Klingon bat'leth sword

My daughter and I are Trekkies, and she asked me for a bat'leth prop for Christmas. I made one out of wood, painted with metallic paint, with leather wrapped handles. Here are the instructions.

Trust in the virtuous and the morality of lying

Helga is well known to be perfectly virtuous. Her best friend Kurt is accused of conspiring to peacefully overthrow a tyrannical government, and will be tortured to death and executed unless Helga can convince government agents that he did no such thing. As a matter of fact, Helga has conclusive first-person evidence that Kurt did no such thing, much as that government deserves to be overthrown.

Suppose that it is sometimes permissible to lie. Then surely lying to save a peaceful conspirator against a tyrannical government would be a paradigm case of permissible lying, and indeed of obligatory lying. Thus, if Helga testifies to government agents that Kurt is not a conspirator, she is doing precisely what a perfectly virtuous person would do if Kurt were a conspirator. Thus if she is known to be perfectly virtuous, her testimony to Kurt's not being a conspirator is unworthy of credence. In fact, her testimony would be more worthy of credence if she were somewhat less virtuous and rigidly opposed to lying in all cases. Thus the permissibility of lying would make the testimony of the virtuous be worthless in a number of high-stakes cases.

But the virtuous are precisely the people whose testimony should carry weight, they are precisely the trustworthy, at least in cases where they are in a position to know whereof they speak. So the conclusion that Helga's truthful testimony about Kurt is worthless is paradoxical. And this paradox gives one reason to reject the premise from which the paradox was derived, namely that it is sometimes permissible to lie.

Despair, virtue and the afterlife

1. If it is physically guaranteed that virtue will perish, despair is the right attitude.
2. If there is no life after death at least for the virtuous, it is physically guaranteed that virtue will perish.
3. Despair is not the right attitude.
4. So there is life after death at least for the virtuous.

Making this argument precise would take some work. One would want to say something about the scope of "virtue" (human virtue? individual virtue?), and one would want to say a little more about "right attitude".

Eternalism and accidents without a subject

A classic objection to transubstatiation, famously pressed by Wycliffe, is that according to the Catholic understanding of the doctrine, the accidents of bread and wine persist even though the substance of bread and wine no longer exists. But in Aristotelian metaphysics, accidents are essentially dependent on their substance.

Eternalism—the view that past and future and present things all exist—provides a neat way for the Catholic to respond to Wycliffe. One can, if one so wishes, hold on to the idea that it is metaphysically necessary that a subject exists if an accident exists. But one denies that it is metaphysically necessary that the subject exists at the same time as the accident. The eternalist then holds that even if the bread and wine have perished at a time t₁ after transubstantation, nonetheless it is true at t₁ that the bread and wine exist, where the "exist" is tenseless. On this view, every accident has a subject in the same world but not always at the same time.

Are parts modes?

There are two variations on Aristotelian ontology. On the sparser version there are substances and their modes (accidents and essences). On the more bloated version there are substances, modes and (proper) parts. I want to argue that the more bloated version should be reduced to the sparser one.

Parts in an Aristotelian ontology are unlike the parts of typical contemporary ontologies. They are not substances, but rather they are objects that depend on the substance they are parts of. At least normally when a part, say a finger, comes to be detached from the substance it is a part of, it ceases to exist—a detached finger is a finger in name only, as Aristotle insists.

This makes the parts of Aristotelian ontology mode-like in their dependence on the whole. Ockham's razor then suggests that rather than supposing three fundamental categories—substances, mode and parts—we will do better to posit that a part is just a kind of mode. Thus, I really do have a heart, but my heart is just much a mode or accident of me—my cardiacality—as my knowing English is. Both my heart and my knowledge of English confer on me certain causal powers and causal liabilities (knowing English makes me liable to having my feelings hurt by uncomplimentary assertions in English!)

This is not an elimination of parts. Some of my accidents are parts and others are not. Which ones? I do not know. Maybe those accidents that occupy space are parts and those accidents that do not are not. My knowing English doesn't occupy space, while my cardiacality is somewhat vaguely but really located located in space.

Perhaps we need a finer distinction, though. Consider the strength of my arm. This isn't a part of me, but it seems to be located in my arm. I suggest that we distinguish between three ways that a mode can get a location. It can (a) inherit a location from a subject, or (b) it can inherit a location from its own modes, or (c) it can be located in its own right. I suggest that a mode is a part if and only if it has a location of type (b) or (c). The strength of my arm inherits its location from its subject—my arm—and hence is not a part. (It's important to the full development of this ontology that modes can nest. Thus, my arm is a mode of me, and the strength of that arm is a mode of this mode. Both I and my arm are subjects of the strength of the arm.)

I think the distinction between type (b) and type (c) parts is worth thinking about. Maybe matter, that mysterious ingredient in Aristotelian ontology, can be identified with type (c) parts?

Being is grounded in fundamental being, and presentism

Assume a bloated ontology, on which there are events, chairs, holes, waves, etc. In defending such a bloated ontology, we should sensibly say that these beings are grounded in what the fundamental beings are and how they are, and so the bloat does not infect fundamental reality.

So far so good. But what if we add presentism into the mix? Imagine two worlds, A and B, that are exactly alike at time t, but in one of them a race is beginning at t and in the other it is ending at t. We may here suppose massive mental dysfunction in B, so that everybody thinks the race is beginning, whereas in fact it has just ended.

Then in A at t, there is an event E that doesn't exist in B at t: the beginning of the race. Suppose t is the present moment and A is actual. Then E is not grounded in what the fundamental beings are and how they are, given presentism, since the fundamental beings and how they are are the same between A and B. In our bloated ontology, there really are events, and E really does exist, though. And so we have a violation of the principle that all beings are grounded in the fundamental beings and how they are.

Some presentists will get out of this by saying that A and B differ in the past, and facts about the past are grounded in what God remembers. Note first that this requires a denial of divine simplicity. For, given divine simplicity, there cannot be two possible worlds that differ only in how God is, since that would imply an intrinsic accidental property in God. Second, we have the standard criticism that this gets things the wrong way around: God remembers the past because it was so, rather than its having been so because God remembers it thus.

The above is a slight tweak on the usual grounding objection to presentism. But I do think that while one might balk at the claim that all truth is grounded in what fundamentally exists and how it is, it is plausible that all being is so grounded.

The best way out for the presentist is to deny the bloated ontology, of course.

A perdurantism without temporal parts

Standard perdurantism holds that we are four-dimensional worms made up of three-dimensional temporal parts. Many of the changing properties that we think of ourselves as having directly, we actually have derivatively from the temporal parts. Thus, I am typing a post in virtue of having a temporal part typing it up, and I am conscious of the screen in front of me in virtue of a temporal part of me being conscious of it.

Standard perdurantism has many problems, for instance:

1. Perdurantism commits one to proper parts, and implausibly thin and hyperplanar ones.
2. Surely I am that entity which is non-derivatively consciously rather than that entity which is derivatively conscious.
3. Perdurantism normally comes along not just with slices, but thicker temporal parts. But then Merricks argues that we cannot know how old we are. For all I know, I might be the temporal part from five minutes ago until now (in which case I am five minutes old), or from ten minutes ago until now (in which case I am ten minutes ld), and so on. Only a handful of temporal parts containing my present stage have the age I think myself to have, so probably I don't have the age I think myself to have.

There is, however, a perdurantism without any of these problems, if one accepts the right kind of trope theory. Suppose I exist at t. Then my existing at t is a trope of me, call it e_t. At least unless t is the first moment of my existence, e_t is an accidental trope: if I perished before t, then I wouldn't have had e_t. (If essentiality of origins holds, then it is an essential property of me that I exist at the first moment of my existence.) Note that I am not committing myself to the controversial thesis that existence is a property. Even if existence isn't a property, it is plausible that existence in a location—i.e., spatial locatedness at x—is a property. And if so, then why shouldn't temporal locatedness at t be a property?

I now suppose that I am a four-dimensional entity that has (where the "has" is not tensed) all of the e_t tropes (where t is a time during my existence): it is true to say that I exist at all these times. But many of the temporally qualifiable predicates, like "is conscious" and "is bent", that apply to me apply in virtue of e_t itself having certain tropes. Thus, I am now bent or conscious in virtue of e_now having a certain bendedness or consciousness trope.

