{"id":845,"date":"2022-05-04T14:15:30","date_gmt":"2022-05-04T19:15:30","guid":{"rendered":"http:\/\/alanrhoda.net\/wordpress\/?p=845"},"modified":"2022-09-22T14:59:27","modified_gmt":"2022-09-22T19:59:27","slug":"todd-ch-4-against-conditional-excluded-middle","status":"publish","type":"post","link":"http:\/\/alanrhoda.net\/wordpress\/2022\/05\/todd-ch-4-against-conditional-excluded-middle\/","title":{"rendered":"Todd (ch.4) \u2013 Against Conditional Excluded Middle"},"content":{"rendered":"

\"\"<\/a>This is part 4 of my ongoing series on Patrick Todd\u2019s recently published book The Open Future: Why Future Contingents are All False<\/em><\/a> (Oxford, 2021). You can find part 1 here<\/a>, part 2 here<\/a>, and part 3 here<\/a>.<\/p>\n

Ch. 3 dealt with will excluded middle (WEM)<\/strong>, the thesis that Fp \u2228 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<\/em> true (and perhaps be true in practice) if one assumes a certain metaphysical model (like a unique actual future<\/em>, or UAF) that rules out cases where ~Fp is true and F~p is false.<\/p>\n

Ch. 4 proceeds in similar fashion, except here the target is conditional excluded middle (CEM)<\/strong>, the thesis that (A > C) \u2228 (A > ~C). In other words, for any given antecedent (A) to a subjunctive<\/em> conditional and for any given consequent (C), either C would <\/em>be true given A, or C would not<\/em> be true given A. Following Todd, I just used ‘>’ to indicate a subjunctive would<\/em> conditional, but one major downside to this notation is that it obscures the distinction between would<\/em> and might<\/em> conditionals. Todd doesn’t give the would<\/em>\/might<\/em> 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 \u2610\u2192 to represent a would-<\/em>conditional and \u25c7\u2192\u00a0 to represent a might<\/em>-conditional, then CEM takes the following form:<\/p>\n

CEM: (A \u2610\u2192 C) \u2228 (A \u2610\u2192 ~C)<\/strong><\/p>\n

As with WEM, Todd argues that CEM is strictly false but can often seem<\/em> 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).<\/p>\n

Throughout Ch. 4 Todd repeatedly draws parallels between WEM and CEM. His stated goal is not so much to\u00a0refute<\/em> 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.<\/p>\n

I. CEM is not a logical truth<\/strong><\/p>\n

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<\/em> with respect to two or more different outcomes. For example (cf. p. 85):<\/p>\n

    \n
  1. If I had flipped this fair coin, it would<\/em> have landed heads.<\/li>\n
  2. If I had flipped this fair coin, it would<\/em> have landed tails.<\/li>\n<\/ol>\n

    If we stipulate that “tails” = any result that is “not heads”, then (b) can be rewritten as<\/p>\n

      \n
    1. If I had flipped this fair coin, it would not<\/em> have landed heads.<\/li>\n<\/ol>\n

      When joined together in a disjunction, (a) and (c) give us an instance of CEM in which neither<\/em> disjunct seems true. This example therefore suggests that CEM is false<\/em> 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).<\/p>\n

      To the extent that CEM seems to be a logical truth, this may be due to a conflation of (A \u2610\u2192 C) \u2228 (A \u2610\u2192 ~C) with A \u2610\u2192 (C \u2228 ~C). In the latter case, the consequent (C \u2228 ~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 \u2610\u2192 (C \u2228 ~C) could be false (except, perhaps, for reasons of relevance<\/em>, but let’s set that aside). But (A \u2610\u2192 C) \u2228 (A \u2610\u2192 ~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 <\/em>determinate<\/em>, 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.<\/p>\n

      II. CEM is based on metaphysical assumptions<\/strong><\/p>\n

      How then might one uphold CEM as true\u2014not as a truth of logic, but as true in practice? As Todd points out (pp. 91\u201394), the situation is parallel to that with WEM. While not will <\/em>(~Fp) and\u00a0will not<\/em> (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<\/em> (~(A \u2610\u2192 C)) and\u00a0would not<\/em> (A \u2610\u2192 ~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\u2013C pair whether C or its negation (~C) would<\/em> obtain given A.<\/p>\n

      Because WEM and CEM depend on such metaphysical commitments, one obviously cannot use WEM and CEM to ju<\/em>stify<\/em> 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.<\/p>\n

