A method of proof

I'm still working on trying to prove the results here with while satisfying as many axioms of conditionals as possible, and ideally with countably additive probabilities. It's not all that easy. Anyway, while thinking about it, I came up with a method of proof that while probably not new, I have never used before.

Suppose we want to close some set A₀ under some finite set of operations with finite arity. One way is to inductively form a sequence of larger and larger sets, A₀⊆A₁⊆A₂⊆..., such that A_n+1 is something like the result of applying the operations to all the tuples of elements of A_n. Then take the union A of the A_n, and that will be our closure, because any finite collection of elements of A will all be in some A_n for a large enough n, and hence the result of applying the operations to that collection will be in A_n+1, and hence in A. So A will be closed under the operations.

But what if our operations have countably infinite arity? An example is probability spaces which we want to be closed under countable unions. In that case, the above method of proof may not be good enough. But there is a cool modification given the Axiom of Choice. By Choice, let β be an ordinal with uncountable cofinality. Use transfinite recursion over α < β to generate a sequence of larger and larger sets A_α, such that A_α is the result of applying the operations to the countable sequences of elements of A_α, and letting A_α, for α a limit ordinal, be the result of applying the operations to the countable sequences of elements of the union of the sets for α'<α. Let A be the union of the A_α. Then a countable sequence of elements of A will all be contained in some one of the A_α because of uncountable cofinality. And then the result of applying the operations to that sequence will be in A_α+1, and hence in A. So A will be closed under the operations.

Deep Thoughts XXXIV

H₂O is nothing but water.

What is unconditional love?

This following is an excerpt from chapter 2 of my One Body book.

One sense of “unconditional” is negative: there are no conditions on account of which one is loving the beloved. This negative sense, however, denies the truism that if someone loves you unconditionally, you can count on his or her love. A love that comes about for no reason at all might, as already noted, equally well disappear for no reason.

Let us, then, hold on to the truism. An unconditional love is one you can count on, no matter what. On the face of it, this makes unconditional love something humanly unattainable. For in our earthly lives, brainwashing and sin are always possible: the continuation of love is never completely certain. There is no present state of earthly love that guarantees a future continuation. It is plainly a myth, though a not uncommon one, that the way two people love each other at the beginning of their relationship determines the future course of the relationship.

The unattainability objection to the possibility of unconditional love understands an unconditional love as one that is certain to last. This would mean that if I said that I love my children unconditionally, I would be presumptuously asserting that my future love will last forever. But we need to distinguish two senses of the claim that one “can count on” the loving continuing. In one sense, something can be counted on provided that you have epistemic certainty of its truth. But there is another sense: we can read “can count on” as “have the right to count on,” in the way that you have the right to count on people to keep their promises to you. But you only have the right, in the relevant sense, to count on my doing something if I owe it to you to do it. Having the right to count on someone to do something is correlated with an obligation on the part of that person. Thus, unconditional love is a present love that the lover is obligated to persevere in no matter what (even if the beloved should no longer desire that perseverance—this is important in the case of children, who have the right to count on their parents loving them even at times when the children might say that they don’t care about the parents’ love).

The obligation to persevere, however, is not enough to make a love unconditional. All parents have the obligation to love their children no matter what, but not all love their children unconditionally. Thus, to say that a love is unconditional if and only if that the obligation to continue loving is certain to be kept would be to make unconditional love unattainable in our earthly lives. But to say that a love is unconditional simply providing that there is an obligation to continue loving, whether or not the lover accepts the obligation, would also not be enough. We need something in between. The notion of commitment gives us what we need. An unconditional commitment to a moral obligation is an unreserved acceptance of the obligation. Such an acceptance does not make certain the fulfillment of the obligation—we do sometimes wrongfully go back on our commitments, after all—but it does set one on the path to fulfillment, and gives others reason to think we will fulfill the commitment. It is worth noting here that probably only an obligation can be accepted unconditionally, unreservedly, because we are unable to predict the future with great certainty, and anything other than a moral obligation may be something that one day we might have a reason to go against.

Unconditional love, thus, includes an obligation and an unreserved acceptance of the obligation.

Can A-theorists believe in time travel?

I used to think that A-theorists cannot consistently believe in time travel. I think I was mistaken. As best as I can reconstruct my line of thought it was this. Time travel requires a distinction between external and internal time. If I go into a time machine, then maybe in five minutes I'll be a thousand years ago. That's a contradiction given non-circular time unless one distinguishes as follows: internally in five minutes I'll be a thousand years ago externally. But now I think that what I must have been thinking was that in five internal minutes my internal present will no longer line up with the world's objective present, since in five internal minutes my internal present will be about thousand years behind the world's external present. I don't know for sure if that's the thought I had, but if it was, it would have been a howler. For on the view, internally in five minutes, I will be at the time at which the world's external present was about a thousand years ago. Or, to put it from the external point of view, a thousand years ago I was five minutes older than I am now (age is measured internally). Even presentists can say that.

To see that this is coherent, consider a theory that takes external time to governed by the A-theory but internal time to be entirely governed by the B-theory. Thus, superimposed on the external A-series of past, present and future, there is an indexical B-series of earlier-for-me and later-for-me, where these relations are perhaps defined by internal causal relations (earlier states causing later ones). There is no more need for these two series to line up than there would be a need for the two series to line up if the external series were a B-series.

