Distributive promises

Suppose I promise my class to grade all the weekly homework within three days. In week four, I fail and am late with grading. If the content of my promise was simply the proposition

1. that I grade all the homework within three days,

then after week four, then no matter how speedy I am with grading the homework, proposition (1) is just plain false. And this means that my promise no longer generates any reason for me to grade the homework in weeks five and onward within three days, which seems wrong. I should, instead, apologize for failing in week four, and work even harder in the succeeding weeks.

I think this is because the promise was distributive. It wasn’t a promise to make proposition (1) true. It was a promise that for each week of homework generated a separate promissory reason to grade that week’s homework within three days.

The normative force of the promise is rather like making a separate promise for each week of class:

2. I promise that in the first week, I will grade the homework in three days. I promise that in the second week, I will grade the homework in three days. … I promise that in the 15th week, I will grade the homework in three days.

But other cases of distributive promises aren’t as neatly handled. Suppose I promise my class:

3. If you come to office hours, I will try to answer any question you might have about logic.

Again, this is distributive. If I refuse to answer a question out of laziness, it doesn’t let me off the hook with regard to the next question. But if I analyze this as a sequence of separate promises, then that sequence has to be infinite:

4. I promise that if you ask me q₁ at t₁, I will try to answer. And if you ask me q₂ at t₁, I will try to answer. … And if you ask me q₁ at t₂, I will try to answer. …

where the list goes through all the possible questions about logic and all the possible times that fall within office hours. Have I really made an infinite number of promises? This seems implausible. Moreover, normally, we think that one cannot make a promise without knowing that one has done so. But I might not know that q₈ is a question about logic or that t₃ is a time within office hours (in fact, I might not even know that t₃ exists—I might think that there are no intervals finer grained than a Planck time, but there might be).

Or I could make a promise to God regarding my treatment of all future as-yet-unconceived persons who have some property. Again, this is distributive: failure in one case does not excuse me from trying in other cases. But if analyzed as a collection of promises about actual future persons, we get the weirdness that what I have promised depends on what I will do. So it would have to be a collection of promises about possible future persons. And it’s not clear that this makes sense except given some controversial metaphysical assumptions, such as the existence of haecceities.

So, I think distributive promises don’t reduce to non-distributive ones.

Maybe, though, one can try to handle the cases with some sort of a doctrine of approximate truth. Perhaps when I promise a proposition, if I am no longer in a position to make the proposition true, I am required to make it as approximately true as I can? I think this kind of a principle will lead to counterintuitive results. For suppose that there is some benefit to you from having p be exactly true, while close approximations to p are harmful to you, while some way of making p very false is much better for you. Then I shouldn’t strive for a close approximation to p. (Think of cases of medicinal dosage, perhaps.)

Knowing how fast to change

Imagine two objects, M and H, where M has the intrinsic causal power of emitting some sort of a pulse once per minute and H has the intrinsic causal power of pulsing once per hour, and suppose M and H are causally separated from the rest of the universe. How do M and H “know” how quickly to pulse?

The problem seems to me to be particularly pressing on Aristotelian theories of time on which time is defined by the changes of objects. Let’s imagine that in addition to M and H there are a thousand identical clocks running in the universe, with all objects causally isolated from each other, and that there is nothing else that is changing besides what I have described. Presumably, then, the changes that define time are the changes in the positions of the clock hands. Then on our Aristotelian theory, to say that H pulses every hour just means that H pulses once per revolution of the minute hand on a typical clock, and to say that H pulses every minute means that H pulses once per revolution of the seconds hand on a typical clock.

But, first, how do M and H know about the movement of the hands of the clocks, so as to keep in sync with them, if all the objects are causally isolated?

And, second, let’s imagine this. God speeds up the clocks one by one by a factor of two: today he speeds up clock C₁, tomorrow he speeds up clock C₂, and so on, until eventually all the clocks have been sped up. After a week, seven clocks, C₁, ..., C₇, have been sped up. This doesn’t matter for the definition of time: seven clocks out of a thousand are negligible, and the time standards are set up by C₈, ..., C₁₀₀₀. According to C₁, ..., C₇, M pulses once per two rotations of the seconds hand, and H once per two rotations of the minutes hand. But according to C₈, ..., C₁₀₀₀, M pulses once per rotation of the seconds hand and H once per rotation of the minutes hand, since these rotations correspond to real minutes and real hours.

