Grammatical possibility

I asked my kids whether a circle that is square is logically possible. My seven-year-old answered in the negative. My ten-year-old said it was impossible, but it was "grammatically possible". I think that's a rather curious kind of modality!

Qualitative probabilities, regularity and nonmeasurable sets

Normally, regularity is formulated as saying that P(A)>0 for every non-empty A. But suppose that instead of working with numerical probability assignments, we work with qualitative probabilities, i.e., probability comparisons. Thus, instead of saying B is at least as likely as A provided that P(B)≥P(A), we might take the relation of being at least as likely as to be primitive, and then give axioms.

Given a theory of qualitative probabilities, it will be possible to define an equiprobability relation ~ such that we can say A~B if and only if A and B are equiprobable. (The typical way would be to say that A~B provided that B is at least as likely as A and A is at least as likely as B.) This relation ~ will satisfy some axioms, but we actually won't need them for the argument. We shall suppose that ~ is defined on some collection of subsets of a sample space, which we will call the measurable sets. Our setup generalizes classical probabilities, as well as hyperreal probabilities, since if we have probability-values, we can say that A~B if and only if P(A)=P(B).

We can plausibly formulate regularity in terms of an equiprobability relation:

• An equiprobability relation ~ is regular if and only if whenever A and B are measurable sets such that A is a proper subset of B, then we do not have A~B.

Now suppose that our sample space is (the circumference of) a circle. Then:

• An equiprobability relation ~ is rotation-invariant if and only if whenever A and B are measurable sets such that B is a rotation of A, then A~B.

Now, we know that given the Axiom of Choice, and given classical probabilities, there is no way of defining probabilities for all subsets of our circle in a rotation-invariant way. Surprisingly, but very simply, if we assume regularity, we need neither classical probability—any equiprobability relation will do—nor the Axiom of Choice. In fact, we will have a countable nonmeasurable set, so when we add regularity to the mix, we have to sacrifice the measurability of sets that are unproblematically measurable using classical measures.

Theorem: There is no equiprobability relation ~ such that (a) all countable subsets of the circle are measurable; (b) the relation is regular; and (c) the relation is rotation-invariant.

Proof: Let u be any irrational number. Let B be the set of all points on the circle at angles 2πnu (to some fixed axis, say the x-axis), for positive integers n. Let A be a rotation of B by the angle 2πu. Then A is a proper subset of B (A contains all the points on the circle at angles 2πnu for n an integer greater than one, and by the irrationality of u that will not include the point at angle 2πu). So if we had regularity, we couldn't have A~B. But if we had rotation-invariance, we would have to have A~B. ∎

The above proof is based on the counterintuitive fact that there is a subset of the circle, i.e., B, that can be rotated to form a proper subset of itself, i.e., A. (This reminds me of the Sierpinski-Mazurkiewicz paradox and other cases of paradoxical decomposition, though it's much more trivial.)

This is, of course, a trivial modification of the Bernstein and Wattenberg inspired argument here.

Could God have become incarnate as a non-person?

The Logos became incarnate as a human being, to save us from our sins. There would have been no similar point to his becoming incarnate as a cat, an oak or a photon? But could he have done so, if he had a purpose to?

Suppose we say "no" to at least one of the three options (cat, oak or photon). Why would we? Assuming we accept that the Logos could have become incarnate as a human being, we would have to suppose some relevant difference between humans and cats, oaks or photons. What the difference is will depend where one draws the possibility-of-incarnation line. If one thinks that God could have become a cat but not an oak, that's presumably because one thinks that sentience is crucial to the possibility of incarnation. And one will presumably then deny that God could have become a photon. If one thinks God could have become a human but not a cat, then presumably one thinks sapience (and I won't worry about the details of what the consists in, say agency or abstract thought) is crucial.

But we human beings don't always exhibit sentience, much less sapience. We don't exhibit sentience for the first weeks of life after conception, and we don't exhibit sapience until at least around one year after birth. Moreover, when unconscious we do not exhibit sentience and need not exhibit sapience (though, maybe, sapience doesn't always require consciousness). In becoming one of us, the Logos would have become a being that wasn't always sentient or sapient. So if one thinks that sentience or sapience is crucial for incarnation, and yet one accepts that the Logos could become a being like we who does not always have sentience or sapience, one has to say that it is something like the potential for sentience or sapience (depending on which view we are considering) that is a necessary precondition for incarnation or that it is sometimes having sentience or sapience that is necessary.

