The Paradox of Charity

We might call the following three statements "the Paradox of Charity":

1. In charity, we love our neighbor primarily because of our neighbor’s relation to God.

2. In the best kind of love, we love our neighbor primarily because of our neighbor’s intrinsic properties.

3. Charity is the best kind of love.

I think this paradox discloses something very deep.

Note that the above three statements do not by themselves constitute a strictly logical contradiction. To get a strictly logical contradiction we need a premise like:

4. No intrinsic property of our neighbor is a relation to God.

Now, let’s think (2) through. I think our best reason for accepting (2) is not abstract considerations of intrinsicness, but particular cases of properties. In the best kind of love, perhaps, we love our neighbor because our neighbor is a human being, is a finite person, has a potential for human flourishing, etc. We may think that these features are intrinsic to our neighbor, but we directly see them as apt reasons for the best kind of love, without depending on their intrinsicness.

But suppose ontological investigation of such paradigm properties for which one loves one’s neighbor with the best kind of love showed that these properties are actually relational rather than intrinsic. Would that make us doubt that these properties are a fit reason for the best kind of love? Not at all! Rather, if we were to learn that, we would simply deny (2). (And notice that plenty of continentally-inclined philosophers do think that personhood is relational.)

And that is my solution. I think (1), (3) and (4) are true. I also think that the best kind of neighbor love is motivated by reasons such as that our neighbor is a human being, or a person, or has a potential for human flourishing. I conclude from (1), (3) and (4) that these properties are relations to God.

But how could these be relations to God? Well, all the reality in a finite being is a participation in God. Thus, being human, being a finite person and having a potential for human flourishing are all ways of participating in God, and hence are relations to God. Indeed, I think:

5. Every property of every creature is a relation to God.

It follows that no creature has any intrinsic property. The closest we come to having intrinsic properties are what one might call “almost intrinsic properties”—properties that are relational to God alone.

We can now come back to the original argument. Once we have seen that all creaturely properties are participations in God, we have no reason to affirm (2). But we can still affirm, if we like:

6. In the best kind of love, we love our neighbor primarily because of our neighbor’s almost intrinsic properties, i.e., our neighbor’s relations only to God.

And there is no tension with (1) any more.

First person survivalship bias?

Suppose I take a nasty fall while biking. But I remain conscious. Here is the obvious first thing for a formal epistemologist to do: increase my credence in the effectiveness of this brand of helmets. But by how much?

In an ordinary case of evidence gathering, I simply conditionalize on my evidence. But this is not an ordinary case, because if things had gone otherwise—namely, if I did not remain conscious—I wouldn’t be able to update or think in any way. It seems like I am now subject to a survivorship bias. What should I do about that? Should I simply dismiss the evidence entirely, and leave unchanged my credence in the effectiveness of helmets? No! For I cannot deny that I am still conscious—my credence for that is now forced to be one. If I leave all my other credences unchanged, my credences will become inconsistent, assuming they were consistent before, and so I have to do something to my other credences to maintain consistency.

It is tempting to think that perhaps I need to compensate for survivorship bias in some way, perhaps updating my credence in the effectiveness of the helmet to be bigger than my priors but smaller than the posteriors of a bystander who had the same priors as I did but got to observe my continued consciousness without a similar bias, since they would have been able to continue to think even were I to become unconscious.

But, no. What I should do is simply update on my consciousness (and on the impact, but if I am a perfect Bayesian agent, I have already done that as soon as it was evident that I would hit the ground), and not worry about the fact that if I weren’t conscious, I wouldn’t be around to update on it. In other words, there is no such problem as survivorship bias in the first person, or at least not in cases like this.

To see this, let’s generalize the case. We have a situation where the probability space is partitioned into outcomes E₁, ..., E_n, each with non-zero prior credence. I will call an outcome E_i normal if on that outcome you would know for sure that E_i has happened, you would have no memory loss, and would be able to maintain rationality. But some of the outcomes may be abnormal. I will have a bit more to say about the kinds of abnormality my argument can handle in a moment.