Strictly speaking, it's not e_now that is bent or conscious, but it has the kind of trope which makes that substance that has the e_now trope be bent or conscious. Compare this: If I am gorging myself, then that happens in virtue of an eating trope itself having a gorging trope. But the eating trope isn't gorging itself. It is I who am gorging myself. So the gorging trope is a trope of eating such that any substance that has the eating trope with the gorging trope gorges itself. The gorging trope, thus, makes me—the substance—be a gorger and makes my eating trope be not a gorger, but gorgingly. This is linguistically tricky.[note 1]

On standard perdurantism, I persist over time in virtue of having temporal parts that exist at various times. On this trope perdurantism, I persist over time in virtue of having temporal locatedness tropes.

On this theory, the temporal locatedness tropes e_t play the role of the temporal parts of standard perdurance. But they aren't parts. So we aren't committed to parts, much less implausibly thin and hyperplanar ones.

We also do not have the problems in (2) and (3). For while I am conscious at t in virtue of e_t having a certain consciousness trope c_t, that consciousness trope doesn't make e_t be conscious. So while I am conscious in virtue of something other than me—namely, e_t—being a certain way, I am not conscious in virtue of something other than me being conscious. Thus, I do not derive my consciousness from the consciousness of anything else, and so I am non-derivatively conscious. I do derive my consciousness from something else being a certain way, but when that something else is a trope of me, that's quite innocent. Thus, (2) is not an issue here.

Nor is (3), for the obvious reason that a fusion of e_t-type tropes, even if there is such a fusion (which I very much doubt), doesn't think. It's substances that think, and they think in virtue of having certain tropes. The tropes don't think, and neither do their fusions.

I don't think this is the whole story. If I were seriously defending this story, I wouldn't say that I have the trope e_t directly. I might say that I have the trope e_t as a trope of my humanity, where my humanity may well be the only trope I have directly (see the paper of mine here).

I don't know if the above story is true. I am a bit sceptical of the thinness and hyperplanarity, as it were, of the e_t tropes—they don't seem to me to be very natural. And I am not 100% sure I want to commit to tropes. But this version of perdurantism might be true.

Note, also, a neat thing. Normally the perdurantist needs to argue why perdurance is preferable to exdurance. But I do not think there is any plausible trope exdurantism paralleling this trope perdurantism.

Objection: The trope e_t is a part of me, so this devolves to a more standard perdurantism.

Response: Maybe in some sense my tropes are parts of me. But they are different sorts of parts from the kinds of parts that standard perdurantism invokes. For tropes depend, at least for their identity, on that which they are tropes of. But the whole is constructed out of the temporal parts on standard perdurantism. So trope perdurantism reverses the order of grounding.

Existence, eternalism, continuous creation and concurrence

The doctrine of continuous creation is something like this:

1. For all x and t, if x is a creature and x exists at t, then x is created or preserved by God at t.

On the other hand, the doctrine of concurrence is something like this:

2. God causally concurs in every instance of creaturely causation,

where the exact details of concurrence need to be spelled out, but it is some sort of causal cooperation in or causal responsibility for the instance of creaturely causation. Given the auxiliary hypothesis that:

3. For all x and t, if x is a creature and x exists at t, then either x is created or preserved by God at t or x is created or preserved by a creature at t,

we have a somewhat handwaving argument from (2) to (1), namely an argument that given (2), when the second disjunct in (3) holds, so does the first. For suppose that x is created or preserved by a creature at t. Then by (2), God concurs in this. But if God concurs in creation or preservation, then God creates or preserves (albeit non-solitarily).

In the literature, (2) is seen as a stronger and more controversial claim than (1), and the above argument vindicates this.

Interestingly, given eternalism, there is an argument—albeit rather handwavy—from (1) to (2).

Start with this observation: Given eternalism, existing in 2014 is rather like existing in Waco. It isn't a case of existence simpliciter, but is simply the possession of a locational property. (Given endurantism, to exist in 2014 is like being wholly present in Waco; given some four-dimensionalist theorys, it's like being partly present in Waco; but on all the theories there is a close analogy.)

Now, according to (1), when x exists at t, God is causally responsible for this. But it is strange and ad hoc given eternalism to think God is causally responsible for temporally locational properties, but not for spatially locational ones, nor for other non-privative properties. So it is very plausible that:

4. If God is causally responsible for every case of a creature's possession of temporal location, then God is causally responsible for every case of a creature's possession of a non-privative accidental property.

But:

5. Every instance of creaturely causation is the causation of the possession of non-privative accidental properties or the causation of existence.

And it seems that it is "metaphysically harder" to cause existence than to cause the possession of an accidental property, so:

6. If God concurs in every case of the creaturely causation of the possession of non-privative accidental properties, then God concurs in every case of the creaturely causation of existence.

From (4), (5) and (6), we conclude that if (1) is true, then given eternalism we have God concurring in every instance of creaturely causation, and so we have (2).

This argument is handwavy, but it does show that it is ad hoc to hold on to (1) but deny (2).

Predictions and presuppositions

It is a prediction of Newtonian physics that two isolated massive bodies will move relative to each other in a conic section. It is a presupposition of Newtonian physics that there is space. It is a prediction of perfect being theology that persons (or at least, good persons) live forever. It is a presupposition of perfect being theology that there are objective values.

The distinction matters epistemologically. Suppose you have significant but non-overwhelming evidence for Newtonian theory but don't notice that the theory presupposes that there is space. And then you learn that there is this presupposition. This should make you both (a) more suspicious of Newtonian theory (after all, the presupposition might be false) and (b) more friendly to the idea that there is space (after all, the inference to best explanation argument for Newtonian physics now extends to the existence of space).

On the other hand, if you don't realize that Newtonian physics implies that two bodies will move in a conic section, and then you come to realize it, that will make you want to run to the observatory to see if the prediction is true. But while you're running to the observatory, your credence in the theory will not have gone down. It will only go down if the prediction is not borne out.

Likewise, suppose that you have significant but non-overwhelming evidence for perfect being theology, and then you realize—somehow you missed it before—that this theology presupposes objective values. If your evidence for objective values is non-overwhelming (I think it's overwhelming) then this realization will make you more suspicious of perfect being theology. On the other hand, if you realize that the theory predicts that persons live forever, that by itself shouldn't make you more suspicious of the theory, just make you look for evidence for and against this prediction.

Yet, you might think this: "Both the prediction and the presupposition is something you get committed to by the theory. Learning that you're committed to more things by a theory makes the theory more top-heavy, and so it should make you more suspicious of the theory."

That's a mistake. Discovering that a scientific theory has predictions you didn't see before doesn't by itself make us more suspicious of the theory (of course, if we have evidence against the predictions, that's a different thing). But the thought highlights a crucial question: What is the difference between the presupposition and the prediction? Both are implied (entailed, or maybe just significantly probabilified) by the theory. The difference isn't logical, but I think explanatory:

• A prediction of a theory is an implication that the theory explains.
• A presupposition of a theory is an implication that the theory does not explain.

From a Bayesian point of view, we can now say a little bit more about what happens when we discover a new prediction or presupposition. In general, when we discover an implication of a theory, we realize that we were mistaken in regard to logical interconnections. When that happens, we need to go back to the drawing board, and re-do our relevant priors.

Now when we learn of a new prediction of a theory, that doesn't affect the complexity of the theory, because only the explanatorily fundamental parts of a theory enter into the complexity of the theory (see this post), and while perhaps complexity isn't the only determinant of priors, I do not see any other relevant one here that would raise the probability. So our priors for the theory stay the same, but now the prediction gets tied to the theory, so that the evidence for the theory ends up typically being evidence for the prediction as well.

On the other hand, when we discover a new presupposition, that makes the explanatorily fundamental parts of the theory more complicated, and so the prior for the theory should go down a little. But at the same time, once we factor in the posterior evidence, then typically the evidence for the theory will transfer to the presupposition. As a result, our prior for the theory goes down, and hence so does the posterior, but the posterior for the presupposition is apt to go up.

Impanation

The doctrine of transubstantiation has two primary components:

• The real presence of Christ's body and blood
• The real absence of bread and wine.

Some objections center on the Real Absence. After all, it looks like bread and wine are present—why would God make our senses deceitful? And why would God destroy the bread and wine? Doesn't nature build on grace? (Quick answers: Senses give prima facie reasons to believe, but in the context of the liturgy as a whole there is no deceit as it is explicitly stated that this is Christ's body and blood. And we are built out of our food, even though our food is destroyed when we eat it.)

One theory that attempts to avoid the Real Absence is impanation. Analogously to how Christ became a human, he now becomes bread and wine. But here is a curious fact about impanation. While it does hold that bread and wine exist after consecration, it has to say that the bread and wine cease to exist after consecration. In other words, the particular piece of bread and the particular sample of wine that were present before consecration cease to exist given consecration, and what we have after consecration is a new piece of bread and a new sample of wine. But if this is right, then impanation really doesn't help much with the two objections to Real Absence. We still have cessation of the existence of bread and wine. And while our sense that bread and wine are present is not mistaken, our sense that it's the same bread and wine as before consecration is mistaken. So impanation doesn't offer much of an advantage with respect to the objections.

