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.

Three versions of proportionality in Double Effect

Bob sees a trolley speeding towards five strangers on a track and can redirect the trolley towards another track which has one stranger on it. Alice, who is herself unable to redirect the trolley, offers Bob a dollar to redirect it. Suppose Bob redirects the trolley solely for the sake of dollar. Bob is clearly a callous individual. But has Bob violated the strictures of the Principle of Double Effect?

Well, Bob has done an action that’s intrinsically good or neutral (redirecting a trolley). The bad effect—the death of the one stranger—was not intended either as an end or as a means, and indeed does not in any way (we assume) contribute to the intended good effect, which is getting a dollar. What remains to check is the proportionality condition.

Here it depends on exactly how the proportionality condition is formulated. There are at least three formulations:

1. The bad effects are not disproportionate to the intended good effects.

2. The bad effects are not disproportionate to the good effects.

3. The bad effects are not disproportionate to those good effects that are not themselves the outcomes of bad effects.

On formulation (1), Bob has violated Double Effect: the death of the one stranger is disproportionate to the sole intended good, namely Bob getting a dollar. On formulations (2) and (3), Bob has not violated Double Effect, since the good effect—the saving of the five—is proportionate (and is not the outcome of a bad effect).

My intuition is that the case supports (1). But I worry that this rides on our desire to get the obviously vicious Bob on some charge or other, and violating Double Effect is the obvious one. But there may be another charge. Bob had a moral duty to the do the following: to redirect the trolley in order to save five lives. He failed to do that. His failure to do that is a morally wrong abstension. So even if (2) or (3) are the right story, we can still get Bob on some moral charge or other.

So I am not sure how far the case helps adjudicate between (1)–(3).

Note one nice thing about (1), though. If we go for (1), we automatically filter out any good effects that are the outcomes of bad effects, since if we intended such good effects, we would be intending a bad means and violating the means condition of Double Effected. So (1) implicitly contains the same restriction as is found in (3).

Cupcakes and trolleys

A trolley is heading towards a person lying on the tracks. Also lying on the tracks is a delicious cupcake. You could redirect the trolley to a second track where there is a different person lying on the tracks, but no cupcake.

Utilitarianism suggests that, as long as you are able to enjoy the cupcake under the circumstances and not feel bad about the whole affair, you have a moral duty to redirect the trolley in order to save the cupcake for yourself. This is morally perverse.

Besides showing that utilitarianism is false, this example shows that the proportionality condition in the Principle of Double Effect cannot simply consist in a simple calculation comparing the goods and bads resulting from the action. For there is something morally disproportionate in choosing who lives and dies for the sake of a cupcake.

What needs a cause

Suppose Alice has existed for an infinite amount of time and now time 0 (in some unit system) has just come. Imagine that between time −1 and time 0, Barbara lived internally a life just like Alice did between time −1 and time 0. But between time −1.5 and −1, Barbara lived a life sped up by a factor of two, exactly like Alice’s life between time −2 and time −1. And between time −1.75 and −1.5, Barbara lived a life sped up by a factor of two, exactly like Alice’s life between time −3 and time −2. And so on, with Barbara coming into existence right after time −2.

Some people think that the principle:

1. Everything that has a beginning has a cause

is significantly more plausible than the principle:

2. Everything contingent has a cause.

Now, (1) requires Barbara to have a cause but does not require this of Alice, while (2) requires both Barbara and Alice to have causes. But internally, there is really no significant difference between Alice’s life and Barbara’s. Thus, someone who thinks that (1) is significantly more plausible than (2) needs to think that external differences—such as Barbara’s past life being metrically finite with respect to external time—might make a difference as to what needs and what does not need a cause.

If, however, we think that external differences do not make a difference with respect to what needs a cause, we should judge Barbara and Alice the same way. And if we judge Barbara and Alice the same way, then it seems that we should not think (1) is significantly more plausible than (2).

There are other minds

Suppose there are n (physically, including neurally) healthy mature humans on earth. Let Q₁, ..., Q_n be their non-mental qualitative profiles: complete descriptions of their non-mental life in qualitative terms. Let H_i be the hypothesis that everything with profile Q_i is conscious. Now, consider the hypotheses:

• M: All healthy mature humans have a mental life.

• N: Exactly one healthy mature human has a mental life.

• Z: No healthy mature human has a mental life.

Assume our background information contains the that there are at least two healthy mature humans. Given that background, the hypotheses are mutually exclusive. Now add that there are n healthy mature humans on earth, where n is in the billions, and that they have profiles Q₁, ..., Q_n, which are all different. What’s a reasonable thing to think now? Well, N is no more likely than M or Z. Conservatively, let’s just suppose they are all equally likely, and hence all have probability 1/3. Furthermore, if N is true, exactly one H_i is true. Moreover all the H_i are just about on par given N, so P(H_i|N)≈1/n for all i, and hence P(H_i&N) is at most about 1/(3n). On the other hand, P(H_i|Z)=0 and P(H_i|M)=1.

