Virtue ethics and peer disagreement

Aristotelian ethics is committed to the claim that the virtuous person knows what actions and habits are virtuous and is justified in holding on to that knowledge, and indeed should hold on to it. There is a deep stability to virtue. This means that an Aristotelian virtuous person ought not adopt a conciliationist response to those who disagree as to what is virtuous, suspending judgment over the disagreed-upon items.

Indeed, one imagines that Aristotle’s virtuous person could say of those who disagree: “They are not virtuous, and hence do not see the truth about moral matters.” Aristotle’s virtuous person would reject the idea that someone who disagrees with them about virtue could be an epistemic peer. Virtuous habits give epistemic access to moral (and not only moral) truth.

Of course, the disagreer may think themselves virtuous as well, and may think the same thing about the virtuous person as the virtuous person thinks about them. But that does not shake the Aristotelian virtuous person.

This means that if Aristotelian virtue ethics is correct, there is a clear thing that a Christian can say about religious disagreement. The Christian thinks faith is a virtue, albeit an infused rather than natural one. As such, faith gives epistemic access, and someone lacking faith is simply not an epistemic peer, since they lack a source of truth. The fact that a person lacking faith thinks they have the virtue of faith should not move the person who actually has the virtue.

Of course, one might turn all this around and use it as an argument against virtue ethics. But I think Aristotle’s picture seems exactly correct as to the kind of firmness of moral knowledge that the virtuous person exhibits, the kind of spine that lets them say, without pride or vanity, to vast numbers of others that they are simply wrong.

Moral bindingness and levels of jurisdiction

In the US, you are sometimes told that something “violates federal law”, and it is said in a way that suggests that violating federal law is somehow particularly bad.

This raises a moral question. I will assume, contrary to philosophical anarchists, that valid and reasonable laws are in some way morally binding. Other things being equal, is it morally worse to violate the laws that operate at broader levels of organization. In the US, an affirmative answer would imply that federal law is morally worse to break than state law, and state law than county law, and county law than city law.

One might think this: the power to make laws belongs to more local levels of organization by delegation from broader levels of organization, and hence violating the laws of a more local jurisdiction is less morally bad. But this argument does not fit with what I understand is the US consitutional system’s idea that sovereignty starts with the states which permanently delegate some of their authority to the federal system. And, in any case, it is not clear why it would be less bad to go against the laws of a more delegated authority: if x delegates some authority to y, then relevant disobedience to y is also disobedience to x.

A perhaps more plausible argument in favor of the laws of broader jurisdictions being morally more strongly binding is that in violating a law, one offends against the body of citizens. With a broader jurisdiction, that body of citizens is larger, and hence the offense is worse. But this can’t be right. It is not morally less bad to commit federal tax fraud in Canada than in the US just because in Canada the population is smaller! (This observation perhaps suggests that if we do adopt the view that violating the law offends against the body of citizens, we should not view the “offense against the body of citizens” as meaning an offense against the citizens taken severally—to offend against a body is different from offending against the body’s constituents taken severally, or else punching a bigger person would be a worse thing than punching a smaller, just because the bigger person’s body has more cells. Or, perhaps, we have to say that the offensiveness of a law breaking is diluted among the citizenry, so that in a larger body, each citizen is less offended against.)

I want to suggest that the idea that it is worse to offend against broader jurisdictions is backwards for multiple reasons:

1. An offense against a narrower jurisdiction is an offense against a body of citizens who are more closely related to one, and hence is a greater breach of the duties of civic friendship.

2. The laws of narrower jurisdictions can be reasonably expected to be on the hwole better fitted to the community, because there is less variation in circumstance within a narrower jurisdiction.

3. One has a greater say in the laws of the laws of the narrower jurisdiction, and hence they better fit with the autonomy of the governed.

4. It is typically less burdensome to choose which narrower jurisdiction one lives under than which wider one: it is easier to move to a different city than to a different country. Therefore, any implied consent to local laws is greater than to wider laws.

These considerations suggest that offending against a narrower body is worse. Interestingly, (3) suggests that in my earlier example of tax fraud in the US and Canada, it is even worse to commit tax fraud in Canada, because doing so violates laws one has a greater say in. That actually sounds right to me, but I do not feel the difference in moral badness is a very big one, so (3) is probably not a major factor (of course, in the special case of tax fraud, a lot of the immorality comes from the immorality of lying, which precedes law).

