Two kinds of Platonism

There are two kinds of Platonism. Both hold that there are properties. But they differ as to the grounding relation that holds between predication and property possession (I will also assume that what goes for properties goes for relations, but sometimes formulate things just in terms of properties for simplicity). Both agree that if there is a property of Fness (there might not be if F is gerrymandered or negative, on sparse Platonisms), then x is F if and only if x instantiates Fness. Deep Platonism further affirms:

1. If there is a property of Fness, then the fact that x is F is grounded in the fact that x instantiates Fness.

Shallow Platonism denies (1). It is likely to instead affirm:

2. If there is a property of Fness, then the fact that x is F partly grounds or explains the fact that x instantiates Fness.

Deep Platonism faces two problems. The first is the Regress Problem. For if Deep Platonism is true, then "instantiates" seems non-gerrymandered and positive, and so it should correspond to a Platonic entity, the relation of instantiation. Then, the fact that x instantiates Fness will be grounded in the fact that x and Fness instantiate instantiation. But this leads to a vicious regress where each instantiation relation is grounded in the next.

The second is the Creation Problem. Everything that exists and is distinct from God is created by God. If the properties are all distinct from each other, then at most one is identical with God, and hence all but at most one property are created by God. But explanatorily prior to creating anything, will have multiple properties such as that he is able to do something and that he knows something. But how can he have those properties when there is at most one property at this point in the explanatory story?

Both problems have Deep Platonist solutions. For instance, one might say that "instantiates" is the unique non-gerrymandered and positive predicate that has no Platonic correspondent, or one might say that (1) has an exception in the case of instantiation. And one might say that God has at most one property, say divinity, and he is identical to that property. (But this, too, seems to lead to exceptions for (1), or perhaps an implausible view of what predicates correspond to properties. For there sure seem to be many other non-gerrymandered and positive predicates, like "is wise" and "is powerful", that apply to God.)

But Shallow Platonism has a particularly neat solution to both problems. There either is no regress, or if there is a regress, it is an unproblematic forward regress: because x is F, x and Fness instantiate instantiation, and because of that x, Fness and instantiation instantiate instantiation, and so on. Forward regresses are not at all problematic. And while it may be explanatorily prior to the creation of properties (or of all but one property) that God is wise, it is not explanatorily prior to the creation of properties that God instantiates wisdom.

Dominating reasons

Some things just aren't reasons for a choice. For instance, the fact that a portion of ice cream has an odd number of carbon atoms is by itself not a reason at all for eating the ice cream, and the fact that I find hot chocolate unpleasant is by itself not a reason to choose the hot chocolate. (The "by itself" qualifier is needed. I might have some instrumental reason for consuming an odd number of carbon atoms, and I might be ascetically training myself to consume what is unpleasant.)

Sometimes, however, something can be a reason for A without being a reason for A rather than B. For instance, that I enjoy hot chocolate to degree 100 is a reason to have hot chocolate. But if I enjoy ice cream to degree 150 on the very same scale, then my enjoying hot chocolate to degree 100 is not by itself a reason to have hot chocolate rather than ice cream. In the absence of other reasons, it would then make no rational sense to choose hot chocolate over ice cream, since my reason for hot chocolate is strictly dominated by my reason for ice cream.

At least roughly speaking:

• Reason R (not necessarily strictly) dominates reason S if and only if S is not at all a reason for choosing an action supported by S over an action supported by R.
• Reason R strictly dominates reason S if and only if R dominates S and S does not dominate R.

And of course reasons can be replaced by sets of reasons here. Then, Buridan's Ass cases are ones where the reasons for each action non-strictly dominate the reasons for the other.

Rational choice between A and B occurs only when one has reason to choose A over B and reason to choose B over A. Thus, rational choice between A and B occurs only when the reasons for neither option dominate the reasons for the other.

Definition: Reasons R and S are incommensurable if and only if neither dominates the other.

Thus, rational choice is possible only given sets of reasons that are incommensurable.

Hour of Code

I think computer programming should be taught from early grades, both in order to expand the mind and to be able to use the computing tools around us more effectively (it's no harder than cursive handwriting, and so much more useful!). And then I came across Hour of Code, which is an attempt to introduce kids to programming in an hour during Computer Science Education Week (Dec. 9-15). I hope to run an afterschool Hour of Code event at my kids' school for grades five and six.

I tested the "Write your first computer program" tutorial on my 11-year-old daughter, and she completed it in half an hour, so it seems right for children her age, though she wasn't deeply excited about it. (But it did frustrate my 8-year-old son.)

My daughter then went for the "Create a holiday card" activity with Scratch, and made an animated Christmas card. Scratch is an event-driven graphical programming environment for kids that reminds me a lot of Hypercard (which I still have a full version of on our Powerbook 190 laptop). Scratch has her hooked. I tried Scratch with her about two years ago, but the computer we were running it on was a bit too old and we didn't see the nice little intro they now have for Hour of Code, so it was frustrating. It helps, of course, that she's done some Mindstorms programming before.

Both of the activities were web-based. They're still in beta, but I highly recommend them for kids. There are lots of other activities there. And it's not too late to volunteer at your kids' school to run an Hour of Code activity for them.