Consider first the "sometimes" view. This presumably requires that the incarnation cannot precede the developmental attainment of sentience or sapience (for the incarnation does so precede, we could imagine it terminating, with the destruction of the finite nature, before that attainment). If sapience is the relevant condition, then we get the view that barring miraculous precociousness, God cannot be incarnate as a newborn, which at least to us Christians will be absurd. If sentience is necessary, then we get the view that, again barring miraculous precociousness, the incarnation couldn't have happened simultaneously with conception. (Interestingly, Aquinas actually goes for miraculous precociousness here—his view that we don't come into existence at conception but a significant amount of time thereafter forced him into holding that Jesus came into existence fully formed in Mary's womb.)

Still, the "sometimes" view just seems implausible. Why would the incarnation require initial exercise of sentience or sapience without the need for exercise of sentience or sapience thereafter?

Now consider the potentiality view. This, too, does not seem all that plausible to me. Presumably the pull of saying that God couldn't become a cat or an oak or a photon is that these beings are so very unlike God. But potentiality is very much unlike God's perfect actuality, too. In the end, I think that once one reflects on the fact that human beings often exhibit neither sentience nor sapience, the pull to thinking the Logos couldn't have become a cat or an oak weakens.

How about a photon? There the relevant difference would be something like life. But again it seems hard to see why life is a necessary condition for an incarnation. There are, plausibly, infinitely many attributes as significant as life that God has and that human beings lack. The gaps between the photon and the oak, the oak and the cat, and the cat and the human are infinitely less than the gap between humans and God, a gap that God can bridge, we have assumed.

The above arguments are not very strong. But I think they do give one a presumption in favor of the view that if God can become incarnate as a human, he can become incarnate as any kind of being.

Is the Internet "the same" as face-to-face social interaction?

I used to claim to think that social interaction by email is not significantly different qua social interaction from face-to-face interaction. But the fact that I typically strongly preferred email interaction to face-to-face interactions is evidence that it's significantly different, and given my introvertive tendencies, it is evidence that it is less social. People like I need significant time "without social interaction" to avoid exhaustion. But writing this post qualifies, even though it is obviously a social activity.

Or maybe it's not correct to characterize introverts as tired out by social interaction. Rather, they are tired out by particular modalities of social interaction. So perhaps there is a response possible to the argument of the preceding paragraph.

"Figuratively"

Consider this great T-shirt slogan (I have no financial ties to the seller, but if you click on it you can buy the shirt with it).

Everyone I've talked to agrees that statements like the one on the T-shirt are an example of literal language.  The wearer is claiming to literally be made figuratively insane.

But here is an oddity. If you say: "Misuse of 'literally' makes me insane", I can say: "Figuratively speaking, that is." My use of "figuratively" attributes figurativeness to your sentence, which sentence is figurative. But in the slogan on the T-shirt, what does "figuratively" attribute figurativeness to? Presumably, the word "insane"? So does the sentence, thus, contain figurative language after all? But the sentence seemed like a piece of literal language. The "insane" is only there in the scope of "figuratively". So does the "figuratively", perhaps, implicitly attribute figurativeness to a different sentence that hasn't actually been uttered, namely the sentence "Misuse of 'literally' makes me insane". Or, more precisely, maybe it attributes figurativeness to the word "insane" as found in that unsaid sentential context? If so, then analyzing actual sentence tokens requires thinking about sentence types or nonactual sentence tokens.

First and second order desires

My fear of dogs brings involves a paradigmatic first-order desire: a desire to avoid the proximity of unsecured dogs. But a desire to avoid the proximity of unsecured dogs also motivates me to avoid activities that have a sufficiently high (which does not need to be high at all!) probability of leading to being in the proximity of unsecured dogs, activities such as walking to work. This, too, is a paradigmatic feature of this first-order desire.

But now one of the activities that would have a sufficiently high probability of leading to being in the proximity of unsecured dogs would be getting rid of my fear and hence desire for avoidance. If I didn't fear dogs, I wouldn't avoid the proximity of unsecured dogs. Thus, the desire to avoid the proximity of dogs motivates me to avoid getting rid of this very desire. But such motivation is paradigmatically the work of a second-order desire. Yet it comes about through exactly the same means-end reasoning by which the desire to avoid the proximity of unsecured dogs motivates me to avoid walking to work.

This isn't an exceptional case. Normally, the possession of a desire for A helps promote getting A. There could be exceptions: a desire to have many friends might not make one a good friend and joy might be the sort of thing that comes most when not pursued. But normally desires help promote what they are desires for—indeed, that's presumably at least a part of why we have desires. But then, when one reflects on this, the desire for A will motivate one to maintain a desire for A.

