Scepticism and causeless events

Suppose that there is no First Cause. Then there can be uncaused events—the coming into existence of the universe is an example, for instance. Now consider the Ultimate Sceptical Hypothesis (USH): you are a nonmaterial being that is the only thing that ever exists; you came into existence the previous moment for no cause at all; and there is no cause of your having the presnet occurrent mental states you now have; and you have just these occurrent mental states and no other states.

Compare, now, USH to what its main nonsceptical alternative is if there is no First Cause. That main alternative will be SN: scientific naturalism, with the initial state of the universe being a brute, unexplained fact. USH is simpler than SN. It is simpler on the crude criterion of entity counting: USH has only one entity, you, while SN has many atoms, galaxies, houses, geckos, etc. However, if we count only unexplained entities, as I suggested in a previous post, USH has only you and your present occurrent mental states (which there aren't many of!), while PN has the universe and its initial state, so maybe we have a tie. But PN is much more descriptively complex: it includes a number of laws of nature with various constants, for instance, as well as a high-energy extremely low entropy initial state. While you just have whatever occurrent mental states you now have—which is not much at all (how much of a thought can you think in an instant). So USH seems to be preferable to PN on grounds of parsimony.

Thus, rejecting a First Cause leads to scepticism.

This is, of course, a variant of a Rob Koons argument.

Music, religion and appreciation

I typically do not appreciate music at all. While there are rare exceptions, music typically leaves me aesthetically cold or annoys me (though there may be a non-aesthetic emotional impact, say of creepy music during a scary part of a movie). This inability to appreciate music is a kind of disability, one that I hope will be gone in heaven (plus the music there will be better), since music seems an important part of the human good of aesthetic appreciation.

I suspect that how I typically feel about music is how many (though not all) non-religious people feel about religion: while it may be good for others, it's just not something one finds oneself getting anything out of. But I think there is a crucial disanalogy. For it is uncontroversial that to be properly benefited by receptive aesthetic goods, like those proper to listening to music or contemplating a painting, one needs to experience them with appreciation. One gets nothing from musical goods without listening to the music, and mere listening gets one nothing of the aesthetic good if one doesn't appreciate. (Though experiencing the art without appreciation can lead to later development of appreciation, and an analogous claim can be true of religious practice.) But according to many of the great religions, many of the goods of participation—say, innate transformations of the soul, the intrinsic value of praising God, etc.—can occur in the absence of experiential appreciation.

There is also another disanalogy. Participating in religious goods isn't exactly analogous to experiencing works of art. Rather, it is analogous both to experiencing and to creating them. And creating works of art is an aesthetic good that perhaps does not require appreciation of the works of art that one is creating. One could have a sculptor who manages to express her artistic vision in incredible ways, but who incorrectly experiences herself as producing junk. The artist need not understand her work.

"Love does not seek its own"

Here is another excerpt from Section 2.2. of my forthcoming One Body book.

One way love is humble is that the actions of love are not focused on agapê itself [...]. There would be something odd about a parent explaining why he stayed up the night with a sick child by saying: “I love my son.” Surely the better justification would be the simpler: “He is my son.” The latter justification puts the parent in a less grammatically prominent spot (“my” instead of “I”), and shows that the focus is on the son. Most importantly, however, the use of “I love my son” as a justification would suggest that if one did not love him, the main reason to stay up the night would be missing. But the main reason to stay up the night is that he is one’s son. That he is one’s son is also a reason to love him as one’s son, and that one loves him may provide one with a further reason to stay up with him. However, the main reason for staying up is not that one loves him; rather, the love, expressed in the staying up, is a response to a reason that one would have independently of the love. Thus, in an important sense, the parent acts lovingly—acts in a way that is at least partly constitutive of love—without acting on account of love. Love’s actions are not focused on love but on the beloved as seen in the context of a particular relationship.

However, to explain why we made some sacrifice for someone to whom we had no blood ties, we might well say, “I love him.” Nonetheless, I suggest, this may be an imperfection—it may be a case of seeking one’s own. Why not instead act on account of the value of the other person in the context of the relationship? It is true that love may be a central part of that relationship, but I want to suggest that love is not the part of the relationship that actually does the work of justifying the sacrifice. For suppose that I stopped loving my friend. Would that in itself take away my obligation to stand by him in his time of need? Certainly not. The commitment I had implicitly or explicitly undertaken while loving him, a commitment that made it appropriate for him to expect help from me, is sufficient for the justification. If I need to advert to my own love, then something has gone wrong.

Besides, there would a circularity in appealing to one’s own present love to justify one’s basic willingness to engage in loving actions for the beloved. For if one were not willing to do loving actions for the other, then one would not be loving the other, and hence a total failure to will to do loving actions for the other would not be a violation of love, for there would be no love there to be violated. Of course, such a failure might well be a violation of one’s duty to love the person (whether arising out of personal commitment, or a general duty to love everyone or some specific duty like those we have to our relatives), but that is a different issue. It is not love, then, that justifies the general willingness to act lovingly, but the value of the other and the kind of relationship that one stands in to the other apart from the fact of love.