Now suppose I learn that Q_m is my profile. Then I learn that H_m is true. That rules out the all-zombie hypothesis Z, and most of the H_i&N conjunctions. What is compatible with my data are two mutually exclusive hypotheses: H_m&N as well as M. It’s easy to check (e.g., with Bayes’ theorem) that my posterior probability for H_m&N will then be approximately at most 1/(n + 1). Thus, the probability that there is another mind is bigger than 0.999999999.

Whether we can argue for M in this way depends on how the priors for M compare to the priors of hypotheses in between M and N, such as the hypothesis that all but seven healthy mature humans have consciousness.

Antisolipsism

I never heard anyone defend this view: "Billions of people exist but I don't."

Van Inwagen's ear

Van Inwagen holds that:

1. All and only things whose activity constitutes a life (properly) compose a whole.

2. Whether a plurality of things composes a whole depends only on their internal relations.

He considers a counterexample to (1) and (2) of the following sort. Let the xs be the particles in van Inwagen outside the right ear.

3. If van Inwagen were to have lost the right ear, the activity of the xs would have constituted a life (his life) and composed a whole (namely, van Inwagen).

4. But in fact, the activity of the xs does not constitute a life, but only partly does so, along with the activity of the right ear particles.

5. However, the internal relations between the xs were he to have lost his right ear would have been the same as they are now.

This is a problem: for by (4) and (1), the xs do not compose a whole, but by (3) they would have had he lost his right ear, and by (5) they would have had the same internal relations then, which contradicts (2).

Van Inwagen attempts to escape this problem by denying (5), saying that the internal relations between the particles in his body in the vicinity of the right ear would be affected by the ear not being there. For they would no longer experience forces from the ear particles.

But let d be the closest distance between a right-ear particle and a van Inwagen particle not in the right ear (i.e., one of the xs). But now if God were to suddenly annihilate the right ear, then it seems that none of the xs would be in any way affected until influences traveling at the speed of light could bridge the distance d. I.e., until d/c (where c is the speed of light) had passed, the xs would be without the ear just as they are with the ear. Hence, if we specify that the time of severance in (3) is less than d/c ago, van Inwagen’s response seems to fail.

One might try to get out of this by invoking (non-Bohmian) quantum mechanics, and saying that all particles have fuzzy positions, and the ear particles overlap positionally with the non-ear particles, so that the disappearance of the ear particles affects the non-ear particles instantly. But the instant part of the effect is slight. We can imagine that the disappearance of the ear is so orchestrated as to never split any molecules or atoms. But particles in different molecules are fairly localized to their respective molecules, and the effect of the tails of the wavefunction on what is going on in a neighboring molecule will presumably be negligible.

Of course, a negligible effect is still an effect. But we could imagine a third scenario: van Inwagen loses his ear, and God miraculously tweaks the movements of the xs in a slight and biologically negligible way during the d/c period so that they behave just as they do in the actual world where the ear is attached. In that scenario, the xs would compose van Inwagen, but they would have exactly the same internal relations as they do in the actual world.

Artifacts and non-naturalism

One of the reasons to be suspicious of artifacts is that it seems magical to think we have the power to create a new object just by thinking about things a certain way while manipulating stuff. If Bob gets some clay and exercise his fingers by randomly kneading it, he doesn’t make a sculpture or any other new object out of it. But if his identical twin Carl intends to shape the clay into a sculpture, and in doing so moves his fingers in exactly the same way that Bob did, and produces exactly the same shape, then—assuming artifacts exist—he creates a new object, a sculpture. It seems magical that our thoughts should affect what object exists in the world, even when the thoughts make no difference to our manipulation of the world.

When I discussed arguments with this in my Mid-Sized Objects graduate seminar, I found, however, that there was a lot of friendliness towards the view that, yes, we are capable of this magic, though some demurred at the word “magic”. And in particular, a student pointed out that we are in the image of a God who can create.

This has made me think that a non-naturalist can think that our thoughts have effects that are not screened by the movements of our bodies. Thus, it could well be that Carl’s thoughts causes the world to be different. For instance, on a hylomorphic view, Carl could have the power to create a scu;tural form for a piece of clay by his thoughts. Or on a variant of Markosian’s brute composition view, Carl could have the power simply to cause a new object composed of the clay.

