Thursday, November 16, 2017

Truth-value open theism

Consider the view that there are truth values about future contingents, but (as Swinburne and van Inwagen think) God doesn’t know future contingents. Call this “truth-value open theism”.

  1. Necessarily, a perfectly rational being believes anything there is overwhelming evidence for.

  2. Given truth-value open theism, God has overwhelming but non-necessitating evidence for some future contingent proposition p.

  3. If God has overwhelming but non-necessitating evidence for some contingent proposition p, there is a possible world where God has overwhelming evidence for p and p is false.

  4. So, if truth-value open theism is true, either (a) there is a possible world where God fails to believe something he has overwhelming evidence for or (b) there is a possible world where God believes something false. (2-3)

  5. So, if truth-value open theism is true, either (a) there is a possible world where God fails to be perfectly rational or (b) there is a possible world where God believes something false. (1,4)

  6. It is an imperfection to possibly fail to be perfectly rational.

  7. It is an imperfection to possibly believe something false.

  8. So, if truth-value open theism is true, God has an imperfection. (6-7)

And God has no imperfections.

To argue for (2), just let p be the proposition that somebody will freely do something wrong over the next month. There is incredibly strong inductive evidence for (2).

A version of the cosmological argument from preservation

Suppose that all immediate causation is simultaneous. The only way to make this fit with the obvious fact that there is diachronic causation is to make diachronic causation be mediate. And there is one standard way of making mediate diachronic causation out of immediate synchronic causation: temporally extended causal relata. Suppose that A lasts from time 0 to time 3, B lasts from time 2 to time 5, and C lasts from time 4 to time 10 (these can be substances or events). Then A can synchronically cause B at time 2 or 3, B can synchronically cause C at time 4 or 5, and one can combine the two immediate synchronic causal relations into a mediate diachronic causal relation between A and C, even though there is no time at which we have both A and C.

The problem with this approach is explaining the persistence of A, B and C over time. If we believe in irreducibly diachronic causation, then we can say that B’s existence at time 2 causes B’s existence at time 3, and so on. But this move is not available to the defender of purely simultaneous causation, except maybe at the cost of an infinite regress: maybe B’s existence from time 2.00 to time 2.75 causes B’s existence from time 2.50 to time 3.00; but now we ask about the causal relationship between B’s existence at time 2.00 and time 2.75.

So if we are to give a causal explanation of B’s persistence from time 2 to time 5, it will have to be in terms of the simultaneous causal efficacy of some other persisting entity. But this leads to a regress that is intuitively vicious.

Thus, we must come at the end to at least one persisting entity E such that E’s persistence from some time t1 to some time t2 has no causal explanation. And if we started our question with asking about the persistence of something that persists over some times today, then these times t1 and t2 are today.

Even if we allow for some facts to be unexplained contingent “brute” facts, the persistence of ordinary objects over time shouldn’t be like that. Moreover, it doesn’t seem right to suppose that the ultimate explanations of the persistence of objects involve objects whose own persistence is brute. For that makes it ultimately be a brute fact that reality as a whole persists, a brute and surprising fact.

So, plausibly, we have to say that although E’s persistence from t1 to t2 has no causal explanation, it has some other kind of explanation. The most plausible candidate for this kind of explanation is that E is imperishable: that it is logically impossible for E to perish.

Hence, if all immediate causation is simultaneous, very likely there is something imperishable. And the imperishable entity or entities then cause things to exist at the time at which they exist, thereby explaining their persistence.

On the theory that God is the imperishable entity, the above explains why for Aquinas preservation and creation are the same.

(It’s a pity that I don’t think all immediate causation is simultaneous.)

Problem: Suppose E immediately makes B persist from time 2 to time 4, by immediately causing it to exist at all the times from 2 to 4. Surely, though, E exists at time 4 because it existed at time 2. And this “because” is hard to explain.

Response: We can say that B exists at time 4 because of its esse (or act of being) at time 2, provided that (a) B’s esse at time 2 is its being caused by E to exist at time 2, and (b) E causes B to exist at time 4 because (non-causally because) E caused B to exist at time 2. But once we say that B exists at time 4 because of its very own esse at time 2, it seems we’ve saved the “because” claim in the problem.