Fortunately, however, often the motivation to maintain a desire for A will not be as strong as the motivation for more direct means to A. This contingent fact makes it easier to rid ourselves of desires that we should not have: for even if the desire is very strong indeed, its motivational force for self-maintenance may not be all that strong, and hence we may be able to induce, through reflection on the perniciousness of that desire, a sufficiently strong motivation not to have that screwed-up desire. Notice, though, that at least sometimes that motivation-to-remove-desire will itself be simply a means-to-end motivation in light of a first-order goal. One doesn't want to die of lung cancer—so one works to remove the remove the desire to smoke.

Goedelian ontological argument

I just posted a PDF of my "A Goedelian ontological argument improved even more" article. The article came out in an anthology by Fr. Szatkowski. Unfortunately it contains an error (which doesn't affect the main philosophical points): see my May 14, 2016 comment below.

One Body: released

Amazon now has One Body: An Essay in Christian Sexual Ethics in stock, though they say they only have seven copies left and more are on the way.

I got my copies yesterday. They look nice. Here's the blurb from Amazon:

This important philosophical reflection on love and sexuality from a broadly Christian perspective is aimed at philosophers, theologians, and educated Christian readers. Alexander R. Pruss focuses on foundational questions on the nature of romantic love and on controversial questions in sexual ethics on the basis of the fundamental idea that romantic love pursues union of two persons as one body.

One Body begins with an account, inspired by St. Thomas Aquinas, of the general nature of love as constituted by components of goodwill, appreciation, and unitiveness. Different forms of love, such as parental, collegial, filial, friendly, fraternal, or romantic, Pruss argues, differ primarily not in terms of goodwill or appreciation but in terms of the kind of union that is sought. Pruss examines romantic love as distinguished from other kinds of love by a focus on a particular kind of union, a deep union as one body achieved through the joint biological striving of the sort involved in reproduction. Taking the account of the union that romantic love seeks as a foundation, the book considers the nature of marriage and applies its account to controversial ethical questions, such as the connection between love, sex, and commitment and the moral issues involving contraception, same-sex activity, and reproductive technology. With philosophical rigor and sophistication, Pruss provides carefully argued answers to controversial questions in Christian sexual ethics.


"This is a terrific—really quite extraordinary—work of scholarship. It is quite simply the best work on Christian sexual ethics that I have seen. It will become the text that anyone who ventures into the field will have to grapple with—a kind of touchstone. Moreover, it is filled with arguments with which even secular writers on sexual morality will have to engage and come to terms." —Robert P. George, Princeton University


"One Body is an excellent piece of philosophical-theological reflection on the nature of sexuality and marriage. This book has the potential to become a standard go-to text for professors and students working on sex ethics issues, whether in philosophy or theology, both for the richness of its arguments, and the scope of its coverage of cases. " —Christopher Tollefsen, University of South Carolina


"Alexander Pruss here develops sound and humane answers to the whole range of main questions about human sexual and reproductive choices. His principal argument for the key answers is very different from the one I have articulated over the past fifteen years. But his argumentation is at every point attractively direct, careful, energetic in framing and responding to objections, and admirably attentive to realities and the human goods at stake." —John Finnis, University of Oxford

An electronic version (PDF) can be purchased directly from the press.

There is such a thing as supererogation

Supererogatory actions are admirable but not obligatory. A sufficient, and perhaps necessary, condition for an action A to be supererogatory is that (a) A is permissible and (b) there is an alternative B to A such that (i) it is permissible to do B instead and (ii) A is more morally praiseworthy than B.

Over the years, I've met people--including myself--who have been troubled by the idea of supererogatory actions and tempted to deny that there is such a thing as supererogation. But here is a pretty conclusive argument that there can be supererogatory actions. You and your friend, both innocent people, are captured by a tyrant. The tyrant sentences your friend to 24 hours of torture. Then the tyrant offers you the option of reducing your friend's torture by any amount of time less than 12 hours. And of course, she notes, any torture taken away from your friend will be given to you, by Public Law Number One: the Preservation of Torment.

Now, many people will say that any taking on of your friend's torture is automatically supererogatory. But I think the sort of people who doubt that there are supererogatory actions won't be impressed--they tend to have a view that morality does indeed sometimes call us to very great sacrifices (and they are right about that, even if they might be wrong about this case).

However, the following is surely true: There is an amount T₁<12 such that it is permissible to reduce the friend's torture by T₁ hours. Indeed, surely, T₁=11.99 qualifies. (Argument: reducing one's friend's torture by 11.99 hours, given the cost that one will suffer that torture oneself, is plainly praiseworthy simpliciter, but only permissible actions are praiseworthy simpliciter.) Let B be the action of reducing one's friend's torture by T₁ hours. Let T₂ be a number such that T₁<T₂<12. Let A be the action of reducing one's friend's torture by T₂ hours and let B be the action of reducing one's friend's torture by T₁ hours. Then, barring further factors not given in the story: (a) A is permissible (it would be odd if it were permissible to reduce one's friend's torture by, say, 11.99 hours but not by 11.999 hours); (b)(i) B is permissible and (b)(ii) A is more morally praiseworthy than B. Thus, A is supererogatory.