Simplicity and multiple universes

In yesterday's post, I offered a criterion for when multiplicity of objects or kinds posited by a theory counts against the simplicity of a theory: namely, when it is a multiplicity of objects or kinds not explained by the theory.

Let's apply this to multiple universe theories, which people do tend to see as offending against simplicity.

Lewis's Modal Realism: Lewis's universes have no explanation of their existence. Their existence is simply a brute fact. Thus, the infinitude of Lewis's universes counts against the simplicity of a theory. Moreover, along with unexplained universes there will be unexplained kinds. Lewis's theory, for instance, implies that there exists a universe where an uncaused griffin exists from the beginning. Likewise, unicorns, Pegasuses, and so on. So Lewis's theory implies the existence of a great diversity of unexplained kinds of things. And that should count against the theory.

Theistic Multiverses: Theistic multiverse theories have an infinity of universes, but these universes are explained by God's goodness in creating all universes that it is worth creating. Thus there is only one unexplained entity in the theory--God--and there is no offense against simplicity.

Physicists' Multiverses: I don't know enough about string-theoretic multiverses to say anything about those here. But inflationary universes that bubble up out of other universes will be exempt from the worry if there is one root universe from which the others come, since then the cost in terms of unexplained entities is the same as that of single universe theories, and there is no offense against simplicity.

Simplicity

Consider two theories.

Theory 1: There is a single asexually reproducing ancestor of all life on earth, Ag, who came into existence at time A and split into two almost genetically identical descendants, Bof and Bok, at time B, and there was no biological life before Ag.

Theory 2: There is a pair of almost genetically identical ancestors of all life on earth, Bof and Bok, who came into existence at time B, and there was no biological life before Bof and Bok.

Both theories fit our observed data equally well, and will always do so, since we have basically no chance of identifying a fossil of Ag. The question I want to ask is which theory is simpler.

An easy thing to say is that Theory 2 is simpler, as it posits one fewer entity, namely Ag, but Theory 1 compensates for its greater complexity through having an additional explanatory merit--it explains the genetic similarity between Bof and Bok, which Theory 2 leaves unexplained. Moreover, the difference in complexity is small, because while Theory 1 posits one more entity, it is an entity of the same kind as Bof and Bok.

But I want to consider a different evaluation: Theory 1 is simpler, because when we consider the simplicity of a theory in terms of entity or kind counting, we only count the entities or kinds not explained by this theory (or, better, entities or kinds weighted by the degree to which they are unexplained by the theory, if explanation comes in degrees). Thus, Theory 1 posits one theory-unexplained entity while Theory 2 posits two theory-unexplained entities. So Theory 1 wins not only on explanatory grounds, but also on simplicity grounds.

Why go for this method of evaluation? First, in cases where the explanation is deterministic, it coheres with information-theoretic compression-based measures of complexity, which is a plus. Second, I think this fits with our intuitions about other examples. Consider two theories about the origin of life on earth.

Theory 3: A meteorite deposited some organic chemicals 4.1 billion years ago that combined to produce life.

Theory 4: A meteorite deposited some organic chemicals 3.9 billion years ago that combined to produce life.

As far as the details I gave of the two theories, there is no difference in complexity. But Theory 3 commits us to way more organisms in the history of the earth--200 million years' worth of organisms. And when we conjoin with evolutionary theory, Theory 3 will commit us to significantly more kinds of organisms as well--over those 200 million years, surely there would be a lot more species. But these added entities (or kinds) should not by themselves count as increasing the complexity of Theory 3 over Theory 4 (or Theory 3 + evolution over Theory 4 + evolution). Why not? The best explanation of why not seems to me to be that these added entities and kinds are easily explained by the theory in question.

The entities that are unexplained by one theory may, of course, be explained by another. It is only entities posited but unexplained by the theory in question that I am considering here. Nor am I saying that this is the only contribution to complexity--there are, no doubt, many others.

All this casts helpful light on the question whether theism or naturalism is simpler. At least it undercuts simple arguments that naturalism is simpler as it posits one fewer entity or even kind of entity.

Kant's argument from gratitude

Kant famously argued that we had a duty to believe in God, as this was necessary for us to fulfill the duty to give thanks for the universe. My understanding is that Kant thought that this was not an argument for the existence of God, but only an argument for the duty to believe in God. But surely it becomes an argument for the existence of God (or at least an agent who caused the universe) when one adds the very plausible premise:

• One only has the duty to give thanks for a product of agency.

Perhaps, though, Kant would want to subjectivize this premise into:

• One only has the duty to give thanks for what one believes to be a product of agency.

But if one says this, then the atheist doesn't have a duty to give thanks for the existence of the universe, and Kant's argument fails. So it seems that either Kant's argument fails (at least in the case of atheists--maybe you could argue that if you believe in God then you have a duty to believe in God, which is an interesting result, but I don't think it's what Kant was trying to argue for) or Kant is wrong that we can't argue for the existence of God or both.

But I am no Kant scholar.