(These same considerations support the principle of subsidiarity.)

So far I have been thinking about geographically defined jurisdictions. But consider a very different jurisdiction: the body of a profession, such as physicians or lawyers or electricians. The standards of such a body have a great deal of moral force. When a doctor says that disclosing some information about a patient violates medical ethics, that carries a great deal of moral force. And yet it really is “just” a violation of the law of a body, because there would be no such moral duty of confidentiality without the standards of the body of physicians (there would be more limited duties of confidentiality, say when the doctor specifically promised the patient not to disclose something). The laws of the professional jurisdictions have a lot of moral force, and it is not implausible that 1-4 are at least partly explanatory of that force.

Mice

It feels like I am constantly fixing computer mice. The most common issue is the wires breaking near the mouse, requiring me to shorten the cable and resolder it to the PCB. I usually also add some glue or heat shrink tubing as strain relief if I haven’t done so already. Switching the household to wireless mice would solve the problem, at the expense of having yet more batteries to deal with.

I started off today with a new fix: the plastic axle from a mouse wheel broke off. I drilled through the mouse wheel and put a toothpick in instead. We’ll see how long that holds up. If it doesn’t, I’ll have to see if I can find a screw or nail of the right diameter instead.

Likely near future task: my daughter complains of a mouse double clicking. When it earlier had that problem, it seemed to me that it was generating an extra click on release, which sounds to me like a debouncing failure. But then I took it apart and put it back together and the problem disappeared, so I didn’t have a chance to fix it. Apparently the problem has come back, but I can’t duplicate it. If and when I duplicate it, I plan to hook it up to an oscilloscope, and play around with capacitors to debounce the release.

I wonder if problems would decrease if we bought more expensive mice.

Comparing the epistemic relevance of measurements

Suppose P is a regular probability on (the powerset of) a finite space Ω representing my credences. A measurement M is a partition of Ω into disjoint events E₁, ..., E_n, with the result of the experiment being one of these events. In a given context, my primary interest is some subalgebra F of the powerset of Ω.

Note that a measurement can be epistemically relevant to my primary interest without any of the events in in the measurement being something I have a primary interest in. If I am interested in figuring out whether taller people smile more, my primary interest will be some algebra F generated by a number of hypotheses about degree to which height and smiliness are correlated in the population. Then, the measurement of Alice’s height and smiliness will not be a part of my primary interest, but it will be epistemically relevant to my primary interest.

Now, some measurements will be more relevant with respect to my primary interest than others. Measuring Alice’s height and smiliness will intuitively be more relevant to my primary interest about height/smile correlation, while measuring Alice’s mass and eye color will be less so.

The point of this post is to provide a relevance-based partial ordering on possible measurements. In fact, I will offer three, but I believe they are equivalent.

First, we have a pragmatic ordering. A measurement M₁ is at least as pragmatically relevant to F as a measurement M₂, relative to our current (prior) credence assignment P, just in case for every possible F-based wager W, the P-expected utility of wagering on W after a Bayesian update on the result of M₁ is at least as big as that of wagering of W after updating on the result of M₂, and M₁ is more relevant if for some wager W the utility of wagering after updating on the result of M₁ is strictly greater.

Second, we have an accuracy ordering. A measurement M₁ is at least as accuracy relevant to F as a measurement M₂ just in case for every proper scoring rule s on F, the expected score of updating on the result of M₁ is better than or equal to the expected score of updating on the result of M₂, and M₁ is more relevant when for some scoring rule the expected score is better in the case of M₁.

Third, we have a geometric ordering. Let H_P, F(M) be the horizon of a measurement M, namely the set of all possible posterior credence assignments on F obtained by starting with P, conditionalizing on one of the possible events in that M partitions Ω into, and restricting to F. Then we say that M₁ is at least as (more) geometrically relevant to F as M₂ just in case the convex hull of the horizon of M₁ contains (strictly contains) the convex hull of the horizon of M₂.

I have not written out the details, but I am pretty sure that all three orderings are equivalent, which suggests that I am on to something with these concepts.

