The sexual, the secret and the sacred

Some ethical truths are intuitively obvious but it is hard to understand the reasons for them. For instance, sexual behavior should be, at least other things being equal, kept private. But why? While I certainly have this intuition, I have always found it deeply puzzling, especially since privacy is opposed to the value of knowledge and hence always requires a special justification.

But here is a line of thought that makes sense to me now. There is a natural connection between the sacred and the ritually hidden recognized across many religions. Think, for instance, of how the holiest prayers of the Tridentine Mass are said inaudibly by the priest, or the veiling of the Holy of Holies in the Temple of Jerusalem, or the mystery religions. The sacred is a kind of mysterium tremendum et fascinans, and ritual hiddenness expresses the mysteriousness of the sacred particularly aptly.

If sexuality is sacred—say, because of its connection with the generation of life, and given the sacredness of human life—then it is unsurprising if it is particularly appropriately engaged in in a context that involves ritual hiddenness.

Note that this is actually more of a ritual hiddenness than an actual secrecy. The fact of sex is not a secret in the case of a married couple, just as the content of the inaudible prayers of the Tridentine Mass is printed publicly in missals, but it is ritually hidden.

I wonder, too, if reflection on ritual hiddenness might not potentially help with the “problem of hiddenness”.

An argument that the moment of death is at most epistemically vague

Assume vagueness is not epistemic. This seems a safe statement:

1. If it is vaguely true that the world contains severe pain, then definitely the world contains pain.

But now take the common philosophical view that the moment of death is vague, except in the case of instant annihilation and the like. The following story seems logically possible:

2. Rover the dog definitely dies in severe pain, in the sense that it is definitely true that he is in severe pain for the last hours of his life all the way until death, which comes from his owner humanely putting him out of his misery. The moment of death is, however, vague. And definitely nothing other than Rover feels any pain that day, whether vaguely or definitely.

Suppose that t₁ is a time when it is vague whether Rover is still alive or already dead. Then:

3. Definitely, if Rover is alive at t₁, he is in severe pain at t₁. (By 2)

4. Definitely, if Rover is not alive at t₁, he is not in severe pain at t₁. (Uncontroversial)

5. It is vague whether Rover is alive at t₁. (By 2)

6. Therefore, it is vague whether Rover is in severe pain at t₁. (By 3-5)

7. Therefore, it is vague whether the world contains severe pain at t₁. (By 2 and 6, as 2 says that Rover is definitely the only candidate for pain)

8. Therefore, definitely the world contains pain at t₁. (By 1 and 7)

9. Therefore, definitely Rover is in pain at t₁. (By 2 and 8, as before)

10. Therefore, definitely Rover is alive at t₁. (Contradiction to 5!)

So, we cannot accept story 2. Therefore, if principle 1 is true, it is not possible for something with a vague moment of death to definitely die in severe pain, with death definitely being the only respite.

In other words, it is impossible for vagueness in the moment of death and vagueness in the cessation of severe pain to align perfectly. In real life, of course, they probably don’t align perfectly: unconsciousness may precede death, and it may be vague whether it does so or not. But it still seems possible for them to align perfectly, and to do so in a case where the moment of death is vague—assuming, of course, that moments of death are the sort of thing that can be vague. (For a special case of this argument, assume functionalism. We can imagine a being of such a sort that the same functioning constitutes it as existent as constitutes it as conscious, and then vagueness in what counts as functioning will translate into perfectly correlated vagueness in the moment of death and the cessation of severe pain.)

The conclusion I’d like to draw from this argument is that moments of death are not the sort of thing that can be non-epistemically vague.

Note that 1 is not plausible on an epistemic account of vagueness. For the intuition behind 1 depends on the idea that vague cases are borderline cases, and a borderline case of severe pain will be a definite case of pain, just as a borderline case of extreme tallness will be a definite case of tallness. But if vagueness is epistemic, then vague cases aren't borderline cases: they are just cases we can't judge about. And there is nothing absurd about the idea that we might not be able to judge whether there is severe pain happening and not able to judge whether there is any pain happening either.