If you think time is discrete, the above example still can be made to work. Suppose for simplicity 11.99 hours is the longest time interval short of 12 hours. Then if you think there is no supererogation, you might think that you're obligated to request that your friend be relieved of 11.99 hours of torture. But as long as the agent in the story doesn't know that 11.99 hours is the longest time interval short of 12 hours there is, she can do something more praiseworthy than requesting the 11.99 hour reduction: she can request 11.999 hours, and as long as she is not certain that 11.99 hours is the most she can get, she thereby risks getting more than 11.99 hours of torture as the cost of trying to relieve more than 11.99 hours, and that's more praiseworthy than just going for 11.99.

Almost necessary beings and the ontological argument

The familiar S5 ontological argument for a necessary being goes:

1. Possibly, there is a necessary being.
2. So, there is a necessary being. (By S5)

Say that a being x is (at least) almost necessary provided that it is necessary that if anything at all exists, then x exists. Then one can also run an S5 ontological-style argument for an almost necessary being;

3. Possibly, there is an almost necessary being.
4. Something exists.
5. So, there is an almost necessary being. (By S5. If an almost necessary being exists at one world, it exists at all worlds at which something exists; but something actually exists.)

It's not quite an ontological argument in that (4) is an a posteriori premise.

Could one support (3) without that also giving an equally good argument for (1)? Maybe.

6. It quasi-perceptually seems to some mystic that love grounds all being.
7. What quasi-perceptually seems to someone is probably possible (or at least conceivable in the two-dimensionalist sense, but that's all we actually need).
8. Necessarily, if x grounds all being, then x grounds all being in all worlds in which something exists.
9. Necessarily, if love grounds all being, then there is a lover who grounds all being.
10. So, probably it's possible that love grounds all being. (6 and 7)
11. So, probably it's possible that there is an almost necessary being. (8, 9)
12. So, probably there is an almost necessary being.

Another start on the problem of evil

According to Socrates the greatest goods and evils are moral ones. Call this the "Socratic thesis". On the Socratic thesis, the worst thing that can befall one is to act culpably wrongly. Now, we may divide up the evils of the world into three classes:

1. Culpable wrongdoings.
2. Harms other than culpable wrongdoings resulting from culpable wrongdoings.
3. Harms neither identical with culpable wrongdoings nor resulting from them.

For instance, if Jones tortures Smith, then Jones suffers a Class 1 evil while Smith suffers a Class 2 evil.

Each of these three classes of evils is very large. I think we can say that if we confine ourselves to evils happening to humans (bracketing the problems of animal suffering and angelic fall): Class 1 is roughly as large as the Class 2 (granted, some culpable wrongdoings result in many harms; but many culpable wrongdoings stay at the level of an evil thought that leads to no harmful action) and also roughly at least as large as Class 3. So roughly, about a third of the evils of the world are in Class 1.

Next notice that we have a theodicy for Class 1 evils: the free will theodicy, in its different versions (straight free will theodicy, soul-building, autonomy, need for love to be a free response, etc.) By the Socratic thesis, we thus have a theodicy for the greatest evils that occur, and these evils are roughly a third of all the evils that occur to humans. This provides us with some inductive reason to think that there is a theodicy for the rest of the evils: if a theodicy can be found for the greatest evils, and indeed for about a third of the evils happening to humans, then the existence of a theodicy for the rest seems more plausible.

Moreover, some versions of the theodicy for Class 1 evils extend to theodicies for many Class 2 evils. First, our free will would be a bit of a sham if it wasn't effective—if evil choices never resulted in in the chosen state of affairs. (This is less plausible for the worst Class 2 evils.) Second, while terribly harms do befall undeserving people, most of the evils that befall are, I suspect, quite deserved. Those evils, then have a justice theodicy, given a freedom theodicy for the actions that deserved them. (This might shift our count of some evils from Class 3 to Class 2, though we might also say that there are evils in Class 3 that do not result from our culpable wrongdoings, but that on account of our culpable wrongdoings weren't prevented by God.)

Supererogation

Supererogation is a difficult concept for me. But there has to be such a thing. If Jones has suffered two hundred weeks of torture to save the lives of two hundred strangers, and then declines the 201st week of torture to save the life of the 201st stranger, Jones does not do wrong. And if he were to accept the torture, he would be acting superegatorily (barring special circumstances).