Prayer and Thomistic accounts of chance and design

On Thomistic accounts of chance and design, God micromanages the outcomes of chancy processes by means of primary causation, ensuring that the processes secondarily cause precisely the results that God wants. (Thomists often say a similar thing about free will, too.) On such accounts we can distinguish between two different ways that God can achieve a result, which I will call the miracle and natural methods. In the miracle method, God suspends the causal powers of the chancy process and directly cause the specific outcome he wants. If he does this on a die toss (I'll assume that die tosses are indeterministic), then the hand tosses the die, but somewhere there will be a break in the natural chain of causes. In the natural method, God causes the the causal powers of the chancy process to cause, in the way proper to them, the specific outcome he wants. Presumably, given that the natural method preserves the value of finite causes' activity, much of the time God providentially acts using the natural rather than the miracle method.

Now suppose that I am about to toss a die. And suppose that I pray, for all the right reasons (say, a good to a friend will result from non-six, and nobody will be harmed by it) and in the right way, that the die should show a non-six, while no one else prays that it should show a six. Moreover, suppose that God in fact does not have any significant counterbalancing reasons in favor of the die showing six. Let C be a complete description of the state of the world--including all the facts about the universe on which God's reasons are based--just before the die toss result. This seems a paradigmatic case for God to be moderately likely to exercise providential control. Moreover, let us suppose with the typical Thomist that almost all the time, excepting cases of particularly spectacular demonstrations, God exercises providential control by the natural method. Suppose then:

• P(God wills non-six | C and no miracle) > 0.95.

And since God wills non-six if and only if non-six occurs on the Thomistic view:

• P(non-six occurs | C and no miracle) > 0.95.

Suppose that in fact three occurs. It is then obviously correct to explain the non-six by adverting to the above 0.95 probability. The question of interest to me is this: Can we also explain the non-six by the fact that natural causes described in C, in isolation from the facts about prayer and the like, had a probability of 5/6 of producing a non-six?

Probability and divine will

The Thomistic reconciliation of design with chance—of which Barr's reconciliation is a special case—holds that, necessarily, each particular chancy event occurs precisely because God causes it with primary causation. Now, if p and q are propositions that have the property that, necessarily, p is true if and only q is true, then P(p)=P(q) and P(p|r)=P(q|r) for any proposition r such that P(r)>0 and, in fact, intuitively, for any possibly true r for which the conditional probabilities are defined. Suppose:

1. P(the coin lands heads)=1/2.

Then:

2. P(God primarily causes the coin to land heads)=1/2.

But why should the probability of God's primarily causing the coin to land heads be exactly 1/2? Indeed, why should there be numerical probabilities of God's choices at all? (Note: It won't help to say that the probabilities are conditional on some background. For whatever background they are conditional on, as long as the background is possible, the conditional probabilities of the coin landing heads and of God primarily causing the coin to land heads will be 1/2.)

If the probabilities are epistemic, there may be less of a problem. For typically we have no reason to think God prefers the coin to land heads than to land tails or vice versa, and so the epistemic probability of his causing it to land heads may be 1/2. (Generalizing this to other cases may be problematic. How would this work for a Poisson or Gaussian distribution? To suppose a Poisson or Gaussian distribution on God's preferences would be weird.)

But if the probability in (1) is merely epistemic, then it isn't going to be useful for explaining why of a hundred tosses about fifty landed heads. Maybe one could still explain it by saying that God's preferences are likely to be randomly or quasi-randomly distributed, because of the great diversity of factors that affect God's choices about different coin flips. But then it is (2) rather than (1) that is the real explanation of why about fifty tosses landed heads: our explanation essentially involves a random distribution on the factors that God's decision is made on the basis of.

A non-Thomist (and by that I just mean someone who doesn't accept this reading of Thomas and the corresponding reconciliation—she might be a Thomist in all sorts of other ways) could say that God doesn't specifically choose which way each coin toss goes, but cooperates with the coin-tosses in a way that does not determine the specific outcome. Of course, God can still work a miracle and specifically choose a coin outcome, but then that outcome will be miraculous (in a weak sense of the word not implying God's self-revelation in the event) and not random. Such a non-Thomist will then say:

3. P(the coin lands heads | no miracle)=1/2

and

4. P(God cooperates in a way that results in the coin landing heads | no miracle)=1/2.

And there is no surprise about (4) since on this kind of a view God's non-miraculous cooperation with chancy processes (or free choices for that matter) does not micro-manage the outcome. But the non-Thomist will then have to work hard to reconcile design with chance.

I do not think this is available to the Molinist: I suspect it only works on simple foreknowledge (or open theist, for that matter) views.

Mike Almeida has told me that he has worried about the coincidence between (1) and (2) as well.

Barr on chance and design

Stephen Barr has an article in First Things where he argues that there is no conflict at all between chance and divine design. The position seems to hinge on two claims:

1. A series of events is chancy if and only if the secondary (i.e., finite, non-divine) causes of the events are independent of one another.
2. God controls series of events by primary causation.

Given (1) and (2), there is no conflict between divine design, since (1) says nothing about dependencies among events induced by primary causation.

I wish Barr's account worked. I'd love for there to be a good account of the interplay of chance and design. But there are a number of serious problems with Barr's proposal.

