note that we could develop parallel arguments for “Duality” in the case of will and might not.… It is the Peircean, if anyone, who maintains that [corresponding will and might not claims embody] a semantic contradiction.

In Ch. 2 (pp. 36–40) Todd objects to Peirceanism because it equates will with causal necessity. While he grants that there is an extensional equivalence between these two concepts—the only will propositions that come out as true on Todd’s account are those describing the future in ways that are in fact causally necessary—he denies that there is an intensional or semantic equivalence between them. I agree with this. Peirceanism is false as a general semantics for will. But what Todd misses is that Duality (of any sort) doesn’t require Peirceanism. All it requires is that will and would be understood as semantic indicators of determinacy—not causal determinism. So Duality doesn’t collapse the semantic / metaphysical distinctions that he wants to maintain—it leaves defenders of WEM and CEM perfectly free to postulate a UAF or a book or primitive counterfactuals that specify a determinate future or determinate hypothetical outcomes without semantically committing themselves to determinism, fatalism, or anything like that. The connection, if any, between an alethically settled future and a causally settled future needs to be argued for on metaphysical grounds—it can’t just be read off the semantics.

V. Counterfactual semantics

The standard semantics for subjunctive conditionals employs the notion of a possible world and relations of accessibility and similiarity / closeness between worlds. Roughly speaking, a ‘possible world’ is a complete way things could be or could have been. To say that world W is ‘accessible’ from world W* is, roughly, to say that if something is ‘possible’ in W, then it’s possible in W* as well. In other words, W* can access those possibilities. Finally, worlds can be more or less ‘close’ or ‘similar’. For example, they might share the same laws of nature and the same history up until a certain point after which they diverge, or they might have completely different laws and a completely different history. In these terms, the standard semantics says something this: A ☐→ C is true in the actual world (α) if and only if all of the closest worlds accessible to α in which A is true are also worlds in which C is true.

Todd proposes a similar semantics but one articulated in such a way as to emphasize the parallels between will and would. In Ch. 2 his proposed semantics for will treats it as a universal quantifier over available futures. The extension (not the semantics) of ‘availability’ is then determined by one’s metaphysical commitments. The idea is to have a semantics for will that is metaphysically neutral between the various models of the future that Todd wants to consider, specifically those that endorse WEM and those that don’t. Likewise, when it comes to counterfactuals, Todd proposes a semantics for would that is neutral between various models of modal space, specifically those that endorse CEM and those that don’t. He calls his semantic proposal ACW (p. 97):

(ACW) A ☐→ C if and only if in all of the closest counterfactually available A-worlds, C.

Would is thus to be understood as a universal quantifier over counterfactually available worlds. The extension (not the semantics) of ‘counterfactual availability’ is determined by one’s metaphysical commitments. Those who deny CEM typically hold, says Todd, that “the truth of a counterfactual is a matter of objective (non-modal) similarity to the actual world” (p. 97). In cases (if there are any) where the closest A-worlds are neutral with respect to both C and ~C, CEM fails. In contrast, those who affirm CEM typically hold that, over-and-above the non-modal facts there are also primitively modal facts or “counterfacts” that are relevant for assessing world-similarity. These primitive modal facts can then serve to “break ties” between C and ~C, thereby saving CEM.

I think this is a really interesting and plausible proposal. Identifying the semantic and metaphysical issues and keeping them distinct is probably Todd’s more important contribution to the debate over open futurism.

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