Necessary coincidences

On standard naturalist views, neither the objective facts of mathematics and morality nor their grounds (e.g., Platonic entities, etc.) have any influence on how matter behaves and hence on how we think. This seems to imply that if our mathematical or moral beliefs happen to be true, that's just a coincidence. But merely coincidentally true belief isn't knowledge (maybe it's Gettiered knowledge). Now consider a response on which:

1. Of biological necessity, we have evolved through unguided natural selection to have mathematical or moral beliefs of type N.
2. Of metaphysical necessity, most mathematical or moral beliefs of type N are true.
3. Therefore, there is no coincidence here and nothing that calls out for further explanation.

(Erik Wielenberg offered a response of roughly this sort last week here.) The response presupposes that there cannot be coincidences between necessary truths that call out for further explanation.

But there can. There are two real numbers, x and y, between 0 and 1 with the following property. If you write them out in binary, divide up the bits into groups of eight, and then put the bits into ASCII code, then you actually find a lot of comprehensible text in each. In particular:

4. In x, there are infinitely many occurrences of "Consider the following proposition:", and each of them is followed by a well-formed arithmetical sentence (say, written in TeX) and a period. In fact, all possible arithmetical sentence thus occur in x.
5. In y, at exactly the same point as each "Consider the following proposition:" string occurs, there instead occurs "That's true" or "That's false."
6. Moreover, "That's true" occurs in y precisely when the proposition given in that place in x is true, and "That's false" occurs when the proposition is false.

But of course, it is necessary that x and y have the binary expansion they do.

Now, if we're given two such numbers x and y, the above is an apparent coincidence that calls out for explanation. And maybe an explanation can be given, say in terms of a selection effect: Perhaps the reason we're considering these two numbers is because a logically omniscient being exhibited them to us, and the being chose the two numbers for these remarkable properties. No surprise then!

But what if turned out that x=π and y=e satisfy (4)-(6)? Then we would consider the above coincidence truly remarkable. We would search for some deep mathematical reason for it. But suppose this search fizzled out and we came to conclude that although, of course, it is necessary that π and e have the properties (4)-(6), e.g., it being necessary that Fermat's Last Theorem occur at location 12848994949494888 in π (I assume it doesn't) and "That's true" in e at the same location and so on, mathematically this is just an incredibly unlikely coincidence. That would be a highly intellectually unsatisfying position. So unsatisfying that we would reach for a metaphysical explanation like Descartes' story about God having designed mathematics or a science fictional one like Carl Sagan's novel about aliens having embedded a message in π. We would have good reason to accept such an explanation if it were offered, and if we rejected there being such an explanation, we would have to say we have just a coincidence.

Thus, we can imagine cases of agreement between necessary mathematical facts which genuinely call out for explanation. And we can imagine concluding that although they call out for explanation, there is none, and hence we have a coincidence. Thus we can imagine a coincidence in the realm of necessary truth.

The Axiom of Choice in some claims about probabilities

I spent the last week trying to get clear on the logical interconnections between a number of results about probabilities that are relevant to formal epistemology and that use a version of the Axiom of Choice in proof, such as:

1. For every non-empty set Ω, there is an ordered field K and a K-valued probability function that assigns non-zero finitely additive probability to every non-empty subset of Ω.
2. For every non-empty set Ω, there is a full finitely additive conditional probability on Ω (i.e., a Popper function with all non-empty subsets normal).
3. The Banach-Tarski Paradox holds: one can decompose a three-dimensional ball into a finite number of pieces that can be moved around and made into two balls of the same size.
4. There are Lebesgue non-measurable sets in the unit interval [0,1].

All of these results require some version of the Axiom of Choice. It turns out that there is a very simple map of their logical interconnections in Zermelo-Fraenkel (ZF) set theory:

• BPI→(1)→(2)→(3)→(4),

where BPI is the Boolean Prime Ideal theorem, a weaker version of the Axiom of Choice.

The proof from BPI to (1) is standard--just let K be an ultrapower of the reals with an appropriate ultrafilter. That from (1) to (2) is almost immediate: just define the conditional probabilities via the ratio formula and take the standard part. Pawlikowski's proof of Banach-Tarski easily adapts to use (2) (officially, he uses Hahn-Banach). Finally, Foreman and Wehrung show in ZF that every subset of Rⁿ is Lebesgue measurable iff every subset of [0,1] is. But it follows from (2) that not every subset of R³ is Lebesgue measurable.

This has important consequences. Without the Axiom of Choice, one can prove that either (a) there are sets that have no regular probabilities no matter what ordered field is chosen for the values and no full conditional probabilities, or (b) the Banach-Tarski Paradox holds and hence there are no rigid-motion-invariant probabilities on regions of three-dimensional space big enough to hold a ball. And in either case, Bayesianism has a problem.