I. The account of chanciness does not work in the case of a single event. Depending on how we read (1), a single event will either trivially count as chancy (since its cause is independent of the causes of all other events in the series, there being no others) or it will never count as chancy. But single events can be chancy (imagine a universe where there is only one quantum collapse happening) or non-chancy.

II. Chance is explanatory, both in gambling and in evolution. But independence of causes has no explanatory force--without probabilities or chances, it generates no useful statistic predictions or explanations (here is a very technical way to make the point). So Barr's account needs something more, something like objective tendencies of the secondary causes that give rise to probabilities. I will assume in some of the following criticisms that something like this has been added.

III. Suppose I go to the casino and I play the slot machine a thousand times, and each time win, due to independent secondary causes, because God so arranged it. It seems absurd to say that I won by chance, and yet on Barr's definition my winning is a matter of chance.

IV. The preceding case shows that it is difficult to see how one can make any probabilistic predictions about a chancy series of events. Consider an infinite sequence of coin tosses where the limiting frequency of heads is 1/2. God can just as easily make this infinite sequence of heads come out in the case of an ordinary fair coin as in the case of a coin heavily biased in favor of heads. When God controls series of events by primary causation—and as far as Barr's position goes, this could be always—it is not clear why we should expect frequencies to match the probabilities arising from the tendencies of secondary causes. The frequencies of events will be precisely what God needs them to be for his purposes. Why think his purposes match the probabilistic tendencies of secondary causes? Now it could be that God wills to ensure that the actual frequencies usually match the secondary causal tendencies, in order that the universe be simpler and more predictable. That is a reasonable hypothesis. But then the it seems that it isn't the secondary causal tendencies that are directly explanatorily of the observed frequencies, but rather the explanation of the observed tendencies is God's will. I.e., in the case of a fair coin, the reason the limiting frequency comes out as heads is because God willed to ensure that the limiting frequency match the secondary causal tendency of the coin. The secondary causal tendency of the coin is still explanatorily responsible for this outcome (because God willed to match the frequency to it), but it isn't causally responsible for this outcome (unless we take an occasionalist analysis of secondary causation).

V. In light of III and IV, no statistical prediction can be made from probabilistic facts about the causal tendencies of secondary causes without an implicit auxiliary hypothesis that God through primary causation willed a particular series of events whose statistical features match the stochastic features of the causes. This is not entirely special to Barr's account—-probably every theistic account requires an implicit auxiliary hypothesis that God works no miracle here. But in Barr's case there is a difference—it's not just a hypothesis that God works no miracle here, since in the case where God makes me win the slot machine a thousand times in a row, on Barr's view no miracle has occurred, just the ordinary chancy operation of secondary causes and God's primary causal oversight. So Barr's view needs two auxiliary hypotheses to generate empirical predictions from scientific data: a no-miracles hypothesis like in every theistic case and a hypothesis of stochastic-to-statistical match.

VI. Random processes need not involve independent causes. Take, for instance Markov chains or exchangeable sequences of random variables.

VII. The elliptical orbits of the planets in our solar system and of the planets in another solar system have independent causes—the gravitational influences of different bodies—and hence by Barr's criterion the two events are chancy. But it's not chance that the orbits are elliptical. Now maybe Barr will count these cases as not independent because they are governed by the same laws of nature. True, they are. But so are paradigmatically chancy events, like the results of successive quantum collapse experiments.

A characterization of naturalism

It's hard to define naturalism. After all, even if there were souls and the like, naturalists could still treat them as natural phenomena.

Maybe a better way to characterize naturalism is that it is the view that objectively speaking in itself there is nothing numinous: Nothing holy or sacred, but only the good or right; nothing sinful or unholy, but only the morally wrong; nothing uncanny or eldritch, but only the unusual or the scary; nothing aweful, but only the impressive; nothing mysterious, but only the puzzling or the strange; nothing fascinating, but only the attractive; nothing sublime, but only the beautiful.

Some naturalists will have an error theory about the holy, sacred, sinful, unholy, uncanny, eldritch, aweful, mysterious and fascinating. Others will say that such that such predicates can be rightly applied, but they indicate in large part our attitudes to these things, rather than indicating the intrinsic characteristics of things that make those attitudes appropriate.

(It is also interesting that even some of the de-sacralized replacements—especially the good, the right, the wrong and the beautiful—are troubling to many naturalists.)

Why infinitesimals are too small to help with infinite lotteries: Part IV

In previous parts, I argued that infinitesimals, or at least hyperreal infinitesimals, are too small to be the outcomes of a countably infinite lottery. Now it is time to extend this result to one hyperreal infinitesimal assignment in an uncountable lottery. Consider, then, the case of a uniform distribution on the interval [0,1) = { x:0≤x<1 }, say induced by a dart being thrown at a linear target. Bernstein and Wattenberg (1969) have shown that there is a hyperreal valued measure on all subsets of [0,1) such that (a) it is finitely additive, (b) it assigns infinitesimal probability to each singleton, and (c) it is almost translation invariant, in the sense that P(A@x) is within an infinitesimal of P(A), where A@x = {y@x : y in A}, and where y@x is addition modulo 1 (so, 0.5@0.7 = 0.2 and 0.2@0.3 = 0.5).