I doubt the following account is in the end right, but I think it is surprisingly defensible (modulo perhaps some minor tweaks):

• An action is supererogatory if and only if it is permissible and less convenient than some available alternative permissible action.

I don't have a good account of what "convenient" means, but "convenience" is meant to convey what one sacrifices when one makes "self-sacrifices". Thus, it is more convenient to endure less pain rather than more; it is more convenient to do the easier rather than the harder thing; it is more convenient to save than to lose one's life (this is an understatement in ordinary English, but I am using "convenience" in a sort of technical sense). But convenience probably won't count some higher goods to self, such as the exercise of virtue, which are gained rather than lost in self-sacrifice. Thus, a self-sacrifice can count as inconvenient even if overall one benefits from it because of the value of the exercise of virtue.

The account above seems to be subject to simple counterexamples. Let's say it's permissible for me to go to the kitchen, and suppose there are two paths—an easier and a harder one. Then surely both paths are permissible, and the harder one is less convenient, but that doesn't make the less convenient one supererogatory!

To respond I note that it is wrong to pointlessly impose burdens on any person—including oneself. (Argument 1: We are to love all of the people that God loves, and love prohibits pointless imposition of burdens. But I am one of the people God loves. So I am not permitted to impose pointless burdens on myself. Argument 2: What is vicious is impermissible. But pointless imposition of burdens on myself is contrary to the virtues of prudence and hence vicious.) Thus if there is no benefit to anybody from taking the harder path, the harder path is not permissible, and hence is not supererogatory. But suppose that there is a benefit to someone from the harder path: maybe I become physically or morally stronger, or maybe someone else benefits in some way. Then as long as the harder path is permissible (if the benefit is too trivial as compared to the burden, it might not be), it does seem to be supererogatory.

I do suspect that this account of supererogation only stands a chance if we have duties to self, but that's not a weakness of it.

Some people doubt that there are any supererogatory actions. On the above account, it is quite plausible that there are. First, we need to note that surely there are cases where we choose between multiple permissible actions. And second we note that it is very likely that among such choices there are going to be cases where the permissible options are not all equally convenient. And then the less convenient ones will be supererogatory.

Note that if convenience is what is given up in self-sacrifice, then every supererogatory action involves self-sacrifice. Now, self-sacrifice is relative to some alternative that does not involve such a sacrifice. We might then rephrase our definition of supererogation as:

• An action is supererogatory if and only if it is permissible and it is a self-sacrifice relative to some permissible alternative.

Go back to my initial case of Jones. If Jones did undergo the 201st week of torture, he would be doing something permissible, but it would also be permissible for him not to undergo that torture. However, undergoing the torture is less convenient. Again, this sounds like an absurd understatement, but in our technical sense of "convenient", it's not. It sounds a lot better in the language of self-sacrifice: Jones' undergoing the torture is permissible and is a self-sacrifice relative to the alternative of not undergoing it.

I think the weakness of the account is it does not make clear why supererogation is particularly praiseworthy. Moreover, even if the account happens to be extensionally correct, I don't think it captures what it is that grounds supererogation.

Uniform measure and nonmeasurable sets, without the Axiom of Choice

Given the Axiom of Choice, there is no translation invariant probability measure on the interval [0,1) (the relevant translation is translation modulo 1). But this fact really does need something in the way of the Axiom of Choice. Moreover, the fact only obtains for countably additive measures. Interestingly, however, if we add the assumption that our measure assigns non-zero (presumably infinitesimal) weight to each point of [0,1), then the non-existence of a translation invariant finitely additive measure follows without the Axiom of Choice. I got the proof of this from Paul Pedersen who thinks he got it from the classic Bernstein and Wattenberg piece (I don't have their paper at hand). I am generalizing trivially.

Theorem: Let P be any finitely additive measure taking values in a partially ordered group G and defined on a collection of subsets of [0,1) such that every countable subset has a measure in G. Suppose P({x})>0 for some x in G. Then P is not translation invariant (modulo 1).

Proof: To obtain a contradiction, suppose P is translation invariant. Then P({x})>0 for every x in [0,1). Let r be any irrational number in (0,1), and let R be the set of numbers of the form nr modulo 1, as n ranges over the positive integers. Let R' be the set of numbers of the form nr modulo 1, as n ranges over the integers greater than 1. Then R' is a translation of R by r, modulo 1. Observe that r is not a member of R' since there is no natural number n greater than 1 such that r=nr modulo 1, since if there were, we would have (n−1)r=0 modulo 1, and hence r would be a rational number with denominator n−1. Thus by finite additivity P(R)=P(R')+P({r})>P(R'). Hence, R is a counterexample to translation invariance, contradicting our assumption.

