Todd (ch.3) – Against “Will Excluded Middle”

By | April 25, 2022

This is part 3 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 and part 2 here.

Todd’s main focus in chapter 3  is a thesis that he calls “will excluded middle” (WEM). Simply put, WEM says that not-will equals will not. In other words, if we let Fp and F~p stand for “It will be the case that p” and “It will not be the case that p”, respectively, then WEM says that F~p is semantically equivalent to ~Fp (“It is not the case that it will be the case that p”). Whether the negation (~) has wide scope as in ~Fp or narrow scope as in F~p, there’s no difference in meaning. Given WEM, corresponding will and will not propositions are contradictories, and their disjunction is therefore a proper instance of the logical law of excluded middle (i.e., necessarily, for all p, either p or ~p).

Thus, if WEM is correct, then <There will be a sea battle tomorrow OR there will not be a sea battle tomorrow> is a logical truth. And if, in addition, bivalence is correct, then necessarily one or the other of that pair of propositions is true, and the other is false. So, given WEM and bivalence, there has to be a “complete, true story” of the future. Many philosophers happily accept that result. Others, like Aristotle, use it as a reason for ditching bivalence. Still other philosophers, like Todd, keep bivalence and ditch WEM.

The burden of Todd’s ch. 3 is to show why it’s reasonable to deny WEM. According to Todd, WEM is not a logical truism, despite the fact that many philosophers think it is. Contrary to those philosophers, ~Fp and F~p are not semantically equivalent. The impression that they are equivalent, argues Todd, is driven not by semantic competence but by metaphysical assumptions about the nature of the future.

In what follows I will explain Todd’s reasoning and argue that he is correct. WEM is false and reasonably denied. One minor debatable issue aside, his ch. 3 hits the nail on the head.

I. WEM depends on a metaphysical model of the future

Why do many scholars endorse WEM? One reason is that it seems to them that pairs of sentences like

  1. It will not rain tomorrow. (F~p)
  2. It is not the case that it will rain tomorrow. (~Fp)

say the same thing.

One immediate problem noted by Todd (pp. 53–54) is that WEM doesn’t work in the case where there is no future. Suppose that time ends at 11:59 pm tonight, before “tomorrow” has a chance to even get started. Since (a) represents it as a positive fact about tomorrow that is it a rain-free day, then (a) can’t be true if there is no tomorrow. (Compare “The present king of France is bald” asserted at a time when there is no king of France.) (b), however, doesn’t assert anything positive about tomorrow. Rather, it simply denies that it is a positive fact that tomorrow is a rainy day. That’s perfectly consistent with there being no tomorrow. (Compare “It is not the case that the present king of France is bald” asserted at a time when there is no king of France.) Thus, if there is no tomorrow, then (b) is true and (a) is false, or at least not-true. This thought experiment suffices to show that (a) and (b) do not have the same truth conditions and therefore are not semantically equivalent. They do not say the same thing. Hence, WEM is false.

But suppose we waive this worry. Suppose we think it somehow metaphysically necessary that time continue forever. Under that assumption—and it is important to be clear that it is an assumption—it may seem that WEM holds. For this reason Todd doesn’t want to put too much emphasis on “end of time” scenarios. He wants to argue that WEM is false even if time is guaranteed to continue indefinitely.

Todd also wants to make sense of why it may seem (in ordinary, non-end-of-time contexts) that (a) and (b) are equivalent. His answer is that in ordinary contexts “it is presupposed that there exists what we might call ‘the actual future'” (p. 54). Let’s set aside for the moment the question of whether this is in fact an apt description of our “ordinary” way of thinking about the future, and simply point out that given this assumption and the continuance of time, WEM does in fact follow by process of elimination (Todd, p. 60):

  1. There is a unique actual future (UAF) in which the future continues for at least n time units and which, for any proposition p, either includes or excludes p in n time units. (Assumption)
  2. Fnp ∨ Fn~p. (Consequence of 1. If the UAF includes p in n time units, then Fnp is true. If the UAF excludes p in n time units, then Fn~p is true.)
  3. It is not the case that the UAF includes p in n time units. (~Fnp)
  4. Therefore, the UAF excludes p in n time units. (Fn~p) (From 2 and 3 by disjunctive syllogism.)

In other words, if we start by assuming there is a determinate fact of the matter as to how the future will be in n time units (premise 1), then in n time units the future must be determinately such that p or determinately such that ~p (premise 2). Since those are the only two possibilities, if we exclude one of them (premise 3), then the remaining possibility must be correct.

In sum, even though ~Fp is not semantically equivalent to F~p, given a metaphysical model of the future that includes a UAF we can reason from ~Fp to F~p. Likewise, it should be obvious that F~p entails ~Fp. (If there will not be a sea battle tomorrow, then it’s not the case that there will be a sea battle tomorrow.) So given a UAF, ~Fp and F~p are practically equivalent. As Todd puts it, the semantic or logical difference in scope between ~Fp and F~p is “suppressed or masked” by the underlying metaphysical model (p. 60).