Various challenges for evolutionary psychology

I took my kids to an event tonight with Eric Wielenberg, and my 11-year-old daughter found herself puzzling about how our emotions could evolve. I tried to convince her that some aspects of our emotional lives that have plausible evolutionary explanations. But she came up with a number of challenges for evolutionary explanations that withstood my critical scrutiny. They are an interesting bunch:

1. The desire to do what is wrong
2. Happiness, in the sense of contentment—think of a cat lying down while being stroked (my example)
3. The drive to achieve things the hard way even when one can get them without effort—wanting the achievement of getting a meal by hunting even if one can get an equally delicious meal without much effort (her example)
4. Mercy towards weaker animals, even ones that we could eat or that could eventually harm us.

Note that while occasional pleasure can definitely contribute to fitness by rewarding behavior, the kind of contentment in (2) might actually be deleterious, by making us less active. One might think of (3) as a form of practicing, but that seems a stretch in a lot of cases. And (4) is particularly puzzling—it's not so hard to come with stories about a lot cases of mercy towards members of our own species, but mercy to other species seems a lot harder. One might try a malfunction story for (1), but my daughter thinks (1) is too widespread to be anything like a disease, and we surely don't want to say that (2)-(4) are faults. I suppose one could try to find a spandrel story about some of these, but I am not sure how convincing these will be and whether they will fit well with the fact that (2)-(4) seem to be components of our flourishing.

Of course (1) is puzzling on a theistic view (whether evolutionary or not), though perhaps the theist can give a view on which it's a distortion of a desire to imitate God, by desiring to be ultimately in charge. On the other hand, (2) and (4) have very nice theistic explanations.

The Axiom of Choice gives and takes away probabilities

Suppose you have assigned coherent (i.e., finitely additive) probabilities to a collection of options, but then you come upon a refinement of this collection of options, including more fine-grained ones. For instance, previously you had assigned probabilities to propositions about which individuals in a population had brown or non-brown eyes. But now you realize you should refine the non-brown-eye group into the blue, green and none of the above groups, as well as considering people's hair-color. It is intuitively plausible that if your initial probabilities about brown versus non-brown eyes were coherent, you should be able to come up with a coherent assignment of probabilities to the refined cases, e.g., by equally dividing up the probabilities of the non-brown eye category between the three newly recognized suboptions. One way to state the above in full generality is this:

1. Whenever P is a finitely additive probability assignment on an algebra F of subsets of a space Ω, and F is a subalgebra of a finer algebra G of subsets of Ω, then P can be extended to G.

Is this true? Well, the Axiom of Choice implies it is and I think this was first proved by Tarski. The Axiom of Choice gives probabilities (or at least implies that they exist).

But the Axiom of Choice also takes away probabilities. One famous case is that of nonmeasurable subsets of an interval, but that's about countably additive stuff, while I am right now talking of finitely additive stuff. One way it does this is by implying the Banach-Tarski paradox:

2. A solid three-dimensional ball can be decomposed into a finite number of subsets which can be moved rigidly to produce two balls of the same size as the original. It follows that no region in three dimensions that has the room to contain a ball has a (finitely additive) probability assignment on all its subsets that is invariant under rigid motions.

Now, for those who, like Bayesians, think epistemology is basically probability theory, (1) is going to be attractive but (2) is going to be paradoxical and repugnant. These thinkers may be tempted to give up the Axiom of Choice in order to deny (2). But I think they are likely to still want (1). And since (1) is known not to actually be equivalent to the Axiom but weaker, there might seem to be some hope.

Question: Can we coherently deny (2) and accept (1) in Zermelo-Fraenkel Set Theory (ZF) without Choice?

It turns out that the hope is vain. For:

Theorem. Claim (1) implies claim (2) in ZF.

For Luxemburg (1969) proved that (1) is equivalent to the Hahn-Banach Theorem, while Pawlikowski (1991) proved that the Hahn-Banach Theorem implies the Banach-Tarski Paradox.

So we cannot get away from this (at least not without even more radical revision to set theory). If we want probability existence results like (1), we must accept probability nonexistence results like (2).

Moral and perfect freedom

Say that moral freedom is the ability to choose between a morally permissible and a morally wrong action. Perfect freedom, on the other hand, is a freedom to choose between morally permissible actions, but with a perfect and infallible directedness at the good of the sort that God and the saints in heaven are said to enjoy.

Morriston (and others before him, like Quentin Smith, but Morriston's piece is particularly well developed) basically offers this dilemma: Either moral freedom is better than than perfect freedom or not. If moral freedom is better, then God has the less valuable kind of freedom, which seems incompatible with God's perfection. If moral freedom is not better, then God should have created beings with perfect freedom, since this way all the evils flowing from our misuse of moral freedom would have been prevented.

I want to make two points. First, the relevant question shouldn't be whether moral freedom is better than perfect freedom, but whether the action of creating beings with moral freedom is better than the action of creating beings with perfect freedom. An action can be better than another, even if its intended effect is no better. For instance, if I promised an editor a paper on modality, and I have the time for only one paper, the action of writing a paper on modality is better than the action of writing a paper on the Trinity, even if the effect of the latter action may be the better.

With this distinction in mind, one notices that there is a difference in value between God's creating a being that inevitably loves him back and his creating a being that gets to choose whether or not to love him back. Even if a being that inevitably loves him back is no better, God's action of inviting someone into communion with him very much has something very significant to be said for it that God's creating someone who will inevitably be in communion with him doesn't.

