# Excluded middle

Some, but not all, open futurists deny excluded middle for future contingents. Thus, they deny that either there will or there will not be a sea battle tomorrow. Here is an inductive argument to the contrary: A sea battle typically requires a conflict between naval powers that has been brewing for some time. Such a conflict is probably not brewing right now. Therefore, probably, there will be no sea battle tomorrow. But if there will be no sea battle tomorrow, then there will or will not be a sea battle tomorrow (disjunction-introduction). Hence, there will or will not be a sea battle tomorrow. But if there will or will not be a sea battle tomorrow, views that deny excluded middle for future contingents are false.

Objection: The argument is merely probabilistic, so all that we have establishes is that probably there will or will not be a sea battle tomorrow.

Response: Indeed. But that there will or will not be a sea battle tomorrow entails (given uncontroversial facts about what is and is not contingent) that excluded middle holds for some future contingents. And hence, probably, the open future view under consideration is false. That is a non-trivial result.

Moreover, we can jack up the probabilities higher. Let p be the proposition that the pattern of the first ten die-pair tosses at Las Vegas over time tomorrow will not be 1-1, 2-2, 1-1, 2-2, 1-1, 2-2, 1-1, 2-2, 1-1, 2-2. Then, P(p) is approximately 1/3610, a very small number. Hence, P(p or not-p) is at least 1-1/3610 which is very close to 1. Hence the probability of the open future view in question is close to zero.

Hi Alex,

Your argument assumes, incorrectly I think, that corresponding 'will' and 'will not' propositions are contradictories. Equivalently, it assumes that 'will not' = 'not will'. Obviously, if this is not so then "there will or will not be a sea battle tomorrow" is not an instance of excluded middle.

The difference between 'will not' and 'not will', as I see it, lies it the scope of 'not'. In the 'not will' case an entire 'will' proposition is negated, thereby giving its contradictory. In the 'will not' case only a predicate is negated. Case in point: "The present king of France will shave his head tomorrow" vs. "The present king of France will not shave his head tomorrow". Arguably, both of these are false on account of presupposition failure. Both seem to assume, falsely, that there is a present king of France. If so, then what we have here are contraries, not contradictories.

If you want to maintain that 'will not' = 'not will', then bear in mind that "It is not the case that there will be a sea battle tomorrow" would be true even if there were no creation at all. Hence it is not equivalent to any proposition which presupposes that there are or have been such things as seas or battles or that there will be times like today and tomorrow.

Hi Alex,

Your argument assumes, incorrectly I think, that corresponding 'will' and 'will not' propositions are contradictories. Equivalently, it assumes that 'will not' = 'not will'. Obviously, if this is not so then "there will or will not be a sea battle tomorrow" is not an instance of excluded middle.

The difference between 'will not' and 'not will', as I see it, lies it the scope of 'not'. In the 'not will' case an entire 'will' proposition is negated, thereby giving its contradictory. In the 'will not' case only a predicate is negated. Case in point: "The present king of France will shave his head tomorrow" vs. "The present king of France will not shave his head tomorrow". Arguably, both of these are false on account of presupposition failure. Both seem to assume, falsely, that there is a present king of France. If so, then what we have here are contraries, not contradictories.

If you want to maintain that 'will not' = 'not will', then bear in mind that "It is not the case that there will be a sea battle tomorrow" would be true even if there were no creation at all. Hence it is not equivalent to any proposition which presupposes that there are or have been such things as seas or battles or that there will be times like today and tomorrow.

Oops, sorry for the double post. My reply didn't appear the first time I submitted it.

Hi Alan,

I don't think your worry addresses the Las Vegas example, which is the more convincing case. That case does use wide-scope negation as Alex formalizes it. That aside, I don't see how narrow scope negation presupposes anything. If you Russellize the definite description you get an entailment that there is a PKF on the narrow scope reading, not a presupposition. But that's a good reason not to read it as narrow scope.

Alex,
In your response to the objection, I wasn't clear on how you moved from it is very likely that (pv~p) to it is very likely that open futurism is false. Or is that what you were saying?