In fact, this suggests an interesting new argument against physicalism, where physicalism is understood as the claim that all causal powers reduce to those of physics. Intuitively, the correct ontology includes more things than van Inwagen’s ontology of particles and organisms and but not all the things from the mereological universalist’s bloated ontology. In particular, intuitively, the correct ontology does include Carl’s new sculpture, but Bob hasn’t produced anything new, and hence the correct ontology seems to require a non-natural “magical” power over composition facts to be found in Carl’s (and presumably, albeit in this context unexercised, Bob’s) mind. And if our ontology is to include, as common-sense would suggest, galaxies, planets, mountains and rocks, we need powers in things to produce such objects—i.e., to ensure that their particulate parts do compose something—and these powers are not to be found in physics.

Markosian’s apparently preferred version of the brute composition view can almost accommodate this. On that version, the composition facts supervene on the arrangement of particles: there are infinitely many necessary truths that specify which arrangements of particles compose. But these necessary truths would include lots of arbitrary parameters (e.g., encoding the difference between some stones that are just lying there and a hillock). We don’t want necessary truths with arbitrary parameters. It is much better if any such arbitrary parameters are relocated to the laws of nature or, better, the causal powers of things.

A gambling puzzle about nonmeasurable events

I have two sealed envelopes, labeled A and B. One contains $3 and the other nothing. You don’t know which is which. I am willing to sell either or both envelopes for $1 each. You have a fixed period of time to inform me whether you are buying neither, both, A, or B, after which time you pay any get to open any envelopes you bought.

Obviously, it makes sense for you to hand me $2 and buy both envelopes and profit by a dollar.

But suppose now that I tell you that I chose which envelope to put the $3 using a saturated nonmeasurable method. For instance, perhaps I chose a subset N of the points on the circumference of a spinner that has the properties that:

1. N is nonmeasurable,

2. the only measurable subsets of N have measure zero, and

3. the only measurable subsets of the complement of N have measure zero,

then I spun the spinner, and if the spinner landed in N, I put the $3 in envelope A, and otherwise in B.

Your purchase options are: Neither, Both, A and B. The probability that the $3 is in A is completely undefined (we should represent the probability as the full interval from 0 to 1) and the probability that the $3 is in B is completely undefined.

It seems then:

1. It’s clearly rationally permissible for you to go for Both.

2. Going for A is neither rationally better nor rationally worse than going for Both. For by going for A, you miss out on B. But the expected utility of purchasing B iscompletely undefined: it is a choice to pay $1 for a gamble that has a completely undefined probability of paying out $3. So, it is completely undefined whether Both is better than A or worse. If Both is permissible, so is A, then.

3. But by similar reasoning it is completely undefined whether going for Neither is better than or worse than going for A. For the expected payoff of A is completely undefined. So, if A is rationally permissible, so is Neither, then.

4. Swapping A and B in the reasoning in (2) shows that B is rationally permissible as well.

So now it seems that all four options are equally permissible. But something has gone wrong here: Clearly, Both beats Neither, and it’s irrational to go for Neither.

I think to get out of the above puzzle, we have to deny the initially plausible principle:

5. If an option is rationally permissible, and another option is neither better nor worse than it, then the latter is also permissible.

Here is another case where this principle needs to be denied. You have a choice between playing Pac Man, or eating one scoop of ice cream, or eating two. Playing Pac Man is neither better nor worse than either one or two scoops of ice cream. Two scoops of ice cream is better than one. It is clearly rationally permissible to play Pac Man. By (5), it’s permissible to eat one scoop of ice cream, then. But that’s not true, since two scoops beats one.

So, let’s deny (5). Now I think the reasonable thing to say is that Neither is irrational, but each of Both, A and B is rationally permissible. But there is still a puzzle in the vicinity. Suppose you are asked about your purchases envelope-by-envelope. First you’re offered the chance to buy A, and then a chance to buy B, and once a deal is declined, it’s gone. You have no rational obligation to buy A. After all, going for B alone is permissible. So, let’s say you decline A. Next you’re asked about B. At this point, A is out of the picture, and the question is whether to pay $1 for a completely undefined probability of getting $3. It’s permissible to decline that. So, you can permissibly decline B as well. So, let’s say you do so. Now by a pair of perfectly rational choices you ended up “doing something stupid”. This is a bit like Satan’s Apple. but with a finite number of choices.

The puzzle above seems familiar. I may have read it somewhere and it stuck in my subconscious.

The great chain of being and the glory of God

There are things with power but no knowledge or moral will: e.g., trees. There are things with power and knowledge but no moral will: e.g., horses. There are things with all three: e.g., human beings.

These fundamental attributes mark radical qualitative differences. I suspect there are infinitely many further possible fundamental attributes besides power, knowledge and moral will. A being that had one more of these attributes would be qualitatively as far above us as we are above horses or as far as horses are above trees. But just as a horse cannot conceive of moral will, and a tree cannot conceive of anything, we cannot conceive of what these further attributes would be. All we can do is speculate that then chain power, knowledge and moral will can be continued indefinitely.