Fusions and organisms

Suppose you believe the following:

1. For any physical objects, the xs, there is a physical object y with the following properties:
   1. each of the xs is a part of y;
   2. it is an essential property of y that it have the parts it does; and
   3. necessarily, if all the actual proper parts of y exist, then y exists as well.

For instance, on the standard version of mereological universalism, it seems we could just take y to be the fusion of the xs. And on some versions of monism, we could take y to be the cosmos.

But it seems (1) is false if organisms are physical objects and if particles survive ingestion. For suppose that there is exactly one x, Alice, who is a squirrel, and at t₁ we find a y that satisfies (1). And now suppose that at t₂ there comes into existence a nut whose simple parts are not already parts of y, and at t₃ this nut has been eaten and fully digested by Alice. Suppose no parts of y have ceased to exist between t₁ and t₃. Then y exists at t₃ by (c), and has Alice as a part of itself (by (a) and (b)), and the simple particles of the nut are parts of y by transitivity as they are parts of Alice. Hence y has gained parts, contrary to (b), a contradiction.

(Note that the argument can be run modally against a four-dimensionalist version of (1).)

The mereological universalist’s best bet may be to deny that fusions satisfy (c). Normally, we think that the only way for a fusion to perish is for one of its proper parts to perish. But there may be another way for a fusion to perish, namely by certain kinds of changes in the mereological structure of the fusion’s proper parts, and specifically by one of the fusion’s proper parts gaining a part that wasn’t already in the fusion.

Here is another problem for (1), though. Suppose that Alice the squirrel is the only physical object in the universe. Now consider a y satisfying (1)(a)–(b). Then y is distinct from Alice because y has different modal properties from Alice: Alice can survive annihilation of one of her claws while y cannot by (b). But this violates the Weak Supplementation mereological axiom, since all of y’s parts overlap Alice. So we cannot combine fusions as normally conceived of (since the normal conception of them includes classical mereology) with organisms.

A way out of both problems is to say that there are two different senses of parthood at issue: fusion-parthood and organic-parthood, and there is no transitivity across them. This is a serious ideological complication.

Oligonism

Monism holds there is only one (or at least one fundamental) thing in reality: the universe. Pluralism, as normally taken, holds there are many. An underexplored metaphysical view is oligonism: the view that there are (at least fundamentally) only a handful of objects in reality, but more than one.

One way to get oligonism is to take the universe of monism and add God while holding that God is not derivative from the universe. But that’s still a monism about created reality, and my interest here is going to be in oligonism about created reality (the non-theist reader can substitute “concrete reality”).

The most promising version of oligonism is one on which the correct physics of the world consists of a handful of fundamental fields (e.g., gravitational, electromagnetic, etc.) and these fundamental fields are the fundamental objects in reality.

Oligonism suffers from an inconvenient complication as compared to monism. The monist can at least say that we have derivative existence as parts of a fundamental whole. The field oligonist cannot, because there is no one fundamental whole that we are parts of. On field oligonism, what we need to say is that each of us is jointly constituted by the arrangement of a handful of fields: I exist in virtue of the gravitational, electromagnetic and other fields having the right sorts of concentrations here.

Maybe, though, one can have one-many parthood relation: x is a part of y, z, w, ... even though x isn't a part of y, or z, or w, but only of all them jointly. Then we could exist as parts of the gravitational, electromagnetic and other fields, without us existing as parts of any one of them. A one-many parthood relation isn't crazy. Take an Aristotelian or van Inwagen view on which living things are the only complex objects. Now we could imagine two organisms, A and B, that each have a symbiotic relationship with a third object C but not with each other, so that we have two symbiotic wholes: AC and BC. Further suppose that only a part of C is involved in AC and a disjoint part of C is involved in BC. Then we could say that C is a part of AC and BC, but isn't a part of either AC or of BC, nor is there a greater whole ABC that contains all of C.

Of course, I don't think oligonism is true. The main reason I don't think that is that I think we are fundamental.

Do inconsistent credences lead to Dutch Books?

It is said that if an agent has inconsistent credences, she is Dutch Bookable. Whether this is true depends on how the agent calculates expected utilities. After all, expected utilities normally are Lebesgue integrals over a probability measure, but the inconsistent agent’s credences are not a probability measure, so strictly speaking there is no such thing as a Lebesgue integral over them.