To put this another way, I took you to be saying it is very likely that either there will or there will not be a sea battle tomorrow to it is very likely that open futurism is false. Is that what you were saying? If so, then what justification is there for that inference? If not, it's not clear to me what your response to the objection is. Maybe you can clarify?

in the first line of my second paragraph, replace "saying" with "moving from". replace the second "saying" (in line 3) with "inferring".

OK, but isn't a more plausible position for the proponent of the open future view to retain LEM but deny the Principle of Bivalence? On most readings of the text, that's the position put forward in De Int. 9--it's not true (or false) that there will be a sea battle tomorrow, nor is it true (or false) that there will not be a sea battle tomorrow, but it is true that either there will or will not be a sea battle tomorrow.

Alex,

This is an interesting argument, but first a question: Which open future view that accepts LEM do you find most plausible? Are there any mandatory readings on the subject?

One way to get out of your argument is to claim that the probability that "there will be a sea battle tomorrow = 0" and that "the probability that there will not be a sea battle tomorrow = 0". If future contingents are guaranteed to be false, then the probability that they are true = 0. Moreover, for any proposition P, if the probability that P is true = 0, then no evidence raises the probability that P is true. Thus, there are no cogent inductive arguments for propositions about future contingents.

"Therefore, probably, there will be no sea battle tomorrow."

So, this claim is false. It is not probable that there will be a sea battle tomorrow. The probability that there will be a sea battle tomorrow = 0.

Alex,
ah, thanks, I see. Hmm... interesting argument.

Alex,

"It's hard to choose, because all these views are so implausible. :-)"

That's my impression too. So I'm wondering what I'm missing.

"Or maybe it's Alan Rhoda's view that LEM and bivalence hold, but it is both false that there will be a sea battle tomorrow and that there will not be a sea battle tomorrow."

I don't see how such a view could work either. How can one maintain that P is false, not-P is false, but also maintain that (P or not-P) is true? Disjunctions are false when their disjuncts are both false, right? Alan, if you're reading, paper reference?

"Surely if something has probability zero, it is almost surely false."

I would say it is certainly false.

"So, on the view you propose, it is almost surely false that there will be a sea battle tomorrow."

Yes, it is certainly false.

"But if so, then it is almost surely true that there won't be one tomorrow."

No, on the view (not mine) that I'm offering, it is also certainly false that there will be a sea battle tomorrow (without Rhoda semantics).

"But if so, then it is almost surely true that there will or won't be one tomorrow."

On this view, this is false as well. It is certainly false that there will or there won't be one tomorrow.

"And if so, then it is almost surely true that LEM holds for the sea battle."

But it's not so. LEM does not hold for the sea battle.

Interesting. I wonder why an open theist would be tempted to give up the LEM, rather than to say merely that all future tensed contingencies are false. Take for example:

(1) WILL
(2) WILL

One might think that (1) and (2) are both false but that LEM is true. LEM commits one to saying that either (1) is true or NOT (1) is true. An open theist who thinks that (1) is false may affirm that NOT (1) is true. What's crucial is that affirming that NOT (1) is true is not the same as affirming that (2) is true. Thus, one might think that NOT (1) is true while also thinking that (2) is false. (I think this was Alan's point.)

Now I realize this doesn't pose a problem for your argument against those open theists who deny LEM. But I am curious if you have any reasons against this verson of open theism. On this view, God could know all true propositions and so be omniscient in the classical sense.

I'm especially curious because PSR + presentism is pushing me through the door of open theism... care to block my next step?

The props above should be:

(1) WILL [there is a sea battle]
(2) WILL [there is not a sea battle]

Alex,

I thought you might say that--give up presentism. I realize we are straying a bit from the original topic, but a few comments about your comment:

1. On my understanding, the A-theory says that the A-properties are not reducible to tenseless B-properties. But I'm inclined to think that A-properties are reducible to the B-properties (see “Presentism and Grounding,” Noûs 41 (2007): 90-109). Ironically, I think a B-theory is only plausible given presentism. But that's another story...