The second point is this. There is a value to loving someone by choice. Now when God and St Francis love each other, each loves the other by choice. Francis chooses to love God, while being able not to. But God likewise chooses to love Francis, while being able not to. Now you might say: "But doesn't God have to love everyone, given that he is love itself?" I agree (though I know some don't): necessarily, if Francis exists, God loves him. But Francis doesn't have to exist—Francis only exists because God chose to create him. Thus God has freedom whether to love Francis, a freedom he exhibits in choosing to create Francis, something he did not have to do.

Now there is a necessary asymmetry here. Since we cannot have a choice about whether God exists, and once God exists, there is the obligation to love him, our choice requires moral freedom: it is a choice between the good of love and the evil of not loving the supremely lovable God. But for God the choice whether to love Francis was at the same time a choice whether to create Francis. This choice does not require moral freedom, since it is not a choice between good and evil, but only good and good-or-neutral.

So on both sides, the relationship between God and Francis involves a freedom to love or not to love Francis. This freedom is valuable and God has it. But in Francis this freedom, of necessity, is moral freedom. So it is not that moral freedom is more valuable than perfect freedom. Rather, it is that in a creature, freedom whether to love God has to be an instance of moral freedom, while in God, freedom whether to love a creature is an instance of perfect freedom.

Objection: But doesn't Morriston's problem come back when we consider the doctrine of the Trinity? The Father and the Son do not choose to love each over not loving each other. The Son is not a creature, and so the Father does not choose to create the Son rather than the Father. Yet, surely, the intra-Trinitarian love is the most perfect kind of love. So wouldn't a creature that has to love God have a better kind of love than one that has a choice about it?

Response 1: A certain symmetry and equality in love are particularly valuable. In the Trinity, we have a symmetry: no Person of the Trinity has the freedom to fail to love another. But we automatically start off with God having a choice whether to be in a relationship of love with a creature, namely through his having a choice whether to create the creature. It makes for deeper equality and symmetry if the creature also has a choice about how to respond to God.

A love relationship that is chosen on one side but not on another is less valuable through the asymmetry. Imagine a woman who chose to have a baby had a drug that would ensure that the child would love her back. She had a choice, to some degree, whether to love the baby. But she refuses the child a choice about whether and how to reciprocate the relationship.

Response 2: To choose to love makes one intimately related to one's love. But in the Trinitarian case, there is an even deeper relation to love: God is identical with his love.

Manipulation, randomness and responsibility

Suppose you chose A over B, but that through minor changes in your circumstances, changes that at most slightly rationally affect the reasons for your decision and that do not intervene in your mental functioning, I could reliably control whether you chose A or whether you chose B. For instance, maybe I could reliably get you to choose B by being slightly louder in my request that you do A, and to choose A by being slightly quieter. In that case your choice is in effect random—the choice is controlled by features that from the point of view of your rational decision are random—and your responsibility slight.

Now suppose you are a friend of mine. To save my life, you would need to make a sacrifice. There is a spectrum of possible sacrifices. At the low end, you need to spend five minutes in my company (yes, it gets worse than that!). At the high end, you and everybody else you care about are tortured to death. With the required sacrifice being at the low end, of course you'd make the sacrifice for your friend. But with the required sacrifice being at the high end, of course you wouldn't. Now imagine a sequence of cases with greater and greater sacrifice. As the sacrifice gets too great, you wouldn't make it. Somewhere there is a critical point, a boundary between the levels of sacrifice you would undertake to save my life and ones you wouldn't. This critical point is where the reasons in favor of the sacrifice and those against it are balanced.

Speaking loosely, as the degree of required sacrifice increases, the probability of your making that sacrifice goes down. The "probability" here is something rough and frequentist, compatible with determinism. If determinism is true, however, in each precise setup around the critical point, there is a definite fact of the matter as to what you would do. And there are two possibilities about your character:

1. You have a neat and rational character, so that for all sacrifices below the critical level, you'd do it, and for all the sacrifices above the critical level, you wouldn't do it.
2. At around the critical value, whether you make the sacrifice or not comes to be determined not by the degree of sacrifice but by irrational factors—what shoes I'm wearing, how long ago you had lunch, etc.

I suspect that in most realistic cases we'd have (2). But on both options, we have effective randomness: your action can be controlled through minor changes in your environment that at most slightly affect your reasons. For instance, in option (1), where you are simply rationally going by the strength of the reasons, the slightest tipping of the scales will do the job—you'll undergo 747.848 minutes of torture but not 747.849. And in option (2), non-rational factors that have only a slight rational effect, or no rational effect, control your aciton. In both cases, your choice can be controlled. By the principle I started the post with, around the critical point you couldn't be very responsible.

But surely you would be very praiseworthy for undertaking a great sacrifice to save my life, especially around the critical point. That the sacrifice is so great that we're very near the point where the reasons are balanced does nothing to diminish your responsibility. If anything, it increases your praiseworthiness. Thus determinism is false.