Two moment presentism

The biggest problem for presentism is the problem of diachronic relations, especially causation. If E is earlier than F and E causes F, then at any given time, this instance of causation will have to either be a relation between two non-existent relata or a relation between one existent and one non-existent relatum, and this is problematic. Here’s a variant on presentism that solves that problem.

Suppose time is discrete, but instead of supposing that a single moment is always actual, suppose that always two successive moments are actual. Thus, if the moments are numbered 0, 1, 2, 3, …, first 0 and 1 are actual, then 1 and 2 are actual, then 2 and 3 are actual, and so on. We then say that the present contains both of the successive moments: the present is not a moment. It is never the case that a single moment is actual, except maybe at the beginning or end of the sequence (those are variants whose strengths and weaknesses need evaluation). Strictly speaking, then, we should label times with pairs of moments: time 1–2, time 2–3, etc. (There are now two variants: on one of them, time 2–3 consists of nothing but the two moments, or it also has an “in between”.)

We then introduce two primitive tense operators: “Just was” and “Is about to be”. Thus, if an object is yellow from times 0 through 2 and blue from time 3 onward, then at time 2–3 it just was yellow and is about to be blue. We can say that an object is F at time 2–3, where Fness is something stative rather than processive, provided that it just was F and is about to be F. We might want to say that it is changing from being F1 to being F2 if it just was F1 and is about to be F2 instead (or maybe there is something more to change than that).

We can now get cases of direct diachronic causation between events at neighboring moments, and because both of the moments are present, our “two-moment presentist” will say that when the two moments are both present, causation is a relation between two existent relata, one at the earlier moment and the other at the later. Of course, there will be cases of indirect diachronic causation to talk about, where some event at time 2 causes an event at time 4 by means of an event at time 3, but the two-moment presentist can break this up into two direct instances of diachronic causation, one of which did/does/will take take place at time 2–3 and the other of which did/does/will take place at time 3–4.

I bet this view is in the literature. It’s too neat a solution to the problem not to have been noticed.

A spatial "in between"

In my last post I offered the suggestion that someone who thinks time is discrete has reason to think that there is something in between the moments—a continuous unbroken (but perhaps breakable) interval.

I think a similar thought can be had about discrete space.

Consideration 1: Imagine that space is discrete, arranged on a grid pattern, and I touch left and right index fingers together. It could happen that the rightmost spatial points of my left fingertip is side-by-side with the leftmost spatial points of my right fingertip, but nonetheless my hands aren’t joined into a single solid. One way to represent this setup would be to say that a spatial point in my left fingertip is right next to a spatial point in my right fingertip, but the interval between these spatial points is not within me.

But positing a spatial “in between” isn’t the only solution: distinguishing internal and external geometry is another.

Consideration 2: Zeno’s Stadium argument can be read as noting that if space and time are discrete, then an object moving at one point per unit of time rightward and an equal length object moving at one point per unit of time leftward can pass by each other without ever being side-by-side. Positing an “in between”, such that objects may be “inbetween places when they are in between times, may make this less problematic.

Wednesday, November 15, 2017

A non-reductive eternalist theory of change

It is sometimes said that B-theorists see change as reducible to temporal variation of properties—being non-F at t1 but F at t2 (the “at-at theory of change”)—while A-theorists have a deeper view of change.

But isn’t the A-theorist’s view of change just something like: having been non-F but now being F? But that’s just as reductive as the B-theorist’s at-at theory of change, and it seems just as much to be a matter of temporal variation. Both approaches have this feature: they analyze change in terms of the having and not having of a property. Note, also, that the A-theorist who gives the having-been-but-now-being story about change is committed to the at-at theory being logically sufficient for change from being non-F to being F.

I think there may be something to the intuition that the at-at theory doesn’t wholly capture change. But moving to the A-theory does not by itself solve the problem. In fact, I think the B-theory can do better than the best version of the A-theory.

Let me sketch an Aristotelian story about time. Time is discrete. It has moments. But it is not exhausted by moments. In addition to moments there are intervals between moments. These intervals are in fact undivided, though they might be divisible (Aristotle will think they are). At moments, things are. Between moments, things become. Change is when at one moment t1 something is non-F, at the next moment t2 it is F, and during the interval between t1 and t2 it is changing from non-F to F.