2. I'll give you two reasons I favor presentism:

A. I don't see what it can mean to say that x exists at a time t other than to say that t entails that x's existence obtains simpliciter. This means that the time at which dinosours roamed the earth doesn't obtain (otherwise, that would contradict the fact that there are no dinosours). By similar reasoning, no past or future times obtain (without contradicting something that obtains). The 'present time' I understand to be whatever time it is that obtains. Thus, only the present time obtains (simpliciter). Another way to this conclusion is to view times as maximal, abstract states of affairs (which I'm prepared to argue for). By virtue of being maximal, only one time can have that special status of obtaining (being present). I'm of course assuming that there is a non-time indexed property of obtaining simpliciter (which strikes me as undeniable).

B. Consider the proposition P that Alex used to be shorter. Now of course one and the same thing cannot be shorter than it is. Therefore, if I reject presentism, I might translate (1) as saying that Alex has the property of being tall-at-t and short-at-t0, where t0 is earlier than t, and t is the time I uttered P (or something like that). But that is a bad translation for two reasons: (i) (P) isn't nearly that complicated; (ii) although Alex may well have the property of being tall-at-t, he also has the simpler property of being tall; and although Alex may well have the property of being short-at-t0, he also used to have the simpler property of being short. Surely, there are non-time indexed properties, and surely Alex exemplifes some of them. But I do not see how someone who denies presentism can accept that. Someone may try to translate (1) as talking about things distinct from me, like my temporal parts. But then that surely won't be a translation.

Those two reasons seem to me much more pressing than the feeling that my life is ebbing away.

Oops: (1) = P above.

Alex: How do you understand the truth of "I am writing this now"

Alex,

"Then" is a tensed property!

Hi Alex,

I realized that you weren't directly critiquing my position. I merely wanted to point out that, from my perspective at least, what you offer as an instance of LEM isn't.

Alex: "The inductive evidence supports the claim that tomorrow there will be at least one car on I-35. Thus, probably, tomorrow there will be at least one car on I-35. But it is uncontroversially contingent that tomorrow there will be at least one car on I-35. That tomorrow there will be at least one car on I-35 together with uncontroversial claims entails the falsity of Alan's view. Thus, Alan's view is probably false, and I should be at least as surprised by its turning out true as by finding I-35 bereft of cars tomorrow."

You won't be surprised to learn that I don't think this argument is sound. :-) Inductive evidence supports the claim that the chance of a car's being on I-35 tomorrow is very high. By the Principal Principle this grounds a correspondingly high credence in "a car is on I-35 tomorrow". But this credence is not the degree of belief that the proposition is true (now). Rather, it is the degree of belief that the proposition comes to be true (tomorrow).

Alex,

Thanks for those comments. Some replies:

A. x is in London means that x is spatially between some of the many things that jointly exemplify being London. I suppose by analogy, we might say that x is at t means that x is "temporally between" some of the things that jointly instantiate t. But what is this "temporally between"? Should I understand it as a spatial relation, or as something else? If a spatial relation, then why call it temporal? If not a spatial relation, then what relation is this? Whatever it is, it's such that necessarily, if it is exemplified, then its relata exist (all relations are like that). And if it's a temporal relation (and not merely spatial), then its relata exist at different times. But unless the relata are necessarily existing things (like abstract times), there will be cases in which the relata don't all exist unless multiple times obtain, which is impossible if times are maximal states of affairs.

Now someone can reply by saying that times are actually concrete segments, some of which are to the "temporal" left of (earlier than) me and others to my "temporal" right (later than). I suppose I understand what is being said here (in some sense of understand). But the idea that times are really concrete regions to my temporal "left" or "right" seems to me no more plausible than the view that "possibilities" are really causally isolated universes. These views equally invite an incredulous stare. I hope I'm not being too uncharitable. :)