God actually has all three of power, knowledge and moral will, and has each to its maximal perfection. If my suspicion about the chain continuing ad infinitum, then all the further attributes in the chain God also has to an infinite degree. (While remaining simple.) But we have no idea what they are.

Arity-increase and heavy-weight Platonism

Here is a curious problem. To give a heavy-weight Platonist analysis of an n-ary predication requires an (n + 1)-ary predication:

1. Alice is green [unary]: Alice instantiates greenness [binary].

2. Alice and Bob are friends [binary]: Alice and Bob instantiate friendship [ternary].

But higher arity predication is more puzzling than lower arity predication. Hence, heavy-weight Platonism explains the obscure in terms of the more obscure.

What got me to thinking about this was exploring the idea that Platonists can curry higher arity relations into lower arity ones. But doing so requires a multigrade “instantiates” predicate, and the curried expression of an n-ary predication seems to require an n-ary use of “instantiates”.

On a function- rather than relation-based Platonism, the issue comes up as follows. To say that the value of an n-ary function f at x₁, ..., x_n is y is (n + 2)-ary predication which gets Platonically grounded by the application of the (n + 1)-ary function apply_n such that apply_n(f,x_1,…,x_n) = f(x₁, ..., x_n).

Multigrade relations

One strategy for avoiding ontological commitment to sets is to deal with pluralities and multigrade relations. Multigrade relations are relations that can be had by a variable number of things. Instead of, say, saying of the books on my shelf that there is a set of them whose total number of pages is exactly 800, one says that there are xs such that each of them is a book on my shelf and the xs stand in the multigrade relation of jointly having 800 pages. Let’s say these books are x₁, x₂ and x₃. Then we express their jointly having 800 pages as:

• Has800Pages(x₁, x₂, x₃).

We do not need a set of them to express this. And the Has800Pages(x₁, ...) predicate flexibly can take as many arguments as one wishes, corresponding to the multigradeness of the property it expresses.

But now consider a different statement: there are two pluralities of books on my shelf, having no books in common, where each plurality has the same total number of pages as the other. Can we make sense of this using multigrade relations instead of sets?

I don’t see how. Let’s say that the plurality x₁, x₂, x₃ and the plurality y₁, y₂ of my books each have the same total number of pages. So we introduce a predicate with variable arity and say:

• HasSameTotalNumberOfPages(x₁, x₂, x₃, y₁, y₂).

But that doesn’t work! For how can we tell if it is says that x₁, x₂, x₃ have the same number of pages as y₁, y₂ rather than saying that x₁, x₂ have the same number of pages as x₃, y₁, y₃?

We could multiply predicates with fixed arity and say:

• TheFirst3HaveTheSameTotalNumberOfPagesAsTheLast2(x₁, x₂, x₃, y₁, y₂).

But that won’t work with quantification, since we don’t know ahead of time how many xs and how many ys we are dealing with.

Maybe we should do this:

• Count(3,x₁, x₂, x₃) and Count(2,y₁, y₂) and SameNumberOfPages(3,x₁, x₂, x₃,2,y₁, y₂)

where the SameNumberOfPages variable arity predicate takes a number, then a plurality of that number of objects, then another number, and then another plurality of that number of objects.

But these kinds of solutions won’t work for infinite pluralities. For instance, suppose we want to say that the xs cause the ys, where there are ℵ₂ xs and ℵ₃ ys. Then I guess we say something like:

• Cause(ℵ₂, x₁, x₂, ..., ℵ₃, y₁, y₂, ...).

There are serious technical problems here, however. I will leave it to the reader to explore these.

Informative characterizations

It is hard to characterize an “informative characterization”. Here is an instructive illustration.

Ned Markosian in his famous brutal composition paper says that an informative, or non-trivial, characterization of when the xs compose something is one that is not synonymous with the statement that the xs compose something. But by that definition, here is a non-trivial characterization of when the xs compose something:

• water is H₂O and the xs compose something.

This statement is not synonymous with the statement that the xs compose something. Nor are the two statements provably equivalent. Nor are they a priori equivalent. But they are metaphysically necessarily equivalent.

Van Inwagen in Material Beings proceeds seemingly more restrictively. He wants a characterization of when the xs compose something that doesn’t use mereological vocabulary. But here is such a characterization:

• the xs have the property expressed by the actual world’s English phrase “compose something”.

This characterization mentions mereological vocabulary, but doesn’t use it. And if we want, we can avoid mentioning mereological vocabulary as well:

• the xs have the property referred to in the second bulleted item in this post in the actual world.

Obviously, none of these characterizations of “compose something” are informative.

Contrastive PSR

In my Principle of Sufficient Reason (PSR) book, I defend a PSR that holds that every contingent truth has an explanation, but I do not defend a contrastive PSR. Many think this is a cop-out.