However, while this is coherent, maybe it undercuts one of the main motivations for the A-theory. For if there is a distinction between internal and external time, as there must be for time travel to be possible, all the changes we actually experience are changes with respect to internal time. In other words, they are B-type changes. But the typical A-theorist thinks B-type changes--it (internally) earlier being one way, and (internally) later another--are not what we experience when we experience "real change". Indeed, if time travel is possible, it is possible to live all of one's life at one external time, but moving through external space. Basically, just imagine that at each moment you travel to some external time t₀, but to a different spatial location in it. Maybe you have a backpack time-machine which is permanently stuck on t₀, but with the spatial locations changing. You'd experience change, because your state at internally earlier times will be different from your state at internally later times. But it would be mere B-type change, since it would all be happening at one and the same objective time.

I suppose one could say that in time-travel scenarios, especially the preceding one of living all of one's life at one external time, our experiences of change become non-veridical, for a condition on the veridicality of our experiences of change is that our internal clock lines up correctly with external time, and time-travel causes a misalignment. Maybe.

But in any case, now that we have the possibility of living all of one's life--a life that presumably could have rich causal interconnections--at one objective external time, just moving "sideways" to new spatial locations, I do think that the motivations for the A-theory decrease. For we see that what matters for the diachronic richness of our lives is that our lives be stretched over internal time, not over external time. It also matters that other people's internal times be sufficiently lined up with ours. But that doesn't call for the A-theory, either.

So, all in all, while A-theorists can believe in time travel, thinking time travel through would undercut much of the motivation for the A-theory.

Induction, growing block theory and open future

According to induction, the future is like the past and present. But the past and present contain real events. Hence, probably, so does the future. Hence, probably, Growing Block theory, on which only the past and present contain real events, is false. Likewise, excluded middle is true for claims about the past and for claims about the present. So, probably, it's true for claims about the future. So, probably, Open Future views are false.

Solipsism, presentism, actualism

Consider three debates: solipsism vs. other minds; presentism vs. eternalism; actualism vs. extreme modal realism. Let say that, like most people, we want to go for other minds and actualism. The sane view is that of course other people exist but unicorns don't. Can we get any guidance from this decision as to the presentism vs. eternalism debate? Is "now" more like "I", in which case we get the hint that we should be eternalists, or is "now" more like "actual", in which case the hint is that we should be presentists?

Here is one important way in which "now" is more like "I". I communicate with people who are other than I. I do not communicate with people who are other than actual. But I do communicate with people who are other than now: I read Plato and maybe even aspire to writing for people yet to be conceived. And even when we communicate with people who are now alive, typically—unless we're speaking at each other at the same time—we do so diachronically. I speak now and they will respond later. I respond now and they spoke earlier. So our conversation reaches across times, just as it reaches across people. But it does not reach across worlds.

Suppose we think of sentences like "I am sitting" as expressing self-locating propositions (I actually probably don't want to think of them like that). Here, then, is a closely related point, inspired (as really is the above stuff) by this article. I tell you on the phone: "I am sitting." In so doing, I express a de se self-locating proposition, a proposition that locates me. But while I express a self-locating proposition, that isn't what I communicate or even try to communicate. For if you accept the self-locating proposition that I am expressing, you will thereby take yourself to be sitting, and that's not what I am trying to communicate. So there is a difference between what I am expressing and what I am communicating: I am expressing a self-locating proposition that I am sitting, but I am communicating the non-self-locating proposition that Alex is sitting. Moreover, the two are closely related. Quite plausibly, in accepting the self-locating proposition that I am sitting, I am also accepting a non-self-locating version of it. It could even be that I express both.

But a similar thing happens with time. I hereby write: Alex is now sitting. In so doing, I express a tensed proposition (apologies for the use of "tensed" for non-linguistic entities), a temporally-locating proposition. But I do not communicate the same proposition to you. For while you might infer that I may still be sitting when you read the message, that's a risky inference of yours, not just what I communicated to you. If you just want to believe what I have informed you of, you will believe something that you may express with words like "Alex was sitting then." So there is a difference between what I expressed and what I communicated.

You might think that the difference is not a difference in kind. After all "Alex was sitting then" itself expresses a temporary proposition because of the past tense "was". But that, I think, is just an artifact of the fact that when you went from my "Alex is now sitting" to your "Alex was sitting then", you didn't just accept something that I communicated. Rather, you took what I communicated and combined it with the fact that my communication temporally precedes your reception of it, a fact you know empirically (but it would not affect my argument if you knew it a priori—it's still a fact over and beyond my communication). Sticking to what I communicate to you, you cannot think more than some proposition like that Alex is sitting at that time (where "that" refers to the time of my utterance).

There are now two options. We could go the presentist route and say that both what I expressed and what I communicated are tensed propositions. On this reading, what I expressed was that Alex is now sitting, but what I communicated was that Alex was, is or will be sitting then. But this doesn't seem to me to be a very attractive theory. For when things go right in communication, I shouldn't be communicating a proposition I didn't express, while to claim that I expressed two tensed propositions, though only one was communicated, seems odd. It makes it sound as if you only half believed what I said.