Let’s think how a Lebesgue integral is defined. If P is a probability measure and U is a measurable function on the sample space, then the expected value of U is defined as:

1. E(U)=∫₀^∞P(U > y)dy − ∫_−∞⁰P(U < y)dy

where the latter two integrals are improper Riemann integrals and where P(U > y) is shorthand for P({ω : U(ω)>y}) and similarly for P(U < y).

Now suppose that P is not a probability measure, but an arbitrary function from the set of events to the real numbers. We can still define the expected value of U by means of (1) as long as the two Riemann integrals are defined and aren’t both ∞ or both −∞.

Now, here is an easy fact:

Proposition: Suppose that P is a function from a finite algebra of events to the non-negative real numbers such that P(∅)=0. Suppose that U is a measurable (with respect to the finite algebra) function such that (a) P(U > y)=0 for all y > 0 and (b) P(U < 0)>0. Then if E(U) is defined by (1), we have E(U)<0.

Proof: Since the algebra is finite and U is measurable, U takes on only finitely many values. If y₀ is the largest of its negative values, then P(U < 0)=P(U < y) for any negative y > y₀, and hence ∫_−∞⁰P(U < y)dy ≥ |y₀|P(U < 0)>0 by (b), while ∫₀^∞P(U > y)dy by (a). □

But then:

Corollary: If P is a function from a finite algebra of events on the samples space Ω to the non-negative real numbers with P(∅)=0 and P(Ω)>0, then an agent who maximizes expected utility with respect to the credence assignment P as computed via (1) and starts with a baseline betting portfolio for which the utility is zero no matter what happens will never be Dutch Boooked by a finite sequence of changes to her portfolio.

Proof: The agent starts off with a portfolio with a utility assignment U₀ where P(U₀ > y)=0 for all y > 0 and P(U₀ < y)=0 for all y < 0, and hence once where E(U₀)=0 by (1). If the agent is in a position where the expected utility based on her current portfolio is non-negative, she will never accept a change to the portfolio that turns the portfolio’s expected utility negative, as that would violated expected utility maximization. By mathematical induction, no finite sequence of changes to her portfolio will turn her expected utility negative. But if a portfolio is a Dutch Book then the associated utility function U is such that P(U < 0)=P(Ω)>0 and P(U > y)=0 for all y > 0. Hence by the Proposition, E(U)<0, and hence a Dutch Book will not be accepted at any finite stage. □

Note that the Corollary does assume a very weak consistency in the credence assignment: negative credences are forbidden, impossible events get zero credence, and necessary events get non-zero credence.

Additionally, the Corollary does allow for the possibility of what one might call a relative Dutch Book, i.e., a change between portfolios that loses the agent money no matter what. The final portfolio won’t be a Dutch Book relative to the initial baseline portfolio, of course.

Note, however, that we don’t need consistency to get rid of relative Dutch Books. Adding the regularity assumption that P(A)>0 for all non-empty A and the monotonicity condition that if A ⊂ B then P(A)<P(B) is all we need to ensure the agent will never accept even a relative Dutch Book. For regularity plus monotonicity ensures that a relative Dutch Book always decreases expected utility as defined by (1). But these conditions are not enough to rule out all inconsistency. For instance, if in the case of the flip of a single coin I assign probability 1 to heads-or-tails, probability 0.8 to heads, probability 0.8 to tails, and probability 0 to the empty event, then my assignment is patently inconsistent, but satisfies all of the above assumptions and hence is neither absolutely nor relatively Dutch Bookable.

How does all this cohere with the famous theorems about inconsistent credence assignments being Dutch Bookable? Simple: Those theorems define expected utility for inconsistent credences differently. Specifically, they define expected utility as ∑_iU_iP(E_i) where the E_i partition the sample space such that on E_i the utility has the constant value U_i. But that’s not the obvious and direct generalization of the Lebesgue integral!

I vaguely recall hearing something that suggests to me that this might be in the literature.

Also, I slept rather poorly, so I could be just plain mistaken in the formal stuff.