      III. Do we “ordinarily” presuppose that there are primitive counterfactuals?<\/strong><\/p>\n

      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<\/p>\n

      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)<\/p><\/blockquote>\n

      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<\/em>\u2014maybe there is such a fact and maybe there isn’t\u2014we often aren’t in a good position to say either way. In cases where the antecedent seems clearly neutral<\/em> with respect to a given consequent and its denial, most people are not only hesitant to endorse either the would<\/em> or the would not<\/em> claim but also hesitant to admit\u2014apart from antecedent commitment to a theory<\/em> like determinism or Molinism\u2014that there already is a fact of the matter as to which consequent would<\/em> result.<\/p>\n

      Todd presents an imagined dialog to support his position (p. 92):<\/p>\n

      Consider a scenario \u2026 in which a sports fan asks another, “God! Do you think we would have won if only Jones had made that catch?” Note the stark difference between the following replies:<\/p>\n

      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 \u2026<\/p>\n

      That’s a perfectly respectable, cooperative reply. But then consider:<\/p>\n

      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 win, and nothing to break the tie. But in such a circumstance, reality simply doesn’t decide \u2026<\/p>\n<\/blockquote>\n

      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<\/em> a fact of the matter concerning who would have won the game” (p. 93).<\/p>\n

      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<\/em> 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<\/em> “chancy” to us, it may not in fact be<\/em> chancy. That’s why the second response seems oddly dogmatic\u2014how 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\u00a0<\/em>that there is not<\/em> a determinate fact of the matter concerning what\u00a0would<\/em> have happened. It doesn’t show that we tend to presuppose that there is<\/em> 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<\/em> there is such a fact and maybe<\/em> there isn’t.<\/p>\n

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

      IV. The would<\/em> \/ might<\/em> connection<\/strong><\/p>\n

      I was very much surprised how little attention (barely over 1 page) Todd gives to the would <\/em>\/ might<\/em> connection. At least since the ground-breaking work of David Lewis on counterfactuals, the relation between would<\/em>-counterfactuals and\u00a0might<\/em>-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\u00a0will<\/em> and\u00a0will not<\/em> future contingent propositions.<\/p>\n

      Lewis endorses a principle that Todd calls “Duality”. This is the idea that\u00a0would<\/em> entails\u00a0might<\/em> but is opposed to\u00a0might not<\/em>, whereas\u00a0would not<\/em> entails\u00a0might not<\/em> and is opposed to\u00a0might<\/em>. More formally,<\/p>\n

      Would\/Might Duality<\/strong>:
      \nwould<\/em> (A \u2610\u2192 C) = not-might-not<\/em> ~(A \u25c7\u2192 ~C)
      \nmight<\/em> (A \u25c7\u2192 C) = not-would-not<\/em> ~(A \u2610\u2192 ~C)<\/p>\n

      Given Duality, CEM is clearly false, as would<\/em> and\u00a0would not<\/em> do\u00a0not<\/em> exhaust the possibilities. In contrast to CEM, Duality implies that both would<\/em> and\u00a0would not<\/em> are false when both might<\/em> and\u00a0might-not<\/em> are true. In other words, instead of<\/p>\n

      (A \u2610\u2192 C) \u2228 (A \u2610\u2192 ~C)<\/p>\n

      we get<\/p>\n

      (A \u2610\u2192 C) \u2228 (A \u2610\u2192 ~C) \u2228 ((A \u25c7\u2192 C) \u2227 (A \u25c7\u2192 ~C))<\/p>\n

      Why endorse Would\/Might Duality? Just think of cases where A seems to be neutral<\/em> with respect to C and ~C. If nothing in reality decides for either C or ~C, then it is reasonable to think that (A \u2610\u2192 C) and (A \u2610\u2192 ~C) are false and that what’s true instead is that if A then C might-and-might-not<\/em> obtain. In other words, C is contingent relative to A.<\/p>\n

      Or think of it this way. Like\u00a0will<\/em>,\u00a0would<\/em> is a determinacy indicator<\/strong>.\u00a0Will<\/em> represents the future as determinate in some respect (e.g., it will<\/em> rain tomorrow). Likewise, would\u00a0<\/em>represents C as a determinate consequence of A (e.g., if I were to toss this coin then it would<\/em>\u2014not just might<\/em>\u2014land heads). But what happens if the future or the hypothetical consequence are not<\/em> determinate. If rain tomorrow is a\u00a0future contingent<\/strong> 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<\/em> rain tomorrow>. Likewise, if the coin’s landing heads upon being tossed is a hypothetical contingent<\/strong>, then it seems like neither <If tossed, this coin\u00a0would<\/em> land heads> nor <If tossed, this coin\u00a0would not<\/em> land heads> are true. Arguably, both are false because what’s actually true is <If tossed, this coin might-and-might-not<\/em> land heads>.<\/p>\n

      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\u00a0would<\/em> \/\u00a0might<\/em> connection? I think it’s because he worries (wrongly) that the endorsing either Would\/Might Duality or the parallel claim about will<\/em> (call it Will\/May Duality)\u00a0<\/i>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\u201395, Todd writes:<\/p>\n