Now, just as in the standard construction of nonmeasurable sets, define the equivalence relation x~y on [0,1) by saying it holds if and only if there is a rational number q such that x@q=y. By the Axiom of Choice, let A₀ be a set that contains exactly one representative from each equivalence class of [0,1) under ~. Let Q be all the rational numbers in [0,1) and let A_q=A₀@q. Then the A_q are a partition of [0,1). If any one of them has non-infinitesimally positive measure, they all do, which violates finite additivity and total measure one. By (b) they must each have infinitesimal measure. But now we see that we can define a lottery on the countably infinite set Q by saying that q is the winner if and only if our uncountable lottery on [0,1) picked out some number in A_q. This lottery assigns an infinitesimal probability to each outcome in Q. But we have seen that a lottery that does that is a lottery that assigns far too small a value to each outcome. So we're still with the problem of infinitesimals being too small.

Why infinitesimals are too small to help with infinite lotteries: Part III

In two preceding parts (I and II), I argued that assigning the same infinitesimal probability to every outcome of a lottery with countably many tickets assigns too small a probability to those outcomes, no matter which infinitesimal was chosen.

Here I want to note that if we're dealing with hyperreal infinitesimals, then one can't get out of those arguments by assigning a different infinitesimal probability to each outcome. In fact, my second argument worked whether or not the same infinitesimal is assigned. The first did need the same infinitesimal to be assigned, but one can generalize. Suppose I assign infinitesimal probability u_n to the nth ticket. Now it turns out that given any countable set of hyperreal infinitesimals, there is an infinitesimal bigger than them all. So, suppose that u is an infinitesimal bigger than all the u_n. Since u would be too small for the probability of the tickets, a fortiori, the u_n will be too small, too.

Two kinds of responsibility revisited

Suppose I ply Pat with drink and then get him to insult you. Then I am not responsible for insulting you, since I didn't insult you. But I am responsible for your getting insulted by Pat as well as for Pat's insulting you, which are harms that I imposed on you and on Pat, respectively.

On its face, the distinction is not available in the case of murder. If I ply Pat with drink and then get him to kill you, then I am responsible for killing you, since I did kill you—by plying Pat with drink and getting him to kill you. But insulting has an essential expressive role such that to get someone to insult you is not the same as to insult you.

But actually the distinction is still there in the case of killing. For there are two token acts of killing: one performed by me and one performed by Pat, with the latter being my means to the former. I am responsible for both, of course. But it is only for the first act of killing that I am responsible in a way that I can express with "I am responsible for killing you." The responsibility for the second is expressed with "I am responsible for your being killed by Pat" and "I am responsible for Pat killing you", which again indicate harms to you and Pat, respectively.

The distinction is that between action responsibility and event responsibility. I have action responsibility for getting Pat to insult or kill you. I have event responsibility for Pat insulting or killing you, and for your getting insulted or killed. I can be event responsible for any event I can intentionally cause. I can only be action responsible for actions of mine.

Now suppose that I get myself drunk in order to get myself to insult you. In that case, I am action responsible for getting myself to insult you and event responsible for your getting insulted and for my insulting you. Am I also action responsible for insulting you? I certainly did insult you. Let's fill out the cases. In my initial example, I got Pat so drunk that he intentionally but non-responsibly insulted you (perhaps Pat didn't know the effects drink has on him). In my final case, suppose that I got myself as drunk when insulting you as Pat was when he insulted you. Thus, were I not responsible for getting drunk, I would not be responsible for insulting you. But I am responsible for getting drunk. And I am clearly effect responsible for my insulting you. But am I also action responsible for insulting you?

I think not. If the answer were affirmative, then there would be an extra instance of action responsibility in the reflexive case where I get myself drunk that isn't present when I get Pat drunk. But it doesn't seem to me that there should be. Here's an intermediate case: I pay Jim to kidnap a random person and get them drunk and have them insult you. I do think an extra instance of action responsibility occurs should Jim happen to randomly kidnap me.

I think the point generalizes. When I am in the sort of state that would render you not action responsible for Aing were I to impose it on you, that token Aing is not one that I am action responsible for (though I might be action responsible for Aing still—see the murder case), even if I am responsible for being in that state. I may very well be event responsible for my Aing, of course.

But I think only incompatibilists can afford to generalize the point. For in a deterministic world, every state I'm in when acting is the sort of state that were I to impose it on you without any responsibility on your part, then you would not be action responsible for the actions the state gives rise to.

Note 1: A related distinction that I've insisted on in the past is between derivative and non-derivative responsibility. The present vocabulary is more neutral. I am inclined to think that action responsibility is always non-derivative and event responsibility is always derivative, but that is a substantive thesis, not the nature of the distinction.

Note 2: The distinction between action and event responsibility is not the Casteneda distinction between my responsibility for my Aing (i.e., my responsibility for Alexander Pruss Aing) and my responsibility for my* Aing. I can have the latter--say, when it is important to me that the insulting be done by the same person as the one getting the insulter drunk, namely me--without yet having action responsibility for Aing.