We can now approach the problem as follows: Prior to the experiment—i.e., prior to the potentially incapacitating observation—you decide rationally what kind of evidence update procedures to adopt. On the normal outcomes, you get to stick to these procedures. On the abnormal ones, you won’t be able to—you will lose rationality, and in particular your update will be statistically independent of the procedure you rationally adopted. This independence assumption is pretty restrictive, but it plausibly applies in the bike crash case. For in that case, if you become unconscious, your credences become fixed at the point of impact or become scrambled in some random way, and you have no evidence of any connection between the type of scrambling and the rational update procedure you adopted. My story can even handle cases where on some of the abnormal outcomes you don’t have any credences, say because your brain is completely wiped or you cease to exist, again assuming that this is independent of the update procedure you adopted for the normal outcomes.

It turns out to be a theorem that under conditions like this, given some additional technical assumptions, you maximize expected epistemic utility by conditionalizing when you can, i.e., whenever a normal outcome occurs. And epistemic utility arguments are formally interchangeable with pragmatic arguments (because rational decisions about wager adoption yield a proper epistemic utility), so we also get a pragmatic argument. The theorem will be given at the end of this post.

This result means we don’t have to worry in firing squad cases that you wouldn’t be there if you weren’t hit: you can just happily update your credences (say, regarding the number of empty guns, the accuracy of the shooters, etc.) on your not being hit. Similarly, you can update on your not getting Alzheimer’s (which is, e.g., evidence against your siblings getting it), on your not having fallen asleep yet (which may be evidence that a sleeping pill isn’t effective), etc., much as a third party who would have been able to observe you on both outcomes should. Whether this applies to cases where you wouldn’t have existed in the first place on one of the items in the partition—i.e., whether you can update on your existence, as in fine-tuning cases—is a more difficult question, but the result makes some progress towards a positive answer. (Of course, it woudn’t surprise me if all this were known. It’s more fun to prove things oneself than to search the literature.)

Here is the promised result.

Theorem. Assume a finite probability space. Let I be the set of i such that E_i is normal. Suppose that epistemic utility is measured by a proper accuracy scoring rule s_i when E_i happens for i ∈ I, so that the epistemic utility of a credence assignment c_i is s_i(c_i) on E_i. Suppose that epistemic utility is measured by a random variable U_i on E_i (not dependent on the choice of the c_j for j ∈ I) for i not in I. Let U(c)=∑_i∈I 1_Ei ⋅ s_i(c_i)+∑_i∉I 1_Ei ⋅ U_i. Assume you have consistent priors p that assign non-zero credence to each normal E_i, and the expectation E^p of the second sum with respect to these priors is well defined. Then the expected value of U(c) with respect to p is maximized when c_i(A)=p(A ∣ E_i) for i ∈ I. If additionally the scoring rules are strictly proper, and the p-expectation of the second sum is finite, then the expected value of U(c) is uniquely maximized by that choice of c_i.

This is one of those theorems that are shorter to prove than to state, because they are pretty obvious once fairly clearly formulated.

Normally, all the s_i will be the same. It's worth thinking if any useful generalization is gained by allowing them to be different. Perhaps there is. We could imagine situtions where depending on what happens to you, your epistemic priorities rightly change. Thus, if an accident leaves you with some medical condition, knowing more about that medical condition will be valuable, while if you don't get that medical condition, the value of knowing more about it will be low. Taking that into account with a single scoring rule is apt to make the scoring rule improper. But in the case where you are conditioning on that medical condition itself, the use of different but individually proper scoring rules when the condition eventuates and when it does not can model the situation rather nicely.

