A reading of 1 Corinthians 14:33b-34a

1 Corinthians 14:33b-34a is one of the “hard texts” of the New Testament. The RSV translates it as:

As in all the churches of the saints, the women should keep silence in the churches.

Besides the fact that this is a hard saying, a textual difficulty is that earlier in the letter, at 11:5, Paul has no objection to women prophesying or praying (it seems very likely that praying would be out loud), though it has been suggested that this was outside of a liturgical context. Nor does later Church practice prohibit women from joining in vocal prayer during the liturgy.

I assume that the second "the churches" means "the churches of Corinth", while the first "the churches" refers to the churches more generally. And yesterday at our Department Bible study, I was struck by the fact that the “As” (Greek hōs) that begins the text can be read as “In the manner of”. On that reading, the first sentence of the hard text does not say that women should keep silent in the Corinthian churches. Rather, it says that women should keep silent in the Corinthian churches in the way and to the extent to which they keep silent in the other churches. In other words, women should only speak up in Corinthian liturgies at the points at which women speak up in non-Corinthian liturgies. This is compatible with women having various speaking roles—but only as long as they have these roles in “all the churches of the saints.”

(Note, however, that some versions punctuate differently, and make “As in all the churches of the saints” qualify what came earlier rather than what comes afterwards. My reading requires the RSV’s punctuation. Of course, the original has no punctuation.)

On this reading, the first sentence of the text is an application of a principle of liturgical uniformity between the churches, and Paul could equally well have said the same thing about the men. But the text suggests to me that there was some particular problem, which we can only speculate about, that specifically involved disorderly liturgical participation by Corinthian women, in addition to other problems of disorderly participation that Paul discusses earlier in the chapter.

The difficulty for my reading is the next sentence, however:

For they are not permitted to speak, but should be subordinate, as even the law says. (1 Cor. 14:34b, RSV)

I would want to read this with “speak” restricted to the kinds of speech not found in the other churches. Perhaps in the other churches, there was no “chatting in the pews”, or socializing during the liturgy (Mowczko in a very nice summary of interpretations notes that this is St. John Chrystostom’s interpretation).

Another interpretation is that “the law” here is Roman law or Corinthian custom (though I don’t know that in Koine Greek “nomos” can still cover custom, like it can in classical Greek), so that Paul is reprising a motif of noting that the Corinthians are behaving badly even by their own cultural standards.

I don’t know that my reading is right. I think it is a little bit more natural to read the Greek as having a complete prohibition on women speaking, but my reading seems to be grammatically permissible, and one must balance naturalness of language with consistency in a text (in this case, consistency with 11:5). And in the case of a Biblical text, I also want an interpretation compatible with divine inspiration.

White lies

Suppose Bob is known by Alice to be an act utilitarian. Then Bob won’t believe when Alice asserts p in cases where Bob knows that by Alice’s lights, if p is false, nonetheless the utility of getting Bob to believe p exceeds the utility of Bob knowing that p is false. For in such cases an act utilitarian is apt to lie, and her testimony to p is of little worth.

Such cases are not uncommon in daily life. Alice feels bad about a presentation she just made. Bob praises it. Alice dismisses the praise on the grounds that even if her presentation was bad, getting her to feel better outweighs the utility of her having a correct estimate of the presentation, at least by Bob’s lights.

Praise from an act utilitarian is of little value: instead of being direct evidence for the proposition that one did well, it is direct evidence for the proposition that it would be good for one to believe that one did well. Now, that it would good for one to believe that one did well is some evidence that one did well, but it is fairly weak evidence given facts about human psychology.

And so in cases where praise is deserved, the known act utilitarian is not going to promote utility for friends as effectively as a known deontologist, since the deontologist’s praise is going to get a lot more credence. Such cases are not rare: it is quite common for human performances to deserve praise and for the agent to be such that they would benefit from being uplifted by praise. While, on the other hand, in cases where praise is undeserved, the known act utilitarian’s praise does little to uplift the spirit.