On this story, the at-at theory gives a necessary condition for changing from non-F to F, but perhaps not a sufficient one. For suppose temporally gappy existence is possible, so that an object can cease to exist and come back. Then it is conceivable that an object exist at t1 and at t2, but not during the interval between t1 and t2. Such an object might be brought back into existence at t2 with the property of Fness which it lacked at t1, but it wouldn’t have changed from being non-F to being F.

But there is a serious logical difficulty with the above story: the law of excluded middle. Suppose that a banana turns from non-blue (say, yellow) to blue over the interval I from t1 to t2. What happens during the interval? By excluded middle, the banana is non-blue or blue. But which is it? It cannot be non-blue on a part of the interval I and blue on another part, for that would imply a subdivision of the interval on the Aristotelian view of time. So it must be blue over the whole interval or non-blue over the whole interval. But neither option seems satisfactory. The interval is when it is changing from non-blue to blue; it shouldn’t already be at either endpoint during the interval. Thus, it seems, during I the banana is neither non-blue nor blue, which seems a contradiction.

But the B-theorist has a way of blocking the contradiction. She can take one of the standard B-theoretic solutions to the problem of temporary intrinsics and use that. For instance, she can say that the banana is neither blue-during-I and nor non-blue-during-I. There is no contradiction here, nor any denial of excluded middle.

What the theory denies is temporalized excluded middle:

  1. For any period of time u, either s during u or (not s) during u

but it affirms:

  1. For any period of time u, either s during u or not (s during u).

A typical presentist is unable to say that. For a typical presentist thinks that if u is present, then s during u if and only if s simpliciter, so that (1) follows from (2), at least if u is present (and then, generalizing, even if it’s not). Such a typical presentism, which identifies present truth with truth simpliciter is I think the best version of the A-theory.

Thinking of time as made up of moments and intervals is, I think, quite fruitful.

Tuesday, November 14, 2017

Freedom, responsibility and the open future

Assume the open futurist view on which freedom is incompatible with there being a positive fact about what I choose, and so there are no positive facts about future (non-derivatively) free actions.

Suppose for simplicity that time is discrete. (If it’s not, the argument will be more complicated, but I think not very different.) Suppose that at t2 I freely choose A. Let t1 be the preceding moment of time.


  1. At t2, it is already a fact that I choose A, and so I am no longer free with respect to A.

  2. At t1, I am still free with respect to choosing A, but I am not yet responsible with respect to A.


  1. At no time am I both free and responsible with respect to A.

This seems counterintuitive to me.

Open theism and divine perfection

  1. It is an imperfection to have been close to certain of something that turned out false.

  2. If open theism is true, God was close to certain of propositions that turned out false.

  3. So, if open theism is true, God has an imperfection.

  4. God has no imperfections.

  5. So, open theism is not true.

I think (1) is very intuitive and (4) is central to theism. It is easy to argue for (2). Consider giant sentence of the form:

  1. Alice’s first free choice on Monday is F1, Bob’s first free choice on Tuesday is F2, Carol’s first free choice on Tuesday is F3, …

where the list of names ranges over the names of all people living on Monday, and the Fi are "right", "not right" and "not made" (the last means that the agent will not make any free choices on Tuesday).

Exactly one proposition of the form (6) ends up being true by the end of Monday.

Suppose we’re back on the Sunday before that Monday. Absent the kind of knowledge of the future that the open theist denies to God, God will rationally assign probabilities to propositions of the form (6). These probabilities will all be astronomically low. Even though Alice may be very virtuous and her next choice is very likely to be right, and Bob is vicious and his next choice is very likely to be wrong, etc., given that any proposition of the form (6) has 7.6 billion conjuncts, the probability of that proposition is tiny.

Thus, on Sunday God assigns miniscule probabilities to all the propositions of the form (6), and hence God is close to certain of the negations of all such propositions. But come Tuesday, one of these negated propositions turns out to be false. Therefore, on Tuesday—i.e., today—there a proposition that turned out false that God was close to certain of. And that yields premise (2).

(I mean all my wording to be neutral between the version of open theism where future contingents have a truth value and the one where they do not.)

Moreover, even without considerations of perfections, being close to certain of something that will turn out to be false is surely inimical to any plausible notion of omniscience.