Proof of Theorem: Let c_i be the result of conditionalizing p on E_i. Then the expectation of s_i(c_i′) with respect to c_i is maximized when (and only when, if the conditions of the last sentence of the theorem hold) c_i′=c_i by propriety of s_i. But the expectation of s_i(c_i′) with respect to c_i equals 1/p(E_i) times the expectation of 1_Ei ⋅ s_i(c_i′) with respect to p. So the latter expectation is maximized when (and only when, given the additional conditions) c_i′=c_i.

Functionalism and multiple realizability

Functionalism holds that two (deterministic) minds think the same thoughts when they engage in the same computation and have the same inputs. What does it mean for them to engage in the same computation?

This is a hard question. Suppose two computers run programs that sort a series of names in alphabetical order, but they use different sorting algorithms. Given the same inputs, are the two computers engaging in the same computation?

If we say “no”, then functionalism doesn’t have the degree of multiple realizability that we thought it did. We have no guarantee that aliens who behave very much like us think very much like us, or even think at all, since the alien brains may have evolved to compute using different algorithms from us.

If we say “yes”, then it seems we are much better off with respect to multiple realizability. However, there is a tricky issue here: What counts as the inputs and outputs? We just said that the computers using different sorting algorithms engage in the same computation. But the computer using a quicksort typically returns an answer sooner than a computer using a bubble sort, and heats up less. In some cases, the time at which an output is produced itself counts as an output (think of a game where timing is everything). And heat is a kind of output, too.

In my toy sorting algorithm example, presumably we didn’t count the timing and the heat as features of the outputs because we assumed that to the human designers and/or users of the computers the timing and heat have no semantic value, but are merely matters of convenience (sooner and cooler are better). But when we don’t have a designer or user to define the outputs, as in the case where functionalism is applied to randomly evolved brains, things are much more difficult.

So, in practice, even if we answered “yes” in the toy sorting algorithm case, in a real-life case where we have evolved brains, it is far from clear what counts as an output, and hence far from clear what counts as “engaging in the same computation”. As a result, the degree to which functionalism yields multiple realizability is much less clear.

Subjective guilt and war

One of the well-known challenges in accounting for killing in a just war is the thought that even soldiers fighting on a side without justice think they have justice on their side, hence are subjectively innocent, and thus it seems wrong to kill them.

But I wonder if there isn’t an opposite problem. As is well-known, human beings have a very strong visceral opposition to killing. Even those who kill with justice on their side are apt to feel guilty, and it wouldn’t be surprising if often they not only feel guilty but judge themselves to have done wrong. Thus, it could well be that soldiers who kill on both sides of a war have a tendency to be subjectively guilty, even if one of the sides is waging a just war.

Or perhaps things work out this way: Soldiers who kill tend to be subjectively guilty unless they are waging a clearly just war. If so, then those who are on a side without justice are indeed apt to be subjectively guilty, since rarely does a side without justice appear manifestly just. And those who are on a side with justice are may very well also be subjectively guilty, unless the war is one of those where justice is manifest (as was the case for the Allies in World War II).

I doubt that things work out all that neatly.

In any case, the above considerations do show that a side with justice has very strong moral reason to make that justice as manifest as possible to the soldiers. And when that is not possible, those in charge should be persons of such evident integrity that it is easy to trust their judgment.

Intrinsic evil

Consider this argument:

1. An action is intrinsically evil if and only if it is wrong to do no matter what.

2. In doing anything wrong, one does something (at least) prima facie bad with insufficient moral reason.

3. No matter what, it is wrong to do something prima facie bad with insufficient moral reason.

4. So in doing anything wrong, one performs an intrinsically evil action.

This conclusion seems mistaken. Lightly slapping a stranger on a bus in the face is wrong, but not intrinsically wrong, because if a malefactor was going to kill everyone on the bus who wasn’t slapped by you, then you should go and slap everybody. Yet the argument would imply that in lightly slapping a stranger on a bus you do something intrinsically wrong, namely slap a stranger with insufficient moral reason. But it seems mistaken to think that in slapping a stranger lightly you perform an intrinsically evil action.