These kinds of ordinary interactions are such a large part of our lives that I think a case can be made that just on the basis of these, by the lights of act utilitarianism, an act utilitarian should either hide their act utilitarianism from others or else should convert to some other normative ethical view (say, by self-brainwashing). Since the relevant interactions are often with friends, and it is unlikely one can hide one’s character from one’s friends over a significant period of time, and since doing so is likely to be damaging to one’s character in ways that even the act utilitarian will object to, this seems to be yet another of the cases where act utilitarianism pushes one not to be an act utilitarian.

Such arguments have been made before in other contexts (e.g., worries that the demandingness of act utilitarianism would sap our energies). They are not definitive refutations of act utilitarianism. As Parfit has convincingly argued, it is logically consistent to hold that an ethical theory is true but that one morally should not believe it. But still we get the conclusion that everybody morally should be something other than an act utilitarian. For if act utilitarianism is false, you surely shouldn’t be an act utilitarian. And if it’s true, you shouldn’t, either.

The above, I think, is more generally relevant to any view on which everyday white lies are acceptable. For the only justifications available for white lies are consequentialist ones. But hiding from one’s friends that one is the sort of person who engages in white lies is costly and difficult, whereas letting it be known undercuts the benefits of the white lies, while at the same removing the benefits of parallel white truths. Thus, we should all reject white lies in our lives, and make it clear that we do so.

Here, I use “white lie” in a sense in which it is a lie. I do not think “Fine” is a lie, white or otherwise, when answering “How are you?” even when you are not fine, because this is not a case of assertion but of a standardized greeting. (There is no inconsistency in an atheist saying “Good-bye”, even though it’s a contraction of “God be with you.”) One way to see this isn't a lie is to note that it is generally considered rude (but sometimes required) to suggest that one's interlocutor lied, there is nothing rude about saying to someone who answered “Fine”: “Are you sure? You look really tired.” At that point, we do move into assertion category. The friend who persists in the “Fine” answer but isn't fine now is lying.

The reportable and the assertible

I’ve just had a long conversation with a grad student about (inter alia) reporting and asserting. My first thought was that asserting is a special case of reporting, but one can report without asserting. For instance, I might have a graduate assistant write a report on some aspect of the graduate program, and then I could sign and submit that report without reading it. I would then be reporting various things (whether responsibly so would depend on how strong my reasons to trust the student were), but it doesn’t seem right to say that I would be asserting these things.

But then I came to think that just as one can report without asserting, one can assert without reporting. For instance, there is no problem with asserting facts about the future, such as that the sun will rise tomorrow. But I can’t report such facts, even though I know them.

It’s not really a question of time. For (a) I also cannot report that the sun rose a million years ago, and (b) if I were to time-travel to the future, observe the sunrise, and come back, then I could report that the sun will rise tomorrow.

And it’s not a distinction with respect to the quantity of evidence. After all, I can legitimately report what I had for dinner yesterday, but it’s not likely that I have as good evidence about that as I do that the sun will rise tomorrow.

I suspect it’s a distinction as to the kind of evidence that is involved. I am a legally bound reporter of illegal activity on campus. But I can’t appropriately report that a violation of liquor laws occurred in the dorms over the weekend if I know it only on the basis of the general claim that such violations, surely, occur every weekend. The kind of evidence that memory provides is typically appropriate for reporting, while the kind of evidence that induction provides is at least typically not.

Interestingly, although I can’t appropriately report that tomorrow the sun will rise, I can appropriately report that I know that the sun will rise tomorrow. This means that the reportable is not closed under obvious entailment.

Lying and consequences

Suppose Alice never lies while Bob lies to saves innocent lives.

Consider circumstances where Alice and Bob know that getting Carl to believe a proposition p would save an innocent life, and suppose that Alice and Bob know whether p is true.

In some cases of this sort, Bob is likely to do better with respect to innocent lives:

1. p is false and Carl doesn’t know Alice and Bob’s character.

2. p is false and Carl doesn’t know that Alice and Bob know that getting Carl to believe p would save an innocent livfe.

For in cases 1 and 2, Bob is likely to succeed in getting Carl to believe p, while Alice is not.

But in one family of cases, Alice is likely to do better:

3. p is true and Carl knows Alice and Bob’s character and knows that they believe that getting Carl to believe p would save an innocent life.

For in these cases, Carl wouldn’t be likely to believe Bob with regard to p, as he would know that Bob would affirm p whether p was true or false, as Bob is the sort of person who lies to save innocent lives, while Carl would surely believe Alice.