But am I right? Does impanation imply that the old bread and wine are absent after consecration? I think so. First, consider the analogy with the Incarnation. The thought was that just as Christ came to be a human being, so too Christ came to be bread and wine. But the human being that Christ came to be did not preexist the incarnation. By analogy, the bread and wine that Christ came to be should not preexist the impanation.

Second, let's call the post-impanation bread B, and the pre-impanation bread A. Then B is Christ (just as the post-incarnation human, Jesus, is Christ). But identity is transitive. So if B is Christ, and A is identical with B, then A is Christ. Which is absurd. So the impanationist needs to deny that A is B.

But perhaps my arguments are a misunderstanding of impanation. For maybe the impanationist doesn't say that Christ becomes bread in the same way that Christ becomes human, but that Christ becomes bread in the same way that Christ becomes flesh. When Christ becomes flesh, there isn't a piece of flesh that is Christ. Rather, Christ comes to be a composite of flesh and soul. Now the analogy between impanation and incarnation forces the idea that there be a Y such that Christ comes to be a composite of bread and Y, analogously to his coming to be a composite of flesh and soul. But it is far from clear what Y would be.

A world abuzz in activity

I sometimes think that the phrase "causal power" doesn't quite convey what the causal powers theorist means or should mean. When I hear the phrase, it makes me think of a mere potentiality, a disposition which needs separate activation. But this is a mistaken image.

A causal power should be seen as active. It is a striving, a conatus. I lie on a bed typing this post. The elasticity of the springs strives to lift me up. The gravitation attraction between my body and the earth strives to press me down. There is not much motion, as the forces have balanced out, but both forces are constantly active. If gravity suddenly disappeared, the bed's springs would eject my upward, and if the springs suddenly disappeared, I would slump down.

The world is abuzz in activity. It trembles with action. Objects strive in various directions in their causal powers, pulling hither and yon (cf. Amjum and Mumford).

Aristotle talks of two levels of potentiality. I am in first potentiality for speaking German—I would have to learn it in order to speak it—and in second potentiality for speaking English—I can do it whenever I wish to. Second potentiality is also first actuality—the general human potential for speaking languages is realized in me in respect of English. And then there is second actuality—I am now in second actuality for writing in English. So we have a three point scale of actuality:

1. first potentiality
2. second potentiality = first actuality
3. second actuality.

Having a causal power, a conatus or a striving is something intermediate between the first and second actuality. When unopposed, the causal power causes second actuality—actual action. When opposed, we just have a tense striving, a pull.

The world, both physical and mental, is a world of tension. Forces push against forces, reasons push against reasons. Or, more precisely, substances with forces or reasons push against the same or other substances with opposed forces or reasons.

Sometimes an image helps develop good metaphysics. That's all the above is: an image.

Love as the whole of morality

After a disaster, there are two people left in the world, A and B. Each fully loves the other, and each fully loves herself. However, they hear noises in the woods occasionally, and come to form a justified false belief that there is a human being in the woods. In fact, there isn't—there is just wind. Now, A comes to love the alleged person, exhibiting the love by leaving food out. On the other hand, B comes to hate the alleged person, exhibiting the hatred by setting traps.

Both A and B love each human being. Yet, clearly, B is doing something wrong. So, it seems, loving each human being is not sufficient for morality.

Yet the New Testament suggests that it is. It seems part of the message of the New Testament that love of God and neighbor fulfills the moral law. And the First Letter of John suggests that one loves God to the extent that one loves one's neighbor.

I think there are two moves here. One is to take the claims in 1 John as applying to typical cases. Typically, one loves God to the extent that one loves one's neighbor. Cases like that of B are too out of the way to be what the author of the text was talking about. And then one can argue that B by failing to love someone that he takes to be a creature of God is failing to love God. So on this view, the foundation of Christian ethics is not just love for neighbor, but either both love for God and love for neighbor, or just love for God, though typically love for neighbor is indicative of this.

The second move would be that to love one's neighbor requires not just that one love each individual neighbor, but that one have a disposition to love all possible people, should they be actual. This seems right to me, actually. For another kind of case would be this. Suppose that the world is down to two people, and they are virulent racists, but they are both of each other's preferred race. They love every human being who exists. But they don't have a disposition to love all possible people.

Space, time and discreteness

A first plausible thesis:

1. Space is discrete if and only if time is discrete.

This is very plausible in any picture like that given by modern physics where space and time, while not on par, are very closely related. It's also intuitive. For suppose space is continuous but time is discrete. Then a small enough object will jump over some intervening space when it moves during a unit of time, which seems strange. Conversely, if space is discrete but time is continuous, then objects always move jerkily: for a bit of time they stand still, then they instantaneously jump to a new point in space, and then they stand still for a little longer. That's strange, too.

Here's a second plausible thesis:

2. A small object can rotate by any small angle without its internal measurements changing much.

But now imagine that space is discrete, with the smallest distance between points being about α. Imagine an object occupying exactly two points x and y spaced out by approximately α. Now swivel that object slightly around the point x without changing internal measurements much. The other point in the object will either continue occupying y, in which case the object hasn't rotated (contrary to (2)), or will occupy some point very close to y (but there isn't any such point, since the minimum spacing is about α), or the object will come to occupy only one point (in which case its internal measurements will change much).

If the above argument isn't clear, just imagine a hexagonal or square grid, an object composed of two points on the grid, and think what it would be to rotate that object by a small angle (smaller than 90 degrees in the case of the square grid and smaller than 60 degrees for the hexagonal one).

So (2) gives us good reason to deny that space is discrete, and then (1) gives us good reason to deny that time is discrete.

But this was too quick. Both my arguments for (1) and my argument that (2) forbids space to be discrete made a crucial assumption, namely that space has a certain fixity to it that it is independent of the objects in space. For suppose that time is continuous but space is discrete. I said that it follows that objects move jerkily. Not so. For the points that the objects occupy could be moving with the objects! Thus an object could move smoothly because the spatial points in it could be moving. The discrete space, then, wouldn't be a regular grid. It would be a mess of points, which shift around as the objects they are in shift. (This doesn't affect the argument that we shouldn't say that space is continuous but time is discrete.)

The same flaw affects my argument based on (2). I was assuming that as I rotate the two-point object, the points x and y stay fixed. But what if points are defined by objects, and so the point y rotates with the object? Again, we wouldn't have a regular grid. We would have an irregular changing grid, where the real points are defined by the objects.

The resulting view of space would be, I think, a version of Aristotle's picture, where space is infinitely divisible but not actually infinitely divided. In the case of our two point object, there could be a point at distance α/10 from y, but there isn't, unless we rotate the object that defines the points.

In other words, Aristotle's account of space is the only discretist view of space that accommodates the intuition that objects can be rotated by small amounts without great distortion. That's pretty neat, I think.

What's the motivation for thinking this is the truth of the matter. Well, causal finitism gives one good reason to think that time is discrete (or at least discrete when we restrict ourselves to a local area of space). The implication from discrete time to discrete space in (1) survives my above criticism of the argument. So we have good reason to think space is discrete. And then the rotation argument yields a version of Aristotle's view.

Deep Thoughts XXXIX

You cannot leave an empty room.

[The idea behind this lovely tautology is from my daughter Clare.]

Compensation

It seems to be widely accepted in the philosophy of religion community that supposing God's merely compensating a person for evils suffered will not make for a good theodicy. Rather, the evils must be defeated in a way that goes over and beyond compensation, that draws a defeating good out of the evil. I think that in many cases this conviction might well be mistaken. Compensation may well be good enough.

Start with this story:

You are a financially well-off Olympic archer. You are about to take your last shot at the Olympics, indeed at the last Olympics you will ever compete at (you have promised your spouse to hang up your bow after these Olympics) and whether you get a gold medal depends on this shot. Getting a gold medal means a lot to you. At the same time, you see in the stands a vicious dog attacking me. You could turn your bow on the dog, and painlessly kill it at this range (nor would it be wrong to do so; the dog will be put down anyway after the vicious attack). There is no other way for the attack to be stopped. But then you would lose your last chance for a gold medal: the dog is not the official target. You also know that what kind of a dog it is and how terrified of dogs I am, so that you know that if that were the whole story, the sufferings that I would endure through being bitten are so great that your gold medal wouldn't be worth it: you would have a duty to shoot the dog. But you also know me well enough to know for sure that I would survive the attack without permanent damage, and that you have a sum of money that you could give me such that both by my own lights and objectively I'd be much better off bitten and compensated than neither bitten nor compensated. You resolve to compensate me financially, take your shot at the target, win your gold medal, write me a large check, and we are both much better off for this.