But it makes sense to ask why it is that

1. The moon is round and I don’t have an odd number of fingers.

The answer is, presumably, that gravity pulled the matter of the moon into a ball and I was sufficiently careful around power tools. And yet it doesn’t make sense to ask why it is that

2. The moon is round rather than my having an odd number of fingers.

This point shows that it makes no sense to have a contrastive Principle of Sufficient Reason of the following form:

3. For all contingent truths p and contingent falsehoods q, there is an explanation of why p rather than q is true.

The only time it makes sense to ask why p rather than q is true is when q is some sort of a “relevant alternative” to p. So the contrastive Principle of Sufficient Reason would have to say something like:

4. For all contingent truths p and contingent falsehoods q, if q is a relevant alternative to p, there is an explanation of why p rather than q is true.

But now note that (4) is way messier than the standard PSR, and depends on an apparently contextual constraint in terms of a “relevant alternative” which feels ill-suited to a fundamental metaphysical principle. So, I do not think a contrastive PSR just is a plausible metaphysical principle.

Two jobs at Baylor

We have two full-time openings in the Baylor Philosophy Department. Both have open AOS and AOC.

One position is at the Associate (tenured) or Assistant Professor (tenure-track) level and the other is at the Assistant Professor (tenure-track) level.

Naturalism and property dualism

It is generally taken that a view on which there are mental properties that do not supervene on the properties of physics is a non-naturalistic view: it is a form of property dualism.

But now imagine that we find out that:

1. There are chemical properties that do not supervene on the properties physics speaks of.

That would be a really exciting discovery, but it wouldn’t be a discovery incompatible with naturalism. The new chemical properties would presumably be just as natural as the physical ones.

So, why would we call non-supervenient mental properties non-natural, if we wouldn’t call non-supervenient chemical properties non-natural? It can’t be just because chemical properties are the province of a science, namely chemistry. For mental properties are the province of a science, too, namely psychology.

While we’re exploring this corner of logical space, consider this view:

2. Chemical properties do not supervene on physical properties, and mental properties do not supervene on physical properties either, but mental properties do supervene on, and even reduce to, physical and chemical properties.

I’ve never met an advocate of (2). It would be a very strange view. But here is one that, I think, is not actually all that strange:

3. Biological properties do not supervene on physical properties, and mental properties do not supervene on physical properties either, but mental properties do supervene on, and even reduce to, biological properties.

I think view (3) is worth thinking about. Most of the people who have tried to reduce the mental have tried to reduce it to the physical, but perhaps a reduction to an irreducible biological level would be more promising.

Ordinary language and "exists"

In Material Beings, Peter van Inwagen argues that his view that there are no complex artifacts does not contradict (nearly?) universal human belief. The argument is based on his view that the propositions expressed by ordinary statements like “There are three valuable chairs in this room” do not entail the negation of the Radical Claim that there are no artifacts, for such a proposition does not entail that there exist chairs.

I think van Inwagen is right that such ordinary propositions do not entail the negation of the Radical Claim. But he is wrong in thinking that the Radical Claim does not contradict nearly universal human belief. Van Inwagen makes much of the analogy between his view and the Copernican view that the sun does not move. When ordinary people say things like “The sun moved behind the elms”, they don’t contradict Copernicus. Again, I think he is right about the ordinary claims, but nonetheless Copernicus contradicted nearly universal human belief. That was why Copernicus’ view was so surprising, so counterintuitive (cf. some remarks by Merricks on van Inwagen). One can both say that when people prior to Copernicus said “The sun moved behind the elms” they didn’t contradict Copernicanism and that they believed things that entailed that Copernicus is wrong.

People do not assert everything they believe. They typically assert what is salient. What is normally salient is not that the sun actually moved, but that there was a relative motion between the rays pointing to the elms and to the sun. Nonetheless, if ordinary pre-Copernicans said “The sun doesn’t stand still”, they might well have been contradicting the Copernican hypothesis. But rarely in ordinary life is there occasion to say “The sun doesn’t stand still.” Because of the way pragmatics affects semantics (something that van Inwagen apparently agrees on), we simply cannot assume that the proposition expressed by the English sentence “The sun moved behind the elms” entails the proposition expressed by the English sentence “The sun doesn’t stand still.”

Something similar, I suspect, is true for existential language. When an ordinary person says “There are three chairs in the room”, the proposition they express does not contradict the Radical Thesis. But if an ordinary person says things like “Chairs exist” or “Artifacts exist”, they likely would contradict the Radical Thesis, and moreover, these are statements that the ordinary person would be happy to make in denial of the Radical Thesis. But in the ordinary course of life, there is rarely an occasion for such statements.