The superior reading, I think, is that I expressed both a tensed and an untensed proposition, and what I communicated was the untensed one.

These things combine. When I say: "I am sitting now", I express three things: a self-locating tensed proposition that I am sitting now, a non-self-locating tensed proposition that Alex (or that guy/gal) is sitting, and a non-self-locating untensed proposition that Alex is (tenselessly) sitting then. But only the last of these do I communicate when I communicate across a relevantly large time delay. But that's all predicated on the view on which there are de se propositions.

A many worlds interpretation of the Copenhagen interpretation

The Everett multiverse interpretation of Quantum Mechanics has two parts. First, there is the dynamics: the wavefunction never collapses. Second, there is an interpretation of the dynamics: superpositions (with respect to the privileged basis) can be thought of as multiple worlds, so that the world constantly splits and splits, and we become more and more multilocated. The dynamics solves the problem of the inelegance in the theory induced by collapse, and the oddness of thinking that observers are special vis-a-vis fundamental physics. The interpretation helps solve the conceptual problem of what superposed states mean.

But what if we stick to the dynamics of the Copenhagen interpretation with measurement-induced collapse, but combine it with a multiple worlds interpretation like in Everett? We admittedly do lose the advantages of a no-collapse theory.

However we can still borrow the Everett solution to the conceptual problem of what superposed states mean. Here's how this might work. Let's say that Sally prepares an electron in a mixed positional state, so it's in a superposition of being in box A and of being in box B, with coefficients such that she has a 1/4 probability of finding the electron in A and a 3/4 probability of finding it in B in subsequent experiments. If the preferred basis is positional, then when she prepares that electron, her world branches into two. In one world, the electron is definitely in box A. In another world, the electron is definitely in box B. But unlike in Everett, when she makes the measurement of the positions, one of these worlds ends, in accordance with the probabilities in the wavefunction, and the wavefunction then collapses. So, for a while, Sally was located in two worlds, but then one of the worlds was terminated. However, there never is a superposition of different states of consciousness or of their physical correlates. So all the worlds we're in look the same.

I think this helps with the problem of making biological and geological claims about what things were like before observers true (just indexed to our world). But there are some serious mathematical issues there, so I can't insist on this part.

The Incarnation, personal identity and time

On the traditional understanding of Christ's Incarnation, Christ has two minds—a human mind and a divine mind—even though he is one person. The two minds have different mental states. In his divine mind, which he has in common with the Father and the Holy Spirit, Christ is omniscient. In his human mind, he is not. There are things that he knows with his divine mind which he does not even believe with his human being, because there are thoughts conceptually beyond the ability of the human mind to think. Yet how could one have two incompatible collections of mental contents like that, and yet be one person? Likewise, Christ divinely wills certain things, say that all reality continue to exist, which he presumably does not will humanly.

But suppose that the past is real—i.e., that either eternalism or growing block is true. Then I, too, am a person with incompatible collections of mental contents. At age 2, I did not even have the concepts needed to grasp the Pythagorean Theorem. Now I know the Theorem to be true. Yet it is the very same person we are talking about here. So the very same person has two incompatible collections of mental states.

But there seems to be a difference. I don't know the Pythagorean theorem at age 2, but I know the Pythagorean theorem at age 40. There is no problem here. But Christ at the same time knows and doesn't know some propositions.

Actually, though, it's not clear that it makes sense to say that Christ at the same time knows and doesn't know some propositions. In his divine nature, Christ is timeless. So perhaps we should say that Christ at age 30 doesn't know p, but Christ timelessly (or "at eternity") knows p.

But that's not exactly my point. The point I want to make is a little subtler and would hold even if God was in time rather than being timeless: the adverbial modifiers "at age 2" and "at age 40" work rather like the modifiers "as human" and "as God". Just as there is no contradiction between my knowing p at age 40 and not knowing p at age 2 (or vice versa), there is no contradiction between Christ's knowing p as God and not knowing it as human.

There is a sense in which the succession of time multiplies our wills and minds: we pursue and believe different things at different times. While I don't want to say that we have literally different wills and minds at different times, the diachronic distribution of our pursuits and beliefs shows that personal identity does not require any strong unity of apperception. At most, according to some theorists, there need to be some interconnections, like those of memory. And other theorists—the ones who are right!—don't even think connections of memory are needed for personal identity.

The analogy between times (in our case) and natures (in the case of Christ) is of course only an analogy. But I think it is potentially a fruitful and underexplored one. Notice that the analogy extends beyond the mental life. Just as Christ is both omnipotent (as God) and weak (as a man), I am both relatively strong (now) and helpless (in infancy).

How well the analogy runs will depend on the theory of persistence over time that one is thinking about. I am inclined to either endurantism or stageless worm theory, so those are the theories on which the above intuitions are based. But one will get a different picture—perhaps no longer orthodox—if one bases the analogy on perdurantism.

Kenotic Christianity and the problem of Christ's resurrected body

According to kenotic Christianity, when Christ became incarnate, he literally ceased to have divine omni-properties. But at the same time, after his resurrection, he was glorified and presumably has regained them.

But what happened to his humanity after his glorification? Either Christ is still human or he is simply divine. If Christ is still human, then it follows that having the divine omni-properties is logically compatible with being human, which undercuts I assume one of the major motivations for kenotic Christianity. If Christ is no longer human, what happened to his resurrected body? Was it resurrected only to be destroyed shortly thereafter? That seems deeply unfitting.