II. On whether our “ordinary” conception of the future includes a UAF

I want press Todd a bit on whether the existence of a UAF really is part of our “ordinary” conception of the future. Is that really our default way of thinking about the future? Todd clearly thinks so:

Only a philosopher—a philosopher!—would think to question the inference from (3) to (4), because only a philosopher would have cause to consider and reject (1) (and thereby (2)). (p. 61)

I don’t have any empirical data to offer on my behalf, but I’m skeptical of this claim. There are at least two ways to argue against WEM. One is to point out that ~Fp is not semantically equivalent to F~p because the latter depends, whereas the former does not, on the continuation of time into the future. That sort of scenario is the kind that “only a philosopher” would countenance. But one can also reject WEM without invoking such a recherché scenario. To refute a disjunction all one needs to do is describe a coherent scenario in which neither disjunct is true. The coherent scenario might be a merely speculative possibility (like the end-of-time scenario) or it might be a highly plausible and commonsensical scenario. I believe that WEM can be refuted with a scenario of the latter sort.

Consider this: Both Fp and F~p represent the future as determinate in some respect. “It will rain tomorrow” represents the presence of rain as a determinate feature of tomorrow. “It will not rain tomorrow” represents the absence of rain as a determinate feature of tomorrow. Unlike the end-of-time scenario, which asks us to consider what happens if there is no tomorrow, what if we instead consider what happens if the future is not now determinate with respect to rain tomorrow. Call this the indeterminate-future scenario. For example, what if the weather forecast is currently in flux and there’s objectively a 50–50 chance of rain tomorrow? In other words, what if it is now indeterminate whether it rains tomorrow? And what if it’s not just epistemically indeterminate (we don’t know how things are going to go) but metaphysically indeterminate because present reality is genuinely indeterministic in that regard? Well, in that case it seems like neither Fp nor F~p could be true—at least not true now, while the future remains indeterminate. What’s now true, rather, is that it might and might not rain tomorrow.

I submit that the indeterminate-future scenario is both plausible and commonsensical, especially in contexts where, due to human free-will or indeterminism of some sort, the disposition of the future appears clearly to be “in flux”. This sort of context is not one that “only a philosopher” would countenance. I’m not aware of any empirical research on this, but I’d wager that if you were to poll people who have not been previously conditioned by philosophical, theological, or scientific ideas that entail a UAF and present them with a seemingly 50–50 indeterministic scenario that quite a lot of them would say that neither Fp nor F~p is true. And this is why I’m not convinced by Todd that the assumption of a UAF really is the “ordinary” or default conception of the future.

III. “Will” as a “neg-raiser” and the role of background models

Regardless of whether UAF and, consequently, WEM is part of our default way of thinking about the future, the scope distinction between ~Fp and F~p sometimes seems irrelevant. Accordingly, we need to explain why this is so. To this end Todd draws on Yale linguist Laurence Horn‘s work. According to Horn, some predicates are “neg-raising” in that, when negated, there is (a) a semantic difference between wide-scope and narrow-scope readings, but (b) in certain contexts the wide-scope reading is pragmatically equivalent to the narrow-scope reading. Todd lists a couple dozen such predicates on p. 59. For example,

Think: I don’t think that p / I think that ~p
Want: I don’t want to do it / I want to not do it
Appear: It doesn’t appear that p / It appears that ~p

Apart from any context, the first (wide-scope) reading in each case seems clearly distinct from the second (narrow-scope) reading. But in certain contexts the wide-scope reading can be suppressed.

In a context in which I am assuming that you’re not simply indifferent … , when you say that you don’t want to come to the party I hear this as an assertion that you want to not come. Indeed, it is extremely difficult to hear I don’t want to come as anything but this stronger assertion. (p. 59)

This phenomenon is called neg-raising by Horn. Why is it called that? I don’t know, but I think it has to do with the fact that narrow-scope negation is logically stronger than wide-scope negation. (It’s stronger because the narrow-scope reading entails the wide-scope reading, but not vice-versa.) If this is the rationale, then I don’t like Horn’s terminology. Raising invokes a very different conceptual metaphor than logical strengthening or scope narrowing do, and it’s confusing since nothing is clearly being “raised”. “Neg-strengthening” or “neg-narrowing” would have conveyed the idea more clearly.