This is a clear case of compensation for evil rather than defeat of evil. At the same time, your failure to stop the attack is justified. I chose this case so that my own biases would go against the justification. I am in fact terrified of dogs. Being bitten again would be a truly horrific experience. Nonetheless, if you gave me a sum of money sufficient to pay off the rest of our mortgage and there was no permanent damage, I think it would be well-worth being bitten. (I don't think I would go for it for half of the mortgage!)

Notice what compensation does here. The good you achieve by allowing me to be bitten—the gold medal—is insufficient to justify your permitting me to be bitten. (If this isn't true, we can tweak the case so it is.) But when you add your resolve to compensate me, and your knowledge that the compensation would be sufficient both objectively and by my own lights, you come to be justified.

An important feature of this story is that the good you achieve—the gold medal—is one to which my sufferings are not a means, and indeed you do not intend my sufferings either as an end or as a means. (Here one remembers Double Effect, of course.) But there is an end that you are pursuing, and your pursuit of this end precludes your preventing the evil.

There are many real-world cases that might well have this structure. Consider Rowe's fawn dying painfully in a forest fire. God could miraculously prevent this, but doesn't, because he wants the laws of nature to have as few exceptions as possible. Now, it's good, I suppose, that the laws of nature have as few exceptions as possible. But this good does not seem to be sufficiently great to justify several days of the fawn suffering. However, if God resolves to compensate the fawn in an afterlife, to a degree such that both objectively and by the fawn's lights (to the extent that the fawn is capable of making the relevant judgments) the fawn will be much better off for having both the suffering and the compensation, then we will have the structure of the archery-dogbite case.

I do not know that the compensation story will work for every case. One worry is that if you foreknew that the dog would bite and were responsible for the dog's presence in the stands in the light of this foresight, then the justification in the compensation story is less clear, even if you don't intend the dog's biting. So there will be relevant questions about determinism, Molinism and the like in the theological cases.

Here's another interesting thing, by the way, about the fawn case. The compensation would not have to come in an afterlife. Suppose:

You are a super-rich archer. You know that sometimes dogs show up and bite people in one area of the stadium. So ahead of time you write sufficiently large checks to all these people such that both objectively and by their lights they would be much better off for having the check and the dogbite than for having neither. And then when it's time to compete, you don't even need to think about the dogs.

This seems quite justifiable as well. So if God sufficiently compensates all deer that are in danger of forest fires ahead of time, all is well, too.

Note, though, that in general pre-compensation works less well for human sufferers than for non-human sufferers. For humans see their lives as a narrative, and the narrative structure and order of events matters a lot as a result. So in the case of a human it is particularly tragic if existence ends in a particularly bad way, no matter how good the earlier parts were. So compensation for evils that happen around the time of death still likely requires an afterlife in the case of humans. (And even in the case of non-human animals, it may be better for God to compensate in an afterlife, since it would require fewer miracles in this world.)

Simplicity, language-independence and laws

One measure of the simplicity of a proposition is the length of the shortest sentence expressing the proposition. Unfortunately, this measure is badly dependent on the choice of language. Normally, we think of the proposed law of nature

• F=Gmm'/r²

as simpler than:

• F=Gmm'/r^{2.000000000000000000000000000000000000001},

but if my language has a name "H" for the number in the exponent, then the second law is as brief as the first:

• F=Gmm'/r^H.

One common move is to employ theorems to the effect that given some assumptions, measures of simplicity using different languages are going to be asymptotically equivalent. These theorems look roughly like this: if c_L is the measure of complexity with respect to language L, then c_L(p_n)/c_M(p_n) converges to 1 whenever p_n is a sequence of propositions (or bit-strings or situations) such that either the numerator or the denominator goes to infinity. I.e., for sufficiently complex propositions, it doesn't matter which language we choose.

Unfortunately, one of the places we want to engage in simplicity reasoning in is with respect to choosing between different candidates for laws of nature. But it may very well turn out that the fundamental laws of physics—and maybe even a number of non-fundamental laws—are sufficiently simple that theorems about asymptotic behavior of complexity measures are of no help at all, since these theorems only tell us that for sufficiently complex cases the choice of language doesn't matter.

Simplicity, language and design

1. Simplicity is best understood linguistically (e.g., brevity of expression in the right kind of language).
2. Simplicity is a successful (though fallible) guide to truth.
3. If (1) and (2), then probably the universe was made for language users or by a language user.
4. If the universe was made for language users, it was made by an intelligent being.
5. If the universe was made by a language user, it was made by an intelligent being.
6. So, probably, the universe was made by an intelligent being.

A moral argument

I've never found the moral argument for morality—except in its epistemic variety—particularly compelling. But now I find myself pulled to find plausible premises (1) and (2) of the following pretty standard argument:

1. Only things that are infinitely more important than me can ultimately ground absolutely overriding rules on me.
2. Rules without ultimate grounding are impossible or not absolutely overriding.
3. I am a finite person.
4. The only things that could be infinitely more important than a finite person are or have among them (a) infinitely many finite persons or (b) an infinite person.
5. Moral rules that apply to me are absolutely overriding.
6. Moral rules that apply to me are not grounded in a plurality including infinitely many finite persons.
7. So, moral rules that apply to me are grounded at least in part in an infinite person.
8. So, there is an infinite person.

The vague thought behind (1) is that rules grounded in something merely finitely more important than me will not be absolutely overriding. After all, it is logically possible that I rise in importance by some large finite amount in my life and then exceed the importance of the ground of moral rules if they are grounded in something of merely finite importance. The vague thought behind (2) is that a regress of grounding in effect leaves things ungrounded, and and ungrounded facts can't be that important to me, because it is beings that are important. Premise (3) is plausible.

I find (4) quite plausible. It's based on the personalist intuition that persons are the pinnacle of importance in reality. Merely Platonic entities, should there be any, while perhaps beautifully structured and infinite in their own way are not important, not unless they are persons as well.

Next, (5) is obvious to me. And (6) seems very plausible. The only plurality of finite persons who could plausibly provide a ground for the moral rules that apply to me is a human community, and there are only finitely many humans. Even if we live in an infinite universe with infinitely many people, the infinitely many aliens surely are not needed to ground the absolute wrongness of degrading a fellow human being.

All that said, I am a dubious about (1). I think there are no reasons other than moral reasons, and so the fact that moral reasons take priority over other reasons is a triviality.

But even within this controversial framework, I am now realizing there is room to ask the question of why some reasons are absolutely conclusive—they should close deliberation no matter what else has been brought to bear. "But A requires intentionally degrading my neighbor" should close deliberation about A: it doesn't matter what reasons there are for A once it becomes clear that A requires intentionally degrading my neighbor.

And that makes something like (1) still plausible. For nothing but a person can be the ultimate ground for a rule whose deliberative importance is so absolutely conclusive—nothing but a person matters enough for this task. Could this person just be my neighbor? Yes—but only if my neighbor is infinitely important, and important in a personal kind of way. This infinite importance can be had in two ways: either my neighbor is an infinite person, or else the infinite importance of my neighbor is derivative from other persons (if it's derivative say from Platonic entities it's not the right kind of importance, for only considerations about persons can bestow the kind of importance that trumps all conflicting considerations about persons). In the latter case we get a regress that is vicious unless there is an infinite person or an infinite number of finite persons grounding the rule. The latter is implausible, so there is an infinite person.

This argument requires deontology, of course.

Let me end by saying that none of this means I am being pulled to Divine Command Metaethics (DCM). DCM is just one among many ways of grounding morality in an infinite person, and it seems to me to be less plausible than other ways of doing so.

Possibility, Aristotelian propositions and an open future

Aristotelians think that tensed sentences like "It is sunny" expressed "tensed propositions" capable of changing in truth value between true and false as the facts alter. The proposition that it is sunny is false today but was true two days ago. Anti-Aristotelians, on the other hand, roughly say that the sentence "It is sunny" expresses the proposition that it is sunny at t₀, where t₀ is the time of utterance, a proposition whose truth value does not vary between true and false as the facts alter.

Most presentists are Aristotelians about propositions, and most open futurists these days seem to be presentists. I will argue, however, that an open futurist should not be an Aristotelian about propositions. I think this means that an open futurist should not be a presentist.

Consider the sentence

1. I will freely put on a pink shirt in one day.

Let p be the proposition expressed by this. Clearly:

2. p is possible.