An interesting special case is when one’s interest is binary, an algebra generated by a single hypothesis H, and the measurements are binary, i.e., partitions into two sets. In that case, I think, a measurement M₁ is at least as (more) relevant as a measurement M₂ if and only if the interval whose endpoints are the Bayes factors of the events in M₁ contains (strictly contains) the interval whose endpoints are the Bayes factors of the events in M₂.

What question should I ask?

Epistemology is heavily focused on the question of evaluating a doxastic state given a set of evidence: is the state rational or irrational, is it knowledge or opinion, etc. This can be useful to the epistemic life of an agent, but there is something else that is at least as useful and does not get discussed nearly as much: the question of how we should go about gathering evidence or, equivalently, what experiments (broadly understood) we should perform.

[The rest of the post was based on a mathematical error and has been deleted. The next post is an attempt to fix the error.]

Has cultural relativism about norms of etiquette really been established?

Imagine a philosopher who argued that the norms of assertion are relative to culture on the grounds that in England we have the norm:

1. Only assert “It’s snowing” when it’s snowing

while in France we have the norm:

2. Only assert “Il neige” when it’s snowing.

This would be silly for multiple reasons. Foremost among these is that (1) and (2) are mere consequences of the norm of assertion:

3. Only assert what it is true.

(Of course, you may disagree that truth is the norm of assertion. You may prefer a knowledge or justified belief or belief or high credence norm. But an analogous point will apply.)

It is widely held that while the norm of assertion is essentially the same across cultures, norms of etiquette vary widely. But the main reason people give for believing that the norms of etiquette vary widely is akin to the terrible argument about norms of assertion I began the post with. People note such things as that in some countries when one meets acquaintances one bows, and in others one waves; or that in some one eats fish with two forks and in others with a fork and knife.

But just as the fact that in England one should follow (1) and in France (2) is compatible with the universality of norms of assertion, likewise the variation in greeting and eating rituals can be compatible with the universality of norms of etiquette. It could, for instance, be that the need to eat fish with two forks in Poland and with a fork and knife in the USA derives simply from a universal norm of etiquette:

4. Express respect for your fellow diners.

But just as one asserts the truth with different words in different languages, one expresses respect for one’s fellow diners with different gestures in different cultures.

Indeed, presumably nobody thinks that the fact that in France one says “Merci” and in England “Thank you” implies a cultural relativism in etiquette. In both cases one is thanking, but the words that symbolize thanks are different. But what goes for words here also applies to many gestures (there may turn out to be universal gestures, like pointing).

One object that among the norms of etiquette there are norms that specify which gestures signify, say, respect or thanks. But a specification of what signifies what is not the specification of a norm. That “Merci” signifies gratitude and that eating fish with two forks signifies respect are not norms, because norms tell us what to do, and these do not.

5. “Merci” signifies thanks

is grammatically not a norm but a statement of fact. We might try to make it sound more like a norm by saying:

6. Signify thanks with “merci”!

But that is bad advice when taken literally. For thanks are not to be signified always, but only when thanks are appropriate. A more correct norm would be:

7. When a service has been done for you, signify thanks with “merci”!

But this is just a consequence of the general norm of etiquette:

8. When a service has been done for you, signify thanks!

together with the fact (5).

So, we see that the mere variation in rituals should not be taken to imply that there is cultural relativity of norms of etiquette.

If there is to be a cultural relativity of norms of etiquette, it will have to be at a higher level. If in some cultures, etiquette requires one to show respect for all fellow diners and at others to show disrespect for some—say, those from an underprivileged group—then that would indeed be a genuine relativity of norms of etiquette.

But it’s not clear that me that in a culture where one is expected to show disrespect to fellow diners in some underprivileged group that expectation is actually a norm of etiquette. Not all social expectations, after all, are actually norms of etiquette, or even norms at all. A norm (of behavior) gives norm-based reasons. But an expectation that one show disrespect to members of an underprivileged group has no reason-giving force at all.

We can imagine a culture where there is no way to symbolize respect for members of an underprivileged group when dining. On the view I wish to defend, such a lack would not exempt one from the duty to show respect to all one’s fellow diners—it would just make it more difficult to do so, because it would require one to create new ways of showing respect (say, by adapting the forms of showing respect to members of privileged groups, much as in some European languages the polite forms of address are derived from forms in which one used to address nobility in less democratic times).