This is not an argument for incompatibilism. I am not arguing here that responsible is incompatible with determinism. I am arguing that having full responsibility around the critical level is incompatible with determinism.

An interesting equivalent to the Hahn-Banach theorem

The Hahn-Banach Theorem (HB) cannot be proved without some version of the Axiom of Choice. (Technically, it's stronger than ZF but weaker than BPI.) A cool fact about HB is that it is sufficient for proving the existence of nonmeasurable sets and even for proving the Banach-Tarski paradox.

In 1969, Luxemburg proved that the Hahn-Banach theorem is equivalent to the claim that every boolean algebra has a (finitely additive) probability measure.

He also proved that the Hahn-Banach theorem is also equivalent to the following interesting claim:

1. For any set Ω and proper ideal N of subsets of Ω (i.e., N is closed under finite unions, any subset of a member of N is a member of N, but not every subset of Ω is in N), there is a (finitely additive) probability measure on all subsets of Ω that is zero on every member of N.

One direction can be proved by using the existence of a probability measure on the quotient boolean algebra 2^Ω/N via Hahn-Banach. The other direction follows (I don't know if Luxemburg did it this way) from this fact which can be proved without the axiom of choice:

2. Any boolean algebra is isomorphic to the quotient of a boolean algebra of sets.

Given the Axiom of Choice, this is a trivial consequence of the Stone Representation Theorem. Without the Axiom of Choice, (2) is a quick consequence of a theorem of Buskes, de Pagter and van Rooij. It can also be proved directly.[note 1]

(This is part of a general observation that some of the things that can be done with ultrafilters can also be done with filters, though the results may be weaker.)

Kaczor's review of One Body

Chris Kaczor has a review of my One Body over at Public Discourse.

Why faith in the testimony of others is loving: Notes towards a thoroughly ethical social epistemology

Loving someone has three aspects: the benevolent, the unitive and the appreciative. (I develop this early on in One Body.) Believing something and gaining knowledge on the testimony of another teaching involves all three aspects of love.

Appreciation: If I believe you on testimony, then I accept you as a person who speaks honestly and reasons well. It is a way of respecting your epistemic achievement. This does not mean that a failure to accept your testimony is always unappreciative. I may appreciate you, but have good reason to think that the information you have received is less complete than mine.

Union: Humans are social animals, and our sociality is partly constituted by our joint epistemic lives. To accept your testimony is to be united with you epistemically.

Benevolence: Excelling at our common life of learning from and teaching one another is a part of our flourishing. If I gain knowledge from you, you thereby flourish as my teacher. Thus by learning from you, I benefit not only myself as learner but I benefit you by making you a successful teacher.

We learn from John Paul II's philosophical anthropology that we are essentially givers and accepters of gifts. In giving, epistemically and otherwise, we are obviously benevolent, but also because it is the human nature to be givers, in grateful acceptance of a gift we benefit, unite with and affirm the giver, thereby expressing all three aspects of love.

A Moorean reason not to believe in an open future

Let A be the best valid argument that has been given for an open future. But I have really excellent reasons to think that it's true that my ears won't turn to diamond over the next hour, reasons clearly stronger than my reasons to think that all of A's premises are true. But if there is an open future, then it's not true that my ears won't turn into diamonds over the next hour (since that depends on indeterministic quantum phenomena). So I have on balance reason to think A is unsound.

Of course, there are other arguments for an open future. But I can pair each one with a fact about the future that I have reason to be a lot more confident in than the truth of the argument's premises.

Open theism and risk

We have many well-justified beliefs about how people will freely act. For instance, I have a well-justified belief that at most a minority of my readers will eat a whole unsweetened lemon today. Yet most of you can. (And maybe one or two of you will.) Notice that a fair amount of our historical knowledge is based on closely analogous judgments. When we engage in historical analysis we base ourselves on knowledge of how people freely act individually or en masse. We know that various historical events occurred because of what we know about how people who report historical events behave--given what we know about human character, we know the kinds of things they are likely to tell the truth about, the kinds of things they are likely to lie about and the kinds of things they are likely to be mistaken about. But it would be strange to claim knowledge about past human behavior and disclaim knowledge about future human behavior when exactly similar probabilistic regularities give us both.

But if open theism is true, then God cannot form such beliefs about the future. For open theists agree that God is essentially infallible in his beliefs: it is impossible for God to hold a false belief. But if God were in a habit of forming beliefs about how people will in fact act, then in at least some possible worlds, and probably in this one as well, God would have false beliefs—it may be 99.99% certain that I won't eat a whole unsweetened lemon today, but that just means that there is a 0.01% chance that I will.

So the open theist, in order to hold on to divine infallibility, must say that God keeps from having beliefs on evidence that does not guarantee truth. Why would God keep himself from having such beliefs, given that they seem so reasonable? Presumably to avoid the risk of being wrong about something.

But now notice that open theism has God take really great risks. According to open theism, in creating the world, God took the risk of all sorts of horrendous evils. The open theist God is not at all averse to taking great risks about creation. So why would he be so averse to taking risks with his beliefs?