Compatibilism on the cheap

Could creatures be free in a world with deterministic laws? Yes, because God could work a miracle each time that it is time for a creature to make a decision, a miracle that exempts the choice from the deterministic laws. So, cheaply, free will and deterministic laws are compatible.

This is obviously too cheap, so probably we had better not define compatibilism as the possibility of creaturely freedom in a world with deterministic laws.

Though, maybe, we could stick to that definition of compatibilism (at least in respect of this problem—there are other problems with it), but insist that if God exists, no law can count as deterministic, because God can override every law.

Presentism and simplicity of laws

Consider the following toy example of a law of nature: All electrons are charged. On an eternalist theory of time, the following expression correctly captures the logical structure of this law.

1. ∀x(Ex→Cx)

On the other hand, on presentism, (1) would only say that all the present electrons are charged. To correctly capture the logical structure of the law that all electrons are charged on presentism, we would need something like:

2. ∀x(Ex→Cx) & ∀t[P(t)→Was_t(∀x(Ex→Cx))] & ∀t[F(t)→Will_t(∀x(Ex→Cx))],

where P(t) and F(t) say that t is past or future, respectively, and the Was and Will operators say what was or will be the case at a particular time. There may be ways of slightly simplifying (2), but whatever we do we need to say that all electrons are charged, all electrons were charged and all electrons will be charged. Presumably, this will also come along with a story on which when a physicist says "All electrons are charged", she is using a locution that should be analyzed as something like a triple conjunction, though she may not realize this.

This should be at least a little embarrassing to the presentist. That all electrons are charged seems to be is a very simple law. But (2) is far from simple. Moreover, the analyses may sometimes be even more complex. Consider a putative fairly simple law that makes reference to two different times, say that

3. exposure to V tends to cause a disease D.

The presentist needs to say something like this:

4. exposure to V will tend to cause D, and past cases of exposure to V tended to have caused, be causing or be about to be causing D, and future cases of exposure to V will tend to cause D.

But the concern is not merely esthetic (though I do think beauty is a guide to truth). Suppose that our evidence is equally well explained by two general claims, one of which is both (a) significantly simpler and (b) significantly logically weaker. Then we should not have much confidence in the explanation that is both more complex and stronger. For instance, suppose that all observed ravens are black. This is equally well explained by two general claims: That all ravens are black and that all ravens and geese are black. The claim that all ravens are black is both significantly simpler and significally weaker. We should not go for the more complex explanation that all ravens and geese are black. (If we have no other evidence, though, we might cautiously accept the explanation that all birds are black, because while it's logically stronger, it's no more complex.)

Very well. Now, what is my evidence about electrons? Let's oversimplify the evidence by saying that the evidence is that all the electrons we've observed carefully enough have been charged. This observation can be equally well explained either by (2) or by:

5. ∀t[P(t)→Was_t(∀x(Ex→Cx))].

But notice that (5) is significantly simpler than (2) and also significantly logically weaker. So we should no more accept (2) than we should accept the claim that all ravens and geese are black on the basis of observation of ravens alone.

In other words, Presentism combined with a plausible thesis about which explanation one should not accept leads to inductive scepticism.

Another explanatory question for A-theorists

Suppose that it is now the first moment of time. This is surely possible. While there might not in fact be a first moment of time, surely there could be. So, it's the first moment of time, and according to the A-theory this is an objective fact about the world. Question:

1. Why is there no past?

The question is pressing. After all, at most times, indeed at all but at most one time, there is a past.

Maybe one could say: "Because God is presently beginning to create." But we can then ask:

2. Why is God only presently beginning to create?

It is natural to answer: "What do you expect? How could he create any earlier, given that there is no past?" But the natural answer "Because there is no past" to (2) won't work since we've claimed that there is no past because God is presently beginning to create.

Or perhaps we can try to answer (1) with: "Because God wasn't creating any earlier." But this doesn't seem informative. If there is no past, then by golly God wasn't creating any earlier. But when we ask why there is no past, we're basically asking why nothing at all was happening earlier.

While theistic answers seem the only hope—perhaps an unreal hope—for answering (1), A-theoretic theists have a pressing need to answer (1). For if theism is true, then all contingent truths had better have at least a partial explanation in terms of God's will. (Maybe in the case of free creaturely action, the divine will explanation is only partial—God created the being in such and such a state—and a fuller explanation needs the creature's choice.) But how could one explain (1) in terms of the divine will?

Still, maybe there is a theistic answer possible. Maybe at the first time, t₀, God wills that there be a future but no past. He could, instead, have willed there to be a past then. That would have involved backwards causation, but there is no absurdity in backwards causation for God.

While this solution seems not unattractive to me, I think most A-theorists are suspicious of backwards causation.

But without backwards causation, I cannot see how (1) could be explained by the divine will—or in any other way.

Panpsychism and quantum mechanics

On the Copenhagen interpretation, observation collapses quantum states. If panpsychism is true, there are constant observations even at the microscopic level. But it seems inconsistent with our empirical data to suppose constant collapse at the microscopic level. So if we are right to accept the Copenhagen interpretation, we should reject panpsychism.