The above argument threatens to eviscerate the traditional Christian distinction between intrinsic and extrinsic evil. What should we say?

Here is a suggestion. Perhaps we should abandon (1) and instead distinguish between reasons why an action is wrong. Intrinsically evil actions are wrong for reasons that do not depend on consideration of consequences and extrinsically evil actions are wrong but not for any reasons that do not depend on consideration of consequences.

Thus, lightly slapping a stranger with insufficient moral reason is extrinsically evil because any reason that makes it wrong is a reason that depends on consideration of consequences. On the other hand, one can completely explain what makes an act of murder wrong without adverting to consequences.

But isn’t the death of the victim a crucial part of the wrongness of murder, and yet a consequence? After all, if the cause of death is murder, then the death is a consequence of the murder. Fortunately we can solve this: the act is no less wrong if the victim does not die. It is the intention of death, not the actuality of death, that is a part of the reasons for wrongness.

So, when we distinguish between acts made wrong by consequences and and wrong acts not made wrong by consequences, by “consequences” we do not mean intended consequences, but only actual or foreseen or risked consequences.

But what if Alice slaps Bob with the intention of producing an on-balance bad outcome? That act is wrong for reasons that have nothing to do with actual, foreseen or risked consequences, but only with her intention. Here I think we can bite the bullet: to slap an innocent stranger with the intention of producing an on-balance bad outcome is intrinsically wrong, just as it is intrinsically wrong to slap an innocent stranger with the intention of causing death.

Note that this would show that an intrinsically evil action need not be very evil. A light slap with the intention of producing an on-balance slightly bad outcome is wrong, but not very wrong. (Similarly, the Christian tradition holds that every lie is intrinsically evil, but some lies are only slight wrongs.)

Here is another advantage of running the distinction in this way, given the Jewish and Christian tradition. If an intrinsically evil action is one that is evil independently of consequences, it could be that such an action could still be turned into a permissible one on the basis of circumstantial factors not based in consequences. And God’s commands can be such circumstantial factors. Thus, when God commands Abraham to kill Isaac, the killing of Isaac becomes right not because of any new consequences, but because of the circumstance of God commanding the killing.

Could we maybe narrow down the scope of intrinsically evil actions even more, by saying that not just consequences, but circumstances in general, aren’t supposed to be among the reasons for wrongness? But if we do that, then most paradigm cases of intrinsically evil actions will fail: for instance, that the victim of a murder is innocent is a circumstance (it is not a part of the agent’s intention).

Trust and scepticism

To avoid scepticism, we need to trust that human epistemic practices and reality match up. This trust is clearly at least a part of a central epistemic virtue.

Now, trusting persons is a virtue, the virtue of faith. But trusting in general, apart from trusting persons, is not. Theism can thus neatly account for how the trusting that is at the heart of human epistemic practices is virtuous: it is an implicit trust in our creator.

Another way out of the metaphysical problem of evil

The metaphysical problem of evil consists in the contradiction between:

1. Everything that exists is God or is created by God.

2. God is not an evil.

3. God does not create anything that is an evil.

4. There exists an evil.

The classic Augustinian response is to deny (4) by saying that evil “is” just a lack of a due good. This has serious problems with evil positive actions, errors, pains, etc.

Here is a different way out. Say that a non-fundamental object x is an object x such that the proposition that x exists is wholly grounded in some proposition that makes no reference to x. Now we deny (3) and replace it with:

5. God does not create anything fundamental that is an evil.

How could God create something non-fundamental that is an evil? By a combination of creative acts and refrainings from creative acts whose joint outcome grounds the existence of the non-fundamental evil, while foreseeing without intending the non-fundamental evil. Of course, this requires the kind of story about intention that the Principle of Double Effect uses.