The open theists who think that there are no facts about the future have an answer here. They will say that my belief that at most a minority of my readers will eat a whole unsweetened lemon today is certainly not true, since the fact alleged does not obtain, and hence that I shouldn't have this belief. Instead, I should have some probabilistic belief, like that present conditions have a strong tendency to result in the nonconsumption of these lemons. My argument here is not addressed to these revisionists.

The fixity of the past and laws

Suppose an ever-truthful dictator tells you that he is interested in the truth value of a proposition p solely about the laws of nature and how the world was like 1000 years ago. He tells you that in an hour he will infallibly find out whether p is true. If so, he will execute you. If not, he will let you go free.

Unless you have a time machine or a God who answers prayers before they are made is on your side, fatalism is surely the appropriate attitude. There is nothing you can do about whether you will be executed.

Next suppose that determinism is true. That shouldn't affect what was just said. Determinism does not create new supernatural powers to affect the past.

Now add that your identical twin is told by the dictator that he will be executed if and only if he scratches his head in an hour.

Finally suppose that your proposition p is the proposition that one thousand years ago the universe was such as to nomically determine that in an hour you will scratch your head, and suppose that the dictator's infallible method for finding out whether p is true is simply to see if you scratch your head. Then you are in the same boat as your twin! Each of you will be killed if and only if he scratches his head. Since fatalism is true for you, it's true for your twin. Thus he can't do anything about his execution, and in particular he has no freedom about scratching his head. And so compatibilism is false.

Powers and strivings

Neo-Aristotelian metaphysics of causation is centered around the notion of powers. I wonder, though, whether the term "power" isn't too passive to convey the notion. In my sexual ethics work, I have used the term "striving".

One's image of a power may be something like a match: it has the power to start a fire, but it is not doing anything until the triggering condition—heat from friction—is applied. On the other hand, one's imagine of a striving may be something like a bow pulled back, with the tension and compression in the bow's limbs actively exerting a force that is closely balanced by the tension in the archer's muscles. The striving in the bow is something active, as can be seen by the way the archer gets tired the longer the string is held back, and only needs the archer's resistance to end in order for it to be released. Or one's image of striving may be the archer slowly and effortfully (but perhaps with an appearance of effortlessness) pulling back the string.

It is clear that metaphysically the cases of the archer pulling back the string and the archer holding the string are alike. The only difference is that in the case of holding the strivings of the limbs and the archer are balanced while in the case of pulling back the strivings are imbalanced. I think that in the end the case of the match is also metaphysically alike. In my One Body book, I say in this connection that an army in readineness is not an idle army. Likewise, the match in readiness is not an idle match. The powers of the molecules in the match are in balance, holding each other back, actively striving against each other like the archer against the bow.

Even in God there is a constant striving. God intrinsically is pure act, says Aquinas, and the eternal Trinitarian relations of procession emphasize this. It is crucial not to think of the Trinitarian relations as something historical, in the way that human parenthood in non-ideal cases can be, but rather as continuing—the Father did not once and for all generate the Son at the beginning of time, but in the timeless eternity of God's dynamism always is begetting him, and the Holy Spirit eternally proceeds as the active love between the Father and the Son. And on the economic side, Leibniz talks of the divine ideas of the various worlds that God can create as competing with one another for God to actualize them. We shouldn't overemphasize that competitiveness, but some emphasis is helpful here. (And this competitiveness in the end fits better into an incommensurability picture of creation than Leibniz's optimalism.)

To a large degree everything I said here is metaphor. But good choice of metaphor guides good philosophizing.

Risk compensation

Rational decision theory predicts that decreasing the riskiness of a activity will tend to increase the prevalence and degree of risky activity. As paragliding is made safer, one expects more people to be engaged in paragliding and those who are engaged in it to do it more intensely. But of course increasing the prevalence and degree of behavior will tend to increase the prevalence of occurrences of the negative outcome that one was decreasing. This is the phenomenon of risk compensation: decreasing the risk of a negative outcome of an activity is to some degree—maybe sometimes completely—compensated for by an increase in the prevalence and degree of engagement in the risky activity. For instance, taxi drivers who have antilock brakes tend to follow the vehicle in front of them more closely.

Suppose that in some case the risk compensation to some safety measure is complete: i.e., the prevalence of the relevant negative outcome (say, crashes or fatalities) is unchanged, due to the compensating increase in the prevalence and degree of the risky behavior. One might think that at this point the safety measure was pointless.

Whether this conclusion is correct depends, however, on what values the risky behavior itself has when one brackets the risk in question. If the risky behavior has positive value when one brackets the risk, the safety measure does in fact achieve something good: an increase in prevalence and degree of valuable but risky behavior with no increase in negative outcome. Paragliding is (I assume) a pleasant way to enjoy the beauty of the earth and to stretch the limits of human ability. An increase in the prevalence of paragliding without an increase of negative outcomes is all to the good.

When the behavior is completely neutral, then the safety measure, however, is simply a waste given complete risk compensation.