This is all largely a function of pragmatics than the precise choice of words. Thus, one can say: “Drive slower. Speed limits exist.” The second sentence does not carry ontological commitment to speed limits.

So, how can we check whether an ordinary person believes that tables and chairs exist? I think the best way may be by ostension. We can bid the ordinary person to consider:

1. People, dogs, trees and electrons.

2. Holes, shadows and trends.

We remind the ordinary person that we say “There are three holes in this road” or “The shadow is growing”, but of course there are no holes or shadows, while there are people (we might remind them of the Cogito), dogs, trees and (as far as we can tell) electrons. I think any intelligent person will understand what we mean when we say there are no holes or shadows. And then we ask: “So, are tables and chairs in category 2 or in category 1? Do they exist like people, dogs, trees and electrons, or fail to exist like holes, shadows and trends?” This should work even if like Ray Sorensen they disagree that there are no shadows; they will still understand what we meant when we said that there are no shadows, and that’s enough for picking out what we meant by “exist”. To put in van Inwagen’s terms, this brief ostensive discussion will bring intelligent people into the “ontology room”.

And I suspect, though this is an empirical question and I could be wrong, once inducted into the discussion, most people will say that tables and chairs exist (and that they have believed this all along). But, van Inwagen should say, this nearly universal belief is mistaken.

This story neatly goes between van Inwagen’s view that ordinary people don’t believe things patently incompatible with the Radical Theory and Merricks’ view that ordinary poeple contradict the Radical Theory all the time. Ordinary people do believe things patently incompatible with the Radical Theory, but they rarely express these beliefs. Most ordinary “there exist” statements—whether concerning artifacts or people or particles—do not carry ontological commitment, and those of us who accept the Radical Theory normally aren’t lying when we say “There are three chairs in the room”. But the Radical Theory really is radical.

Creation and artifacts

Analytic metaphysics is widely thought a dry discipline. I want to show how it could be used to connect with some deeply devotional theological claims.

Here is a valid argument:

1. If artifacts exist, we created them.

2. Only God creates.

3. So, artifacts don’t exist.

This argument suggests that there can be a deeply devotional connection to the arguments of those metaphysicians, like Merricks and van Inwagen, who deny the existence of artifacts.

Here is another devotional line of thought towards this. Some radical theologians say that God doesn’t exist. They do this to emphasize the radical difference between God and creatures. But they do so wrong. The right way to emphasize this difference is to say that we don’t exist. (Recall how God is said to have told St. Catherine of Siena: “I am he who is and you are she who is not.”) Only God exists.

So, the things that God creates don’t exist—at least not in the same sense in which God exists. By analogy, it should be no surprise if the things we make don’t exist—at least not in the same sense that we exist.

Objection 1: We can create organisms in the lab, and organisms surely exist.

Response: Maybe we should say that their life comes from God.

Objection 2: The distinction between God’s creating and our making is sufficiently accounted for by noting that God creates ex nihilo and we make things out of preexistent stuff.

Response: God doesn’t always create ex nihilo. He made Adam out of the dust of the earth. And anyway the more differences we see between God and us, the more God’s transcendence is glorified.

Giant numerals stopwatch app

I do two activities where it is nice to have a stopwatch one can see from a distance: lap swimming and lap (indoor) rock climbing. I found one stopwatch app with large numerals, but they weren't large enough for my taste, so I ended up writing my own with the absolutely biggest digits I could (and lots of customizability and various extra features, like a start countdown).

For any Android users who want it, it's here. Source code is here. No ads, no in-app purchases. The app keeps track of the time even if it's navigated away from, and doesn't use any CPU time then. It should even continue keeping track of time if the device reboots or runs out of battery, though perhaps with some loss of precision. The lone permission in the app is to let it run on boot (to adjust the time in case it's running through a reboot).

For climbing, I leave my phone on the ground and easily read it from 50 feet up. For swimming, I put the phone in a freezer bag, and stand it up by the edge of the pool. In foggy dark goggles, I can read it from several yards away, which is good enough for pacing. (Our pool only has one analog wall clock which I can't read in foggy goggles.) Having time feedback as I swam encouraged me to improve two of my swim times (maybe it helped with pacing, too).

Curiously, in an early version I found that there is some bug in the font renderer on my phone: at maximum size, some digits were left blank on the screen (perhaps nobody anticipated drawing text so large that if it had descenders they wouldn't fit on the screen). So I wrote a python script that used a font library to convert digits, colons and minus signs to java code with the Android Path class.

The landscape screenshot shows it running in fullscreen mode with the classic German DIN 1451 highway sign font (and aspect ratio preservation turned off for even bigger digits). The second is with a standard Android Roboto font.

Anselm's ontological argument

Here is my favorite version of the “existence is not a property” objection to Anselm’s first ontological argument.