I suppose one could get out of this by saying that only sapient observation collapses things. But that would be weird indeed. It would mean that the conscious states of infants and dogs are utterly different from ours, because while we observe only collapsed states, they would have the dubious privilege of having superposed observational states.

Instead of the Copenhagen interpretation, we might be able to run this argument against panpsychism from the plausible postulate that it is impossible for there to be superposed states of consciousness conjoined with physicalism. For by physicalism, conscious states will be physical states. Plausibly, if panpsychism is true, the conscious states of fundamental particles will vary depending on the state of the particle, or at least the identity of the particle. But then superpositions of microstates will result in superpositions of conscious states, if conscious states are phsycial states. But superpositions of conscious states are impossible, we have supposed.

Who cares? After all, panpsychism is crazy!

Well, it may be crazy, but it could also be that physicalism leads to panpsychism. And if so, then an argument against panpsychism would provide an argument against physicalism.

McTaggart on the unreality of time

One version of McTaggart's argument for the unreality of time is:

1. If time is real, there is change.
2. If there is change, there is fundamental change, and it is the change of events or instants from future, to present, to past.
3. There is no fundamental change of events or instants from future, to present, to past.
4. So there is no change.
5. So time is not real.

(Here, "past", "present" and "future" are understood as incompatible. Thus, "past" and "future" mean wholly past and wholly future.) I am dubious of (2). But I think (3) can be defended in the following way, inspired by C. D. Broad (I think).

We need three principles:

6. If something undergoes non-Cambridge change from being F to being non-F, then at some time it exists and is F and at some time it exists and is non-F.
7. Cambridge change is never fundamental.
8. Everything that exists is present, and neither past nor future, when it exists.

Thus:

9. Nothing exhibits a non-Cambridge change from being future to being present, or from being present to being past, or from being future to being past. (6 and 8)
10. Hence nothing fundamentally changes from future, to present, to past. (7 and 9)

I think the contemporary A-theorist should deny (2). And most do.

We almost surely don't live in a multiverse where every possibility is realized exactly once

Consider a class C of possible universes where, at t₀, exactly 100 independent random processes are activated, and nothing else random happens. Suppose there are no qualitative differences between universes in C other than those due to the differences in the outcomes of the processes. Suppose, further, that each process can result in either a "heads" or a "tails", with equal probability 1/2.

There are, thus, exactly 2¹⁰⁰ different types of possible universes in C (where we say that universes are of different type provided that they're not exactly alike).

Suppose we live in a multiverse that contains exactly n universes from C. If n<2¹⁰⁰, then there are some possible types of universes not represented in our multiverse—there are some combinations of heads and tails in a C-type universe that don't occur in our multiverse. If n>2¹⁰⁰, then there are some types of C-universes that are multiply represented in our multiverse.

So if every type of universe is represented exactly once, there are exactly 2¹⁰⁰ C-universes in the multiverse, and each has a different heads-tails profile. But how likely is it that each would have a different heads-tails profile? Assuming that what happens in different universes is stochastically independent, this is just a version of the birthday problem. If n=2¹⁰⁰, then the probability that each of the n universes has a different one of the n profiles is n!/nⁿ, which according to Stirling's formula is something like exp(−2¹⁰⁰). That's a very, very tiny number. And since classes like C can be found for any number of random processes, not just 100, it follows that the probability that every type of universe is exemplified exactly once in the multiverse is smaller than every positive real number, i.e., it's zero or infinitesimal.

So if we live in a multiverse, almost surely either some possible types of universes aren't instantiated or some are multiply instantiated or both.

Becoming

A-theorists talk of something called "becoming" which they say B-theorists have no room for. I don't really understand what this is. I am inclined to say that

1. x becomes F if and only if there is a time at which x is non-F and a later time at which x is F.

If so, then there is becoming—-indeed, objective becoming—-on the B-theory. Now, I think most A-theorists will agree that (1) is a necessary truth. So where's the disagreement on becoming?

Maybe, the disagreement lies in this. Although (1) is a necessary truth, nonetheless becoming is something more than just being non-F and later being F. This "something more" is necessarily there whenever something is earlier non-F and later F, but it is nonetheless something extra. (Just as God's knowing that the sky is blue is something more than just the sky being blue, but that "something more" is always there whenever the sky is blue, since God knows that the sky is blue necessarily if and only if the sky is blue.) But I really don't know what this "something more" is. I feel here like van Inwagen on substitutional quantification.

Well, that's not quite right. For it may be that (1) isn't exactly right. Suppose x has gappy existence. It exists from morning until noon and then from dinner until midnight. From morning until noon x is non-F and from dinner until midnight x is F. Does x become F? I could imagine someone saying "No" (and indeed a colleague did say just that). If not, then (1) may not be right.

Maybe one thinks gappy existence is impossible. (But why not? It seems no harder than being a spatially scattered individual, and if we're made of precisely located particles, which sure seems comprehensible, we're that.) But then consider this case. At every time that (in some unit system) is an irrational number, x is non-F and at every time that is a rational number, x is F. Then x satisfies (1). But does x ever become F? If so, when? At any time at which it is F, it also was F at an infinite number of earlier but arbitrarily close times. (This argument requires time to be a continuum, which I fear it's not.)