Are cases of type (1) and (2) more or less common than cases of type (3)?

I suppose standard cases where an aggressor at the door is asking whether a prospective victim is in the house may fall under category (1) when the aggressor knows that they are known to be an aggressor and will fall under category (2) when the aggressor doesn’t know that they are known to be an aggressor (Korsgaard discusses this case in a paper on Kant on lying).

On the other hand, category (3) includes some death penalty cases where (a) the life of the accused depends on some true testimony being believed and (b) the testifier is someone likely to think the accused to be innocent independently of the testimony (say, because the accused is a friend). For in such a case, Bob would just give the testimony whether it’s true or false, while Alice would only give it if it were true (or at least she thought it was), and so Bob’s testimony carries no weight while Alice’s does.

Category (3) also includes some cases where an aggressor at the door knows the character of their interlocutor in the house, and knows that they are known to be an aggressor, and where the prospective victim is not in the house, but a search of the house would reveal other prospective victims. For instance, suppose a Gestapo officer is asking whether there are Jews in the house, which there aren’t, but there are Roma refugees in the house. The Gestapo officer may know that Bob would say there aren’t any Jews even if there were, and so he searches the house and finds the Roma if Bob is at the door; but he believes Alice, and doesn’t search, and the Roma survive.

Roughly, the question of whether Alice or Bob’s character is better consequentialistically comes down to the question whether it is more useful, with respect to innocent life, to be more believable and always honest (Alice) or to be less believable and able to lie (Bob).

More on grounding of universals

The standard First Order Logic translation of “All As are Bs” is:

1. ∀x(A(x)→B(x)).

Suppose we accept this translation and we further accept the principle:

2. Universal facts are always partially grounded in their instances.

Then we have the oddity that the fact that all ravens are black seems to be partially grounded in my garbage can being black. Let R(x) and B(x) say that x is a raven and black, respectively, and let g be my garbage can. Then an instance of ∀x(R(x)→B(x)) is R(g)→B(g), and the latter material conditional is definable as ¬R(g)∨B(g). But a disjunction is grounded in its true disjuncts, and hence this one will be grounded in B(g) (as well as in ¬R(g)).

There are three things to dispute here: the translation (1), the grounding principle (2), and the claim that a material conditional is grounded in its consequent whenever that consequent is true. Of these, I am most suspicious of the translation of the two-place universal quantifier and the grounding principle (2).

Grounding of universals and partial grounding

It is common to claim that:

1. The fact that everything is F is partially grounded in the fact that a₁ is F and in the fact that a₂ is F and so on, for all the objects a_i in the world.

But this can’t be right if partial grounds are parts of full grounds. For suppose you live in a world with only two objects, a and b, which are both sapient. Then everything is sapient, and by (1) it follows that:

2. The fact that everything is sapient is partially grounded in a being sapient and in b being sapient.

But suppose partial grounds are parts of full grounds. The facts that a is sapient and b is sapient are not a full ground of the fact that everything is sapient, because the full grounds of a fact entail that fact, and a being sapient and b being sapient doesn’t entail that everything is sapient (since it’s possible for a to be sapient and b to be sapient and yet for there to exist a c that is not).

So we need to be able to add something to the two particular sapience facts to get full grounds. The most obvious thing to add is:

3. Everything is a or b.

Clearly fact (3) together with the facts that a is sapient and b is sapient will entail that everything is sapient.

But applying (1) to (3), we get:

4. Fact (3) is partially grounded in the facts that a is a or b and that b is a or b.

But, once again, if partial grounds are parts of full grounds, then we need a fact to add to the two facts on the right hand side of the grounding relation in (4) such that together these facts will entail (3). But the obvious candidate to add is:

5. Everything is a or b.

And that yields circularity.

So it seems that either we should reject the particular-grounds-universal principle (1) or we should reject the principle that partial grounds are parts of full grounds.

Here is a reason for the latter move. Maybe we should say that God’s creating me is partially grounded in God. But that’s merely a partial grounding, since God’s existence doesn’t entail that God created me. And it seems that the only good candidate for a further fact to be added to the grounds so as to entail that God created me would be my existence. (One might try to add the fact that God willed that I exist. But by divine simplicity, that fact has to be partly constituted by my existence or the like.) But my existence is grounded in God’s creating me, so that would be viciously circular.