Why infinitesimals are too small to help with infinite lotteries: Part II

Suppose someone described a lottery with three tickets, where it was certain that some ticket won, and the probability of each ticket winning was 1/100. We could note that the description violates additivity. Or, more intuitively, we could say that while lottery would make sense with, say, win probabilities 1/3, 1/3 and 1/3, the claimed 1/100 is just way too small a probability, because it is so much smaller than a set of probabilities that do make sense for a lottery with these many tickets.

We can say the same thing about infinite countable lotteries with infinitesimal outcome probabilities. Suppose the proposed probability of each individual ticket winning is some infinitesimal u. Now consider a perfectly probabilistically sensible and unparadoxical, albeit unfair, infinite lottery with individual win probabilities 1/2, 1/4, 1/8, 1/16, .... That lottery makes perfect sense. But our alleged infinitesimal probability lottery has the same number of tickets, but assigns to each one an infinitely smaller probability, since u is infinitely smaller than 1/2ⁿ. And so our alleged infinitesimal probability lottery assigns much too small a probability to each ticket.

A failure of the free market?

I am no economist, so take this with a big pinch of salt. I am writing this as someone who loves to tinker and fix stuff. I have noticed that often consumer products have a simple mode of failure that could have been prevented at a very small incremental cost. For instance, a plastic bracket is used instead of a metal one or not enough plastic is used. Or a wire is too thin and breaks when cables are flexed too many times (two nights ago, one of the remote control wires for my beloved Logitech Z-2200 speakers broke for a second time; looking at the web, this may be a common mode of failure, at least for the Z-2300, which has the same remote control; I fixed it, but ended up dripping solder over the circuit board, and after cleanup, it may not be quite the same).

The additional cost of metal brackets, more plastic, thicker wires and similar simple upgrades would typically be no more than 5% of the total price, and might extend the length of life of the product by a factor of two. This would be a good thing for the consumer and the environment. But there is little in the way of incentive for this, except in the case of a few kinds of items (major appliances, motor vehicles, boats, etc.), since such things are well hidden from the consumer. Brand-loyalty might help here: consumers might notice that X's products last a long time. But this may be counteracted profitwise by the fact that if X's products last a long time, consumers buy replacements less often.

Maybe, though, there is no added utility from that 5% price increase, because maybe such a high percentage of consumers upgrade before the item breaks down that there is no net benefit to consumers.

And, no, I am not advocating for government regulations here: that's likely to result in even worse consequences.

Diachronic Dutch Books

You have a Dutch Book (DB) against you at t provided that, given your credences at t, you would assent to each of a set of bets such that you're guaranteed to lose on balance if you assent to them all.

This morning, I was thinking about cases where people are offering diachronic DB argument.

Suppose you rationally change your mind about p, adjusting your credence between today and tomorrow, say from 1/4 to 3/4, in the light of new evidence, all duly according to Bayes. My initial thought was that there is then a diachronic DB against you in the following sense: there is a pair of bets such that if one is offered today and another tomorrow, you will accept both and be guaranteed to lose overall. (For instance, today, you will accept the deal that you will pay three dollars if p and get a dollar if not p, and tomorrow you will accept the deal that you will pay three dollars if not p and get a dollar if p. But then you're going to lose two dollars whether or not p is true.)

But that was careless of me. A Dutch Book would do better to be defined as a set of bets that you're individually rational in accepting and that are sure to lose you money given the information you have. But you don't have a guarantee that you will change your mind about p from 1/4 to 3/4 in this case. (This is at the heart of the diachronic DB argument that has been given for the Reflection Principle.)

Is there anything to be learned from my case above, other than to be more careful in thinking about DBs? Maybe. Consider your situation during the second bet, the one tomorrow. You accept that bet. In accepting the bet, you bring it about that by your present lights you are bringing it about that you have played a game that you are sure to have overall lost. So one lesson of this is that it is not irrational to bring it about that you have played a game that you are sure to have lost. Moreover, this case suggests that there is a crucial temporal or causal directionality to DB-based arguments. DB arguments have been used to argue that you should now adopt any credences you know for sure you will rationally have (with some provisos). But one had better not use DBs to argue that you should now adopt any credences you know for sure you rationally had: that way lies stasis.

One Body: update on release date

I have an update from the publisher about my One Body Christian sexual ethics book. They are expecting advances on December 1, plus or minus a week, and the official release date right now is December 20. I expect your best bet for getting it soonest, unless you're a reviewer, is to preorder from Amazon.

To whet appetites, here's an excerpt on appetites from Chapter 3 on Desire:

Although anybody who is hungry desires food, one can desire food without feeling hungry—for instance, because one recognizes intellectually that one ought to eat at a given time. Hunger is thus a species of desire for food. As hunger, its content may be rather more vague than one’s desire for food. Thus, while one might desire to eat a particular food or with a particular person, hunger simply calls out for nutrition. It is relatively blind and may be more based in our animal biology than our intellectual faculties.