It makes no sense to talk of the greatness of a nonexistent being except hypothetically as the greatness it would have if it existed. When we compare the greatness of things, we compare what the things would be like if they existed. Thus, when we say that Thor is greater than Hermes, we mean something like this: if Thor existed, he would be greater than Hermes would be if Hermes existed. And to exist is to exist in reality.

But now take the crucial claim in Anselm’s argument that it is greater for x to exist in mind and in reality than just in mind. This claim is simply false when we understand it in the hypothetical way. For we need to compare the greatness of the x that exists in mind and reality to the greatness that the x that exists only in mind would have if it existed in reality. But that’s the same greatness!

Anselm’s second argument makes no such slip, for it is based on a comparison between contingent and necessary existence, and that comparison survives the criticism.

Ethics and complexity

Here is a picture of ethics. We are designed to operate with a specific algorithm A for generating imperatives from circumstances. Unfortunately, we are broken in two ways: we don’t always follow the generated imperatives and we don’t always operate by means of A. We thus need to reverse engineer algorithm A on the basis of our broken functioning.

In general, reverse-engineering has to be based on a presumption of relative simplicity of the algorithm. However, Kantian, utilitarian ethics and divine command ethics go beyond that and hold that A is at base very simple. But should we think that the algorithm describing the normative operation of a human being is very simple? The official USA Fencing rule book is over 200 pages long. Human life is more complex than a fencing competition. Why should we think that there are fundamental rules for human life that can be encompassed briefly, from which all other rules can be derived without further normative input? It would be nice to find such brief rules. Many have a hope of finding analogous brief rules in physics.

We haven’t done well in ethics in our attempts to find such brief rules: the Kantian and utilitarian projects make (I would argue) incorrect normative claims, while the divine command project seems to give the wrong grounds for moral obligations.

It seems not unlikely to me that the correct full set of norms for human behavior will actually be very complex.

But there is still a hope for a unification. While I am dubious whether one can find a simple and elegant set of rules such that all ethical truths can be derived from them with no further normative input, there may be elegant unifying ethical principles that nonetheless require further normative input to generate the complex rules governing human life. Here are two such options:

• Natural Law: Live in accordance with your nature! But to generate the rules governing human life requires the further information as to what your nature requires, and that is normative information.

• Agapic ethics: Love everyone! But one of the things that are a part of love is adapting the form of one’s love to fit the the persons and circumstances (fraternal love for siblings, collegial love for colleagues, etc.), and the rules of “fit” are extremely complex and require further normative input.

Eleven varieties of contrastive explanation

In connection with free will, quantum mechanics or divine creation it is useful to talk about contrastive explanation. But there is no single generally accepted concept of contrastive explanation, and what one says about these topics varies depending on the chosen concept.

To that end, here is a collection of definitions of contrastive explanation. They all have this form:

• r contrastively explains why p rather than q if and only if r explains why (p and not q) and [insert any additional conditions].

They vary depending on the additional conditions to be inserted. Here are some options for these:

1. No additional conditions.

2. r makes p more likely than q.

3. r cannot explain q.

4. r wouldn’t explain q if q were true instead of p.

5. r wouldn’t explain q as well as it now explains p if q were true instead of p.

6. q wouldn’t be explained by r or by any proposition with r’s actual grounds if q were true instead of p.

7. q wouldn’t be explained by r or by any proposition with r’s actual grounds as well as r now explains p if q were true instead of p.

8. the conjunction of everything explanatorily prior to p makes p more likely than q.

9. r entails (p and not q).

10. r entails the truth of p.

11. r entails the falsity of q.

It is not possible to normally have contrastive explanations of indeterministic free choices or quantum events in senses 9–11, and probably sense 8, but it is possible (with an appropriately metaphysical theory of free choice or quantum events) in senses 1-7. As for the case of contingent divine creative decision, things depend on divine simplicity. Without divine simplicity, contrastive explanations are possible in senses 1–7. Interestingly, if divine simplicity is true, then it is not possible to have contrastive explanations of contingent divine creative decisions in senses 6 or 7.

In what I said above, I assumed that the explanandum cannot be a part of the explanans. If following Peter Railton one drops this condition, then contrastive explanation of all three phenomena (with or without divine simplicity) becomes possible in all the senses.

Lesson: When one talks about contrastive explanation, one needs to define one’s terms.

Acknowledgments: I am grateful to Christopher Tomaszewski for in-depth discussion that led me to recognize the important difference between 4–5 and 6–7. And the Railton point is basically due to a remark by Yunus Prasetya.

Substances are not parts of substances

Here is a quick and simple argument for the Aristotelian axiom that substances are not parts of substances.