(Also, the negation of p is possible.)

According to the anti-Aristotelian open futurist, p is the proposition that I will put on a pink shirt on day d₀+1, where d₀ is November 14, 2014. The anti-Aristotelian open futurist holds that on November 14, p is not true, but that on November 14 it may become true. So the anti-Aristotelian open futurist has a nice way of accounting for (2). While it's impossible that today p is true, it is possible that p be true tomorrow, and that's enough to make p possible.

But the Aristotelian open futurist is in trouble. For on her view, on November 15, p doesn't tell us about how things are on November 15, but about how things are on November 16: it's a tensed proposition that on any day says how things will be on the next. But on no day is it true that on the next day I will freely put on a pink shirt, if open futurism is true. And open futurism isn't just a contingent thesis. So given open futurism:

3. It is impossible that p ever be true.

(And what cannot ever be true cannot become true either, since if something were to become true, it would then be true.) But surely:

4. If it is impossible that p ever be true, then p is not possible.

And that contradicts (2).

Freedom and theodicy

Invoking free will has always been a major part of theodicy. If God has good reason to give us the possibility to act badly, that provides us with at least a defense against the problem of evil. But to make this defense into something more like a theodicy is hard. After all, God can give us such pure characters that even though we can act badly, we are unlikely to do so.

I want to propose that we go beyond the mere alternate-possibilities part of free will in giving theodicies. The main advantage of this is that the theodicy may be capable of accomplishing more. But there is also a very nice bonus: our theodicy may then be able to appeal to compatibilists, who are (sadly, I think) a large majority of philosophers.

I think we should reflect on the ways in which one can limit a person's freedom through manipulation of the perfectly ordinary sort. Suppose Jane is much more attractive, powerful, knowledgeable and intelligent than Bob, but Jane wants Bob to freely do something. She may even want this for Bob's own sake. Nonetheless, in order not to limit Bob's freedom too much, she needs to limit the resources she uses. Even if she leaves Bob the possibility of acting otherwise, there is the ever-present danger that she is manipulating him in a way that limits his freedom.

I think the issue of manipulation is particularly pressing if what Jane wants Bob to do is to love her back. To make use of vastly greater attractiveness, power, knowledge and intelligence in order to secure the reciprocation of love is to risk being a super-stalker, someone who uses her knowledge of the secret springs of Bob's motivations in order to subtly manipulate him to love her back. Jane needs to limit what she does. She may need to make herself less attractive to Bob in order not to swamp his freedom. She may need to give him a lot of time away from herself. She might have reason not to make it be clear to him that she is doing so much for him that he cannot but love her back. These limitations are particularly plausible in the case where the love Jane seeks to have reciprocated is something like friendship or, especially, romantic love. And Scripture also presents God's love for his people as akin to marital love, in addition to being akin to parental love (presumably, God's love has no perfect analogue among human loves).

So if God wants the best kind of reciprocation of his love, perhaps he can be subtle, but not too subtle. He can make use of his knowledge of our motivations and beliefs, but not too much such knowledge. He can give us gifts, but not overload us with gifts. He may need to hide himself from us for a time. Yes, the Holy Spirit can work in the heart all the time, but the work needs to be done in a way that builds on nature if God is to achieve the best kind of reciprocation of his love.

I think there are elements of theodicy here. And a nice bonus is that they don't rely on incompatibilism.

The Incarnation is also an important element here—I am remembering Kierkegaard...

A Metaphysicality Index

A grad student was thinking that Platonism isn't dominant in philosophy, so I looked at the PhilPapers survey and indeed a plurality of the target faculty (39%) accepts or leans towards Platonism. Then I got to looking at how this works across various specializations: General Philosophy of Science, Philosophy of Mind, Normative Ethics, Metaethics, Philosophy of Religion, Epistemology, Metaphysics, Logic / Philosophy of Logic and Philosophy of Mathematics. And I looked at some other views: libertarianism (about free will), theism, non-physicalism about mind, and the A-theory of time.

Loosely, the five views I looked at are "metaphysical" in nature and their denials tend to be deflationary of metaphysics. I will say that someone is "metaphysical" to the extent that she answers all five questions in the positive (either outright or leaning). We can then compute a Metaphysicality Index for an individual, as the percentage of "metaphysical" answers, and then an average Metaphysicality Index per discipline.

Here's what I found. (The spreadsheet is here.) I sorted my selected M&E specialities from least to most metaphysical in the graph.

On each of the five questions, the Philosophers of Science were the least metaphysical. This is quite a remarkably un-metaphysical approach.

With the exception of Platonism, the Philosophers of Religion were the most metaphysical. (A lot of Philosophers of Religion are theists and may worry about the fit between theism and Platonism, and may think that God's ideas can do the work that Platonism is meant to do.)

Unsurprisingly, the Metaphysicians came out pretty metaphysical, though not as metaphysical as the Philosophers of Religion. (And this isn't just because the Philosophers of Religion believe in God by a large majority: even if one drops theism from the Metaphysicality Index, the Philosophers of Religion are at the top.

Interestingly, the Philosophers of Mathematics were almost as metaphysical as the Metaphysicians (average Metaphysicality Index 29.2 vs. 29.8). They were far more Platonic than anybody else. I wonder if Platonism is to Philosophy of Mathematics like Theism is to Philosophy of Religion. The Philosophers of Mathematics were also more theistic and more non-physicalistic than any group other than the Philosophers of Religion.

It's looking to me like the two fields where Platonism is most prevalent are Logic (and Philosophy of Logic) and Philosophy of Mathematics. This is interesting and significant. It suggests that on the whole people do not think one can do mathematics and logic in a nominalist setting.

For the record, here's where I stand: Platonism: no; Libertarianism: yes; God: yes; Non-physicalism: yes; A-theory: no. So my Metaphysicality Index is 60%.

Ex nihilo nihil fit, and presentism

According to presentism, events come out of nothing (the future), have a flash of reality as they are briefly present, and then pass back into nothing (the past). But nothing comes out of nothing. So, it seems, presentism is false.

I wonder if the above argument equivocates on "comes out of nothing".

A Thomistic creationism of sorts

Suppose that God made physical stuff (say, particles) be arranged just like in a butterfly, but he did not give (either directly or by some general policy) a butterfly form. Then we would have something that looks just like a butterfly. And to the extent that butterfly behavior is ultimately predictable just from physics, that bunch of physical stuff would behave like a butterfly. There wouldn't be a butterfly there. In fact, there wouldn't be one thing there: just a bunch of physical stuff.

Now, we are not yet in a position to know how much of the physical behavior of organisms—especially non-human ones—is predicted by the physics. Let us suppose, however, that it turns out that all the physical behavior of non-human animals is predicted by the physics. (Humans have free will, and that's a different business.)

Now let me tell a story. I don't think the story is actually true, though it seems basically[note 1] logically possible:

God created a physical world and had some chemical stuff come together in a way that "reproduced". And then evolution took over, and bundles of physical stuff that were better at survival and reproduction reproduced more, until we had bundles of physical stuff shaped like algae, trilobites, trees, dinosaurs, birds, horses and apes. But there never were any algae, trilobites, trees, dinosaurs, birds, horses or apes. Finally, not too long ago on a cosmic scale, there came to be two bundles of physical stuff that were physically rather like humans. By this point, there was no physical stuff shaped like trilobites or dinosaurs, but there was physical stuff shaped like algae, trees, birds, horses and apes. And God then said: "Let there be algae, trees, birds, horses, apes and many other organisms", and he created forms which informed the algaelike, treelike, birdlike, horselike and apelike bundles of physical stuff, and all sorts of other bundles. Thereupon, there were birds, horses and apes, though things didn't look any different. Finally God said: "Let there be humans", and he created forms which informed the human-like bundles of physical stuff. And there were humans.

This story is fully compatible with naturalistic evolution. Indeed, the only bar to the possibility of this story would be a vitalism on which physical stuff does not behave like living organisms. On this story, there literally never have been any dinosaurs. But there will have been bundles of physical stuff arranged dinosaurwise, and that's all many a biologist thinks a dinosaur is anyway.

But since bunches of physical stuff can't be conscious—they need soul, i.e., form, for that—then on this story there was no consciousness before human beings came on the scene. This is theologically attractive in that it enables us to hold that suffering came into the world through human sin. For we can continue the logically possible story thus:

When God created the forms of all the organisms, he miraculously arranged things so that no organism would suffer, miraculously making a harmonious state. And he put humans in charge of this delicately balanced system. Humans, however, quickly came to freely reject God's sovereignty in the system that he put them in charge of, and he reluctantly removed his miraculous protection from the system in deference to the authority he granted humans. And so humans and other animals came to suffer.