Are desires really different from wishes?

It is tempting to conflate what is worth desiring with what is worth pursuing. But there seem to be cases where things are worth desiring but not worth pursuing:

1. Having a surprising good happen to you completely gratuitously—i.e., without your having done anything to invite it—seems worth desiring but the pursuit of it doesn’t seem to make sense.

2. If I have published a paper claiming a certain mathematical result, and I have come to realize that the result is false, it seems to make perfect sense to desire that the result be true, but it makes no sense to pursue that.

The standard response to cases like 1 and 2 is to distinguish wishes from desires, and say that it makes sense to wish for things that it makes no sense to pursue, but it does not make sense to desire such things.

But consider this. Suppose in case 2, I came to be convinced that God has power over mathematics, and that if I pray that the result be true, God might make it be true. Then the affective state I have in case 2 would motivate me to pray. But the nature of the affective state need not have changed upon coming to think that God has power over mathematics. Thus, either (a) I would be motivated to pray by a mere wish or else (b) wishes and desires are the same thing. But the wish/desire distinction does not fit with (a), which leaves (b).

I suppose one could claim that a desire just is a wish plus a belief that the object is attainable. But that makes desires be too gerrymandered.

Three places for beauty in representational art

There seem to be three senses in which beauty can be found in a piece of representational art:

1. The piece represents something as beautiful.

2. The piece in and of itself is beautiful.

3. The task of representing is performed beautifully.

One can have any one of the three without the others. For instance, the one-line poem “The kitty was pretty” satisfies 1 but fails 2 and 3. Though, to be precise, I think sense 1 is not a real case of something being beautiful, but only of something being represented as beautiful. The kitty could be ugly and yet described as pretty.

I think 3 is particularly interesting. It opens up the way for works of art that are in themselves not beautiful and that do not represent beauty, but which do a beautiful job of representing their objects (Sartwell says that Picasso’s Guernica may be beautiful; I think my aspect 3 of the beauty of representational art may explain this). Note that “beautiful” here does not merely mean “accurate”, as the case of my one-line poem shows, since that poem may represent the beauty of a cat with perfect accuracy, but there is very little of the beautiful about how it accomplishes this.

Fundamental bearers of aesthetic properties

I am finding myself frustrated trying to figure out whether the fundamental bearers of aesthetic properties are mental states or things out in the world. When I think about the fact that there does not seem to be any significant difference between the beauty of music that one actually listens to with one’s ears versus “music” that is directly piped to the auditory center of the brain, that makes me think that the fundamental bearers of aesthetic properties are mental states.

But on the other hand, when I think about the beauty of character exhibited by a Mother Teresa, I find it hard to think that it is my mental states—say, my thoughts about Mother Teresa—that bear the fundamental aesthetic properties. If I thought that it was my mental states that are the bearers of aesthetic properties, then I would think that a fictional Mother Teresa is just as beautiful as a real one. But it seems to me that a part of the beauty of the real Mother Teresa is that she is real.

Perhaps the fundamental bearers of aesthetic properties vary. For music and film, perhaps, the fundamental bearers are mental states: the experiences one paradigmatically has when listening and viewing (but which one could also have by direct brain input). For the characters of real people, perhaps, the fundamental bearers are the people themselves or their characters. For the characters of fictional people, perhaps, the fundamental bearers are mentally constituted (in the mind of the author or that of the audience or both).

Maybe the beauty of a real person is a different thing from the beauty of a fictional character. This kind of makes sense. For we might imagine an author who creates a beautiful work of literature portraying a nasty person: the nasty person qua fictional character is beautiful, but would have been ugly in real life, perhaps.

But I hate views on which we have such a pluralism of fundamental bearers of a property.

Conciliationism and natural law epistemology

Suppose we have a group of perfect Bayesian agents with the same evidence who nonetheless disagree. By definition of “perfect Bayesian agent”, the disagreement must be rooted in differences in priors between these peers. Here is a natural-sounding recipe for conciliating their disagreement: the agents go back to their priors, they replace their priors by the arithmetic average of the priors within the group, and then they re-updated on all the evidence that they had previous got. (And in so doing, they lose their status as perfect Bayesian agents, since this procedure is not a Bayesian update.)