In any case, Todd proposes that we apply Horn’s apparatus to the future tense. “Will”, he suggests, is a “neg-raiser” (p. 60). In each case of neg-raising, he notes, a contextually-supplied model rules out a mere wide-scope but non-narrow-scope reading. In the above-quoted example, for instance, the model contains the assumption that “you’re not simply indifferent” with respect to coming to the party. Obviously if we set that possibility aside, then there no longer remains any semantic space between “you don’t want to come” and “you want to not come”. Likewise, with respect to “will” and “will not”, if we start with a model that assumes a UAF or at least that the future is determinate in the relevant respects, then the semantic space between the wide-scope “not-will” and the narrow-scope “will not” vanishes.

I agree with Todd’s conclusion that it is only by way of a background metaphysical model that the inference from ~Fp to F~p sometimes seems correct. Furthermore, generalizing from specific cases where the ~Fp to F~p inference looks legit to WEM requires a background model that includes a UAF. Why? Because for the inference to work the possibility that the future is either non-existent (end-of-future scenario) or indeterminate (indeterminate-future scenario) has to be suppressed. The existence of a UAF, a fully determinate and indefinitely ongoing future, ensures that neither scenario obtains.

IV. Additional observations

Todd’s chapter is lengthy and dense. We have already covered the main lessons of the chapter, namely, (a) that WEM is false as a purely semantic claim (~Fp and F~p are not logically equivalent), and (b) that in certain pragmatic contexts, or because we have adopted a background metaphysical model like UAF, the semantic distinction between ~Fp and F~p is suppressed. To keep this blog review to a manageable length, I will merely summarize the rest of the chapter in bullet-point form.

  • Will as a modal term (pp. 70–72). The future tense marker will seems to function like a modal operator. But it is characteristic of such operators to have scope distinctions with respect to negation. For example, ~(necessarily p) ≠ necessarily ~p. The former includes, while the latter excludes, the status of contingency. Similarly, ~Fp ≠ F~p because the former includes, while the latter excludes, the possibility that the future is open with respect to whether p occurs. If we ignore scope distinctions in the former case, then we lose sight of contingency. If we ignore them in the latter case, then we lose sight of the openness of the future.
  • Openness of the future and bivalence (p. 72). Why not cash out the openness of the future by denying bivalence rather than by rejecting WEM? Answer: because there is no need to do so and because it makes the openness of the future much harder to express. Consider a parallel with contingency. We could insist that ~(necessarily p) = necessarily ~p and then define “contingency” as the state when neither disjunct of necessarily p ∨ necessarily ~p is either true or false, thereby denying bivalence. But it is much more straightforward to recognize the scope distinction and say instead that “contingency” is the state when both necessarily p and necessarily ~p are false. If we’re not tempted to deny bivalence to deal with contingency, then why would we deny it to deal with the openness of the future? The cases are parallel.
  • Conflating Fp ∨ F~p with F(p ∨ ~p) (pp. 73–75). The impression that Fp ∨ F~p is a logical truism plausibly stems from a conflation of that disjunction with F(p ∨ ~p). The latter is indeed a truism—a consequence of the logical law of excluded middle. But the former isn’t a truism because it depends on a coherently deniable metaphysical model (i.e., that there is a UAF).
  • Indeterminacy, vagueness, and the open future (pp. 76–79). As discussed in my previous post, some philosophers characterize the open future in terms of indeterminacy in order to assimilate the openness of the future to cases of vagueness. Just as it is indeterminate, say, whether a given man is bald or not-bald, it may be thought to be indeterminate whether this future is the UAF. But, counters Todd, the kind of indeterminacy in these two cases is fundamentally different. Vagueness-indeterminacy has to do with borderline cases, semantic indecision, and fuzzy boundaries. But even if we stipulate precise meanings for “sea-battle”, “occurs”, and “tomorrow” to remove all vagueness, the question of the openness of the future with respect to whether a sea battle occurs tomorrow remains. Openness-indeterminacy is thus not a matter of vagueness but of causal contingency.
  • What about WAS-excluded middle? (pp. 79–82). Philosopher Stephen Cahn has argued for WEM by arguing that because “there was a sea-fight” and “there was not a sea-fight” are contradictories, by parity of cases “there will be a sea-fight” and “there will not be a sea-fight” should also be understood as contradictories. In response, Todd denies that WAS-excluded middle is a logical truism. Like WEM, it seems to be a truism only because of an underlying metaphysical model. In this case, the metaphysical assumption is that there has been a unique actual past (UAP). Nearly everyone accepts UAP, but it is a metaphysical assumption, not a mere truth of logic. (This seems right to me. Consider Bertrand Russell’s famous 5-minute hypothesis—that the world has existed more than 5 minutes, or for any minutes, rather than having just appeared ex nihilo with built-in pseudo-memories is not something that can be ruled out by mere logic.)

In sum, aside from my minor qualm about whether a UAF is part of our default conception of the future, I think Todd’s ch. 3 is a tour de force. He makes a strong case that WEM is not a truth of logic and thus not a matter of pure semantic competence, but rather an idea underwritten by a metaphysical model (UAF) that can be coherently denied.

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