So one might question (1), though I am happy to bite the bullet in both cases. Maybe one should insist that to become F, you need to be non-F over an interval of times and be F over a succeeding interval of times, or something like that. But such modifications are neutral between the A- and B-theories.

Perhaps, though, the A-theorist's insistence on becoming is not so much about objects becoming a certain way, as about truths changing. The B-theorist is committed to propositions not changing in truth value. Many (but not all) A-theorists think propositions change in truth value. Thus,

2. There is becoming if and only if there is a proposition p that is true at one time and not true at another.

The B-theorist will then reject that there is becoming in this way. But notice that no it no longer seems appropriate to speak, as some A-theorists do, of doing justice to our "experience of becoming." For while we may experience the sky turning from blue to purple, we do not experience, except in a very theory-dependent way, the proposition <The sky is blue> turning from true to false. That's a change (though presumably a Cambridge one) in a Platonic entity.

Presentism and vagueness

I've been playing with thoughts of the following sort. It can be vague whether something—say, a pain, an itch or a blueberry bush—is still in existence. If presentism is true, then cases where it's vague whether something is still in existence are cases where it's vague whether something exists. Thus it's somewhat harder for the presentist to avoid vagueness about existence than for the non-presentist. If one thinks that there is no vagueness about existence, this will be a problem for presentism.

An argument for a version of the Axiom of Choice

This is an argument for the Axiom of Choice where the sets we're choosing from are all subsets of the real numbers. The argument needs the notion of really independent random processes. Real independence is not just probabilistic independence (if you're not convinced, read this). I don't know how to characterize real independence, but here is a necessary condition for it. If S is a collection of really independent processes producing outcomes, and for each s in S, U_s is a non-empty subset of the range of s (where the range of a process here is all the outcomes that it can generate) then it is metaphysically possible that each member s of S produces an outcomes in U_s. (This need not hold for merely probabilistically independent processes.)

Now, let U be a set of disjoint non-empty subsets of the real numbers R. Let N be the cardinality of U. It is surely metaphysically possible to have N really independent random processes each of which has range R. For instance, one might have a multiverse with N universes, in each of which there is a random process that produces a particle at a normally distributed point from the emitter, and the outcome of the process can be taken to be the x-coordinate of where the particle is produced.

Now, there is a one-to-one correspondence between the members of U and the random processes. If r is one of the random processes, let U_r be the member of U that corresponds to it (after fixing one such correspondence). By real independence, it is metaphysically possible that for all r, the outcome of r is in U_r. Take a world w where this is the case. In that world, the set of outcomes of our processes will contain exactly one member from each member of U, and hence will be a choice set. But what sets of real numbers there are surely does not differ between worlds (I can imagine questioning this, though). So if in w there is a choice set, there actually is a choice set.

Granted, this only gives us the Axiom of Choice for subsets of the reals. But that's enough to generate the Banach-Tarski, Hausdorff and Vitali non-measurable sets. It's paradoxical enough.

An argument against the possibility of instantaneous causation

Instantaneous causation is causation where the cause and effect both occur at the same instant. It's a species of simultaneous causation, with the added condition that the events are instantaneous.

Suppose that instantaneous causation is possible. Then the following are compossible characters: the Judge, who instantaneously stamps death warrants, and the Executioner, who executes the person listed on the warrant in such a way that the very instant that the death warrant is stamped, the person is dead. Moreover, the Executioner takes no orders from dead people: she only executes people if the Judge was alive at the instant the warrant was stamped, and she executes no one else.

There is no metaphysical absurdity if the Judge stamps a warrant for your death—there is "merely" an injustice. But what if the Judge stamps a warrant for his own death?

Then, instantaneously, the Executioner executes the Judge. But then the Judge wasn't alive to stamp the warrant (and if he stamped it posthumously, then that doesn't count). But with no warrant stamped, the Executioner didn't do anything. And so the Judge both is and is not executed, which is absurd.

Now, we might conclude from this just that it's not possible for the Judge to stamp a warrant for his own death. And we could tell stories similar to banana-peel stories from the Grandfather Paradox: if the Judge were to go to stamp his own death warrant, he'd slip on a banana peel, or the stamp would be out of ink, or some other such thing would happen. But many philosophers are unsatisfied with such stories. It sure seems like it's no harder in principle (though it may be psychologically harder, though only if he knows it's his) for the Judge to stamp his own warrant than anyone else's.

It seems that a particularly good way to explain the impossibility of the setup, without any banana peels, is that instantaneous causation is not possible. In any case, people who think the Grandfather Paradox establishes the impossibility of time travel should think that this argument establishes the impossibility of instantaneous causation.

But what about the intuition one might have that instantaneous causation is possible? Here is a suggestion. Let the date of an event E be the temporal duration between the beginning of the universe and the event. (If the universe has no beginning, choose some other base for dates, with dates before it being negative.) Then our intuition that instantaneous causation is possible can have some justice done to it by saying that it is possible to have causation where the cause and effect have the same date, even though they are at different instants. These instants, then, have no duration between them. Thus, we could have the Judge and Executioner story work like this. There is a duration T (say, in seconds) from the beginning of the universe at which there is an instant, a, at which the Judge stamps his death warrant. And with no temporal gap, no duration in between, there is another instant b, at which the Judge is dead, also duration T after the beginning of the universe. (And between a and b there will be other instants, such as the instant when the Executioner sees the stamping and the instant when she initiates the causal process that kills the Judge. Quite possibly, in this story time is not dense.)