Monday, November 13, 2017

Flying rings

My five-year-old has been really enjoying our Aerobie Pro flying disk, but it has too much range to use at home or in a backyard. The patent has expired, so I designed a 3D-printable version with a similar airfoil profile and customizable diameter and wing-chord. The inner one is 100mm diameter (20mm chord), and can be used indoors. Here are the files.

Open theism and utilitarianism

Here’s an amusing little fact. You can’t be both an open theist and an act utilitarian. For according to the act utilitarian, to fail to maximize utility is wrong. It is impossible for God to do the wrong thing. But given open theism, it does not seem that God can know enough about the future in order to be necessarily able to maximize utility.

Thursday, November 9, 2017

Proportionality in Double Effect is not a simple comparison

It is tempting to make the final “proportionality” condition of the Principle of Double Effect say that the overall consequences of the action are good or neutral, perhaps after screening off any consequences that come through evil (cf. the discussion here).

But “good or neutral” is not a necessary condition for permissibility. Alice is on a bridge above Bob, and sees an active grenade roll towards Bob. If she does nothing, Alice will be shielded by the bridge from the explosion. But instead she leaps off the bridge and covers the grenade with her body, saving Bob’s life at the cost of her own.

If “good or neutral” consequences are required for permissibility, then to evaluate the permissibility of Alice’s action it seems we would need to evaluate whether Alice’s death is a worse thing than Bob’s. Suppose Alice owns three goldfish while Bob owns two goldfish, and in either case the goldfish will be less well cared for by the heirs (and to the same degree). Then Alice’s death is mildly worse than Bob’s death, other things being equal. But it would be absurd to say that Alice acted wrongly in jumping on the grenade because of the impact of this act on her goldfish.

Thus, the proportionality condition in PDE needs to be able to tolerate some differences in the size of the evils, even when these differences disfavor the course of action that is being taken. In other words, although the consequences of jumping on the grenade are slightly worse than those of not doing so, because of the impact on the goldfish, the bad consequences of jumping are not disproportionate to the bad consequences of not jumping.

On the other hand, if it was Bob’s goldfish bowl, rather than Bob, that was near the grenade, the consequences of jumping would be disproportionate to the consequences of not jumping, since Alice’s death is disproportionately bad as compared to the death of Bob’s goldfish.

Objection: The initial case where Alice jumps to save Bob’s life fails to take into account the fact that Alice’s act of self-sacrifice adds great value to the consequences of jumping, because it is a heroic act of self-sacrifice. This added increment of value outweighs the loss to Alice’s extra goldfish, and so I was incorrect to judge that the consequences are mildly negative.

Response: First, it seems to be circular to count the value of the act itself when evaluating the act’s permissibility, since the act itself only has positive value if it is permissible. And anyway one can tweak the case to avoid this difficulty. Suppose that it is known that if Alice does not jump on the grenade, Carl who is standing beside her will. And Carl only owns one goldfish. Then whether Alice jumps or not, the world includes a heroic act. And it is better that Carl jump than that Alice, other things being equal, as Carl only has one goldfish depending on him. But it is absurd that Alice is forbidden from jumping in order that a man with fewer goldfish might do it in her place.

Question: How much of a difference in value can proportionality tolerate?

Response: I don’t know. And I suspect that this is one of those parameters in ethics that needs explaining.

A simple "construction" of non-measurable sets from coin-toss sequences

Here’s a simple “construction” of a non-measurable set out of coin-toss sequences, i.e., of an event that doesn’t have a well-defined probability, going back to Blackwell and Diaconis, but simplified by me not to use ultrafilters. I’m grateful to John Norton for drawing my attention to this.

Let Ω be the set of all countably infinite coin-toss sequences. If a and b are two such sequences, say that a ∼ b if and only if a and b differ only in finitely many places. Clearly ∼ is an equivalence relation (it is reflexive, symmetric and transitive).

For any infinite coin-toss sequence a, let ra be the reversed sequence: the one that is heads wherever a is tails and vice-versa. For any set A of sequences, let rA be the set of the corresponding sequences. Observe that we never have a ∼ ra, and that U is an equivalence class under ∼ (i.e., a maximal set all of whose members are ∼-equivalent) if and only if rU is an equivalence class. Also, if U is an equivalence class, then rU ≠ U.