Thus, consider George Shaw’s erroneous (initial) error that there are no platypuses. God creates George Shaw. He creates Shaw’s belief. He creates platypuses. The belief isn’t an evil. The platypuses aren’t an evil. The combination of the belief and the platypuses is an error. But the combination of the two is not a fundamental entity (even if the belief and the platypuses are). God can intend the belief to exist and the platypuses to exist without intending the combination to exist.

A variant virtue ethic centered on virtues and not persons

I’ve been thinking a bit about the virtue ethical claim that the right (i.e., obligatory) action is one that a virtuous person would do and the wrong one is one that a virtuous person wouldn’t do. I’ve argued in my previous posts that this is a problematic claim, since given either naturalism or the Hebrew Scriptures, it is possible for a virtuous person to do something wrong.

Maybe instead of focusing on the person, the virtue ethicist can focus on the virtues. Here is an option:

1. An action is wrong if and only if it could not properly (non-aberrantly) flow from the relevant virtues.

This principle is compatible with a virtuous person doing something wrong, as long as that wrong thing doesn’t flow from virtue.

The “properly” in (1) is an “in the right way” condition. Once we have allowed, as I think we should, that a virtuous person can do the wrong thing, we should also allow that a wrong action can flow from virtue in some aberrant way. For instance, we can imagine a wholly virtuous person falling prey to a temptation to brag about being wholly virtuous (and instantly losing the virtue, of course). The bragging flows from the virtue—but aberrantly.

A down-side of (1) is that it is a pretty strong condition on permissibility. One might think that there are some permissible morally neutral actions which can be done by a perfectly virtuous person but which do not flow from their virtue. If we accept (1), then in effect we are saying that there are no morally neutral actions. I think that is the right thing to say.

The big problem with (1) is the “properly”.

Naturalists shouldn't be virtue ethicists

Virtue ethics is committed to this claim:

1. A choice of A is wrong if and only if a person who had the relevant virtues explanatorily prior to having chosen A and was in these circumstances would not have chosen A.

But (1) implies this generalization:

2. A person who has the relevant virtues explanatorily prior to a choice never chooses wrongly.

In my previous post I argued that Aristotelian Jews and Christians should deny (2), and hence (1).

Additionally, I think naturalists should deny (1). For we live in a fundamentally indeterministic world given quantum mechanics. If a virtuous person were placed in a position of choosing between aiding and insulting a stranger, there will always be a tiny probability of their choosing to insult the stranger. We shouldn’t say that they wouldn’t insult the stranger, only that they would be very unlikely to do so (this is inspired by Alan Hajek’s argument against counterfactuals).

And (2) itself is dubious, unless we have such a high standard of virtue that very few people have virtues. For in our messy chaotic world, very little is at 100%. Rare exceptions should be expected when human behavior is involved.

(Perhaps a dualist virtue ethicist who does not accept the Hebrew Scriptures could accept (1) and (2), holding that a virtuous soul makes the choices and is not subject to the indeterminacy of quantum mechanics and the chaos of the world.)

There is a natural way out of the above arguments, and that it so to change (1) to a probabilistic claim:

3. A choice of A is wrong if and only if a person who had the relevant virtues explanatorily prior to having chosen A and was in these circumstances would be very unlikely to have chosen A.

But (3) is false. Suppose that Alice is a virtuous person who has a choice to help exactly one of a million strangers. Whichever stranger she chooses to help, she does no wrong. But it is mathematically guaranteed that there is at least one stranger such that her chance of helping them is at most one in a million (for if p_n is her chance of helping stranger number n, then p₁ + ... + p_1000000 ≤ 1, since she cannot help more than one; given that 0 ≤ p_n for all n, it follows mathematically that for some n we have p_n ≤ 1/1000000). So her helping a particular such stranger is very unlikely to be chosen, but isn’t wrong.

Or for a less weighty case, suppose I say something perfectly morally innocent to start off a conversation. Yet it is very unlikely that a virtuous person would have said so. Why? Because there are so very many perfectly morally innocent ways to start off a conversation, it is very unlikely that they would have chosen the same one I did.