Note 1: On the assumption that the half-open intervals are all measurable and the measurable sets form an algebra (the standard case), translation invariance modulo 1 follows from ordinary translation invariance within the interval, namely the condition that P(A)=P(A+x) whenever both A and A+x={y+x:y in A} are subsets of [0,1).

Note 2: The proof above shows that if P({x})>0 for every x in [0,1), then the set of all positive integral multiples of any fixed irrational number (modulo 1) is nonmeasurable. It is interesting to note that this nonmeasurable set is actually measurable using standard Lebesgue measure. Thus, by enforcing regularity using infinitesimals, one is making some previously measurable sets nonmeasurable if one insists on translation invariance.

Note 3: Bernstein and Wattenberg construct a hyperreal valued measure that is almost translation invariant: the difference between the measure of a set and of a translation of the set is infinitesimal.

Another argument about simplicity

In an earlier post, I defended the idea (which Trent Dougherty also came up with independently and earlier) that only theory-unexplained entities, or kinds of entities, count against the simplicity of a theory. Here is another argument for this. Start with these two principles:

1. If theories T₁ and T₂ are otherwise equally evidenced and explanatorily powerful, but T₁ is simpler, then T₁ is more epistemically likely to be true than T₂.
2. The Principal Principle: Epistemic probabilities should (except in exceptional cases) be set to equal objective chances when the latter are available.

Now imagine that there is a powerful physical theory, T₀, according to which there is a special type of particle, U, that can only be produced through an exceedingly unlikely combination of events, so unlikely that it is unlikely that in the lifetime of the world the particle would ever be produced outside the lab. Scientists build the extremely expensive piece of apparatus to produce the particle. The apparatus is so expensive that it is unlikely it would ever be built again. But a rogue scientist gets hold of the apparatus at night and hooks up a bomb that will destroy the apparatus, and all results of any experiment, in five minutes. She also hooks an indeterministic fair coin flipper to the apparatus, so that if the coin comes up heads, U particle production is triggered, and if comes up tails, U particle production is not triggered. Consider now two theories:

• T_H: T₀ is true, no U particles ever get produced except perhaps in a moment by this apparatus, heads will come up, and a U particle will be produced by the apparatus.
• T_T: T₀ is true, no U particles ever get produced except perhaps in a moment by this apparatus, tails will come up, and no a U particle will be produced by the apparatus.

By a very plausible application of the Principal Principle, since the chances of heads and tails are equal as the coin is fair:

3. P(T_T)=P(T_H).

But if the number of explained kinds of objects counts against simplicity, then T_T is simpler than T_H, since according to T_T reality includes an extra kind of particle, the U particle. (If one doesn't think reality includes the future, run this thought experiment retrospectively after the explosion.) So by (2), then, P(T_T)>P(T_H). But this contradicts (3). Thus, by modus tollens, the number of explained kinds of objects does not count against simplicity.

Parental duty

Another excerpt from my forthcoming One Body book, this time from the discussion of gamete donation (challenge to the reader: find the relevance of this to gamete donation):

Now, it is not merely the duty of the parents to bring it about that the children are cared for and appropriately educated morally, religiously and academically. Rather, it is the duty of the parents to care for and educate the child—i.e., to do it themselves. In caring for and educating the child, parents will make use of the help of others, including that of family members, friends, and professionals. How much the parents can rely on the help of others before they have failed in their duty of caring for and educating the child will depend on the circumstances.
There are thus two aspects of the parental duty: (a) caring for and educating, and (b) ensuring that the child is cared for and educated. In other words, there is the aspect of parental activity and the aspect of results. These two aspects need to be balanced prudently, and, moreover, balanced with other duties the parents may have; how they are balanced will depend on particular circumstances. In no cases will it be desirable and rarely will it be possible for the parents directly to care for and educate the child in all respects with the help of no one else. Moral education, for instance, requires contact with virtuous people of a significant variety of different characters, not just the parents. Academic education should typically include education in subjects in which the parents lack competency. The need to work to earn money to provide for the child can force the parents to delegate a significant degree care to a third party.
Here is an observation worth making. In most couples, there will be specialization. Thus, the mother might be working long hours to earn the money needed to diaper, feed, clothe, and house the child, while the father might be changing the diapers, feeding, clothing, and otherwise taking care of the child for most of the day. It might seem that in such cases, each parent will be neglecting an aspect of the parental responsibility to himself or herself care for and educate the child. But we can respond to this by noting that parents should be friends of each other, and bringing in an idea from Aristotle’s Nicomachean Ethics. Aristotle considers what value there in having good friends. He observes that friends share a life, a friend is “another self,” and one can be active through one’s friend’s activity: what the friend does virtuously is something that accrues to oneself.