Finally, if the risky behavior has negative value even when one brackets the particular risky outcome, then in the case where the risk compensation is complete, the safety measure is counterproductive. It does not decrease the negative outcome it is aimed at, but by increasing the prevalence of an otherwise unfortunate activity it on balance has a negative outcome. For instance, suppose bullfighting is an instance of immoral cruelty to animals. Then apart from the risks to the bullfighter, the activity has negative value: it harms the animal and damages the soul of the person. If a safety measure for prevention of goring then were compensated for by an increase in prevalence, the safety measure would have on balance a negative outcome: there would be no decrease in gorings but there would be an increase in immoral and harmful activity.

Moreover, in cases where the risky behavior has independently negative value, a safety measure can have on balance negative effect even when the risk compensation is quite modest. Suppose that (I am making up the numbers) gorings occur in 10% of bullfights and cruelty to bulls in 80%. Suppose, further, that cruelty to bulls is at least as bad as goring (since it not only harms the bull but more importantly it seriously damages the soul of the cruel person). Then a safety measure that decreases the probability of goring by a half but results in a modest 10% increase in the prevalence of bullfighting will have on balance negative effect. For suppose that previously there were 1000 bullfights, and hence 100 gorings and 800 instances of cruelty. Now there will be 1100 bullfights, and hence (at the new rate) 55 gorings and 880 instances of cruelty. We have prevented 45 gorings at the cost of 80 instances of cruelty, and that is not worth it.

Much of the public discussion of risk compensation and safety measures centers on sex, and particularly premarital sex. We should typically expect some behavioral change in the direction of risk compensation given a safety measure. If one thinks premarital sex to be itself typically valuable, then even given total risk compensation one will think the safety measure to be worthwhile. If one thinks premarital sex to be value-neutral, then as long as the risk compensation is incomplete (i.e., the decrease in the risks due to the safety measure is not balanced by the increase in prevalence), one will think the safety measure to be worthwhile (at least as long as the costs of the safety measure are not disproportionate). But if one thinks premarital sex to have negative moral value, then one may well think a safety measure to be counterproductive even if the risk compensation is incomplete—as in my imaginary bullfighting cases.

I think public discussion of things like condoms and sex education could be significantly improved if participants in the discussion were all open and clear about the fact that we should expect some degree of risk compensation—that's just decision theory[note 1]—and were mutually clear on what value they ascribe to the sexual activity itself, independently of the risks in question.

In these kinds of cases, it sounds very attractive to say: "Let's focus on what we all agree on. Being gored, getting AIDS and teen pregnancy are worth preventing." But a public policy focused successful at improving the outcomes we have consensus on can still be on balance harmful, as my (made up) bullfighting example shows.

Treatment and enhancement

Let's grant that my ability to use my hands is normal. Suppose a world-class violin maker loses the little finger on her non-dominant hand. This slightly impedes her ability to use her hands. But the incredible amount of gross and fine motor skills that a top violin maker needs to have exceed my own merely normal skills to such a degree that she is going to do better in any non-gerrymandered manual activity (wiggling ten fingers is gerrymandered!) than I.

But my own abilities are normal. So if her abilities exceed mine, how can hers fail to be normal? Yet it seems clear that to the extent that reattachment of the finger would be treatment rather than enhancement, even though it takes someone whose abilities are above normal, and raises her even higher above what is normal.

So we should not define the kind of abnormalcy or disability that calls for medical treatment in terms of a below-normal degree of overall function. For overall function is affected by compensation—the violin-maker's manual skills compensate for her genuine loss. Rather, we must look at something like local function, the function of a particular bodily subsystem. And here it is clear that when she lost her finger, she lost the full use of a subsystem. Disability is the loss of the full functioning of a subsystem, not necessarily of the whole.

But now here we have an interesting thing. An operation that destroys the functioning of a bodily subsystem that otherwise would have functioned properly, even if it does not adversely affect—or maybe even enhances—overall functioning of the person, nonetheless is producing a disability. Now a physician should be a healer. Sometimes to heal one must destroy a subsystem—amputating a gangrenous limb is an example. Even in those cases, the destruction is a moral reason for the physician not to do the operation, though a reason that may well be outweighed (as it is in the gangrenous limb case) by the need for healing.

But this is far more problematic when the destruction of a subsystem is not done in order to heal the system as a whole, even if in some way the person as a whole benefits. Suppose Sam has a job that consumes all his waking hours and involves no contact with people, and his normal interest in social relationships makes him less good at his job. Moreover, suppose the sad economic realities are such that he has no hope of another job. He is going to live a life of loneliness and unfulfilled sociality. Should we give him drugs that would destroy his sociality? Such drugs would improve his life, after all. Yes: but they would do so by destroying a subsystem. And their positive effect would not be a form of healing—at most a form of enhancement at adaptation to unfortunate circumstances. So there is strong—I think conclusive—moral reason why a physician should not give Sam the drugs.

And the same line of thought applies in a much more controversial, because more realistic, case: sterilization.