Likewise, we can try to distinguish libido from the desire for sex. Libido would be a biologically-based appetite for sex, and this would be a species of the desire for sex. A person can desire sex for a variety of reasons, and libido need not enter in at all. No valuation is implied here. Nonlibidinous desires for sex may sometimes be better and sometimes worse than libidinous ones: one might libidinously desire to fulfill the couple’s joint emotional need for union, or one might nonlibidinously desire to make a conquest, or one might libidinously desire to humiliate the other, or one might nonlibidinously desire to comfort one’s beloved.

Looking forward, we will see that the desire for real union in erotic love includes a desire for sexual intercourse. It does not follow, however, that libido is an essential aspect of erotic love. First of all, it is not clear that the desire for union has to be present for love to be there. Love is defined by action and will, and it may be sufficient that one aims at or strives for union (or maybe aims at or strives for union for its own sake), without one actually desiring it. Or it may be that desire is the same thing as one’s will being aimed at some goal, in which case all one requires for a desire for union is that one’s will be directed at union, and not that one have any libido.

Secondly, kinds of love are distinguished by the kind of union sought and by the aspects of the other person that are appreciated. With or without libido, one can appreciate the same aspects of the other person and seek the same kind of union. Of course, libido can make it easier to appreciate the other’s sexual aspects, and can make possible some particular ways of experiencing this appreciation and enhancing the experience of union, and, at least in the male, some libido might be a biologically necessary precondition for the full union (we could likewise imagine an animal that could not swallow when it was not hungry).

Libido can come and go, while a striving, aiming and/or desire for union remains. In fact, when libido is absent, a person might desire sexual union and therefore desire to have libido in order to better experience this union. Moreover, it seems that libido is not in and of itself the desire for union that is found in romantic love. For the desire for union that love includes is always a desire to unite with the other as with a person, whereas libido probably lacks this recognition of the personal element. It is, at most, a component of the way that a desire for love’s union may exhibit itself on a given occasion, though it never constitutes the whole of that desire, nor is it an essential component.

A simple argument that the PSR is necessarily true or necessarily false

Aron Zavaro, in correspondence, supplied me with the following simple central idea for this argument: If the Principle of Sufficient Reason (PSR) were contingently true, there would be no explanation of why it is true.

So the PSR is either necessarily true or necessarily false.

A conditional probability argument for zero probability events

Suppose a dart is tossed at a circular target centered on the point 0, in such a way that it has uniform probability of landing anywhere within a meter of 0, and cannot land anywhere else. Let Z be the event that the dart lands at 0. Let E be the event that the dart lands within half a meter of 0.

Suppose, for a reductio, that P(Z)>0. Presumably, P(Z) is going to be some infinitesimal. Now consider P(Z|E). Intuitively this should be the same as the probability P₂(Z) that a second dart that has uniform probability of landing within half a meter of 0, and unable to land anywhere else, lands at 0. But I claim that P₂(Z)=P(Z). For surely when you have a uniform distribution over a target, and you scale the target up or down, the probability of landing at the center should remain the same. But now P(Z|E)=P₂(Z)=P(Z). But if P(Z)>0 then P(Z|E)=P(ZE)/P(E)=P(Z)/P(E). Moreover, P(E)=1/4 (since E has a quarter of the area of the radius 1 circle). Thus P(Z|E)=4P(Z). Thus 4P(Z)=P(Z). And this can only be true if P(Z)=0.

Suppose you challenge my scaling invariance claim. Then you will have to think that there is a non-constant mathematical function f from lengths to probabilities such that f(r) is the probability of hitting the exact center of a target of radius r. The above argument then shows f(1/2)=4f(1). Generalizing the above argument, we can see that f will have to satisfy the formula f(ar)=f(r)/a² for all a>0.

But what determines this function? The distances that are plugged into f are genuine-distances-with-units. Units for distances depend conceptually on laws of nature: think of the Planck Length or of the meter, which is 1/299792458 of the distance light travels in a vacuum in a second. Does f then depend on the laws of nature in some empirically nonverifiable way? And isn't it amazing that there be this function from distances to infinitesimals that tells us what exactly the probability of hitting the center of a target of one meter is?

Why infinitesimals are too small to help with infinite lotteries: Part I

Let N be the natural numbers 0, 1, 2, 3, .... Let [n] be the first n natural or hypernatural numbers (the hypernaturals are an extension of the naturals that include infinite "natural" numbers; the reciprocals of such infinite naturals are infinitesimal). Thus, [5] = {0,1,2,3,4}.

Suppose we want to model a fair lottery on N by using hyperreal infinitesimals. (What I say should extend to other constructions of infinitesimals.) Let the relevant probability be P₁. Thus, P₁({n})=u for some infinitesimal u>0. Various paradoxes follow: P₁ is non-conglomerable (a corollary of this), this non-conglomerability can be made to spill out into other domains of investigation (see this) and we can't define expected values without violating domination (Paul Pedersen has proved this this summer).

But nevermind the paradoxes. I want to get at one thing that is fundamentally wrong about this approach: no matter which infinitesimal u we've picked, u is infinitely too small.