I am not sure if there is cultural variation in norms of etiquette. But if there is, that variation will not be proved by shallow differences between rituals, and may not even follow from deeper variation, such as a culture where it is not appropriate to thank one’s subordinates for work well done. For in the case of deeper variation, it could simply be that some in some cultures violation of certain norms of etiquette is nearly universal, and there are no accepted ways to show the relevant kind of respect.

In fact, it could even be the case that there is only one norm of etiquette, and it is culturally universal:

9. Signify respect to other persons you interact with in ways fitted to the situation.

If this is right, then social rules designed to show disrespect, no matter how widespread, are not norms of etiquette.

Reasons from the value of true belief

Two soccer teams are facing off, with a billion fans watching on TV. Brazil has a score of 2 and Belgium has a score of 0, and there are 15 minutes remaining. The fans nearly unanimously think Brazil will win. Suddenly, there is a giant lightning strike, and all electrical devices near the stadium fail, taking the game off the air. Coincidentally, during the glitch, Brazil’s two best players get red cards, and now Belgium has a very real chance to win if they try hard.

But the captain of the Brazilian team yells out this argument to the Belgians: “If you win, you will make a billion fans have a false belief. A false belief is bad, and when you multiply the badness by billion, the result is very bad. So, don’t win!”
Great hilarity ensues among the Belgians and they proceed to trounce the Brazilians.

The Belgians are right to laugh: the consideration that the belief of a billion fans will be falsified by their effort carries little to no moral weight.

Why? Is it that false belief carries little to no disvalue? No. For suppose that now the game is over. At this point, the broadcast teams have a pretty strong moral reason to try to get back on the air in order to inform the billion fans that they were mistaken about the result of the game.

In other words, we have a much stronger reason to shift people’s beliefs to match reality than to shift reality to match people’s beliefs. Yet in both cases the relevant effect on the good and bad in the world can be the same: there is less of the bad of false beliefs and more of the good of true beliefs. An immediate consequence of this is that consequentialism about moral reasons is false: the weight of moral reasons depends on more than the value of the consequences.

It is often said that belief has a mind-to-world direction of fit. It is interesting that this not only has repercussions for the agent’s own epistemic life, but for the moral life of other parties. We have much more reason to help others to true belief by affecting their beliefs than by affecting the truth and falsity of the content of the beliefs.

Do the Belgians have any moral reason to lose, in light of the fact that losing will make the fans have correct belief? I am inclined to think so: producing a better state of affairs is always worthwhile. But the force of the reason is exceedingly small. (Nor do the numbers matter: the reason’s force would remain exceedingly small even if there are trillions of fans because Earth soccer was famous through the galaxy.)

There is a connection between the good and the right, but it is quite complex indeed.

Two spinners and infinitesimal probabilities

Suppose you do two independent experiments, A and B, each of which uniformly generates a number in the interval I = [0, 1).

Here are some properties we would like to have on our probability assignment P:

1. There is a value α such that P(A = x)=P(B = x)=α for all x ∈ I and P((A, B)=z)=α² for all z ∈ I².

2. For every subset U of I² consisting of a finite union of straight lines, P((A, B)∈U) is well-defined.

3. For any measurable U ⊆ I², if P((A, B)∈U|A = x)=y for all x ∈ I, then P((A, B)∈U)=y.

4. For any measurable U ⊆ I², if P((A, B)∈U|B = x)=y for all x ∈ I, then P((A, B)∈U)=y.

5. The assignment P satisfies the axioms of finitely additive probability with values in some real field.

Here is an interesting consequence. Let U consist of two line segments, one from (0, 0) to (1, 1/2) and the other from (0, 1/2) to (1, 1). Then every vertical line in I² intersects U in exactly two points. This is measurable by (2). It follows from (1) and (5) that P((A, B)∈U|A = x)=2α for all x ∈ I. Thus, P((A, B)∈U)=2α by (3). On the other hand, every horizontal line in I² meets U in exactly one point, so P((A, B)∈U|B = x)=α by (1) and P((A, B)∈U)=α by (4). Thus, 2α = α, and so α = 0.

In other words, if we require (1)-(5) to hold, then the probability of every single point outcome of either experiment must be exactly zero. In particular, it is not possible for the probability of a single point outcome to be a positive infinitesimal.