HHS contraception mandate

I am one of the signatories of an amicus brief by Catholic theologians and ethicists in the Gilardi case against the HHS contraception mandate. The DC Circuit Court recently ruled against the HHS in this case. The main line of thought in our brief was that a Catholic employer's providing coverage for contraception makes the employer cooperate in the employee's use of contraception. Now, Catholic moral thought not only takes (marital) contraception to be wrong, but also takes cooperation in someone else's sin to be morally problematic. Whether the cooperation is not just problematic but wrong depends on questions about the degree and kind of cooperation involved.

Reflecting on these issues makes me think there is a second really crucial thing going on, besides making employers complicit in employees' sin (and we touch on this in the brief, though I think it can be developed). One of the reasons for the HHS mandate is precisely to encourage women to use contraception (this is certainly not denied). But this means that the employer is made complicit in what the employer conscientiously takes to be the government's morally wrongful promotion of wrongdoing. And this cooperation is even closer, and thus far even more morally problematic, than cooperation with the employee's use of contraception. For the employer here acts as the government's instrument in its policy of promotion of contraception. Note that promotion does not require success: in this case, governmental promotion of contraception occurs whether or not any employees actually use the contraception.

Invariance of Popper functions under symmetries

Popper functions are primitive finitely additive conditional probabilities—i.e., P(A|B) is the fundamental quantity, and P(A) is the defined quantity. Now, in some situations we have expect our probabilities to be invariant under some group G of symmetries. For instance, if we're shooting an idealized dart at a circular target and aiming at the center, our idealized method of shooting might be rotationally invariant so that the probability of hitting some region A will be the same as the probability of hitting rA where r is some rotation about the center. (In real life, this need not be so. For instance, in archery, one might have bigger error in the vertical direction than in the horizontal direction, or vice versa, depending on one's skills.) We might also think that similarly there is invariance under reflections about lines through the center.

With unconditional probabilities, we can just formulate these invariance condition as: P(gA)=P(A) for all symmetries g in G and all (measurable) regions A. But how to formulate this for conditional probabilities?

There are two natural definitions:

• P is weakly invariant if and only if P(A|B)=P(gA|gB) for all A, B and g.
• P is strongly invariant if and only if (a) whenever A∪gA⊆B, we have P(A|B)=P(gA|B) and (b) whenever A⊆B∩gB, we have P(A|B)=P(A|gB).

Fact: Conditions (a) and (b) in the definition of strong invariance are equivalent.

Personally, I find weak invariance to be the more intuitive condition, though strong invariance has some intuitive pull. It's an interesting question how the two are related.

One interesting special case is where G is generated by symmetries of finite order. A symmetry g has finite order provided that there is a finite number n such that gⁿ is the identity—i.e., applying it n times gets you back to where you started. For instance, rotation by an angle of 360/n degrees where n is a non-zero integer has finite order—you do this |n| times and you're back where you started. And all reflections have finite order.
Fact: If every symmetry in G can be written as a combination of symmetries of finite order, then weak invariance implies strong invariance.

For instance, while most rotations in the plane don't have finite order (only ones by a rational-number angle do), any rotation in the plane can be generated by combining two reflections. Thus, in our circular target case, where we are looking at invariance under reflections and rotations, weak invariance implies strong invariance.

Armstrong in this paper claims that weak invariance implies strong invariance in general (Prop. 1.3). Unfortunately Armstrong's proof is incomplete. And well it might be. For yesterday I came up with a super-simple case showing:
Fact: Weak invariance does not imply strong invariance.

It would be interesting to characterize cases where weak invariance does imply strong invariance. Two general cases are known to me. One is where the symmetries are generated by symmetries of finite order. The second is where the conditional probabilities are defined by the ratio formula starting with a regular probability (one that assigns non-zero probability to each empty set).

Error theory

Error theorists in ethics think that claims like "Murder is wrong" are all false. But this seems to me to be a self-undermining position. For if there were no true moral claims, our moral predicates would have no meaning. They would simply be nonsense.

The irrationality of undue scepticism

One might think that sceptical tendencies are intellectually safe. It's clearly irrational for one's level of belief to exceed one's level of evidence. But it does not seem harmful to be more cautious, and hence to make one's level of belief be less than one's level of evidence.

However, if one's levels of belief are classical consistent probabilities, then when one's degree of belief in p is lower than what the evidence yields, one's degree of belief in not-p will be higher than what the evidence yields. And that is surely bad.

One might think it's not so bad as long as one's degree of belief in not-p stays well below 1/2. After all, in that case, one isn't believing not-p, and hence none of one's beliefs is irrational. Yes, but such errors are apt to add up. Suppose there are twelve independent propositions p₁,...,p₁₂ that one believes at the 0.75 level, instead of the 0.95 that the evidence supports. Then one's degree of belief in their conjunction will be (0.75)¹²=0.03, instead of the (0.95)¹²=0.54 that the evidence yields. And hence the sceptic will believe the negation of the conjunction of the 12 propositions to degree 0.97, instead of having a level of belief in the conjunction of only 0.46, as per the evidence. Excess of caution leads to excessive credulity.

That's true on classical sharp numerical probabilities. The sceptic may better off on interval-valued probabilities. But even so, depending on how one interprets the intervals, there may be a similar kind of criticism available.

The above underlines something I heard Bob Brandom say: one needs to have a reason to be a sceptic about something.