Let C be the set of all unordered pairs {U, rU} where U is an equivalence class under ∼. (Note that every equivalence class lies in exactly one such unordered pair.) By the Axiom of Choice (for collections of two-membered sets), choose one member of each pair in C. Call the chosen member “selected”. Then let N be the union of all the selected sets.

Here are two cool properties of N:

  1. Every coin-toss sequence is in exactly one of N and rN.

  2. If a and b are coin-toss sequences that differ in only finitely many places, then a is in N if and only if b is in N.

We can now prove that N is not measurable. Suppose N is measurable. Then by symmetry P(rN)=P(N). By (1) and additivity, 1 = P(N)+P(rN), so P(N)=1/2. But by (2), N is a tail set, i.e., an event independent of any finite subset of the tosses. The Kolmogorov Zero-One Law says that every (measurable) tail set has probability 0 or 1. But that contradicts the fact that P(N)=1/2, so N cannot be measurable.

An interesting property of N is that intuitively we would think that P(N)=1/2, given that for every sequence a, exactly one of a and ra is in N. But if we do say that P(N)=1/2, then no finite number of observations of coin tosses provides any Bayesian information on whether the whole infinite sequence is in N, because no finite subsequence has any bearing on whether the whole sequence is in N by (2). Thus, if we were to assign the intuitive probability 1/2 to P(N), then no matter what finite number of observations we made of coin tosses, our posterior probability that the sequence is in N would still have to be 1/2—we would not be getting any Bayesian convergence. This is another way to see that N is non-measurable—if it were measurable, it would violate Bayesian convergence theorems.

And this is another way of highlighting how non-measurability vitiates Bayesian reasoning (see also this).

We can now use Bayesian convergence to sketch a proof that N is saturated non-measurable, i.e., that if A ⊆ N is measurable, then P(A)=0 and if A ⊇ N is measurable, then P(A)=1. For suppose A ⊆ N is measurable. Suppose that we are sequentially observing coin tosses and forming posteriors for A. These posteriors cannot ever exceed 1/2. Here is why. For a coin toss sequence a, let rna be the sequence obtained by keeping the first n tosses fixed and reversing the rest of the tosses. For any any finite sequence o1, ..., on of observations, and any infinite sequence a of coin-tosses compatible with these observations, at most one of a and rna is a member of N (this follows from (1) and the fact that ra ∈ N if and only if rna ∈ N by (2)). By symmetry P(A ∣ o1...on)=P(rnA ∣ rn(o1...on)) (where rnA is the result of applying rn to every member of A). But rn(o1...on) is the same as o1...on, so P(A ∣ o1...on)=P(rnA ∣ o1...on). But A and rnA are disjoint, so P(A ∣ o1...on)+P(rnA ∣ o1...on)≤1 by additivity, and hence P(A ∣ o1...on)≤1/2. Thus, the posteriors for A are always at most 1/2. By Bayesian convergence, however, almost surely the posteriors will converge to 1 or 0, respectively, depending on whether the sequence being observed is actually in A. They cannot converge to 1, so the probability that the sequence is in A must be equal to 0. Thus, P(A)=0. The claim that if A ⊇ N is measurable then P(A)=1 is proved by noting that then N − A ⊇ rN (as rN is the complement of N), and so by the above argument with rN in place of N, we have P(N − A)=0 and thus P(A)=1.

Tuesday, November 7, 2017

Why might God refrain from creating?

Traditional Jewish and Christian theism holds that God didn’t have to create anything at all. But it is puzzling what motive a perfectly good being would have not to create anything. Here’s a cute (I think) answer:

  • If (and only if) God doesn’t create anything, then everything is God. And that’s a very valuable state of affairs.

Adding infinite guilt

Bob has the belief that there are infinitely many people in a parallel universe, and that they wear numbered jerseys: 1, 2, 3, …. He also believes that he has a system in a laboratory that can cause indigestion to any subset of these people that he can describe to a computer. Bob has good evidence for these beliefs and is (mirabile!) sane.

Consider four scenarios:

  1. Bob attempts to cause indigestion to all the odd-numbered people.

  2. Bob attempts to cause indigestion to all the people whose number is divisible by four.

  3. Bob attempts to cause indigestion to all the people whose number is either odd or divisible by four.

  4. Bob yesterday attempted to cause indigestion to all the odd-numbered people and on a later occasion to all the people whose number is divisible by four.