However, after 993 days, clocks C₁, ..., C₉₉₃ are running at the same rate and setting the time standards. The clocks C₉₉₄, ..., C₁₀₀₀ are the outliers. So, now, M pulses once per rotation of the seconds hands and H once per rotation of the minutes hands of C₁, ..., C₉₉₃. And M pulses twice per rotation of the seconds hands of C₉₉₄, ..., C₁₀₀₀, and H twice per rotation of their minutes hands. So at some point in the process of speeding up clocks C₈, ..., C₁₀₀₀, M’s pulses go from once per two rotations of the seconds hand of C₁, ..., C₇ to once per rotation, and similarly for H. But how can the speeding up of clocks other than C₁, ..., C₇ affect the correlations between H and M when everything is causally isolated?

(Note that talk of “speeding up clock C_n” makes sense even if the clocks jointly define time. If n = 1, we can understand it as speeding it up relative to clocks C₂, ..., C₁₀₀₀. If n > 1, we can understand it as bringing clock C_n in sync with clock C_n − 1. What may be a bit ambiguous is what “day n” means, but nothing hangs on the details of how to resolve that.)

But even without an Aristotelian theory of time it is puzzling how M and H “know” how quickly to pulse.

I think there is a nice solution that lets one keep a part of the Aristotelian theory of time: (external) time is not just the measure of change, but the measure of change in causally interconnected things. There is no common timeline between M, H and the clocks when there are no causal connections between them. This, I think, requires the rejection of any theory on which there is a metaphysically deep “objective present”.

Another move is to deny that there can be intrinsic causal powers that make reference to metric external time.

Continuous choices

Suppose at at noon, Alice is relaxed in an armchair listening to music, but at any given time she is capable of choosing to get up, walk over to the kitchen and make herself a sandwich for lunch, which it’s time for. For fifteen minutes she continues listening to the music and then gets up at 12:15. It seems that she is continually responsible for her continuing to sit until 12:15, and then she is responsible for getting up.

Here is one realistic question about what happened between 12:00 and 12:15:

1. Did Alice make a vast number of choices, one at every moment until 12:15, to remain seated, and then at 12:15 a choice to get up?

In favor of a positive answer, it is difficult to see how she could be responsible for not getting up at a given time if she did not choose not to get up.

But a positive answer seems psychologically implausible. Indeed, it doesn’t seem like Alice would be enjoying the music if every moment she had to positively choose to stay.

Also, let’s think about what the reasons weighing in on each choice would be. On the one hand, there is a very weak reason to get up now. It’s a weak reason because getting up the next moment would be just as good hunger-wise. On the other hand, there is a very weak reason to keep sitting in order to enjoy music between this moment and the next. It’s a weak reason because the amount of music involved is very small. Choices on the basis of such very weak reasons are hard to make. These reasons would be hard to weigh. And when making choices between hard to weigh reasons, it seems that the chances of going for either option should be of the same order of magnitude. But if Alice were to make a vast number of choices between getting up and staying between, say, 12:00 and 12:10, with each choice having roughly the same order of magnitude of probability, then it was very unlikely that all these choices were choices to stay.

I find the responsibility argument pretty persuasive, though. Maybe, though, the right story that balances psychological plausibility with intuitions about responsibility is this: Alice made a small number of choices between 12:00 and 12:15. Most of these choices were a choice whether to think harder about whether to get up or just let the status quo continue “for a while”. Most of the time, she chose just to let the status quo roll on. At a time t during which the status quo was “just rolling on”, Alice’s responsibility for not getting up was derivative from her choice to stop thinking about the question. Sometimes, however, Alice decided to think harder about whether to get up. Finally, she thought harder, and got up.

Since the number of choices is smaller on this story, it doesn’t interfere as much with the enjoyment. There is some interference, but that’s realistic. And since the number of choices is smaller, the probabilities of each option can be of the same order of magnitude without this creating any problems.