Since the average of consistent probability functions is a consistent probability function, we maintain consistency. Moreover, the recipe is a conciliation in the following sense: whenever the agents previously all agreed on some posterior, they still agree on it after the procedure, and with the same credence as before. Whenever the agents disagreed on something, they now agree, and their new credence is strictly between the lowest and highest posteriors that the group assigned prior to conciliation.

Here is a theory that can give a justification for this natural-sounding procedure. Start with natural law Bayesianism which is an Aristotelian theory that holds that human nature sets constraints on what priors count as natural to human beings. Thus, just as it is unnatural for a human being to be ten feet tall, it is unnatural for a human being to have a prior of 10⁻¹⁰⁰ for there being mathematically elegant laws of nature. And just as there is a range of heights that is natural for a mature human being, there is a range of priors that is natural for the proposition that there are mathematically elegant laws.

Aristotelian natures, however, are connected with the actual propensities of the beings that have them. Thus, humans have a propensity to develop a natural height. Because of this propensity, an average height is likely to be a natural height. More generally, for any numerical attribute governed by a nature of kind K, the average value of that attribute amongst the Ks is likely to be within the natural range. Likely, but not certain. It is possible, for instance, to have a species whose average weight is too high or too low. But it’s unlikely.

Consequently, we would expect that if we average the values of the prior for a given proposition q over the human population, the average would be within the natural range for that prior. Moreover, as the size of a group increases, we expect the average value of an attribute over the group to approach the average value the attribute has in the full population. Then, if I am a member of the group of disagreeing evidence-sharing Bayesians, it is more likely that the average of the priors for q amongst the members of the group lies within the natural human range for that prior for q than it is that my own prior for q lies within the natural human range for q. It is more likely that I have an unnatural height or weight than that the average in a larger group is outside the natural range for height or weight.

Thus, the prior-averaging recipe is likely to replace priors that are defectively outside the normal human range with priors within the normal human range. And that’s to the good rationally speaking, because on a natural law epistemology, the rational way for humans to reason is the same as the normal way for humans to reason.

It’s an interesting question how this procedure compares to the procedure of simply averaging the posteriors. Philosophically, there does not seem to be a good justification of the latter. It turns out, however, that typically the two procedures give the same result. For instance, I had my computer randomly generate 100,000 pairs of four-point prior probability spaces, and compare the result of prior- to posterior-averaging. The average of the absolute value of the difference in the outputs was 0.028. So the intuitive, but philosophically unjustified, averaging of posteriors is close to what I think is the more principled averaging of priors.

The procedure also has an obvious generalization from the case where the agents share the same evidence to the case where they do not. What’s needed is for the agents to make a collective list of all their evidence, replace their priors by averaged priors, and then update on all the items in the collective list.

Musical beauty and virtual music

We have beautiful music at home on a hard drive. But wait: the arrangement of magnetic dipoles on a disc is not musically beautiful! So it seems inaccurate to say that there is music on the hard drive. Rather, the computer, hard drive, speakers and the orientations of magnetic dipoles jointly form a device that can produce the sound of beautiful music on demand.

One day, however, I expect many people will have direct brain-computer interfaces. When they “listen to music”, no sounds will be emitted (other than the quiet hum of computer cooling fans, say). Yet I do not think this will significantly change anything of aesthetic significance. Thus, the production of musical sounds seems accidental to the enjoyment of music. Indeed, we can imagine a world where neither composers nor performers nor audiences produce or consume any relevant sounds.

Perhaps, then, we should say that what is of aesthetic significance about my computer, with its arrangements of magnetic dipoles, is that it is a device that can produce musical experiences.

But where does the musical beauty lie? Is it that the computer (or the arrangement of magnetic dipoles on its drive) is musically beautiful? That seems wrong: it seems to be the wrong kind of thing to be musically beautiful. Is it the musical experiences that are musically beautiful? But that seems wrong, too. After all, a musical performance—of the ordinary, audible sort—can be musically beautiful, and yet it too gives rise to a musical experience, and surely we don’t want to say that there are two things that are musically beautiful there.