Approximatable laws

Some people, most notably Robin Collins, have run teleological arguments from the discoverability of the laws of nature.

But I doubt that we know that the laws of nature are discoverable. After all, it seems we haven’t discovered the laws of physics yet.

But the laws of nature are, surely, approximatable: it is within our power to come up with approximations that work pretty well in limited, but often useful, domains. This feature of the laws of nature is hard to deny. At the same time, it seems to be a very anthropocentric feature, since the both the ability to approximate and the usefulness are anthropocentric features. The approximatability of the laws of nature thus suggests a universe whose laws are designed by someone who cares about us.

Objection: Only given approximatable laws is intelligence an advantage, so intelligent beings will only evolve in universes with approximatable laws. Hence, the approximatable laws can be explained in a multiverse by an anthropic principle.

Response: Approximatability is not a zero-one feature. It comes in degrees. I grant that approximatable laws are needed for intelligence to be an advantage. But they only need to be approximatable to the degree that was discovered by our prehistoric ancestors. There is no need for the further approximatability that was central to the scientific revolution. Thus an anthropic principle explanation only explains a part of the extent of approximatability.

Humean accounts of modality

Humean accounts of modality, like Sider’s, work as follows. We first take some privileged truths, including all the mathematical ones, and an appropriate collection of others (e.g., ones about natural kind membership or the fundamental truths of metaphysics). And then we stipulate that to be necessary is to follow from the collection of privileged truths, and the possible that whose negation isn’t necessary.

Here is a problem. We need to be able to say things like this:

1. Necessarily it’s possible that 2+2=4.

For that to be the case, then:

2. It’s possible that 2+2=4

has to follow from the privileged truths. But on the theory under consideration, (2) means:

3. That 2 + 2 ≠ 4 does not follow from the privileged truths.

So, (3) has to follow from the privileged truths. Now, how could it do that? Suppose first that the privileged truths include only the mathematical ones. Then (3) has to be a mathematical truth: for only mathematical truths follow logically from mathematical truths. But this means that “the privileged truths”, i.e., “the mathematical truths”, has to have a mathematical description. For instance, there has to be a set or proper class of mathematical truths. But that “the mathematical truths” has a mathematical description is a direct violation of Tarski’s Indefinability of Truth theorem, which is a variant of Goedel’s First Incompleteness Theorem.

So we need more truths than the mathematical ones to be among the privileged ones, enough that (3) should follow from them. But it unlikely that any of the privileged truths proposed by the proponents of Humean accounts of modality will do the job with respect to (3). Even the weaker claim:

4. That 2 + 2 ≠ 4 does not follow from the mathematical truths

seems hard to get from the normally proposed privileged truths. (It’s not mathematical, it’s not natural kind membership, it’s not a fundamental truth of metaphysics, etc.)

Consider this. The notion of “follows from” in this context is a formal mathematical notion. (Otherwise, it’s an undefined modal term, rendering the account viciously circular.) So facts about what does or does not follow from some truths seem to be precisely mathematical truths. One natural way to make sense of (4) is to say that there is a privileged truth that says that some set T is the set of mathematical truths, and then suppose there is a mathematical truth that 2 + 2 ≠ 4 does not follow from T. But a set of mathematical truths violates Indefinability of Truth.

Perhaps, though, we can just add to the privileged truths some truths about what does and does not follow from the privileged truths. In particular, the privileged truths will contain, or it will easily follow from them, the truth that they are mutually consistent. But now the privileged truths become self-referential in a way that leads to contradiction. For instance:

5. No x such that F(x) follows from the privileged truths.

will make sense for any F, and we can choose a predicate F such that it is provable that (5) is the only thing that satisfies F (cf. Goedel’s diagonal lemma). Now, if (5) follows from the privileged truths, then it also follows from the privileged truths that (5) doesn’t follow from the privileged truths, and hence that the privileged truths are inconsistent. Thus, from the fact that the privileged truths are consistent, which itself is a privileged truth or a consequence thereof, one can prove (5) doesn’t follow from the privileged truths, and hence that (5) is true, which is absurd.