Cognoscenti of these kinds of arguments will recognize (3) and (4) as special cases of conglomerability, and are likely to say that we cannot expect conglomerability when dealing with infinitesimal probabilities. Maybe so: but (3) and (4) are only a special case of conglomerability, and they feel particularly intuitive to me, in that we are partitioning the sample space I² on the basis of the values of one of the two independent random variables that generate the sample space. The setup—say, two independent spinners—seems perfectly natural and unparadoxical, the partitions seem perfectly natural, and the set U to which we apply (3) and (4) is also a perfectly natural set, a union of two line segments. Yet even in this very natural setup, the friend of infinitesimal probabilities has to embrace a counterintuitive violation of (3) and (4).

Naturalism and lovability

1. If naturalism is true, Stalin is not lovable.

2. Everyone is lovable.

3. So, naturalism is not true.

Here, by “lovable”, I don’t mean that it is possible to love the person, but that it is not inappropriate to do so.

Premise 2 follows the intuition that it is permissible for every parent to love their children. It also follows from the more controversial claim that everyone should love everyone.

The intuition behind premise 1 is something like this: Stalin’s actions were so horrible that the only plausible hypotheses on which he is lovable are that there is some deeply mysterious and highly valuable metaphysical fact about his being, such as that he is in the image and likeness of God, or that his Atman is Brahman, a fact incompatible with naturalism. For if all we have are the ordinary naturalistic goods in Stalin, these goods are easily outweighed by the horrors of his wickedness.

Impairment and saving lives

Bob and Carl are drowning and you can save only one of them. Bob is a human being in the prime of life, physically and mentally healthy, highly intelligent, and leading a happy and fulfilling life as a physicist committed to lifelong celibacy. To look at him, Carl is Bob’s identical twin. Carl has the same physical and mental powers as Bob, and leads a very similar happy and fulfilling life as a physicist committed to lifelong celibacy.

But there is one crucial difference that you know about, but Carl does not. Carl is actually a member of a superintelligent humanoid alien species. However, due to an unfortunate untreatable genetic condition, Carl suffers from a severe intellectual impairment, having merely the intelligence of a highly intelligent human. In order that Carl might avoid the stigma of the impairment, his parents had some highly sophisticated surgery done on him to make him fit into human society, and arranged for him to be adopted by a human family and raised as a human. No one except for you on earth will ever know that Carl isn’t human. You know because you happened to see the aliens arranging this (but you haven’t told anyone, because you don’t want people to think you are crazy).

Should you save Bob or Carl from drowning? My intuition is that if the above is all that you know, you have no reason to prefer saving one over the other. If one of them is slightly more likely to be saved by you (e.g., they are slightly closer to you), you should go for that one, but otherwise it’s a toss-up.

But notice that if you save Carl, there will be more natural evil in the world: There will be a severe intellectual impairment, which won’t be present if you choose to save Bob instead. It seems pretty plausible that:

1. If you have a choice between two otherwise permissible courses of action, which result in the same goods, but one of them results in exactly one additional evil, you have a moral reason to choose the course of action that does not result in the evil.

Thus, it seems, you should save Bob.

So there is something paradoxical here. On the one hand, there seems to be no reason to pick Bob over Carl. On the other hand, the plausible general ethical principle (1) suggests you should pick Bob.

How can we get out of this paradox? Here are two options.

First, one could say that impairment is not an evil at all. As long as Carl leads a fulfilling life—even if it is merely fulfilling by human standards and not those of his species—his impairment is no evil. Indeed, we might even take the above story to be a reductio ad absurdum of an Aristotelian picture of species as having norms attached to them with it being a harm to one to fall short of these norms.

Second, one argue that principle (1) does not actually apply to the case. For there is a difference of goods in saving Carl: you are saving a member of a superintelligent species, while in the case of saving Bob, you are saving a mere human. For this to fit with the intuition that it’s a toss-up whether to save Bob or Carl, it has to be the case that what the superintelligence of his species adds to the reasons for saving Carl is balanced by what his abnormally low intelligence subtracts from these reasons.

Of these options, I am more attracted to the second. And the second has an interesting and important consequence: "mere" membership in a natural kind can have significant value. This has important repercussions for the status of the human fetus.