Perhaps a Platonic answer works well here: Maybe it is some Platonic entities that are trulymusically musically beautiful, and sometimes their beauty is experienced in and through an audible performance and sometimes directly in the brain?

Another possibility I am drawn to is that there is a property that isn’t exactly beauty, call it beauty^*, which is had by the musical experiences in the mind. And it is this property that is the aesthetically valuable one.

And of course what goes for musical beauty goes for visual beauty, etc.

Natural law: Between objectivism and subjectivism

Aristotelian natural law approaches provide an attractive middle road between objectivist and subjectivist answers to various normative questions: the answers to the questions are relative to the kind of entity that they concern, but not to the particular particular entity.

For instance, a natural law approach to aesthetics would not make the claim that there is one objective beauty for humans, klingons, vulcans and angels. But it would make the absolutist claim that there is one beauty for Alice, Bob, Carl and Davita, as long as they are all humans. The natural lawyer aestheticist could take a subjectivist’s accounts of beauty in terms, of say, disinterested pleasure, but give it a species relative normative twist: the beautiful to members of kind K (say, humans or klingons) is what should give members of kind K disinterested pleasure. The human who fails to find that pleasure in a Monet painting suffers from a defect, but a klingon might suffer from a defect if she found pleasure in the Monet.

Supervenience and omniscience

Problem: It seems that if God necessarily exists, then the moral automatically supervenes on the non-moral. For, any two worlds that differ in moral facts also differ in what God believes about moral facts, and presumably belief facts are non-moral. This trivializes the mechanism of supervenience for theists.

Potential Solution: Divine simplicity makes God’s beliefs about God-external facts be externally constituted. Thus, a part of what makes it true that God believes that there are sheep are the sheep. If so, then perhaps a part of what makes it true that God believes a moral fact is that very moral fact. Thus, God’s beliefs about moral facts are partly constituted by the moral facts, and hence are not themselves non-moral.

Anti-reductionism and supervenience

In the philosophy of mind, those who take anti-reductionism really seriously will also reject the supervenience of the mental on the non-mental. After all, if a mental property does not reduce to the non-mental, we should be able to apply a rearrangement principle to fix the non-mental properties but change the mental one, much as one can fix the shape of an object but change its electrical charge, precisely because charge doesn’t reduce to shape or shape to charge. There might be some necessary connections, of course. Perhaps some shapes are incompatible with some charges, and perhaps similarly some mental states are incompatible with some physical arrangement. But it would be surprising, in the absence of a reduction, if fixing physical arrangement were to fix the mental state.

Yet it seems that in metaethics, even the staunchest anti-reductionists tend to want to preserve the supervenience of the normative on the non-normative. That is surprising, I think. After all, the same kind of rearrangement reasoning should apply if the normative properties do not reduce to the non-normative ones or vice versa: we should be able to fix the non-normative ones and change the normative ones at least to some degree.

Here’s something in the vicinity I’ve just been thinking about. Suppose that A-type properties supervene on B-type properties, and consider an A-type property Q. Then consider the property Q^B of being such that the nexus of all B-type properties is logically compatible with having Q. For any Q and B, having Q^B is necessary for having Q. But if Q supervenes on B-type properties, then having Q^B is also sufficient for having Q. Moreover, Q^B seems to be a B-type property in our paradigmatic cases: if B is the physical properties, then Q^B is a physical property, and if B is the non-normative properties, then Q^B is a non-normative property. (Interestingly, it is a physical or non-normative property defined in terms of mental or normative properties.)

But now isn’t it just as weird for a staunch anti-reductionist to think that there is a non-normative property that is necessary and sufficient for, say, being obligated to dance as it is for a staunch anti-reductionist to think there is a physical property that is necessary and sufficient for feeling pain?

Supervenience and natural law

The B-properties supervene on the A-properties provided that any two possible worlds with the same A-properties have the same B-properties.

It is a widely accepted constraint in metaethics that normative properties supervene on non-normative ones. Does natural law meet the contraint?

As I read natural law, the right action is one that goes along with the teleological properties of the will. Teleological properties, in turn, are normative in nature and (sometimes) fundamental. As far as I can see, it is possible to have zombie-like phenomena, where two substances look and behave in exactly the same way but different teleological properties. Thus, one could have animals that are physically indistinguishable from our world’s sheep, and in particularly have four legs, but, unlike the sheep, have the property of being normally six-legged. In other words, they would be all defective, in lacking two of their six legs.