How the law needs to be written in the heart

In ethics, we seek a theory of obligation whose predictions match our best intuitions.

Suppose that explorers on the moon find a booklet with pages of platinum that contains an elegant collection of moral precepts that match our best intuitions to an incredible degree, better than anything that has been seen before. When we apply the precepts to hard cases, we find solutions that, to people we think of as decent, seem just right, and the easy cases all work correctly. And every apparently right action either follows from the precepts, or turns out to be a sham on deeper reflection.

This would give us good reason to think the precepts of the booklet in fact do sum up obligations. But now imagine Euthyphro came along and gave us this metaethical theory:

1. What makes an action right is that it follows from the content of this booklet.

Euthyphro would be wrong. For even though (1) correctly gives a correct account of what actions are in fact right, the right action isn’t right because it’s written in the booklet. (Is it written in the booklet because it’s right? Probably: the best theory of the booklet’s composition would be that it was written by some ethical genius who wrote what was right because it was right.)

Why not? What’s wrong with (1)? It seems to me that (1) is just too extrinsic to us. There is no connection between the booklet and our actions, besides the fact that the actions required by the booklet are exactly the right ones.

What if instead the booklet were an intrinsic feature of human beings? What if ethics were literally written in the human heart, so that microscopic examination of a dissected human heart found miniature words spelling out precepts that we have very good reason to think sum up the theory of the right? Again, we should not go for a Euthyphro-style theory that equates the right with what is literally written in the heart. Yet on this theory the grounds of the right would be literally intrinsic to us—and they could be essential to us, if we wish: further examination could show that it is an essential feature of human DNA that it generates this inscription. This would give us reason to think that human beings were designed by an ethical genius, but not that the ground of the right is the writing in the heart.

The lesson is this, I think. We want the grounds of the right to be of the correct sort. Being metaphysically intrinsic to us is a necessary condition for this, but it is not sufficient. We want the grounds of the right to be “close to us”: closer than our physical hearts, as it were.

But we also don’t want the grounds of the right to be too close to us. We don’t want the right to be grounded in the actual content of our desires or beliefs. We are looking for grounds that exercise some sort of a dominion over us, but not an alien dominion.

The more I think about this, the more I see the human form—understood as an actual metaphysical component intrinsic and essential to the human being—as having the exactly right balance of standoffish dominion and closeness to provide these grounds. In other words, Natural Law provides the right metaethics.

And the line of thought I gave above can also be repeated for epistemological normativity. So we have reason to think the Natural Law provides the right metaepistemology as well.

A tension in some theistic Aristotelian thinkers

Here is a tension in the views of some theistic Aristotelian philosophers. On the one hand, we argue:

1. The mathematical elegance and discoverability of the laws of physics is evidence for the existence of God

but we also think:

2. There are higher-level (e.g., biological and psychological) laws that do not reduce to the laws of physics.

These higher-level laws, among other things, govern the emergence of higher-level structures from lower-level ones and the control that the higher-level structures exert over the lower-level ones.

The higher-level laws are largely unknown except in the broadest outline. They are thus not discoverable in the way the laws of physics are claimed to be, and since no serious proposals are yet available as to their exact formulation, we have no evidence as to their elegance. But as evidence for the existence of God, the elegance and discoverability of a proper subset of the laws is much less impressive. In other words, (1) is really impressive if all the laws reduce to the laws of physics. But otherwise, (1) is rather less impressive. I’ve never never seen this criticism.

I think, however, there is a way for the Aristotelian to still run a design argument.

Either all the laws reduce to the laws of physics or not.

If they all reduce to the laws of physics, pace Aristotelianism, we have a great elegance and discoverability design argument.

Suppose now that they don’t. Then there is, presumably, a great deal of complex connection between structural levels that is logically contingent. It would be logically possible for minds to arise out of the kinds of arrangements of physical materials we have in stones, but then the minds wouldn’t be able to operate very effectively in the world, at least without massively overriding the physics. Instead, minds arise in brains. The higher-level laws rarely if ever override the lower-level ones. Having higher-level laws that fit so harmoniously with the lower-level laws is very surprising a priori. Indeed, this harmony is so great as to be epistemically suspicious, suspicious enough that the need for such a harmony makes one worry that the higher-level laws are a mere fiction. But if they are a mere fiction, then we go back to the first option, namely reduction. Here we are assuming the higher level stuff is irreducible. And now we have a great design argument from their harmony with the lower-level laws.