Now, prescinding from the realism behind the discussion of (1), we can ask the also interesting question:

2. Could it be that both (a) time is continuous and (b) Alice literally makes a choice to remain seated at every single moment of time between 12:00 and 12:15?

The answer, I think, is negative. For consider a choice at t. Alice would be choosing between the good of slightly more music and the good of slightly earlier relief of hunger. But how long as the “slightly more” and “slightly earlier”? Zero temporal length! For if time is continuous, and Alice is choosing at every moment, zero length of time elapses between choices. Indeed, there is no sense to the idea of “between choices”. So Alice would be choosing between zero-value goods. And that doesn’t make rational sense.

Probabilistic propensities and the Aristotelian view of time

Consider an item x with a half-life of one hour. Then over the period of an hour, it has a 50% chance of decaying, over the period of a second it only has a 0.02% chance of decaying. Imagine that x has no way of changing except by decaying, and that x is causally isolated from all outside influences. Don’t worry about Schroedinger's cat stuff: just take what I said at face value.

We are almost sure that x will one day decay (the probability of decaying approaches one as the length of time increases).

Now imagine that everything other than x is annihilated. Since x was already isolated from all outside influences, this should not in any way affect x’s decay. Hence, we should still be almost sure that x will one day decay. Moreover, since what is outside of x did not affect x’s behavior, the propensities for decay should be unchanged by that annihilation: x has a 50% chance of decay in an hour and a 0.02% chance of decay in a second.

But this seems to mean that time is not the measure of change as Aristotle thought. For if time were the measure of change, then there would be no way to make sense of the question: “How long did it take for x to decay?”

Here is another way to make the point. On an Aristotelian theory of time, the length of time is defined by change. Now imagine that temporal reality consists of x and a bunch of analog clocks all causally isolated from x. The chances of decay of x make reference to lengths of time. Lengths of time are defined by change, and hence by the movements of the hands of the clocks. But if x is causally isolated from the clocks, its decay times should have nothing to do with the movements of the clocks. If God, say, accelerated or slows down some of the clocks, that shouldn’t affect x’s behavior in any way, since x is isolated. But an Aristotelian theory of time, it seems, such an isolation is impossible.

I think an Aristotelian can make one of two moves here.

First, perhaps the kinds of propensities that are involved in having an indeterministic half-life cannot be had by an isolated object: such objects must be causally connected to other things. No atom can be a causal island. So, even though physics doesn’t say so, the decay of an atom has a causal connection with the behavior of things outside the atom.

Second, perhaps any item that can have a half-life or another probabilistic propensity in isolation from other things has to have an internal clock—it has to have some kind of internal change—and the Aristotelian dictum that time is the measure of change should be understood in relation to internal time, not global time.

Emotions and naturalism

On occasion, I’ve heard undergraduates suggest that naturalism faces a problem with emotions. They feel that a mere computational system would not have emotional states.

One might take this to be a special case of the problem of qualia, and I think it has some plausibility there. It is indeed hard to see how an emotionless Mary would know what it’s like to be scared or in love. Is it harder than in the case of ordinary sensory qualia, like that of red? I don’t know.

But I think it’s more interesting to take it to be a special case of the problem of intentionality or content. Emotions are at least partly constituted by intentional (quasi?) perceptual states with normative content: to be scared involves perceiving reality as containing something potentially bad for one and being in love involves perceiving someone as wonderful in some respects.

The standard materialist story about the content of perceptual states is causal: a perception of red represents an object as reflecting or emitting light roughly of a certain wavelength range because the perception is typically triggered by objects doing this. But on standard naturalist stories do not have room for normative properties to play a causal role. Post-Aristotelian scientific explanations are thought not to invoke normative features.

There is, of course, nothing here to worry an Aristotelian naturalist who believes that objects have natures that are both normative and causally explanatory.

Over the past year, I’ve been coming to appreciate the explanatory power of the Aristotelian story on which the very same thing grounds normativity and provides a causal explanation.

Is the past changing all the time?

The past is unchangeable. But if the A-theory is true, then past events constantly objectively get older and older. That seems to be a kind of objective change. So, the A-theory is false.