Models of libertarian agency, and some more on divine simplicity

Here is a standard libertarian picture of free and responsible choice. I am choosing between two non-mental actions, A and B. I deliberate on the basis of the reasons for A and the reasons for B. This deliberation indeterministically causes an inner mental state W(A), which is the will or resolve or intention to produce A. And then W(A) causes, either deterministically or with high probability, the extra-mental action A.

Now notice two things. First, notice that my production of the state W(A) is itself something I am morally responsible for. Imagine that I have resolved myself to gratuitously insult you. If it turns out that my vocal chords are paralyzed, my resolve W(insult) is itself enough to make me guilty.

Second, note that that my production of W(A) could involve the production of a prior second-order state of will or resolve, a willing W(W(A)) to will to produce A. For there are times when it’s hard to resolve ourselves to do something, and in those cases we might resolve ourselves to resolve ourselves first. But at the same time, to avoid an infinite regress, we should not adopt a view on which every time we responsibly produce something, we do so by forming a prior state of willing or resolve or intention. In light of this, although my production of W(A) could involve the production of a prior second-order state W(W(A)), it need not do so. In fact, phenomenologically, it seems more plausible to think that in typical cases of free choice, we do not go to the meta level of producing W(W(A)). We only go to the meta level in special cases, such as when we have to “steel” ourselves to gain the resolve to do the action.

Thus we have seen that, assuming libertarianism, it is possible for me to be responsible for indeterministically producing a state of affairs W(A) without producing a prior state of willing or resolving or intending in favor of W(A). The state W(A) is admittedly an inner mental state. But the responsibility for W(A) does not seem to have anything to do with the innerness of W(A). We are responsible for W(A) because our deliberation indeterministically but non-aberrantly results in W(A).

Here is a question: Could there be cases where we have libertarian-free actions where instead of our deliberation indeterministically non-aberrantly resulting in W(A), and thereby making us responsible for W(A) as well as A, our deliberation directly indeterministically and non-aberrantly results in the extra-mental action A, without an intervening inner mental state W(A) that deterministically or with high probability causes A, but with us nonetheless being responsible for A?

Once we have admitted—as a libertarian has to, on pain of a regress of willings—that we can be responsible for producing a state of affairs without a prior willing of that state of affairs, then it seems hard to categorically deny the possibility of us producing an extra-mental state of affairs responsibly without an intervening prior willing. And in fact phenomenology fits quite well with the hypothesis that we do that. We do many things intentionally and responsibly without being aware of a willing, resolve or intention to do them. If we stick to the initial libertarian model on which there must be an intervening mental state W(A), we have to say that either the state W(A) is hidden from us—unconscious—or that these actions are only free in a derivative way. Neither is a particularly attractive hypothesis. Why not, simply, admit that sometimes deliberation results in an extra-mental action that we are responsible for without an intervening willing, resolve or intention?

Well, I can think of one reason:

1. It seems that you can only be responsible for what we do intentionally, and we cannot do something intentionally without intending something.

But note that if this reason undercuts the possibility of our responsibly directly doing A without an intervening act W(A) of intention, it likewise undercuts the possibility of our responsibly directly producing W(A) without an intervening W(W(A)) act, and sets us on a vicious regress.

I actually think (1) can be accepted. In that case, when we directly responsibly produce W(A), the intentionality in the production of W(A) is constituted by the non-aberrant causal connection between deliberation and W(A), rather than by some regress-engendering intention-for-W(A) prior to W(A). And the occurrence of W(A) means that we are intending something, namely A.

But what would it be like if we were to directly responsibly produce A, without an intervening act of intention W(A)? How would that be reconciled with (1)? Again, the intentionality of the production of A would be constituted by the non-aberrant causal connection between deliberation and A. And the content of the intention would supervene on the actual occurrence of A as well as on the reasons favoring A that were instrumental in the deliberation. (There are some complications about excluded reasons. Maybe in those cases deliberation can have an earlier stage where one freely decides whether to exclude some reasons.)

Call the cases where we thusly directly responsibly produce an extra-mental action A cases of direct agency.

A libertarian need not believe we exhibit direct agency. Perhaps we always have one level of resolve, willing or intention as an inner mental state. But the libertarian should not be dogmatic here, given the above arguments.