This story is also interesting in that it is yet another way to reconcile naturalistic evolution with the not dogmatically required but still somewhat attractive theological idea that all the organisms there are were directly created by God. For the story makes clear in a Thomistic setting how naturalistic evolution only explains how we got to have the physical stuff shaped and behaving like algae, trees, birds, horses and apes. God's creation is needed to make this stuff into actual algae, trees, birds, horses and apes: forms need to be put in. (Compare how in Genesis we are told that God made Adam from the "dust of the earth", i.e., physical stuff.)

The main reason I don't like this story has to be with my being an eternalist. I think past (and future) objects are real. And I think reality will be more wonderful if it really contains trilobites and dinosaurs, not just physical stuff arranged trilobitewise or dinosaurwise. So while the above story is basically logically possible, I don't think it's actually true, because it seems likely that a God whose goodness spreads itself out creatively would be likely to create forms for physical stuff arranged trilobitewise and dinosaurwise.

But there might, nonetheless, be aspects of the story that we can adopt. In the story, creation was safeguarded by God's taking bundles of physical stuff and giving them form. We can posit this in a more commonsensical evolutionary story. Maybe the gametes of two dinosaur parents involve some mutation that makes the offspring be particularly birdlike. Then God can simply prevent the offspring from being informed by a dinosaur form and then instead make the offspring be informed by a bird form. This is still direct creation of birds from previously existing physical matter. And while higher animals prior to the first human sin were conscious, we can suppose that God in his love miraculously prevented any instance of conscious suffering that wouldn't be on balance good for the animals, say by miraculously preventing pain qualia or in some other way. But this divine miraculous intervention perhaps ended (though perhaps not!) when Adam and Eve rejected God's rule over the earth they were put in charge of.

Moral sainthood and the afterlife

1. A moral saint can respond in a saintly way to everything the wicked can do to her.
2. If there is no afterlife, then a moral saint cannot respond in a saintly way to being instantly murdered.
3. The wicked can murder the moral saint.
4. So, there is an afterlife.

From properties to sets

If we have abundant properties in our ontology, do we need to posit a second kind of entities, the sets?

Properties are kind of like sets. If P is a property, write x∈P if and only if x has P. A whole bunch of the Zermelo-Fraenkel axioms then are quite plausible. But not all. The most glaring failure is extensionality. The property of being human and the property of being a member of a globally dominant primate species have the same instances, but are not the same property.

We can get extensionality by a little trick and an axiom. Assume the following Axiom of Choice for Properties:

1. If R is any symmetric and transitive relation, then there is a property P such that (a) if x has P, then x stands in R to itself, and (b) for all x if x stands in R to itself, there exists a unique y such that x stands in R to y and y has P.

Like the ordinary Axiom of Choice, this is a kind of principle of plenitude. Apply (1) to the relation C of coextensionality that holds between two properties if and only if they have the same instances. This generates a property S₁ that is had only by properties and is such that for any property P there exists exactly one property Q such that P and Q are coextensive and Q has S₁. In other words, S₁ selects a unique property coextensive with a given property.

To a first approximation, then, we can think of those entities that have S₁ as sets. Then every set is a property, but not every property is a set. We certainly have extensionality, with the usual restriction to allow for urelements (i.e., extensionality only applies to sets). All the other axioms of Zermelo-Fraenkel with urelements minus Separation, Foundation and Choice are pretty plausibly true (they follow from plausible analogues for properties on an abundant view of properties). We get Choice for sets for free from (1).

Unfortunately, we cannot have Separation, however. For the property S₁ is coextensive to some set U by our assumptions. And the members of U will just be the instances of S₁, i.e., all the sets. And so we have a universal set, and of course a universal set plus Separation implies Comprehension, and hence the Russell Paradox.

So matters aren't so easy. The Axiom of Foundation is also not so clear. Might there not be a self-instancing property?

Thus the above simple approach gives us too many sets. But there is a solution to this problem, and this is simply to postulate the following second axiom about properties:

2. There is a property S₂ of properties such that (a) concreteness has S₂, and (b) all the axioms of Zermelo-Fraenkel Set Theory with Urelements minus Extensionality are satisfied when we stipulate that (i) a set is anything that has S₂ and (ii) A∈B if and only if A is an instance of B.

This axiom is fairly plausible, I think.

Now suppose that S₁ is as before, and let S₂ be any property satisfying (2). Then let S be the conjunction of S₁ and S₂. It is easy to see that if we take our sets to be those properties that have S, we will have all of Zermelo-Fraenkel with Choice and Urelements (ZFCU). Or at least so it seems to me—I haven't written out formal proofs, and maybe I need some further plausible assumptions about what abundant properties are like.

Of course, we cannot expect S₁ and S₂ to be unique. So there will be multiple candidates for sets. That's fine with me.

The big question is whether (1) and (2) are true. But if the theoretical utility of positing sets is a reason to think sets exist, then theoretical utility plus parsimony plus the reasons to believe in properties are a reason to think (1) and (2) are true.

A disconnect between lay and philosophical pro-choicers

Without having done any scientific survey, I get the impression that philosophical pro-choicers tend to agree with philosophical pro-lifers on positive answers to the questions:

1. Does a fetus have basically the same intrinsic moral standing as a normal newborn baby?
2. Does the life of members of the human biological species begins at or around conception?

(In some cases, (1) will need to be qualified to: "fetus with brain states", and then the following discussion will need to be restricted to somewhat later abortions.) Of course, the pro-choice and pro-life philosophers disagree on the implications of these positive answers. Thus, pro-choice philosophers who give a positive answer to (1) will either say that killing a normal newborn is permissible or that it is wrong for reasons other than its intrinsic moral standing (e.g., the hurt to adults in our society). And pro-choice philosophers working on abortion tend to distinguish between us and members of our biological species, holding that we are constituted by and not identical with members of our biological species.

I also suspect, again without any scientific survey, that lay pro-choicers by and large answer (1) and (2) with "no". Moreover, I suspect that many of them think that (1) and (2) are crucial disputed questions in the discussion of abortion. In fact, it may be that quite a number of them think that abortion is permissible because the answers to (1) and (2) are negative and even accept the conditional:

3. If the answers to (1) and (2) are positive, then abortion is at least typically impermissible.

If so, then the position of lay pro-choicers is apt to be unstable. It is predicated at least in part on negative answers to (1) and (2), whereas the relevant experts—philosophers working on abortion, whether pro-choice or pro-life—tend to agree that the answers are positive.

Like I said, these are just anecdotal impressions. It would be valuable to have research on both lay and philosophical pro-choicers to see if these impressions are correct or not. Suppose it turns out that my anecdotal impressions turn out to be correct. Then the disconnect between lay and philosophical pro-choicers suggests that even if the philosophical debate is at a stalemate, there are ways for the social debate to move.

A funny discrete view of time

A number of my posts are exercises in philosophical imagination rather than serious philosophical theories. These exercises can have several benefits, including: (a) they're fun, (b) they expand the range of possibilities to think about and thus might contribute to a new and actually promising approach, and (c) they potentially contribute to philosophical humility by making us question whether the views that we take more seriously are actually better supported than these. This is one of those posts.

Suppose that time is discrete and made up of instants. However instead of saying that always some instant is present, we now allow for two possibilities. Sometimes an instant is present. But sometimes presently we are between instants. When an instant is present, there is a present moment. When an instant is not present, when we are between instants, there is a present interval, bounded by the last past instant and the first future instant.

Why posit that sometimes we are between instants? Because this lets us get out of Zeno's paradox of the arrow. Zeno notes that at no instant is the arrow moving, because at no instant does it occupy two places, and so the arrow never moves. But now that we have two possibilities, that of an instant being present and of an interval being present, we see that Zeno's inference from

1. At no instant is the arrow moving

to

2. The arrow never moves

uses the implicit assumption that we are always at an instant. But if sometimes instead of being at an instant we are between them, we are at an interval, then the inference fails. And indeed when instead of a present moment we have a present interval, we can say that the arrow really is moving in the present—it is in two places in the present, in one place in the last past instant and in another in the first future instant.

So we have positions when an instant is present and velocities when an interval is present.

Of course there are other ways out of the Zeno paradox of the arrow, the best of which is to adopt the at-at theory of motion. But it's nice to have other solutions besides the usual ones.

The traveling minds interpretation of indeterministic theories

I'm going to start by offering a simple way—likely not original, but even if so, not very widely discussed—of turning an indeterministic physical theory into a deterministic physical theory with an indeterministic dualist metaphysics. While I do not claim, and indeed rather doubt, that the result correctly describes our world, the availability of this theory has some rather interesting implications for the mind-body and free will and determinism debates.