Unreleasable promises would be useful

Alice promises Bob to impose on him some penalty should Bob do a certain wrong. Bob does the wrong, and points out to Alice that imposing the penalty is some trouble to Alice, and that Bob is happy to release Alice from the promise.

If the promisee can always release the promiser from a promise, then in a case like this Bob may be exactly right. Deterrence thus would sometimes work better if one can make a promise that the promisee cannot release one from.

Of course, the fact that a normative power would be useful does not mean that the normative power exists. I doubt one can make a promise to another that the other cannot release one from.

One might, however, be able to vow the deterrent penalty to God. Or maybe just promise it to a third party (society?) who has no incentive to release one from the promise.

Punishment is not a strict requirement of justice

There is no strict duty to reward a person who has done a supererogatory thing. Otherwise, engaging in generosity would be a way of imposing a duty on others.

But punishment is the flip side of reward. Hence, there is no strict duty to punish a person who has done a wrong.

Of course, supererogatory action makes a reward fitting, and likewise wrong action makes a punishment fitting. But in neither case is the retributive response strictly required by justice.

A problem for some views of a temporal God

Among those who think that God is in time, there are two views:

1. God has existed for an infinite amount of time

2. God came into time a finite amount of time ago when he created the world.

The second view is held by William Lane Craig. On this view, God isn’t essentially temporal: he wouldn’t have been in time if he didn’t create time or temporal beings.

It’s occurred to me that those who accept the first view have the serious problem of getting out of the time-of-creation problems: Why did God create the world when he did, instead of earlier or later? And why did he wait an infinite amount of time before creating?

St Augustine’s answer that time starts with creation doesn’t work for those who accept (1).

Supposing creation itself to be omnitemporally eternal only solves the problem with (1) if one additionally accepts a relationalist B-theory of time. For otherwise there is still the question why God’s omnitemporally eternal creation process isn’t all shifted temporally by a year forward or backward in time.

Theories of time and truth-supervenes-on-being

Truth supervenes on being is the thesis that if two worlds have the same entities, they are otherwise the same. I just realized something that should be pretty obvious. One cannot hold on to all three of the following:

• A-theory

• eternalism

• truth supervenes on being.

For according to eternalism, at any two different times, the facts about what exists are the same. So if truth supervenes on being, at any two different times, all facts are the same—and in particular the facts about what time is objectively present will be the same, which contradicts A-theory.

In other words, just as the best version of presentism (that of Trenton Merricks) rejects that truth supervenes on being, so does the best version of the moving spotlight theory. Moreover, closed-future growing blockers—and, in particular, classical theist growing blockers—will also want to reject that truth supervenes on being since substantive truths about the future won’t supervene on being given growing block.

All this suggests that we are left with only two major theories of time available to those who accept that truth supervenes on being:

• B-theoretic eternalism

• growing block with an open future.

Presentism, change and ontology

Presentism says that only present things exist. This by itself cannot explain the nature of change.

For assume an ontology on which, necessarily, everything that exists is either a concrete substance or a necessary abstract object. Consider a world w₁ where all the concrete substances exist for all time, but some of them are changing their properties, e.g., shape. Notice that the presentist, the growing blocker and the eternalist all agree about what exists at w₁, since no entity comes into or out of existence on this ontology, and hence the differences between the three theories are irrelevant to w₁. Yet, w₁ is a world with change.

Hence presentism by itself cannot explain change.

Perhaps someone who thinks that presentism is needed to explain change should opt for a trope ontology rather than a substance-and-abstracta or substance-only ontology. For then they can say that at w₁, entities—namely, tropes—come into and out of existence.

"On the same grounds"

Each of Alice and Seabiscuit is a human or a horse. But Alice is a human or a horse “on other grounds” than Seabiscuit is a human or a horse. In Alice’s case, it’s because she is a human and in Seabiscuit’s it’s because he’s a horse.

The concept of satisfying a predicate “on other grounds” is a difficult one to make precise, but I think it is potentially a useful one. For instance, one way to formulate a doctrine of analogical predication is to say that whenever the same positive predicate applies to God and a creature, the predicate applies on other grounds in the two cases.