      My complaint against CEM is\u00a0metaphysical<\/em>, not semantic. However, the primary arguments considered in the literature against CEM seem to me to be\u00a0semantic<\/em> arguments\u2014roughly, “might” arguments from the truth of a principle sometimes called “Duality”.\u2026 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\u00a0will<\/em> and\u00a0might not<\/em>.\u2026 It is the\u00a0Peircean<\/em>, if anyone, who maintains that [corresponding\u00a0will<\/em> and\u00a0might not<\/em> claims embody] a semantic contradiction.<\/p><\/blockquote>\n

      In Ch. 2 (pp. 36\u201340) Todd objects to Peirceanism because it equates\u00a0will<\/em> with causal necessity<\/em>. While he grants that there is an extensional equivalence<\/em> between these two concepts\u2014the only will<\/em> propositions that come out as true<\/em> on Todd’s account are those describing the future in ways that are in fact causally necessary\u2014he denies that there is an intensional<\/em>\u00a0or semantic<\/em> equivalence between them. I agree with this. Peirceanism is false as a general semantics for will<\/em>. But what Todd misses is that Duality (of any sort) doesn’t require Peirceanism<\/strong>. All it requires is that will<\/em> and would<\/em> be understood as semantic indicators of determinacy<\/em>\u2014not causal determinism. So Duality doesn’t collapse the semantic \/ metaphysical distinctions that he wants to maintain\u2014it leaves defenders of WEM and CEM perfectly free to postulate a UAF or a book or primitive counterfactuals that specify a determinate<\/em> future or determinate<\/em> hypothetical outcomes without semantically<\/em> 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\u2014it can’t just be read off the semantics.<\/p>\n

      V. Counterfactual semantics<\/strong><\/p>\n

      The standard semantics for subjunctive conditionals employs the notion of a\u00a0possible world<\/em> and relations of accessibility <\/em>and similiarity <\/em>\/ closeness <\/em>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\u00a0access<\/em> 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 \u2610\u2192 C is true in the actual world (\u03b1) if and only if all of the closest worlds accessible to \u03b1 in which A is true are also worlds in which C is true.<\/p>\n

      Todd proposes a similar semantics but one articulated in such a way as to emphasize the parallels between will<\/em> and\u00a0would<\/em>. In Ch. 2 his proposed semantics for\u00a0will<\/em> treats it as a universal quantifier over available<\/em> 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<\/em> 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<\/em> 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):<\/p>\n

      (ACW)<\/strong> A \u2610\u2192 C if and only if in all of the closest counterfactually available A-worlds, C.<\/p>\n

      Would<\/em> is thus to be understood as a universal quantifier over\u00a0counterfactually available<\/em> 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\u00a0objective (non-modal) similarity<\/em> to the actual world” (p. 97). In cases (if there are any) where the closest A-worlds are neutral<\/em> 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 <\/em>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.<\/p>\n

      I think this is a really interesting and plausible proposal. Identifying the semantic and metaphysical issues and keeping them distinct<\/em> is probably Todd’s most important contribution to the debate over open futurism.<\/p>\n","protected":false},"excerpt":{"rendered":"

      This is part 4 of my ongoing series on Patrick Todd\u2019s 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 \u2228 F~p (i.e., that for\u2026 Read More »<\/a><\/span><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[159,63,41,155],"tags":[160,161,163,143,162],"class_list":["post-845","post","type-post","status-publish","format-standard","hentry","category-conditional-excluded-middle","category-modality","category-semantics","category-will-excluded-middle","tag-conditional-excluded-middle","tag-counterfactuals","tag-subjunctive-conditionals","tag-will-excluded-middle","tag-would-vs-might"],"_links":{"self":[{"href":"http:\/\/alanrhoda.net\/wordpress\/wp-json\/wp\/v2\/posts\/845","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/alanrhoda.net\/wordpress\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/alanrhoda.net\/wordpress\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/alanrhoda.net\/wordpress\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/alanrhoda.net\/wordpress\/wp-json\/wp\/v2\/comments?post=845"}],"version-history":[{"count":16,"href":"http:\/\/alanrhoda.net\/wordpress\/wp-json\/wp\/v2\/posts\/845\/revisions"}],"predecessor-version":[{"id":989,"href":"http:\/\/alanrhoda.net\/wordpress\/wp-json\/wp\/v2\/posts\/845\/revisions\/989"}],"wp:attachment":[{"href":"http:\/\/alanrhoda.net\/wordpress\/wp-json\/wp\/v2\/media?parent=845"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/alanrhoda.net\/wordpress\/wp-json\/wp\/v2\/categories?post=845"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/alanrhoda.net\/wordpress\/wp-json\/wp\/v2\/tags?post=845"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}