Start with any indeterministic theory that we can diagram as a branching structure. The first diagram illustrates such a theory. The fat red line is how things go. The thin black dotted lines are how things might have gone but didn't. At each node, things might go one way or another, and presumably the theory specifies the transition probabilities—the chances of going into the different branches. The distinction between the selected branches and the unselected branches is that between the actual and the merely possible.

The Everett many-worlds interpretation of Quantum Mechanics then provides us with a way of making an indeterministic theory deterministic. We simply suppose that all the branches are selected. When we get to a node, the world splits, and so do we its observers. All the lines are now fat and red: they are all taken. There are some rather serious probabilistic problems with the Everett interpretation—it works best if the probabilities of each branch coming out of a node are equal, but in general we would not expect this to be true. Also, there are serious ethics problems, since we don't get to affect the overall lot of humankind—no matter which branch we ourselves take, there will be misery on some equally real branches and happiness on others, and we can do nothing about that.

To solve the probabilistic problems, people introduce the many-minds interpretation of the many-worlds interpretation. Each person has infinitely many minds. When we get to a branch point, each mind indeterministically "chooses" (i.e., is selected to) an outgoing branch according to the probabilities in the physics. Since there are infinitely many of these minds, at least in the case where there are finitely many branches coming out of a node we will expect each outgoing branch to get infinitely many of the minds going along it. So we're still splitting, and we still have the ethics problems since we don't get to affect the overall lot of humankind—or even of ourselves (no matter which branch we go on, infinitely many of our minds will be miserable and infinitely many will be happy).

But now I want to offer a traveling minds interpretation of the indeterministic theory. On the physical side, this interpretation is just like the many-worlds interpretation. It is a dualist interpretation like the many-minds one: we each have a non-physical mind. But there is only one mind per person, as per common sense, and minds never split. Moreover our minds are all stuck together: they always travel together. When we come to a branching point, the physical world splits just as on the many-worlds interpretation. But the minds now collectively travel together on one of the outgoing branches, with the probability of the minds taking a branch being given by the indeterministic theory.

In the diagram, the red lines indicate physical reality. So unlike in the original indeterministic theory, and like in the many worlds interpretation, all the branches are physically real. But the thick red lines and the filled-in nodes, indicate the observed branches, the ones with the minds. (Of course, if God exists, he observes all the branches, but here I am only talking of the embodied observers.) On the many-worlds interpretation, all the relevant branches were not only physically real, but also observed. Presumably, many of the unobserved branches have zombies: they have an underlying physical reality that is very much like the physical reality we observe, but there are no minds.

The traveling minds interpretation solves the probability problems. The minds can travel precisely according to the probabilities given by the physics. Traveling minds as generated in the above way will have exactly the same empirical predictions as the original indeterministic theory. (In particular, one can build traveling minds from a Copenhagen-style consciousness-causes-collapse interpretation of Quantum Mechanics, or a GRW-style interpretation.)

Traveling Minds helps a lot with the ethics problem that many-worlds and many-minds faced. For although physical reality is deterministically set, it is not set which part of physical reality is connected with the minds. We cannot affect what physical reality is like, but we can affect which part of physical reality we collectively experience. And that's all we need. Note that "we" here will include all the conscious animals as well: their minds are traveling as well. In fact, as a Thomist, I would be inclined to more generally make this a "traveling forms" theory. Thus the unselected branches not only have zombies, but they have physical arrangements like those of a tree, but it's not a tree but just an arrangement of fields or particles because it lacks metaphysical form. But in the following I won't assume this enhanced version of the theory.

Now while I don't endorse this theory or interpretation—I don't know if it can be made to fit with hylomorphic metaphysics—I do want to note that it opens an area of logical space that I think a lot of people haven't thought about.

Traveling minds is an epiphenomenalist theory (no mind-to-physics causation) with physical determinism, and is as compatible with the causal closure of the physical as any physicalist theory (it may be that physicalist theories themselves require a First Cause; if so, then so will the traveling minds theory). Nonetheless, it is a theory that allows for fairly robust alternate possibilities freedom. While you cannot affect what physical reality is like, you can affect what part of physical reality we collectively inhabit, and that's almost as good. We have a solution to the mind-to-world causation problem for dualism (not that I think it's an important problem metaphysically speaking).

I expect that I and other philosophers have incautiously said many things about things like epiphenomenalism, determinism and causal closure that the traveling minds theory provides a counterexample to. For instance, while traveling minds is a version of epiphenomenalism, it is largely untouched by the standard objections to epiphenomenalism. For instance, one of the major arguments against epiphenomenalism is that if minds make no causal difference, then I have no reason to think you have a mind, since your mind makes no impact on my observations. But this argument fails because it assumes incorrectly that the only way for your mind to make an impact on my observations is by affecting physical reality. But your mind can also make an impact on my observations by leaving physical reality unchanged, and simply affecting which part of physical reality we are all collectively hooked up to.

Particles

I used to worry for Aristotelian reasons about the particles making up my body. The worry went something like this: Elementary particles are fundamental entities. Fundamental entities are substances. But no substance has substances as parts. The last is, of course, a very controversial bit. However there are good Aristotelian reasons for it.

But I shouldn't have worried much. Elementary particles are not all that likely to be fundamental entities. Quantum mechanics, after all, allows all sorts of superpositions between different particles. But substances either simply exist or simply don't. In the superposition case, they don't simply exist. So they simply don't. But I would expect that the superposition case is more the rule than the exception (if only with small coefficients for all but one one state). I guess we could think that when the wavefunction is in a pure state with respect to the existence of a particle, the particle then pops into existence, and when the state becomes mixed, it pops right out. But notice that the physics behaves in much the same way when we have a pure state and when we have a mixed state that is to a very high approximation pure. So whatever explanatory role the particles play when they pop into existence can be played, it seems, by the wavefunction itself when the particles aren't around. This suggests that the wavefunction is the more explanatorily fundamental entity, not the particles. Of course, the above relies on denying the Bohmian interpretation of quantum mechanics. But it's enough, nonetheless, to establish that elementary particles aren't all that likely to be fundamental entities. And hence they aren't all that likely to be substances.

Of course, it may be that the things that are fundamental physical entities will turn out to be just as problematic for the Aristotelian as the particles were...

The simplest way to run an infinite fair lottery?

I've posted two ways to run an infinite fair lottery (this and this). There is also a very simple way. Just take infinitely many people and have them each independently toss an indeterministic fair coin. If you're lucky enough that exactly one person rolls heads, that's the winner. Otherwise, the lottery counts as a failure. The probability of failure is high—it's one—but nonetheless success should be causally possible. And if you succeed, you've got what is intuitively an infinite fair lottery.

My earlier thought experiments requires a version of the Axiom of Choice. This version doesn't, but the earlier ones has the merit of working always or almost always. However, for the purposes of generating paradoxes and supporting causal finitism this version might be good enough.

A note to fellow mathematicians: Any mathematician reading this and some of my other posts on infinite fair lotteries is apt to be frustrated. There is a lot that isn't rigorous here. But I'm not doing mathematics. One can perhaps best think of what I'm doing as a physicsy thought experiment. When I think of independent indeterministic coin flips, take these as actual causally-independent physical processes, e.g., each indeterministic coin flip happening in a different island universe of an infinite multiverse. I am fully aware, for instance, that the stuff I say in this post isn't fully modeled by the standard Kolmogorovian probability theory. For instance, an infinite sequence of i.i.d.r.v.'s X_n with P(X_n=1)=P(X_n=0)=1/2 need not have any possible state such that exactly one of the variables is 1, depending on how the i.i.d.r.v.'s are constructed. That's an artifact of the fact that probabilistic independence as normally defined is not a sufficient model of genuine causal independence (see here). I am also assuming that permutation symmetries in the space of coin flips persist even when we consider nonmeasurable or null sets. Again that's going beyond the mathematics, but justified as a physicsy thought experiment. If we put each coin flip in a relevantly similar separate universe of a multiverse, then of course everything should be intuitively invariant under permutations of the coins. Probabilities understood vaguely as measures of rational believability go beyond the mathematical theory of probability.

Antipresentism

Presentists think that the past and future are unreal but the present is real. I was going to do a tongue-in-cheek post about an opposed view where we have the past and future but no present. But as I thought about it, the position grew a little on me philosophically, at some expense of the tongueincheekness. Still, please take all I say below in good fun. If you get a plausible philosophical view out of it, that's great, but it's really just an exercise in philosophical imagination.