Our phenomenology suggests that we do exhibit direct agency, and indeed do so quite commonly. And if God is simple, and hence does not have contingent inner states, all of God’s indeterministic free actions are cases of direct agency.

In fact, independently of divine simplicity, we may have some reason to prefer the direct agency model in the case of God. Consider why it is that sometimes we go to the meta level of W(W(A)): we do so because of the weakness of our wills, we have to will ourselves to will ourselves to produce A. It seems that a perfect being would never have reason to go to the meta level of W(W(A)). So, the remaining question is whether a perfect being would ever have reason to go to the W(A) level. I think there is some plausibility in the idea that just as going to the W(W(A)) level is a sign of weakness, a sign of a need for self-control, going to the W(A) level is also a sign of imperfection—a sign that one needs a tool, even if an intra-mental tool, for the production of A. It seems plausible, thus, that if this is possible and compatible with freedom and responsibility, a perfect being would simply directly produce A (where A is, say, the action of the being’s causing horses to exist). And I have argued that it is possible, and it is compatible with freedom and responsibility.

Divine simplicity and responsibility for creation

Libertarians believes something like this:

1. If an agent has primary responsibility for a state of affairs A, then the agent gained that resposibility by intentionally indeterministically causing A.

Here, primary responsibility is distinguished from the kind of derivative responsibility that someone who freely got drunk (and hence has primary responsibility for getting drunk) has for the accident caused while drunk.

Next, add this plausible thesis:

2. If an agent gains primary responsibility for a state of affairs A by intentionally causing A, then the agent has primary responsibility for an intention relevantly connected to A.

Now, combine (1) and (2). Suppose Alice has primary responsibility for A. Then she had an intention I = I₁ relevantly connected to A that she had primary responsibility for. Applying (1) and (2) to I₁, we conclude that Alice had primary responsibility for an intention I₂ relevantly connected to I, and primary responsibility for an intention I₃ relevantly connected to I₂, and so on, ad infinitum.

But of course we don’t have an infinite chain of intentions. So we must break out of the chain. I think there is only one way to do so: at some point, I_n = I_n + 1. In other words, the intention relevantly connected to I_n just is I_n once again: there is no further intention. Rather, one’s intention to produce I_n is just constituted by I_n.

This means that it is possible for one to be primarily responsible for a state of affairs A—say an intention I_n—when the intention with which one caused A is itself at least partly constituted by A.

Now, I take it that something like this is what happens when a simple God creates something: God’s intention to create horses is partly constituted by horses, rather than simply by some inner state of God’s.

There is a lesson here: The primary worry that one has about a simple God’s contingently creating things—namely, that a simple God cannot have contingent inner states—is mirrored by a parallel worry about a libertarian agent’s production of intentions.

Absence of evidence

It seems that the aphorism “Absence of evidence is not evidence of absence” is typically false.

For if H is a hypothesis and E is the claim that there is evidence for H, then E raises the probability of H: P(H|E)>P(H). But then (as long as P(E)>0, as Bayesian regularity will insist), it mathematically follows that P(∼H|∼E)>P(∼H). Thus the absence of evidence is evidence for the falsity (“absence”) of the hypothesis.

I think there is only one place where one can challenge this argument, namely the claim:

1. If there is evidence for H, then the fact that there is evidence for H is itself evidence for H.

First, let’s figure out what (1) is saying. I think the best reading is that it presupposes some kind of notion of a body of first-order evidence—maybe all the stuff that human beings have ever observed—and says that if the actual contents of that body of first-order evidence supports H, then the fact that that body supports H itself supports H.

Here is a way to make this precise. We suppose there is some random variable O whose value (not real valued, of course) is all first-order observations humans ever made. Let W be the set of all possible values that O could take on. For simplicity, we can take W to be finite: there is a maximum number of observations a human can make in a lifespan, a finite resolution to each observation, and a maximum number of human beings who could have lived on earth. Let o₀ be the actual value that O has. Let W_H = {o ∈ W : P(H|O = o)>P(H)}.

Assuming we have Bayesian regularity, we can suppose O = o has non-zero probability for each o ∈ W_H. Then the claim that there is evidence for H is itself evidence for H comes to this:

2. P(H|O ∈ W_H)>P(H).

And it is easy to check that this follows by finite conglomerability from the fact that P(H|O = o)>P(H) for each o ∈ W_H.