B. I suppose some properties turn out to be more complex than they initially seem. But I suspect this happens for properties we don't fully grasp. I also suspect that sometimes we come to analyze certain properties in terms of co-extensive ones that are more complex but also not identical to the properties we originally had in mind. There are probably also times when we start with a determinable property and then come to grasp one of its more complex determinates. Here again the property we grasp later isn't the same as the original one we had in mind. At any rate, I feel nearly certain that I grasp certain geometric properties sufficiently well to tell that they aren't time-indexed. I also feel nearly certain that there are (or at least could be) things that change from having one of these geometric properties to having a different incompatible one. I can't see how to make sense of this if presentism isn't true (assuming that exemplification isn't time-indexed either). That's just me. :)

Alex:

I don't follow your definition of C(p). Specifically, I don't understand the phrase "at the time that it is a proposition about p". Did you mean to say "at the time proposition p is about"?

At any rate, you claim that "If p is probable, then Knows(God,p) is equally probable, since God necessarily knows all truth."

You seem to assume that the probability that p equals the probability that p is true and thus equals the probability that God knows p. I disagree with that assumption. Let p be "E occurs tomorrow". To have a high credence in p is not ipso facto to have a high credence that p is true; rather, it is to have a high credence that "E occurs" comes to be true tomorrow.

Case in point. The weather center has informed me that there is a 30% chance of rain tomorrow. I have no other relevant information. Hence, my credence that it rains tomorrow is 30%. But this is not my credence that it is true now that it rains tomorrow. The chance that any proposition is true now is either zero or one--either it is true or it isn't. The chance of a proposition's being true can't be 30%. So if I'm going to match my credence to my best estimate of the chance (as the Principal Principle urges), then my credence in "it rains tomorrow" has to be my best estimate of the chance that "it rains" comes to be true tomorrow.

In sum, let the chance be 99.99% that, tomorrow, a car is on I-35. It is then true now, and known to God, that the chance of a car's being on I-35 tomorrow is 99.99%. God also has a corresponding credence that "a car is on I-35" comes to be true tomorrow. But this is not equal to God's credence that "a car is on I-35 tomorrow" is now true, for the chance of that has got to be either zero or one--either it is now true or it isn't.

Alex:

I reject 3. I think it's apparent plausibility stems from conflating "Tp iff p" with an indexed analogue, "T(i)p iff p(i)" (read: "It is true at i that p iff p-at-i."). The latter is undeniably true, but not sufficient for your argument.

You might, perhaps, suggest that they come to the same thing on the grounds that the actual world, @, contains a complete history and that truth at @ = truth simpliciter. But if you go that route then you beg the question against open futurism. No open futurist will concede that there is an actual world which includes a complete history. Indeed, it is the very essence of open futurism to deny that.

Alex,

Just for the record, I'm actually inclined to reject relativity theory (of the metaphysically loaded sort, such as Minkowski's version) because it seems to require that the standard geometric properties of concrete objects (e.g., triangularity) are relative to a reference frame, which I take to be impossible (absurd even :)). I account for the same scientific data in a different way.

Your remarks on the incarnation are clever. Sometime I hope to discuss them further, but perhaps in person would be easier...

A final note: I suppose I can understand eternalism if I imagine that there's really just one big time (or state of affairs) that obtains (is 'present' in the real sense of present), which is built up out of things that stand in "temporal" relations of the sort you described. (Though now I want to complain that "change" within this state of affairs isn't genuine.)

Interesting comment, Alex. I sense there are multiple things to explain, and I may not have in mind all the same items you have in mind. But here's a stab:

When [God is presently creating] was true, it was explained by [God wants to create a good world, if no good world has been created], which is a necessary truth. The explanation doesn't entail its explanandum, of course, but I suggest that it's still a satisfying explanation.

You may ask, "But what explains why God created THEN rather than 5 minutes earlier?" I answer, "Because it was impossible to create 5 minutes earlier, since necessarily, there is no such time as 5 minutes before the first action."

You may ask, "But what explains why God created THEN rather than 5 minutes later?" I answer, "Either because God had good reasons for his first action to be one of creation, or because there was a series of prior actions (perhaps divine thoughts), such that the latest member of that series, when it obtains, motivates creation, and the earliest member of that series was explained by God's having good reasons for his first action to be of the relevant sort."