One way to think about antipresentism is to imagine the eternalist's four-dimensional universe, but then to remove one slice from it. Thus, we might have 1:59 pm and 2:01 pm, but no 2:00 pm. Put that way, the view isn't particularly attractive. Still, I do wonder why it would be more unattractive to remove just one time slice than to remove everything but that one time slice as the presentist does. It would, of course, be weird for the antipresentist to say that events first exist in the future, then pop out of existence just as one would have thought that they would come to be present, and then pop back into existence in the past. But perhaps no weirder than events coming out of nothing and going back into nothing, as on presentism. This way to think about antipresentism makes it a species of the A-theory.

But the antipresentisms I want to think about are ones that might be compatible with the B-theory. Start with the famous puzzles of Zeno and Augustine about the now. Augustine worried about the infinite thinness of the now. Zeno on the other hand worried about the fact that there are no processes in the now; there is no change in the now since within a single moment all is still.

One way of taking these ideas seriously is to see the present as an imaginary dividing line between the past and the future. There is in fact no dividing line: there is just the past and the future. (I think Joseph Diekemper's work inspired this thought.)

We might, for instance, instead of thinking of times as instants think of the basic entities as temporally extended events or time intervals, not made out of instantaneous events or moments. An event or interval might be past, or it might be future, or—like the writing of this post—it might be both past and future. (Thus, "past" and "future" is taken weakly: "at least partly past" and "at least partly future".) Some events or time intervals have the special property of being both past and future. We can stipulate that those events or time intervals are present. But they aren't real because they are present. They're just lucky enough to have two holds on reality: they are past and they are present. (In this framework, the presentist's claim that only present events are real sounds very strange. For why should reality require both pastness and futurity—why wouldn't one be enough?) There are no events or time intervals that are solely present.

There is a natural weakly-earlier-than relation e on events. If we had instants of time, we would say that EeF if and only if some time at which E happens is earlier than some time at which F happens. But that's just to aid intuition. Because there are noever instantaneous events, every event is weakly earlier than itself: e is reflexive. It is not transitive, however. The antipresentist theory I am sketching takes e to be primitive. There is also a symmetric temporal overlap relation o that can be defined in terms of e: EoF if and only if EeF and FeE.

If we like, we can now introduce abstract times. Maybe we can say that an abstract time is a maximally pairwise overlapping set of time intervals (or of events, if we prefer). We can say that t₁ is earlier than t₂ provided that some element of t₁ is strictly earlier than some element of t₂ (where E is strictly earlier than F provided EeF but not FeE). I haven't checked what formal properties this satisfies—I need to get ready for class now (!).

How to make an infinite fair lottery out of infinitely many coin flips

This is a technical post arising from a question Rob Koons asked me.

An infinite sequence of fair and independent coin flips determines a sequence of zeroes and ones (e.g., zero = tails, one = heads). Let Ω be the set of all infinite sequences of zero/one sequences, equipped with the probability measure P corresponding to the fair and independent coin flips.

Notice an invariance property capturing at least part of the independence and fairness assumption. If ρ_n is the operation of flipping the nth element in the sequence, and ρ_nA for a subset A of Ω is the set obtained by applying ρ_n to every sequence in A, then P(ρ_nA)=P(A) whenever A is measurable. Moreover, intuition extends this idea beyond the measurable sets: A and ρ_nA are always going to be probabilistically on par.

Let Ω₀ be the subset of Ω consisting of those sequences that have only finitely many ones in them. There is a natural one-to-one correspondence between Ω₀ and the natural numbers N. Suppose a=(a₀,a₁,...,a_k,0,0,0,...) is a member of Ω₀. Then let N(a) be the natural number whose binary digits are a_k...a₁a₀. Conversely, given a natural number n with binary digits a_k...a₁a₀, let n* be the sequence (a₀,a₁,...,a_k,0,0,0,...) in Ω₀. Thus, we can interpret the members of Ω₀ as binary numbers written least significant digit first.

For any members a and b of Ω, write a#b for the sequence whose nth element is the sum modulo 2 (xor) of the nth elements of a and b. For a subset B of Ω, let a#B = { a#b : b∈B }. We can think of a#B as a twist of B by a. If a is in Ω₀, I will call it a finite twist. Any finite twist can be written as a finite sequence of flips ρ_n, where the positions n correspond to the non-zero digits in the sequence we twist by. Thus, if A is measurable, a finite twist of it will have the same probability as A does, and even if A is not measurable, a finite twist will be intuitively equivalent to A.

Say that a~b if and only if a and b differ in only finitely many places. Thus, a~b if and only if a#b is a member of Ω₀. This is an equivalence relation. By the Axiom of Choice, there is a set A₀ such that for every b in Ω, there is a unique a in A₀ with a~b. (Thus, A₀ contains exactly one member of each equivalence class.) For any natural number n other than 0, let A_n=n*#A₀ and it's easy to check that this equation holds for n=0 as well.

It's easy to see that the A_n are disjoint and their union is all of Ω. They are disjoint because if a is in n*#A₀ and m*#A₀, then a=n*#b and a=m*#c for b and c in A₀. It follows that b~c. But A₀ contains only one member from each equivalence class, so b=c, and so n*#b=m*#b, from which it obviously follows that n*=m* and so n=m. Their union is all of Ω, because if b is in Ω, and a is the unique member of A₀ such that a~b, then N(a#b)*#a=(a#b)#a=b (by obvious properties of addition modulo 2), and so b is a member of A_N(a#b).

But all the A_n are going to be intuitively probabilistically on par: they are each a finite twist of A₀.

Our lottery is now obvious. Given a random sequence of coin flips, we take its representation a in Ω and choose the unique number n such that a is in A_n.

This is really the Vitali-set construction applied directly to sequences of coin flips. Note that along the way we basically showed that Ω has nonmeasurable subsets. For the sets A_n cannot be measurable with respect to P, since they would all have equal probability, and so by countable additivity they would have to have probability zero, which would violate the total probability axiom.

The construction in this post is more complicated than the one here, I guess, but it has the advantage that it always works, while that construction only worked with probability 1.

A divine command and an open future

I'm piling on to the argument here.

Suppose God creates Adam and Eve, and gives them eternal life. He then commands them that:

1. They freely pray for at least a minute on each of the infinitely many Sabbaths starting with day t₇ (the day after their creation).

This seems a reasonable command. But it is unreasonable to command something that the agent cannot ever make true. And on open future views, it is impossible for (1) ever to be true. For at any time, (1) depends on future free choices. So on open future views, the command (1) is unreasonable. And that's a problem for open future views.

Yet another infinite population problem

There are infinitely many people in existence, unable to communicate with one another. An angel makes it known to all that if, and only if, infinitely many of them make some minor sacrifice, he will give them all a great benefit far outweighing the sacrifice. (Maybe the minor sacrifice is the payment of a dollar and the great benefit is eternal bliss for all of them.) You are one of the people.

It seems you can reason: We are making our decisions independently. Either infinitely many people other than me make the sacrifice or not. If they do, then there is no gain for anyone to my making it—we get the benefit anyway, and I unnecessarily make the sacrifice. If they don't, then there is no gain for anyone to my making it—we don't get the benefit even if I do, so why should I make the sacrifice?

If consequentialism is right, this reasoning seems exactly right. Yet one better hope that it's not the case that everyone reasons like this.

The case reminds me of both the Newcomb paradox—though without the need for prediction—and the Prisoner's Dilemma. Like in the case of the Prisoner's Dilemma, it sounds like the problem is with selfishness and freeriding. But perhaps unlike in the case of the Prisoner's Dilemma, the problem really isn't about selfishness.

For suppose that the infinitely many people each occupy a different room of Hilbert's Hotel (numbered 1,2,3,...). Instead of being asked to make a sacrifice oneself, however, one is asked to agree to the imposition of a small inconvenience on the person in the next room. It seems quite unselfish to reason: My decision doesn't affect anyone else's (I so suppose—so the inconveniences are only imposed after all the decisions have been made). Either infinitely many people other than me will agree or not. If so, then we get the benefit, and it is pointless to impose the inconvenience on my neighbor. If not, then we don't get the benefit, and it is pointless to add to this loss the inconvenience to my neighbor.

Perhaps, though, the right way to think is this: If I agree—either in the original or the modified case—then my action partly constitutes the a good collective (though not joint) action. If I don't agree, then my action runs a risk of partly constituting a bad collective (though not joint) action. And I have good reason to be on the side of the angels. But the paradoxicality doesn't evaporate.

I suspect this case, or one very close to it, is in the literature.