This does some justice to our intuition that there can be instantaneous causation. It's not quite instantaneous causation, but it's causation with no temporal extension, no temporal gap.

Acknowledgment: I got the warrant-stamping from Jon Kvanvig. It works better rhetorically than the instantaneous writing of the warrant that I initially had in mind.

A quick way to question conjunctive accounts

Suppose someone proposes a philosophical account of the form:

• x is F if and only if x is G₁ and x is G₂ and x is G₃.

There is a quick way to question this that I think works most of the time. Just query the proponent: "What if the three conditions on the right hand side are satisfied merely coincidentally?"

The proponent can only give one of two answers while maintaining the biconditional: "Yes, x is still F when the conditions are satisfied merely coincidentally" or "The conditions are of such a nature that they cannot be satisfied merely coincidentally."

But it is implausible that that a coincidental satisfaction of conditions should suffice for a natural concept. Thus, if coincidental satisfaction of the conditions is sufficient for x to be F, pretty likely Fness is a stipulative rather than natural concept.

On the other hand, if the proponent insists that the conditions were so crafted that they cannot be satisfied coincidentally, it is likely that one of two possibilities is the case. The first is that the proponent lacks philosophical imagination, and you just need to think a little bit about how to make the conditions be satisfied coincidentally, and then you'll have a counterexample on hand. Just reflect a bit on Gettier-type cases, and if you're clever you should be able to find something. The second possibility is that the conditions are weaselly by including something like "relevantly" or "non-aberrantly". Here is an example of weaselly conditions:

• x knows p if and only if p is true and x believes p and x is justified in believing p and the anti-Gettier condition is met for x with respect to p.

These conditions cannot be satisfied coincidentally because the anti-Gettier condition is telling us that the first three conditions are satisfied non-coincidentally. But of course this is weaselly, since we aren't told at all about the kind of non-coincidentally that's required. Every coincidence is a non-coincidence from some point of view. So, really, such weaselly conditions need to tell us not just that the conditions are satisfied non-coincidentally, but that they are satisfied relevantly non-coincidentally.

Moreover, in the above example there is a pretty good chance that the weaselly final condition entails the other three. For what it says is basically that the other three conditions are satisfied in an un-Gettiered way! I think this isn't uncommon with weaselly conditions.

Note: In the example, one could try to formulate the weaselly condition as the denial of Gettiering: "and x is not Gettiered with respect to p". Then the weaselly condition wouldn't entail the other three. But then the resulting conditions would be too strict. For suppose that x has two sources of data on p. One source gives knowledge. The other gives Gettiered knowledge. Then x knows p but x is Gettiered with respect to p.

Philosophical accounts whose right hand sides are of the form

• x is F if and only if ∃y(G₁(x,y) and G₂(x,y) and G₃(x,y))

can use the quantification to avoid coincidentality sometimes, but often are subject to a similar criticism.

Responsibility and reasons

It is not unusual to say that responsibility for an action requires that the action be done for a reason. Compatibilists particularly insist on this. Now, I think there are no actions without reasons, but I don't know that responsibility has that much to do with this.
Consider the psychopath who acts at the expense of others. To evaluate her responsibility, we do not look at the reasons she had for her action as much as at the reasons she had against it. If she was entirely unaware of the moral reasons against her action, we are apt to count her as not culpable, regardless of whether she had reasons for doing as she did. If she was aware of the moral reasons but unmoved by them--or, better, incapable of being moved by them--we are unsure about her culpability. But the reasons for doing the action don't matter, as long as we are sure of the negative fact that she didn't have good reasons.
Suppose, perhaps per impossibile (perhaps action without reasons is impossible--I think that), that someone acted for no reason at all (on a whim? or are whims reasons?) in a way that went against the conclusive moral reasons she had. As long as she was aware of and sufficiently moved by these reasons against her action, we surely would count her culpable for her action. Again, what counts are the reasons she had against her action, not so much as the ones for her action.
It is different for praiseworthy actions than for culpable ones, though. For an action to be praiseworthy, the action may be done for the right reasons, while for it to be culpable it must be done against the right reasons. Nonetheless, even for praiseworthy actions the reasons against that action matter. Suppose I have such an excess of money that I barely feel any reason to hold on to a thousand dollars. Then my thousand dollar donation is barely praiseworthy (though I may be praiseworthy for my ungreedy feelings). The widow, though, who had great reason to hold on to her mite is very much praiseworthy.

A thought about Molinism

Ordinary subjunctive conditionals "were A to hold, B would hold" tell us about how B depends on A. But if that's what Molinist conditionals did, then they would undercut freedom on incompatibilist grounds. So Molinist conditionals aren't the same as ordinary subjunctive conditionals. But if they aren't the same, then it is difficult to see how they are introduced in a meaningful way. Moreover, the Molinist conditionals are treated as if they were ordinary for the purposes of divine decision theory. So this is a problem.