There might be cases where we expect infinite conglomerability to be lacking. In those cases (1) would be dubious. Here is one such case. Suppose Alice and Bob each get a ticket from a fair infinite jar with tickets numbered 1,2,3,…. Alice looks at her ticket. Bob doesn’t look at his yet, but knows that Alice has looked at hers. Bob notes that whatever number Alice has seen, it is nearly certain that his number is bigger (there are infinitely many numbers bigger than Alice’s number and only finitely many smaller ones). Thus, Bob knows that the evidence available to humans supports the thesis that his number is bigger than Alice’s. But Bob’s knowing this is not actually evidence that his number is bigger than Alice’s, for until Bob actually observes one or the other number, he is in the same evidential position as before Alice looked at her ticket—and at that point, it is obvious that it’s not more likely that Bob’s ticket has a bigger number than Alice’s.

But apart from weird cases where conglomerability fails, (1) is true, and so absence of evidence is evidence of absence, assuming we have enough Bayesian regularity.

Perhaps a charitable reading of the aphorism that absence of evidence isn’t evidence of absence is just that absence of evidence isn’t always significant evidence of absence. That seems generally correct.

A tension in some of my recent work

Here is a tension in some recent work of mine. In Chapter 6 of Infinity, Causation, and Paradox, I argue that (a) the Axiom of Choice for countable sets of reals (ACCR) is true, and (b) this version of the Axiom of Choice plus causal infinitism implies a nasty paradox, so we should accept causal finitism instead. The argument for ACCR makes use of the premise that mathematical entities exist necessarily. But in “Might All Infinities Be The Same Size?”, I argue that for all we know, some mathematical entities exist contingently. Thus, the latter paper undercuts the argument of Chapter 6 of the book.

Fortunately, the argument of Chapter 6 of the book looks like it might be fixable. The argument for ACCR proceeded as follows:

1. For any set of non-empty countable sets of reals, it is metaphysically possible that there is a choice function.

2. If possibly there is a choice function, then necessarily there is a choice function.

The argument for (1) is elaborate, but it is (2) that the considerations in my article block.

But we can try to proceed as follows. The paradox in Chapter 6 of the book requires a choice function for a particular collection of non-empty countable sets of reals (reals generatable by a certain infinitary coin-tossing process). By (1), there is a possible world w′ where that particular collection of sets does have a choice function. So it seems all we need to do is to run the paradox in w′, and we should be done.

There are probably other areas in Chapter 6 where some tweaking (or more than that!) is needed to make things work with mathematical contingentism, and hence my cautious wording.

A superpower

Imagine Alice claimed she could just see, with reliability, which unprovable large cardinal axioms are true. We would be initially sceptical of her claims, but we could imagine ways in which we could come to be convinced of her having such an ability. For instance, we might later be able to prove a lot of logical connections between these axioms (say that axiom A₁₂ implies axiom A₁₄) and then find that Alice’s oracular pronouncements matched these logical connections (she wouldn’t, for instance, affirm A₁₂ while denying A₁₄) to a degree that would be very hard to explain as just luck.

Suppose, then, that we have come to be convinced that Alice has the intuitive ability to just see which large cardinal axioms are true. This would be some sort of uncanny superpower. The existence of such a superpower would sit poorly with naturalism. An intuition like Ramanujan’s about the sums of series could be explained by naturalism—we could simply suppose that his brain unconsciously sketched proofs of various claims. But an intuition about large cardinal axioms wouldn’t be like that, since these axioms are not provable.

Now as far as we know, there is no one exactly like Alice who just has reliable intuitions about large cardinal axioms. But our confidence in the less abstruse axioms of Zermelo-Fraenkel set theory—intuitive axioms like the axiom of replacement—commits us to thinking that either we in general, or those most expert in the matter, are rather like Alice with respect to these less abstruse axioms. The less abstruse axioms are just as unprovable as the more abstruse ones that Alice could see. Therefore, it seems, if Alice’s reliable intuition provided an argument against naturalism, our own (or our experts’) intuition about the more ordinary axioms, an intuition which we take to be reliable, gives us an argument against naturalism. Seeing the axiom of replacement to be true is just as much a superpower as would be Alice’s seeing that, say, measurable cardinals exist (or that they do not exist).