The “on other/same grounds” operator can be used in two different ways. To see the difference, consider:

1. Alice is Alice or a human.

2. Bob is Alice or a human.

In one sense, these hold on the same grounds: (1) is grounded in Alice being human and (2) is grounded in Bob being human. In another sense, they hold on different grounds: for the grounds of (1) also include Alice’s being Alice while the grounds of (2) do not include Bob’s being Alice (or even Bob’s being Bob).

Stipulatively, I’ll go for the weaker sense of “on the same grounds” and the stronger sense of “on different grounds”: as long as there is at least one way of grounding “in the same way”, I will count two claims as grounded the same way. This lets me say that Christ knows that 2 + 2 = 4 on the same grounds as the Father does, namely by the divine nature, even though there is another way in which Christ knows it, which the Father does not share, namely by humanity.

Even with this clarification, it is still kind of difficult to come up with a precise account of “on other/same grounds”. For it’s not the case that the grounds are literally the same. We want to say that the claims that Bob is human and that Carl is human hold on the same grounds. But the grounding is literally different. The grounds of the former is Bob’s possession of a human nature while the grounds of the latter is Carl’s possession of a human nature. Moreover, if trope theory is correct, then the two human natures are numerically different. What we want to say is something like this: the grounds are qualitatively the same. But how exactly to account for the “qualitatively sameness” is something I don’t know.

There is a lot of room for interesting research here.

Grace and theories of time

1. All grace received is given through Christ’s work of salvation.
2. Christ’s work of salvation happened in the first centuries AD and BC.
3. One cannot give something through something that does not exist.
4. Abraham received grace prior to the first century BC.
5. So, Abraham’s grace was given through Christ’s work of salvation.
6. So, it was true to say that Christ’s work of salvation exists even when it was yet in the future.
7. So, presentism and growing block are false.

Junia/Junias and the base rate fallacy

I think it would be useful to apply more Bayesian analyses to textual scholarship.

In Romans 16:7, Junia or Junias is described as “famous among the apostles”. Without accent marks (which were not present in the original manuscript) it is not possible to tell purely textually if it’s Junia, a woman, or Junias, a man. Moreover, “among the apostles” can mean “as being an apostle” or “to the apostles”. There seems to be, however, some reason to think that the name Junia is more common than Junias in the early Christian population, and the reading of “among” as implying membership seems more natural, and so the text gets used as support for women’s ordination.

This post is an example of how one might go about analyzing this claim in a Bayesian way. However, since I am not a Biblical scholar, I will work with some made-up numbers. A scholarly contribution would need to replace these with numbers better based in data (and I invite any reader who knows more Biblical scholarship to write such a contribution). Nonetheless, this schematic analysis will suggest that even assuming that there really were female apostles, it is more likely than not that Junia/s is one.

Let’s grant that in the early Christian population, “Junia” outnumbers “Junias” by a factor of 9:1. Let’s also generously grant that the uses of “famous among” where the individual is implied to be a member of the group outnumber the uses where the individual is merely known to the group by a factor of 9:1. One might think that this yields a probably of 0.9 × 0.9 = 0.81 that the text affirms Junia/s to be an apostle.

But that would be to commit the infamous base rate fallacy in statistical reasoning. We should think of a text that praises a Junia/s as “famous among the apostles” as like a positive medical test result for the hypothesis that the individual praised is a female apostle. The false positive rate on that test is about 0.19 given the above data. For to get a true positive, two things have to happen: we have to have Junia, probability 0.9, and we have to use “among” in the membership-implying sense, probability 0.9, with an overall probability of 0.81 assuming independence. So the false positive rate on the test is 1 − 0.81 = 0.19. In other words, of people who are not female apostles, 19 percent of them will score positive on tests like this.

But we have very good reason to think that even if there were any female apostles in the early church, they are quite rare. Our initial sample of apostles includes the 12 apostles chosen by Jesus, and then one more chosen to replace Judas, and none of these were women. Thus, we have reason to think that fewer than 1/13 of the apostles were women. So let’s assume that about 1/13 of the apostles were female. If there were any female apostles, they were unlikely to be much more common than that, since then that would probably have been more widely noted in the early Church.