This suggests that natural law theories depend on a metaphysics that rejects the supervenience of the normative. But I think that is too quick. For in an Aristotelian metaphysics, the teleological properties are not purely teleological. A sheep’s being naturally four-legged simultaneously explains the normative fact that a sheep should have four legs and the non-normative statistical fact that most sheep in fact have four legs. For the teleological structures are not just normative but also efficiently causal: they efficiently guide the embryonic development of the sheep, say.

In fact, on the Koons-Pruss reading of teleology, the teleological properties just are causal powers. The causal power to ϕ in circumtances C is teleological and dispositional: it is both a teleological directedness towards ϕing in C and a disposition to ϕ in C. And there is no metaphysical way of separating these aspects, as they are both features of the very same property.

Our naturally-six-but-actually-four-legged quasi-sheep, then, would differ from the actual world’s sheep in not having the same dispositions to develop quadrapedality. This seems to save supervenience, by exhibiting a difference in non-normative properties between the sheep and the quasi-sheep.

But I think it doesn’t actually save it. For the disposition to develop four (or six) legs is the same property as the teleological directedness to quadrapedality in sheep. And this property is a normative property, though not just normative. We might say this: The sheep and the quasi-sheep differ in a non-normative respect but they do not differ in a non-normative property. For the disposition is a normative property.

Perhaps this suggests that the natural lawyer should weaken the supervenience claim and talk of differences in features or respects rather than properties. That would allow one to save a version of supervenience. But notice that if we do that, we preserve supervenience but not the intuition behind it. For the intuition behind the supervenience of the normative on the non-normative is that the normative is explained by the non-normative. But on our Aristotelian metaphysics, it is the teleological properties that explain that actual non-normative behavior of things.

Properties, relations and functions

Many philosophical discussions presuppose a picture of reality on which, fundamentally, there are objects which have properties and stand in relations. But if we look to how science describes the world, it might be more natural to bring (partial) functions in at the ground level.

Objects have attributes like mass, momentum, charge, DNA sequence, size and shape. These attributes associate values, like 3.4kg, 15~kg m/s north-east, 5C, TTCGAAAAG, 5m and sphericity, to the objects. The usual philosophical way of modeling such attributes is through the mechanism of determinables and determinates. Thus, an object may have the determinable property of having mass and its determinate having mass 3.4kg. We then have a metaphysical law that prohibits objects from having multiple same-level determinates of the same determinable.

A special challenge arises from the numerical or vector structure of many of the values of the attributes. I suppose what we would say is that the set of lowest-level determinates of a determinable “naturally” has the mathematical structure of a subset of a complete ordered field (i.e., of something isomorphic to the set of real numbers) or of a vector space over such a field, so that momenta can be added, masses can be multiplied, etc. There is a lot of duplication here, however: there is one addition operator on the space of lowest-level momentum determinates and another addition operator on the space of lowest-level position determinates in the Newtonian picture. Moreover, for science to work, we need to be able to combine the values of various attributes: we need to be able to divide products of masses by squares of distances to make sense of Newton’s laws of gravitation. But it doesn’t seem to make sense to divide mass properties, or their products, by distance properties, or their squares. The operations themselves would have to be modeled as higher level relations, so that momentum addition would be modeled as a ternary relation between momenta, and there would be parallel algebraic laws for momentum addition and position addition. All this can be done, one operation at a time, but it’s not very elegant.

Wouldn’t it be more elegant if instead we thought of the attributes as partial functions? Thus, mass would be a partial function from objects to the positive real numbers (using a natural unit system) and both Newtonian position and momentum will be partial functions from objects to Euclidean three-dimensional space. One doesn’t need separate operations for the addition of positions and of momenta any more. Moreover, one doesn’t need to model addition as a ternary relation but as a function of two arguments.