An Aristotelian account of proper parthood (for integral parts)

Here it is: x is a proper part of y iff x is informed by a form that informs y and x's being informed by that form is derivative from y's being informed by it.

Shape and parts

Alice is a two-dimensional object. Suppose Alice’s simple parts fill a round region of space. Then Alice is round, right?

Perhaps not! Imagine that Alice started out as an extended simple in the shape of a solid square and inside the space occupied by her there was an extended simple, Barbara, in the shape of a circle. (This requires there to be two things in the same place: that’s not a serious difficulty.) But now suppose that Alice metaphysically ingested Barbara, i.e., a parthood relation came into existence between Barbara and Alice, but without any other changes in Alice or Barbara.

Now Alice has one simple part, Barbara (or a descendant of Barbara, if objects “lose their identity” upon becoming parts—but for simplicity, I will just call that part Barbara), who is circular. So, Alice’s simple parts fill a circular region of space. But Alice is square: the total region occupied by her is a square. So, it is possible to have one’s simple parts fill a circular region of space without being circular.

It is tempting to say that Alice has two simple parts: a smaller circular one and a larger square one that encompasses the circular one. But that is mistaken. For where would the “larger square part” come from? Alice had no proper parts, being an extended simple, before ingesting Barbara, and the only part she acquired was Barbara.

Maybe the way to describe the story is this: Alice is square directly, in her own right. But she is circular in respect of her proper parts. Maybe Alice is the closest we can have to a square circle?
Here is another apparent possibility. Imagine that Alice started as an immaterial object with no shape. But she acquired a circular part, and came to be circular in respect of her proper parts. So, now, Alice is circular in respect of her proper parts, but has no shape directly, in her own right.

Once these distinctions have been made, we can ask this interesting question:

• Do we human beings have shape directly or merely in respect of our proper parts?

If the answer is “merely in respect of our proper parts”, that would suggest a view on which we are both immaterial and material, a kind of Hegelian synthesis of materialism and simple dualism.

Classical probability theory is not enough

Here’s a quick argument that classical probability cannot capture all probabilistic phenomena even if we restrict our attention to phenomena where numbers should be assigned. Consider a nonmeasurable event E, maybe a dart hitting a nonmeasurable subset of the target, and consider a fair coin flip that is causally isolated from E. Let H and T be the heads and tails results of the flip. Then let A be this disjunctive event:

• (E and H) or (not-E and T).

Intuitively, event A clearly has probability 1. If E happens, the probability of A is 1/2 (heads) and if E doesn’t happen, it’s also 1/2 (tails). (The argument uses finite conglomerability, but it is also highly intuitive.)

So a precise number should be assigned to A, namely 1/2. And ditto to H. But we cannot have these assignments in classical probability theory. For if we did that, then we would also have to assign a probability to the conjunction of H and A, which is equivalent to the conjunction of E and H. But we cannot assign a probability to the conjunction of E and H, because E and H are independent, and so we would have a precise probability for E, namely P(E)P(H)/P(H)=P(E&H)/P(H), contrary to the nonmeasurability of E.

Simple dualism and animals

According to simple dualism, our immaterial souls are the bearers of our mental states and we are these souls. We have bodies, but the bodies are not parts of us. We are wholly immaterial.

If the motivation for simple dualism is that only an immaterial soul can have mental states, then we should think something similar about higher animals like dogs and octopuses. Thus, in Rover the dog just as in Alice the human, the soul is the bearer of mental states, and the body is not a part of the soul. Now, the name “Alice” on simple dualism refers to the soul, so that “Alice is in pain” means that she is the bearer of the pain and “Alice has a broken leg” means that the leg associated with Alice is broken (compare: “Alice has a broken bicycle”) rather than that a part of Alice is broken. Surely, “Rover is in pain” and “Rover has a broken leg” mean something very close to “Alice is in pain” and “Alice has a broken leg”, respectively. Thus, “Rover” on simple dualism also refers to the soul.