Christians and Jews should not be Aristotelian virtue ethicists

If virtue ethics is correct:

1. An choice is wrong if and only if a person with the relevant virtues and in these circumstances wouldn’t have made that choice. (Premise)

If Aristotelian virtue ethics is correct:

2. An adult lacking a virtue is defective. (Premise)

But:

3. Humans became defective because of the choice of Adam and Eve to eat the forbidden fruit. (Premise)

And it seems that:

4. Adam and Eve were adult humans when they chose to eat the forbidden fruit. (Premise)

Thus it seems:

5. When Adam and Eve chose to eat the forbidden fruit, they were not lacking relevant virtues. (By 2–4)

6. Thus, persons (namely Adam and Eve!) with the relevant virtues and in their circumstances did choose to eat the forbidden fruit. (By 5)

7. Thus, their choice to eat the forbidden fruit wasn’t wrong. (1 and 6)

8. But their choice was wrong. (Premise)

9. Contradiction!

Here is one thing the classic virtue ethicist can question about this argument: the derivation of (5) depends on how we read premise (1). We could read (1) as:

10. A choice of A is wrong if and only if a person who had the relevant virtues explanatorily prior to having chosen A and was in these circumstances would not have chosen A

or as:

11. A choice of A is wrong if and only if a person who had the relevant virtues while having chosen A and was in these circumstances would not have chosen A.

If we opt for (10), the derivation of (5) works, and the argument stands. But if we opt for (11) then we can say that as soon as Adam and Eve chose to eat the fruit, they no longer counted as virtuous.

Could the virtue ethicist thus opt for (11) in place of (10)? I don’t think so. It seems central to virtue ethics that the right choices are ones that result from virtue. And that is what (10) captures. To a great extent (11) would trivialize virtue ethics, in that obviously in doing a bad thing one isn’t virtuous.

Ethics and multiverse interpretations of quantum mechanics

Somehow it hasn’t occurred to me until yesterday that quantum multiverse theories (without the traveling minds tweak) undercut half of ethics, just as Lewis’s extreme modal realism does.

For whatever we do, total reality is the same, and hence no suffering is relieved, no joy is added, etc. The part of ethics where consequences matter is all destroyed. There is no point to preventing any evil, since doing so just shifts which branch of the multiverse one inhabits.

At most what is left of ethics is agent-centered stuff, like deontology. But that’s only about half of ethics.

Moreover, even the agent-centered stuff may be seriously damaged, depending on how one interprets personal identity in the quantum multiverse.

Consider three theories.

On the first, I go to all the outgoing branches, with a split consciousness. On this view, no matter what, there will be branches where I act well and branches where I act badly. So much or all of the agent-centered parts of ethics will be destroyed.

On the second, whenever branching happens, the persons in the branches are new persons. If so, then there are no agent-centered outcomes—if I am deliberating between insulting or comforting a suffering person, no matter what, I will do neither, but instead a descendant of me will insult and another descendant will comfort. Again, it’s hard to fit this with the agent-centered parts of ethics.

The third is the infinitely many minds theory on which there are infinitely many minds inhabiting my body, and whenever a branching happens, infinitely many move into each branch. In particular, I will move into one particular branch. On this theory, if somehow I can control which branch I go down (which is not clear), there is room for agent-centered outcomes. But this is not the most prominent of the multiverse theories.

Forwards causal closure and dualist theories of perception

A standard dualist theory of perception goes like this:

1. You sense stuff physically, the data goes to the brain, the brain processes the data, and out of the processed data produces qualia.

There is a lot of discussion of the “causal closure” of the physical. What people generally mean by this is that the physical is causally backwards-closed: the cause of a physical thing is itself physical. This is a controversial doctrine, not least because it seems to imply that some physical things are uncaused. But what doesn’t get discussed much is a more plausible doctrine we might call the forwards causal closure of the physical: physical causes only have physical effects. Forwards causal closure of the physical is, I think, a very plausible candidate for a conceptual truth. The physical isn’t spooky—and it is spooky to have the power of producing something spooky. (One could leave this at this conceptual argument, or one could add the scholastic maxim that one cannot cause what one does not in some sense have.)