Conditional probability and nonmeasurable sets

Let P be Lebesgue measure on the three-dimensional cube [0,1]³. Assume the Axiom of Choice. Then there will be P-nonmeasurable sets (e.g, as there is no finitely additive (much less countably additive, rigid-motion-invariant probability measure on all subsets of [0,1]³ by the Banach-Tarski paradox. Now let F be the Lebesgue measurable subsets of [0,1]³. One might then hope that one can define P(X|Y) for all X and Y in F, as long as Y is non-empty.

Turns out that one can't, at least if one expects finite additivity and rigid-motion invariance. The reason for that is that if we let Y be (the surface of) a sphere in [0,1]³, then all subsets of Y will be in F, but there is no finitely additive rotation invariant probability measure on all subsets of a sphere, by the Hausdorff paradox. This problem disappears if we restrict ourselves to the Borel subsets of [0,1]³, but the extension to the Lebesgue measurable ones is epistemologically very plausible—obviously any subset of a null set should be a null set.

But here's a funny thing. While I haven't checked all the details—it's grading time so I can only give this so much thought—it turns out that one can sensibly define P(X|Y) for all X in F and all non-measurable Y. The easiest case is where Y is maximally non-measurable, i.e., all its measurable subsets have null measure and all its measurable supersets have full measure. In that case, one can simply define P(X|Y)=P(X). Moreover, one can naturally extend this measure to all X's in F_Y, where F_Y is the smallest σ-field generated by F and Y (basically by setting P(X∩Y|Y)=P(X) and P(X∩Y^c|Y)=0 for X in F).

This means that the Popper function approach to conditional probability on which P(X|Y) is defined for all X and Y in a single field or σ-field is not general enough, at least if we want P(X|[0,1]³) to be defined for all Lebesgue measurable X's. For in fact it seems we have more freedom as to what Y's we get to plug in and less as to what X's.

Limiting God to solve the problem of evil

Long ago, I remember reading with great curiosity Rabbi Kushner's Why Bad Things Happen to Good People? How disappointing that Kushner's intellectual answer seemed to be that God isn't omnipotent. (His practical answer not to worry about the question but just to do good is much better.) The idea of limiting divine attributes in part to answer the problem of evil has recently had some defense (e.g., here and in the work of open theists), so I guess it's time to blog the objection to Kushner—which applies to the others as well—that I had when I read him, with some elaboration.

Basically, the objection is that as long as God remains pretty good, pretty smart (he was smart enough to create us!) and powerful enough to communicate with us (Kushner at least accepts this), then serious cases of the Problem of Evil remain. Moreover, these cases do not seem significantly easier to solve than the cases of the Problem of Evil that were removed. Consequently, the intellectual benefit with regard to the Problem of Evil is small. And the intellectual loss with regard to the simplicity of the theory is great—the theory that God has all perfections is far simpler.

Start by considering a deity whose goodness is unlimited but whose knowledge and power are fairly limited.

Consider, first, the problem of polio. This is certainly a horrendous evil. And the limited deity could have alleviated a significant portion of the problem hundreds of years earlier simply by whispering into some people's ears how to make a vaccine—surely any deity smart enough to create this world would be smart enough to figure out how to make vaccines. Maybe the limited deity couldn't have prevented all cases, in the way that an unlimited God could. But given that neither did the wholesale prevention happen nor did the partial prevention by vaccines happen as early as it could have.

Consider, second, the many cases where innocent people suffered horrendously at the hands of attackers, where the attack could have been prevented if the people had been warned. Even a deity of limited power and knowledge should be able to see, for instance, that the Gestapo are talking about heading for such-and-such a house, and could then warn the occupants. (I am not saying that such warnings were never given—for all I know, they were in a number of cases. But I am saying that there are many cases where apparently they were not.)

Moreover, even if one limits the goodness of the deity, and only claims that he is pretty good, the problem remains. For unless the deity had a very serious reason not to tell people about vaccines and not to warn the innocent victims of horrendous attacks, it seems plausible that the deity did something quite bad in refraining from helping, so bad as to be incompatible with being pretty good. (If the deity had a reason that fell a little short of justifying the refraining, then that might be compatible with being pretty good; but a reason would have to be pretty serious for it to fall only a little short of justifying the refraining when the evils are so horrendous.) So even if one thinks that the deity has limited power and knowledge and is only pretty good, the problem of finding very serious reasons for the deity's non-interference remains.

Granted, the problem is diminished, especially if one has decreased the belief in divine goodness. But notice that the decrease in belief in divine goodness is the most religiously troubling aspect of a limited God doctrine. And even that does not make the problem go away.