Causal probability and counterfactuals

The causal probability of an event B on an event A is cP_A(B)=∑_KP(K)P(B|AK), where the Ks are a partition based on the relevant dependency hypotheses compatible with A. (Compare to P(B|A)=∑_KP(K|A)P(B|AK).) A standard proposal in the literature is that

1. the degree of the assertibility of an indicative "If A, then B" is equal to the conditional probability P(B|A) of B on A.

Consider the parallel thesis that

2. the degree of assertibility of a subjunctive conditional or counterfactual A→B is equal to the causal probability cP_A(B) of B on A.

This thesis would unify the Stalnaker and Lewis (and Skyrms) approaches to causal decision theory as closely as possible. For according to the Stalnaker version, the causal expected value of an option A is:

3. EV(A) = ∑_BU(BA)P(A→B),

where the sum is over a partition based on outcomes. On the Lewis/Skyrms approach, it will be:

4. EV(A) = ∑_BU(BA)cP_A(B).

Now, if A→B has truth value, then the degree of asssertibility of A→B is equal to P(A→B), and hence by (2) we have P(A→B)=cP_A(B). And so the two formulae are equivalent. If, on the other hand, A→B has no truth value, then P(A→B) in (3) makes no sense. But we can replace it with Assertibility(A→B), which is basically the most natural replacement for P(A→B) when A→B has no truth value, and the revised (3) will come to the same thing as (4). So that's nice.

Notice, however, that this approach may not be compatibility with Molinism. For according to Molinism, God knows some conditionals of free will A→B, where B is a free action and A is a maximally specific set of antecedents, for sure. If P is God's probabilities, then in such cases:

5. 1=cP_A(B)=∑_KP(K)P(B|AK).

But because A is maximally specific, it will be compatible with only one relevant dependency hypothesis, say K₀, describing how B depends on A. So 1=P(K₀)P(B|AK₀). It follows that P(B|AK₀)=1 and P(K₀)=1. But now we see that there is a dependency hypothesis K₀ such that, together with A, it probabilistically necessitates B. But that can't be acceptable to a libertarian.

Why did God become man?

One way to put the question of the theory of the atonement might seem be to ask with Anselm: "Why did God become man?"

I do not think that that is the right way, though. For God is omnirational. This means that whenever he acts, he is moved by all the unexcluded good reasons for the action. But almost any good that is achieved by a morally permissible action will provide God with an unexcluded good reason for that action. Very high up among the goods achieved by God through the Incarnation is ensuring that we can be forgiven with justice. But there are many other goods achieved by the Incarnation: Aquinas lists nine other great goods. And besides the great goods, there are far lesser but nonetheless genuine goods, such as cooperating with Joseph in the making of tables. Plausibly, there are many goods in between the really great ones and the far lesser ones, such as providing Israel with a king, bringing the nations to Israel in worship, and healing certain lepers through a face-to-face interaction.

One might try to run some kind of a distinction between the goods wrought in the Incarnation, so as to delineate the goods that are the proper subject of a theory of the atonement or the proper answers to "Why did God become man?" Let's try a few such distinctions.

First, there is a distinction between those goods that could be achieved without God becoming man and those goods that can only be achieved by God becoming man. But this won't draw the distinguishing line in the right place. First, pace Anselm, it is far from clear that we couldn't be forgiven with justice simply by divine fiat. Certainly, Aquinas thought we could. It would be fitting for the penalty to be paid by Christ on our behalf, but it would not be unjust for God simply to release us. Second, some of the goods lower down on the list require the Incarnation. Thus, the good of God's cooperating with Joseph in carpentry (this cooperation makes me think of Enoch walking with God, but in the case of Joseph there is a greater literalness) requires an Incarnation. (God can, by his omnipotent power, create tables ex nihilo, but such creation is not carpentry.) Likewise, there is a value to God healing the lepers through a face-to-face interaction of a sort that can only be had by means of the Incarnation.

Second, perhaps we could distinguish between reasons that are such that if they were present on their own, God would still have decided on the Incarnation, and reasons that were only contributory. Thus, while omnirationality implies that God became man in part in order to work with Joseph, surely if no other goods were realized by the Incarnation than God's engaging in carpentry with Joseph, then there would be no Incarnation. But why think that? God could have done it. Maybe it would be an unlikely scenario, but surely a possible one. Why accept the counterfactual of free will that had God had working with Joseph as the only reason to be Incarnate then he wouldn't have done it? Such a counterfactual seems meaningless (or trivially false) to me, and even Molinists need not extend their view to counterfactuals of divine freedom.

Third, perhaps we can distinguish reasons such that were they absent, the Incarnation would not have occurred. But again we get into dubious counterfactuals about divine decisions. Take one of the "big" reasons for the Incarnation. I see no reason to accept the counterfactual that had that reason been absent, there would have been no Incarnation. God might still have become incarnate for the other reasons.

There is, of course, a distinction as to the weight of the reasons. The "big" reasons are much better reasons. But this distinction is one of degree.

So it seems to me that the answer to the question of why God became man is simply a list. Aquinas gives ten items to put on the list. There are many more. We can prioritize the items on the list, of course. But each of the items on the list will be a reason that God was acting in the light of, since God acts in the light of all the unexcluded good reasons in favor of his action.

This does not imply that there is no such thing as a theory of the atonement, just that the question "Why did God become man?" doesn't delineate the theory precisely enough.