There is a second reason to admit functions as first-class citizens into our metaphysics, and this reason comes from intuition. Properties make intuitive sense. But I think there is something intuitively metaphysically puzzling about relations that are not merely to be analyzed into a property of a plurality (such as being arranged in a ball, or having a total mass of 5kg), but where the order of the relata matters. I think we can make sense of binary non-symmetric relations in terms of the analogy of agents and patients: x does something to y (e.g. causes it). But ternary relations that don’t reduce to a property of a plurality, but where order matters, seem puzzling. There are two main technical ways to solve this. One is to reduce such relations to properties of tuples, where tuples are special abstract objects formed from concrete objects. The other is Josh Rasmussen’s introduction of structured mereological wholes. Both are clever, but they do complicate the ontology.

But unary partial functions—i.e., unary attributes—are all we need to reduce both properties and relations of arbitrary finate arity. And unary attributes like mass and velocity make perfect intuitive sense.

First, properties can simply be reduced to partial functions to some set with only one object (say, the number “1” or the truth-value “true” or the empty partial function): the property is had by an object provided that the object is in the domain of the partial function.

Second, n-ary relations can be reduced to n-ary partial functions in exactly the same way: x₁, ..., x_n stand in the relation if and only if the n-tuple (x₁, ..., x_n) lies in the domain of the partial function.

Third, n-ary partial functions for finite n > 1 can be reduced to unary partial functions by currying. For instance, a binary partial function f can be modeled as a unary function g that assigns to each object x (or, better, each object x such that f(x, y) is defined for some y) a unary function g(x) such that (g(x))(y)=f(x, y) precisely whenever the latter is defined. Generalizing this lets one reduce n-ary partial functions to (n − 1)-ary ones, and so on down to unary ones.

There is, however, an important possible hitch. It could turn out that a property/relation ontology is more easily amenable to nominalist reduction than a function ontology. If so, then for those of us like me who are suspicious of Platonism, this could be a decisive consideration in favor of the more traditional approach.

Moreover, some people might be suspicious of the idea that purely mathematical objects, like numbers, are so intimately involved in the real world. After all, such involvement does bring up the Benacerraf problem. But maybe we should say: It solves it! What are the genuine real numbers? It's the values that charge and mass can take. And the genuine natural numbers are then the naturals amongst the genuine reals.

God, probabilities and causal propensities

Suppose a poor and good person is forced to flip a fair and indeterministic coin in circumstances where heads means utter ruin and tails means financial redemption. If either Molinism or Thomism is true, we would expect that, even without taking into account miracles:

1. P(H)<P(T).

After all, God is good, and so he is more likely to try to get the good outcome for the person. (Of course, there are other considerations involved, so the boost in probability in favor of tails may be small.)

The Molinist can give this story. God knows how the coin would come out in various circumstances. He is more likely to ensure the occurrence of circumstances in which the subjunctive conditionals say that tails would comes up. The Thomist, on the other hand, will say that God’s primary causation determines what effect the secondary creaturely causation has, while at the same time ensuring that the secondary causation is genuinely doing its causal job.

But given (1), how can we say that the coin is fair? Here is a possibility. The probabilities in (1) take God’s dispositions into account. But we can also look simply at the causal propensities of the coin. The causal propensities of the coin are equibalanced between heads and tails. In addition to the probabilities in (1), which take everything including God into account, we can talk of coin-grounded causal chances, which are basically determined by the ratios of strength in the causal propensities. And the coin-grounded causal chances are 1/2 for heads and 1/2 for tails. But given Molinism or Thomism, these chances are not wholly determinative of the probabilities and the frequencies in repeat experiments, since the latter need to take into account the skewing due to God’s preference for the good.

So we get two sets of probabilities: The all-things-considered probabilities P that take God into account and that yield (1) and the creatures-only-considered probabilities P_c on which:

2. P_c(H)=P_c(T)=1/2.

Here, however, is something that I think is a little troubling about both the Molinist and Thomist lines. The creatures-only-considered probabilities are obviously close to the observed frequencies. Why? I think the Molinist and Thomist have to say this: They are close because God chooses to act in such ways that the actual frequencies are approximately proportional to the strengths of causal propensities that P_c is based on. But then the frequencies of coin toss outcomes are not directly due to the causal propensities of the coin, but only because God chooses to make the frequencies match. This doesn’t seem right and is a reason why I want to adopt neither Molinism nor Thomism but a version of mere foreknowledge.