Moreover, the sorts of things one can then plausibly say about the remaining problems of evil are things that, I suspect, the traditional theist can say as well about this and many other cases. Perhaps God does not prevent all attacks on innocent people (for all we know, he prevents many) because he wants humans to have effective freedom of will. Perhaps he wants to give victims opportunities for forgiveness of their aggressors in an afterlife. Perhaps God does not prevent disease because he wants us to help our neighbor and to develop medical science to this purpose. Or to give us an opportunity to join him on the cross in redeeming humankind. Or perhaps God prevents many evils, but his purposes do not allow him to prevent all, and some arbitrary line-drawing is needed. I am not saying that these answers are sufficient (though I think some contain a kernel of something right), but only that they can be equally used in the case of a limited and unlimited God, and in the case of an unlimited God such answers may well have rather general applicability.

Reducing sets

I find sets to be very mysterious candidates for abstract entities. I think it's their extensionality that seems strange to me. And anyway, if one can reduce entities to entities that we anyway want to have in our ontology, ceteris paribus we should. I want to describe a three-step procedure—with some choices at each step—for generating sets. I will use plural quantification quite a lot in this. I am assuming that one can make sense of plural quantification apart from sets.

Step 1: The non-empty candidates. The non-empty candidates, some of which will end up counting as sets in the next step, will be entities that stand in "packaging" relation to a plurality of objects, such that for any plurality, or at least for enough pluralities, there is a candidate that packages that plurality. There are many options for the non-empty candidates and the packaging relation.

Option A: Plural existential propositions, of the form <The Xs exist>, where a plural existential proposition p packages a plurality, the Xs, provided that it attributes existence to the Xs and only to the Xs.

Option B: Plural existential states of affairs (either Armstrong or Plantinga style), i.e., states of affairs of the Xs existing, where a plural existential state of affairs e packages a plurality, the Xs, provided that it is a state of affairs of the Xs existing. I got this option from Rob Koons.

Option C: This family of options generates the candidates in two sub-steps. The first is to have candidates that stand in a packaging relation to individuals, such that each candidates packages precisely one individual. Call these "singleton candidates". For brevity if x is a singleton candidate that packages y, I will say x is a singleton of y. The second step is to take our non-empty candidates to be mereological sums of singleton candidates, and to say that a mereological sum m packages the Xs if and only if m is a mereological sum of Ys such that each of the Ys is a singleton of one of the Xs and each of the Xs is packaged by exactly one of the Ys. We need the singleton packaging relation to satisfy the condition (*) that a mereological sum of singletons of the Xs has no singletons as parts other than the singletons of the Xs. (In particular, no singleton of y can be a part of any singleton of x if x and y are distinct.)

We get different instances of Option C by considering different singleton candidates. For instance, we could have the singleton candidates be individual essences, and a singleton candidate then packages precisely the entity that it is an individual essence of. I got this from Josh Rasmussen. Or we might use variants of Options A and B here: maybe a proposition attributing existence to x or a state of affairs of x existing will be our candidate singleton. (Whether the state of affairs option here differs from Option B depends on whether the state of affairs of a plurality existing is something different from the mereological sum of the states of affairs of the individuals in the plurality existing.)

There are many other ways of packaging pluralities.

Step 2: The empty candidate. We also need an empty candidate, which will be some entity that differs from the non-empty candidates of Step 1. Ideally, this will be an entity of the same sort as the non-empty candidates. For instance, if our non-empty candidates are propositions, we will want our empty candidate to be a proposition, say some contradictory proposition.

Step 3: Pruning the candidates. The basic idea will be that x is a member of a candidate y if and only if y is one of the non-empty candidates and y packages a plurality that has x in it. But the above is apt to give us too many candidates for them all to be sets. There are at least two reasons for this. First, on some of the options, there won't be a unique candidate packaging any given plurality. For instance, there might be more than one proposition attributing existence to the same plurality. Thus, the propositions <The Stagirite and Tully exist> and <Aristotle and Cicero exist> will be different propositions if Millianism is false, but both attribute existence to the same plurality. Second, some of the candidates will be better suited as candidates for proper classes than for sets and some candidates may be unsuitable either as sets or as proper classes. For instance, there might be a proposition that says that the plural existential propositions exist. Such a proposition packages all the candidates, including itself, and will not be a good set or proper class on many axiomatizations.

Feeling cold? Apply to our graduate program!

Are you thinking of grad school and feeling cold? As your December days get colder and darker, you may want to reflect on the warm weather in Waco. It was over 80F today.

Of course, the really great thing is the warmth of the graduate student community.

We've extended our deadline for the Baylor Philosophy PhD program this year until January 2.


The anole photo is from today, from the path by the river on campus (or just off campus?).  The butterflies are from Thanksgiving, though I saw a number today, too.

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?