Furthermore, Rover might be Alice’s pet. And the kind of interspecies affection that might exist between Rover and Alice requires that Rover be the right kind of thing to have affections and other mental states, and so, once again, “Rover” must refer to the soul.

But of course we also say that Rover is a dog. The simple dualist now has two options. The first is to take literally the statement that Rover is a dog, and conclude that dogs—and presumably other higher animals—are immaterial souls (if Rover is immaterial and Rover is a dog, then Rover is an immaterial dog; and Rover surely does not differ radically from other higher animals). Thus, strictly speaking, biologists don’t primarily study dogs and octopuses but rather their bodies, and we have never seen any higher animal.

The second option is to deny that Rover is literally a dog. This presumably requires denying that we are literally homo sapiens. Rather, “Rover is a dog” is to be understood as shorthand for “Rover ensouls a dog.”

Neither option looks attractive. I conclude that Rover is not a soul, and neither is Alice.

Shuffling an infinite deck of cards

Suppose I have an infinitely deep deck of cards, numbered with the positive integers. Can I shuffle it?

Given an infinite past, here is a procedure: n days ago, I perfectly fairly shuffle the top n cards in the deck.

When one reshuffles a portion of an already perfectly shuffled finite deck of cards, the full deck remains perfectly shuffled. So, the top n cards in the infinitely deep deck are perfectly shuffled for every finite n.

Can we argue that the thus-shuffled deck generates a countably infinite fair lottery, i.e., that if we pick cards off the top of the deck, all card numbers will be equally likely? At the moment I don’t know how to argue for that. But I can say that we get what I have called a countably infinite paradoxical lottery, i.e., one when any particular outcome has zero or infinitesimal probability.

For simplicity, let’s just consider picking the top card off the deck and consider a particular card number, say 100. For card 100 to be at the top of the deck, it had to be in the top n cards prior to the shuffling on day −n for each n. For instance, on day −1000, it had to to be in the top 1000 cards prior to the shuffling. The subsequent 1000 shufflings together perfectly shuffle the top 1000 cards. Thus, the probability that card 100 would end up at the top is 1/1000, given that it was in the top 1000 cards on day −1000. But it may not have been. So, all in all, the probability that card 100 would end up at the top is at most 1/1000. But the argument generalizes: for any n, the probability that card 100 would end up at the top is at most 1/n. Hence, the probability that card 100 would end up at the top is zero or infinitesimal.

If taking an infinite amount of time to shuffle is too boring, you can also do this with a supertask: one minute ago you shuffle the top card, 1.5 minutes ago you shuffle the top two cards, 1.75 minutes ago you shuffled the top three cards, and so on. Then you did the whole process in two minutes.

All the paradoxes of fair countably infinite lotteries reappear for any paradoxical countably infinite lottery. So, the above simple procedure is guaranteed to generate lots of fun paradoxes.

Here is a fun one. Carl shuffles the infinite deck. He now offers to pay Alice and Bob $20 each to play this game: they each take a card off the top of the deck, and the one with the smaller number has to pay $100 to the one with the bigger number. Alice and Bob happily agree to play the game. After all, they know the top two cards of the deck are perfectly shuffled, so they think it’s equally likely that each will win, and hence each calculates their expected payoff at 0.5×$100 − 0.5×$100 + $20 = $20. He puts them in separate rooms. As soon as each sees their own card (but not the other's), he now offers a new deal to them: if they each agree to pay him $80, he’ll broker a deal letting them swap their cards before determining who is the winner. Alice sees her card, and knows there are only finitely many cards with a smaller number, so she estimates her probability of being a winner at zero or infinitesimal. So she is nearly sure that if she doesn’t swap, she’ll be out $100, and hence it’s obviously worth swapping, even if it costs $80 to swap. Bob reasons the same way. So they each pay Carl $80 to swap. As a result, Carl makes $80+$80−$20−$20=$120 in each round of the game.

Causal Finitism, of course, says that you can’t have an infinite causal history, so you can’t have done the infinite number of shufflings.