You may ask, "What explains why, among all the abstract times, any given one is present rather than any of the other ones?" I answer, "t is present because five minutes ago [assuming there was a five minues ago], a certain other time was true, and that other time was such that its being true explains why in five minutes, t would be true. A regress results, but it terminates in the first time which was true in part because of a certain necessary truth concerning God's desires."

What else must I explain?

You aren't going to let me escape easily, are you. :) I would define eternal has having no beginning and no end to its existence, and then I would analyze a beginning of x's existence as a transition from a state of affairs (time) in which x doesn't exist to one in which x does. You've offered me reasons (helpful ones) to not go with this analysis, but you see here my motivation for it... :)

I suppose I do have to say that God, though eternal, is 15 billion years old. I remember when I first told me mom about that--she did find that strange....

Alex,

Well, maybe my view is unusual. :)

My thought concerning the prospect of an infinite future has been that as long as there is change, there is time, and furthermore, even if there is no change, there will be time--otherwise, it would be impossible for everything to freeze and then unfreeze again, and it seems to me that that isn't impossible. So, once time starts, no one can do anything to keep it from continuing forever. But even if time could stop, the thought that God would delete or freeze everything to stop time ("permanently", of course) seems to me contrary to God's character.

Perhaps I should add that all this (concerning eternity) is more a tentative hypothesis of mine than a committed belief.

Hey Alex,

It occurred to me that my view (hypothesis) about eternity may not be quite so "unusual" given a tenseless presentism (where the A-properties are analyzed in terms of tenseless properties; see “Presentism and Grounding,” Noûs 41 (2007): 90-109). For then prior to God's creation, God tenselessly exists. And since I suggested the flow of time is ontologically grounded in God's act of creation, prior to that act, there is a sense in which we might call God "timeless" prior to creation. And perhaps for that very reason it would be inappropriate to assign God an age--to say he's as old as the flow of time. I think this puts my view in close proximity with Bill Craig's.

Perhaps you'll have various objections to this sort of view, but at least it isn't wholly unorthodox.

My understanding is that for Crisp a tenseless proposition is one that one can grasp or conceive of without thereby grasping an A-property. And by that account, the proposition that X is true counts as tenseless because one can grasp it without thereby grasping is presently true. My own hypothesis is that when I say X is presently true, what I'm saying is that it is in/entailed by that time which is true simpliciter. So, I analyze 'presently true' in terms of just plain 'is true'.

Those are all good points, Alex.

I guess it's not really clear to me what 'tenseless is' is supposed mean, but for now I suspect it's equivalent to 'tensed is'. You pose a counter-example: "Napolean is (tenseless) defeated at Waterloo." This you suggest is true, but not presently true (Napolean isn't presently defeated at Waterloo). In reply, I suspect that historical narrative expressed in the present tense is like fiction read as if true. There is an implicit "at that time" or "in that world" clause before the whole thing. In the case of Napolean, we'd be saying "This story (proposition) was the case: Napolean is defeated at Waterloo."

I don't know how tenseless statements work in a tenseless language. If someone says "x IS red" in their tenseless language, are they trying to say that x is red at some time? Or are they merely trying to say that x is red (simpliciter) without reference to time? If it is the later, then I would explain that the difference between 'x IS red' and 'x is presently red' is that only the later is a claim about x's being red at a particular time, namely at that time that IS true. Do you see a problem with this account?

OK, I suppose the problem with my account (above) would be if when someone in a tenseless language says 'X IS red' (without any time reference), it is compatible with what he says that X's being red is something that will happen in the future (or did happen in the past). I don't know enough about tensless languages to answer this, but if you say this is so, I will trust you.

I meant "it is compatible with what he says that X's being red is something that will happen in the future and isn't now happening."

Well, we've certainly strayed from the law of excluded middle. :) Thanks for following me along this path.

You've raised many things for me to think about. Rather than offer a further comment (which is tempting), for now I'll close with a verse that might fit better with your perspective:

"Jesus answered, "before Abraham was born, I am!" (John 8:58).

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