Here's why. If u>0 is infinitesimal, 1/u is an infinite hyperreal. Let K be an infinite hypernatural number such that K<<1/u (for instance, let K be the hypernatural closest to the square root of 1/u), where we say that a<<b for positive real or hyperreal numbers a and b if and only if b/a is infinite (i.e., a is infinitely smaller than b). Note that then u<<1/K. Now consider a second fair lottery, this time on the set [K]. It is clear that for the second lottery the probability of getting any particular outcome P₂({n}) for n in [K] should be 1/K, if we are dealing with hyperreal probabilities. None of the paradoxes follow if we do this [as long as we deal only in internal functions and internal partitions --note added later]: this assignment is both intuitive and stands up to scrutiny.

But now let n be any (finite) natural number. Then P₁({n})=u<<1/K=P₂({n}). But this is unacceptable, because P₁ correctly represents a fair lottery on N and N is a proper subset of [K] (since K is infinite, we have n<K for all n in N). Thus, u is supposedly the individual outcome probability for a lottery on N, but u would be infinitely too small for the individual outcome probabilities in the case of a lottery on the (much) larger state space [K]. Thus, a fortiori, u is infinitely too small for representing the individual outcomes of a fair lottery on N.

And this is true for any infinitesimal u. So the lesson is that the individual outcome probabilities for a fair lottery on N must be bigger than every infinitesimal. But they must also be smaller than every positive real number, since otherwise they will add up to more than 1. So there is nothing for them to be. Such a lottery is, simply, not a probabilistically coherent concept.

Infinitesimals are too small

I am getting to the following intuition as to why it's not appropriate to resolve zero-probability problems, like the problem of the probability of a particular number being picked in the case of an infinite lottery. The intuition is that the infinitesimals are too small to do the job: infinitesimals are too small to bridge the gap between zero and one, even if you have infinitely many of them. Wish I had time to go into why I am getting this intuition, but I need to grade assignments on mathematical induction instead.

Infinity and probability

You are one of a countably infinite number of blindfolded people arranged in a line. You have no idea where in the line you are. Each person tosses a independent fair die, but doesn't get to see or feel the result.

Case 1: What is the probability you tossed a six?

Obviously 1/6.

Case 2: You are informed that, surprisingly, all the even-numbered people tossed sixes, and all the odd numbered people tossed something else. What is the probability you tossed a six?

It sure seems like it's no longer 1/6. For suppose it's still 1/6. Then when you learned that all the even-numbered people tossed sixes and the odd-numbered ones didn't, you thereby received evidence yielding probability 5/6 that you were one of the odd-numbered ones. But surely you didn't. After all, if there were ten people in the line and you learned that all but the tenth tossed a six, that wouldn't give you probability 5/6 that you were the tenth!

But as long as infinitely many people tossed six and infinitely many didn't (and with probability one, this is true), there is always some ordering on the people such that relative to that ordering you can be correctly informed that every second person tossed a six. That puts the judgment in Case 1 into question.

Moreover, why should the alternating ordering in Case 2 carry any weight that isn't already carried by the fact which you know ahead of time (with probability one) that there are equally many sixes as non-sixes?

But if the judgment in Case 1 is wrong, then were we to find out we are in an infinite multiverse, that would undercut our probabilistic reasoning which assumes we can go from intra-universe chances to credences, and hence undercut science. Thus, a scientific argument that we live in an infinite multiverse would be self-defeating.

I don't exactly know what to make of arguments like this.

A way of grading

I've settled on this method for grading weekly undergraduate papers. I write short comments in the body of the paper. For longer comments, I write a circled letter in the margin of the hard copy, and type a comment for that letter in a file of comments on the week's papers (e.g., "a. This sentence is incomprehensible" in the comments file and a circled "a" in the relevant place in the paper). I can copy and paste comments if the same ones are deserved. I also type overall comments and a grade in that file. I then copy and paste the typed comments into email as soon as I've graded each paper, and send to the student. The hard copy, which explains which letters refer to what, is handed back next class.

A nice thing about this method is that students get a grade and some comments faster, though not all comments will be clear without the context of the paper. I also keep the major comments on my computer which is good for letters of recommendation. Another nice thing is that I no longer waste paper printing comments, as I used to. A fully electronic workflow, I suppose, would be a further step, but I am not sure I am ready for it.

Desire and rationality

I claim:

1. Either (a) one cannot desire what one does not believe to be worth having or (b) it is possible to have no reason, even prima facie, to bring about something one desires.

In either case the view that desires are non-doxastic reason-giving states is false.

A simple case is where you are told you will avoid great punishment if you induce in yourself a desire to eat mud, which you antecedently know there is no point to doing. You then induce in yourself this desire. If (a) is false, you should be able to do this while maintaining your belief that there is no point to the action. But you do not gain a reason to do the action in this case: you only have a reason to desire it, not a reason to do it.

Objection: There is inner turmoil when you have an unsatisfied desire for eating mud.

Response: This isn't a reason to eat mud, as much as a reason to end having a desire to eat mud. Eating mud is sometimes a means to this, but not always. We could imagine a situation where the only ways available to eat mud are ways that wouldn't make the desire go away (e.g., you know you have a psychological quirk that makes the desire not go away, or maybe you know an alien will ensure that as soon as you swallow mud, you start believing you're eating chocolate, so you will never form the belief that you have eaten mud, though you will have).