By forwards closure, on the standard dualist theory, the brain is not a physical thing. This is a problem. It is supposed to be one of the advantages of the standard dualist theory that it is compatible with property dualism on which people are physical but have non-physical properties. But if the brain is not physical, there is no hope for people to be physical! Personally, I don’t mind losing property dualism, but it sure sounds absurd to hold that the brain is not physical.

Recently, I have been thinking about a non-causal dualist theory that goes like this:

2. You sense stuff physically, the data goes to the brain, the brain processes the data, and the soul “observes” the brain’s processed data. (Or, perhaps more precisely, the person "feels" the neural data through the soul.)

To expand on this, what makes one feel pain is not the existence of a pain quale, but a sui generis “observation” relation between the soul and the brain’s processed data. This observation relation is not caused by the data, but takes place whether there is data there or not (if there isn’t, we have a perceptual blank slate). The soul is not changed intrinsically by the data: the “observation” of a particular datum—say, a datum representing a sharp pain in a toe—is an extrinsic feature of the soul. Note that unlike the standard theory, this up-front requires substance dualism of some sort, since the observing entity is not physical given the sui generis nature of the “observation” relation.

The non-causal dualist theory allows one to maintain forwards closure of the physical and the physicality of the brain. For the brain doesn’t cause a non-physical effect. The brain simply gets “observed”.

It is however possible that the soul causes an effect in the brain—for instance, the “observation” relation may trigger quantum collapse. Thus, the theory may violate backwards closure. And that’s fine by me. Backwards closure does not follow conceptually from the concept of the physical—a physical thing doesn’t become spooky for having a spooky cause.

There is a difficulty here, however. Suppose that the soul acts on the “observed” data, say by causing one to say “You stepped on my foot.” Wouldn’t we want to say that the brain data correlated with the pain caused one to say “You stepped on my foot”?

I think this temptation is resistable. Ridiculously oversimplifying, we can imagine that the soul has a conditional causal power to cause an utterance of “You stepped on my foot” under the condition of “observing” a certain kind of pain-correlated neural state. And while it is tempting to say that the satisfied conditions of a conditional causal power cause the causal power to go off, we need not say that. We can, simply, say that the causal power goes off, and the cause is not the condition, but the thing that has the causal power, in this case the soul.

On this story, if you step on my foot, you don’t cause me to say “You stepped on my foot”, though you do cause the condition of my conditional causal power to say so. We might say that in an extended sense there is a “causal explanation” of my utterance in terms of your stepping, and your stepping is “causally prior” to my utterance, even though this causal explanation is not itself an instance of causation simpliciter. If so, then all the stuff I say in my infinity book on causation should get translated into the language of causal explanation or causal priority. Or we can just say that there is a broad and a narrow sense of “cause”, and in the broad sense you cause me to speak and in the narrow you do not.

I think there is a very good theological reason to think this makes sense. For we shouldn’t say that our actions cause God to act. The idea of causing God to do anything seems directly contrary to divine transcendence. God is beyond our causal scope! Just as by forwards closure a physical thing cannot cause a spiritual effect, so too by transcendence a created thing cannot cause a divine effect. Yet, of course, our actions explain God’s actions. God answers prayers, rewards the just and punishes the unrepentant wicked. There is, thus, some sort of quasi-causal explanatory relation here that can be used just as much for non-causal dualist perception.

Online talk: A Norm-Based Design Argument

Thursday November 11, 2021, at 4 pm Eastern (3 pm Central), the Rutgers Center for Philosophy of Religion and the Princeton Project in Philosophy of Religion present a joint colloquium: Alex Pruss (Baylor), "A Norm-Based Design Argument".