Fulfilling requests

One of the most moving stories in Rosenbaum’s deeply moving Holocaust and the Halakhah tells of how one can be a great moral hero even when acting out of mistaken conscience. A man in a concentration camp comes to his rabbi with a problem. His son has been scheduled to be executed. But it is possible to bribe the kapo to get him off the death list. However, the kapo have a quota to fill, and if they let off his son, they will kill another child. Is it permissible to bribe the kapo knowing that this will result in the death of another child? The rabbi answers that, of course, it is permissible. The man goes away, but he is not convinced. He does not bribe the kapo. Instead, he concludes that God has called him to the great sacrifice of not shifting his son’s death onto another. The father finds a joy in the sacrifice amidst his mourning.

The rabbi was certainly right. The father’s conscience presumably was mistaken (unless God specifically spoke to him and required the sacrifice). Yet the father is a moral hero in acting from this mistaken conscience. (Here are two relevant features of this case. First, while he was mistaken, he was not mistaken in a way that shows moral callousness—on the contrary, he is obviously a man of moral sensitivity. Second, while he was mistaken in thinking the sacrifice was morally required, nonetheless the sacrifice was—I think—at least permissible.)

The analytic philosopher will see this as a variant of a trolley case (with some complications, such as that the deaths were mediated by the free agency of the kapo). It is permissible to redirect the trolley away from one’s child and towards a stranger’s child. This is another way in which the proportionality condition in the Principle of Double Effect is not a utilitarian calculation: the agent has a proportional reason to save their own child even when it is foreseen (but not intended) to cost another’s their life.

But at the same time it would not be permissible to redirect the trolley away from one stranger’s child towards another stranger’s child. Such redirection would be a grotesque toying with lives. It would be a needless and callous making of oneself into a cause of another’s death, even if unintentionally.

Here, however, is a case that puzzles me. Suppose Alice’s child is on the track the trolley is speeding towards, and a stranger’s child is on another track. Alice is physically incapable of redirecting the trolley but Bob is capable of it. Alice and both children are strangers to Bob. Would it be permissible for Alice to ask Bob to redirect the trolley?

Here is an argument to the contrary. It is impermissible for Bob to redirect the trolley from one stranger to another: that is just playing with lives. But it is impermissible to request someone to perform an impermissible action. Hence, it is impermissible to ask Bob to redirect the trolley.

That seems mistaken. The case of asking Bob to redirect the trolley need not be that different from begging the kapo to take one’s child off the death list, depending on the details of the latter story. So what is going on?

I think there are at least two ways to justify Bob’s acquiescence to the request and hence Alice’s making of the request:

1. Once Alice asks Bob to redirect the trolley, Alice is no longer a stranger to Bob. There is a way in which Bob in receiving her request can become an agent of Alice’s, and hence those that Alice cares for become ones that he has a special reason to care for.

2. On receipt of the request, Bob has two options coming with distinctive incommensurable reasons. The first is not to redirected, with the reason being promote equality, in this case equality between children who don’t have a parent in place to speak up for them and ones who do. The second is to fulfill the request of an anguished parent to save their child. Both reasons are grave, and it is permissible for him (other things being equal) to act on either reason. Requests really do add weight to reasons.

There is another complicating factor. I do have the intuition that if Bob is an employee in charge of the trolley, he should do nothing. The reason is this. Insofar as he is in charge of the trolley, Bob has a role duty of mitigating damage done by the trolley. It is generally good policy that such a role come along with a significant independence from outside influences, such as bribes or even requests. So, in that case, Bob should act as if he did not receive the request. But if he did not receive any request, he shouldn’t do anything, for it is better not to become the cause of the child’s death—as one would if one redirected.

Here is a variant case. There are three tracks. The trolley is on track A with five people. The other two tracks, B and C, have one person each, and Alice is asking Bob not to redirect to track B, as her child is there. Bob has to redirect to either track B or C, but everything other than Alice’s request is equal between these tracks. Here it seems to me that Bob should flip a coin (if there is time; if not, just act as randomly as he can) if he is an employee. And if he is not an employee, then he has a choice to accede to Alice’s request or flip a coin.