Moreover, not everyone that Paul praises are apostles. “Apostle” is a very special position of authority for Paul, as is clear from the force of his emphases on his own status as one. Let’s say that apostles are the subjects of 1/3 of Pauline praises (this is something that it would be moderately easy to get a more precise number on).

Thus, the chance that a randomly chosen person that Paul praises is a female apostle even given the existence of female apostles is only about (1/13)×(1/3) or about three percent.

If we imagine Paul writing lots and lots of such praises, there will be a lot of Junia/s mentioned as “famous among the apostles”, some of whom will be male, some female, and some of whom will be apostles and some not.
All of these are the “positive test results”. Of these positive test results, the 97% percent of people praised by Paul who aren’t female apostles will contribute a proportion of 0.19 × 97%=18% of the positive test results. These will be false positives. The 3% people who are female apostles will contribute at most 3% of the positive test results. These will be true positives. In other words, among the positive test results, approximately the ratio 18:3 obtains between the false and true positives, or 6:1.

In other words, even assuming that some apostles are female, the probability that Junia/s is a female apostle is at most about 14%, once one takes into account the low base rate of women among apostles and apostles among those mentioned by Paul.

But the numbers above are made-up. Someone should re-do the analysis with real data. We need four data points:

• Relative prevalence of Junia vs. Junias in the early Christian population.

• Relative prevalence of the two senses of “famous among” in Greek texts of the period.

• Reasonable bounds on the prevalence of women among apostles.

• Prevalence of apostles among the subjects of Pauline praise.

And without such numbers and Bayesian analysis, I think scholarly discussion is apt to fall into the base rate fallacy.

Is eternalism compatible with the actualization of potentiality?

Every so often, someone claims to me that there is a difficulty in reconciling the Aristotelian idea of the actualization of potential with eternalism, the view that past, present and future are equally real. I am puzzled by this question, because I can’t see the difficulty. On the contrary, there is a tension between presentism, the view that only present things exist, and this Aristotelian thesis:

1. Some present events are the actualization of a no-longer present potentiality.

2. A non-existent thing is not actualized.

3. Therefore, some no-longer present potentialities exist.

4. Therefore, something that is no longer present exists.

5. Therefore, presentism is false.

One might say: Yes, the potentiality doesn’t exist, but it did exist, and it was actualized. But then:

6. Some present potentialities are actualized in not yet present events.

7. A non-existent thing does not actualize anything.

8. So, there exist some not yet present things.

9. So, presentism is false.

Of course, this is the old problem of transtemporal relations for presentism as applied to the actualization relation.

So, what about the question whether eternalists can have actualization of potentials? Here may be the problem. On eternalism plus Aristotelianism, it seems that the past unactualized potential exists even though it is now actualized. This seems to be a contradiction: how can an unactualized potential be actualized?

A first answer is that a potential is actualized at a time t provided that its actualization exists at t. Thus, the potential is unactualized at t₁ but actualized at a later time t₂, because its actualization exists at t₂ but not at t₁. But, the objector can continue, by eternalism at t₁ isn’t it the case that the actualization exists? Yes: but the eternalist distinguishes:

10. It is true at t₁ that B exists.

11. B exists at t₁.

Claim (11), for spatiotemporal objects, means something like this: the three-dimensional spacetime hypersurface corresponding to t = t₁ intersects B. Claim (10) means that B exists simpliciter, somewhere in spacetime (assuming it’s a spatiotemporal object). There is no contradiction in saying that the actualization doesn’t exist at t₁, even though it is true at t₁ that it exists simpliciter.

The second answer is that Aristotelianism does not need actualizations of unactualized potentials. Causation is the actualization of a potential. But Aristotle and Aquinas both believed in the possibility of simultaneous causation. In simultaneous causation, an event B is the actualization of a simultaneous potential A. At the time of the simultaneous causation, nobody, whether presentist or eternalist, can say that B is the actualization of unactualized potential, since then the potential would be actualized and unactualized at the same time. Thus, one can have causation, and actualization of potential, where the potential and the actualization are simultaneously real, and hence where the actualization is not of an unactualized potential. The eternalist could—but does not have to—say that transtemporal cases are like this, too: they are actualizations of a potential, but not of an unactualized potential.