The location will be https://rutgers.zoom.us/s/95159158918

"The whole is bigger than the part"

Some people don’t like Cantorian ways of comparing the sizes of sets because they want to have a “whole is bigger than the (proper) part” principle, denying which they consider to be counterintuitive.

Suppose that there is a relation ≤ which provides a way of comparing the sizes of sets of real numbers (or just the sizes of countable sets of real numbers) such that:

1. the comparison satisfies the “the whole is bigger than the part” principle, so that if A is a proper subset of B, then A < B

2. there are no incommensurable sets: given any A and B, at least one of A ≤ B and B ≤ A holds

3. the relation ≤ is transitive and reflexive.

Then the Banach-Tarski paradox follows from (a)–(c) without any use of the Axiom of Choice: there is a way to decompose a ball into a finite number of pieces and move them around to form two balls of the same size as the original. And Banach-Tarski feels like a direct violation of the "whole is bigger" principle!

Thus, intuitive as the “whole is bigger” principle is, the price of being able to compare the sizes of sets of real numbers in conformity with the principle is quite high. I suspect that most people who think that denying the “whole is bigger” principle also think Banach-Tarski is super problematic.

For our next observation, let’s add one more highly plausible condition:

4. the relation ≤ is weakly invariant under reflections of the real line: for any reflection ρ, we have A ≤ B if and only if ρA ≤ ρB.

Proposition: Conditions (a)–(d) are contradictory.

So, I think we should deny that, in the context of comparing the number of elements of a set, the whole is bigger than the proper part.

Proof of Proposition: Write A ∼ B iff A ≤ B and B ≤ A. Then I claim we have A ∼ ρA for any reflection ρ. For otherwise we either have A < ρA or ρA < A by (b). If we have A < ρA, then we also have ρA < ρ²A by (d), and since ρ²A = A, we have ρA < A, a contradiction. If we have ρA < A, then we have ρ²A < ρA by (d), and hence A < ρA, again a contradiction.

Since any translation τ can be made out of two reflections, it follows that A ∼ τA as well. Let τ be translation by one unit to the right. Then {0, 1, 2, ...} ∼ τ{0, 1, 2, ...} = {1, 2, 3, ...}, which contradicts (a).

Infinite Dedekind finite sets

Most paradoxes of actual infinities, such as Hilbert’s Hotel, depend on the intuition that:

1. A collection is bigger than any proper subcollection.

A Dedekind infinite set is one that has the property that it is the same cardinality as some proper subset. In other words, a Dedekind infinite set is precisely one that violates (1).

In Zermelo-Fraenkel (ZF) set theory, it is easy to prove that any Dedekind infinite set is infinite. More interestingly, assuming the consistency of ZF, there are models of ZF with infinite sets that are Dedekind finite.

It is easy to check that if A is a Dedekind finite set, then A and every subset of A satisfies (1). Thus an infinite but Dedekind finite set escapes most if not all the standard paradoxes of infinity. Perhaps enemies of actual infinity, should thus only object to Dedekind infinities, not all infinities?

However, infinite Dedekind finite sets are paradoxical in their own special way: they have no countably infinite subsets—no subsets that can be put into one-to-one correspondence with the natural numbers. You might think this is absurd: shouldn’t you be able to take one element of an infinite Dedekind finite set, then another, then another, and since you’ll never run out of elements (if you did, the set wouldn’t be finite), you’d form a countably infinite sequence of elements? But, no: the problem is that repeating the “taking” requires the Axiom of Choice, and infinite Dedekind finite sets only live in set-theoretic universes without the Axiom of Choice.

In fact, I think infinite Dedekind finite sets are much more paradoxical than a run-of-the-mill Dedekind infinite sets.

Do we learn anything philosophical here? I am not sure, but perhaps. If infinite Dedekind finite sets are extremely paradoxical, then by the same token (1) seems an unreasonable condition in the infinite case. For Dedekind finitude is precisely defined by (1).