In each scenario, Bob has done something very bad, indeed apparently infinitely bad: he has attempted infinite mass sickening.

In scenarios 1-3, other things being equal, Bob’s guilt is equal, because the number of people he attempted to cause indigestion to is the same—a countable infinity.

But now we have two arguments about how bad Bob’s action in scenario 4 is. On the one hand, in scenario 4 he has attempted to sicken the exact same people as in scenario 3. So, he is equally guilty in scenario 4 as in scenario 3.

On the other hand, in scenario 4, Bob is guilty of two wrong actions, the action of scenario 1 and that of scenario 2. Moreover, as we saw before, each of these actions on its own makes him just as guilty as the action in scenario 3 does. Doing two wrongs, even two infinite wrongs, is worse than just doing one, if they are all of the same magnitude. So in scenario 4, Bob is guiltier than in scenario 3. One becomes the worse off for acquiring more guilt. But if 4 made Bob no guiltier than 3 would have, it would make Bob no guiltier than 1 would have, and so after committing the first wrong in 4, since he would already have the guilt of 1, Bob would have no guilt-avoidance reason to refrain from the second wrong in 4, which is absurd.

How to resolve this? I think as follows: when accounting guilt, we should look at guilty acts of will rather than consequences or attempted consequences. In scenario 4, although the total attempted harm is the same as in each of scenarios 1-3, there are two guilty acts of will, and that makes Bob guiltier in scenario 4.

We could tell the story in 4 so that there is only one act of will. We could suppose that Bob can self-hypnotize so that today he orders his computer to sicken the odd-numbered people and tomorrow those whose number is divisible by four. In that case, there would be only one act of will, which will be less bad. It’s a bit weird to think that Bob might be better off morally for such self-hypnosis, but I think one can bite the bullet on that.

Evidence that I am dead

I just got evidence that I am dead, in an email that starts:

Dear expired [organization] member,
You might think this is pretty weak evidence. Maybe "expired" doesn't mean "dead" here. But the email continues:
Thank you for your past support of [organization]. Your membership has recently expired, and we would like to take this opportunity to urge you to renew your membership.
But last year I acquired a life membership...

Sorry, I couldn't resist sharing this.

From a dualism to a theory of time

This argument is valid:

  1. Some human mental events are fundamental.

  2. No human mental event happens in an instant.

  3. If presentism is true, every fundamental event happens in an instant.

  4. So, presentism is not true.

Premise (1) is widely accepted by dualists. Premise (2) is very, very plausible. That leaves (3). Here is the thought. Given presentism, that a non-instantaneous event is happening is a conjunctive fact with one conjunct about what is happening now and another conjunct about what happened or will happen. Conjunctive facts are grounded in their conjuncts and hence not fundamental, and for the same reason the event would not be fundamental.

But lest four-dimensionalist dualists cheer, we can continue adding to the argument:

  1. If temporal-parts four-dimensionalism is true, every fundamental event happens in an instant.

  2. So, temporal-parts four-dimensionalism is not true.

For on temporal-parts four-dimensionalism, any temporally extended event will be grounded in its proper temporal parts.

The growing block dualist may be feeling pretty smug. But suppose that we currently have a temporally extended event E that started at t−2 and ends at the present moment t0. At an intermediate time t−1, only a proper part of E existed. A part is either partly grounded in the whole or the whole in the parts. Since the whole doesn’t exist at t−1, the part cannot be grounded in it. So the whole must be partly grounded in the part. But an event that is partly grounded in its part is not fundamental. Hence:

  1. If growing block is true, every fundamental event happens in an instant.

  2. So, growing block is not true.

There is one theory of time left. It is what one might call Aristotelian four-dimensionalism. Aristotelians think that wholes are prior to their parts. An Aristotelian four-dimensionalist thinks that temporal wholes are prior to their temporal parts, so that there are temporally extended fundamental events. We can then complete the argument:

  1. Either presentism, temporal-parts four-dimensionalism, growing block or Aristotelian four-dimensionalism is true.

  2. So, Aristotelian four-dimensionalism is true.