1. The parts of substances are at least partly grounded in the substances.

2. Substances are not even partly grounded in other things.

3. Therefore, substances are not proper parts of other substances.

I suppose (1) is probably just as controversial as (3).

Aristotelian metaphysics and global physics

Too much of the contemporary ontological imagination is guided by the idea that the fundamental physical stuff in the world is discrete particles. Yet this is clearly dubious, since quantum mechanics (on non-Bohmian interpretations) suggests that the world is full of superpositions of states with different numbers of particles, while if discrete particles really exist, there had better be a well-defined number of them. Quantum mechanics instead suggests an ontology of the physical world where there is exactly one entity, “the Global Wavefunction”, whose physical state can be aptly represented as a vector in an infinite-dimensional vector space. And even if we didn’t have quantum mechanics’ vector-based approach on the table, we still wouldn’t be in an epistemic position to know that the right physics is based on particles rather than fields.

An ontology of material objects that composes these objects out of particles is held hostage to a particle-based physics that may well not be true. It would be best if one could work on the ontology of material objects without presupposing an answer to the question whether fundamental physical reality is field-like, vector-like or particle-like. I do not know if this is tenable. If it’s not, then the ontology of material objects needs to be done conditionally: If fundamental physical reality is of this sort, then material objects are like this.

Interestingly, some metaphysical problems may become easier given a non-particulate physical substratum. For instance, one of the hardest problems for a contemporary Aristotelian metaphysics has been the problem of what happens to particles that get incorporated into a substance, in light of the axiom that a substance cannot be composed of substances. But if we do not see fundamental physical reality as made of apparently substantial particles, the problem dissolves.

Today I want to sketch two Aristotelian approaches that take globalized vector- and field-approaches seriously. On the vector- and field-approaches, fundamental physical reality consists of a mere handful of entities: a single vector-like entity or several (hopefully no more than a dozen, and ideally only one) field-like entities. But being Aristotelian, we will think there are at least billions of substances: every organism is a substance. If these substances are to be related to fundamental physical entities, billions of them will have to be related to the same fundamental physical entities.

The ordinary substances on my stories will be organisms. There are billions of them. In addition to the ordinary substances, there are extraordinary substances: one for each of the handful of fundamental physical entities (fields or a vector).

My stories now diverge. On the first story, the billions of ordinary substances each encode and ground local features of the global fundamental physical entities. On a field version of the story, you encode and ground the features that the global fields have where you are located and your dog encodes and grounds the features that the global fields have where your dog is located (I am less clear on how to describe the vector version). This is not enough. For there aren’t enough organisms in the universe to ground all of the richness of the global fundamental physical entities: too much of the universe is lifeless. Thus, I propose that there are additional substances located where the organisms are not, and the features of these substances ground the rest of the features of the global fundamental physical entities. One way to run this story is to say that there is one of these additional substances per global fundamental physical entity, and each grounds the features of its corresponding global fundamental phsyical entity away from organisms. These additional substances are like swiss cheese, with the holes being filled with organisms like people and dogs.

On this version of the Aristotelian story—which can be varied in a number of ways—the global fundamental physical entities are not metaphysically fundamental. They are grounded in the many substances of the world.

On the second story, the global fundamental physical entities are substances. They are global substances. These global substances interact with the ordinary substances (there are many ways to spell out this interaction). We can now identify the matter of an ordinary substance x either with x’s powers and liabilities for interaction with the global substances or with the plurality of these global substances qua interacting with x.

There are many options here. Much detail to be worked out. Some options may be inferior to others, but I doubt in the end we will come to a single clearly best option.

A measure of sincerity

On a supervaluationist view of vagueness, a sentence such as “Bob is bald” corresponds to a large number of perfectly precise propositions, and is true (false) if and only if all of these propositions are true (false). This is plausible as far as it goes. But it seems to me to be very natural to add to this a story about degrees of truth. If Bob has one hair, and it’s 1 cm long, then “Bob is bald” is nearly true, even though some precisifications of “Bob is bald” (e.g., that Bob has no hairs at all, or that his total hair length is less than 0.1 cm) are false. Intuitively, the more precisifications are true, the truer the vague statement:

1. The degree of truth of a vague statement is the proportion of precisifications that are true.

But for technical reasons, (1) doesn’t work. First, there are infinitely many precisifications of “Bob is bald”, and most of the time the proportion of precisifications that are true will be ∞/∞. Moreover, not all precisifications are equally good. Let’s suppose we somehow reduce the precisifications to a finite number. Still, let’s ask this question: If Bob is an alligator is Bob bald? This seems vague, even though the precisifications of “Bob is bald” that require Bob to be the sort of thing that has hair seem rather better. But for any precisification that requires Bob to be a hairsute kind of thing, there is one that does not. And so if Bob is an alligator, he is bald according to exactly half of the precisifications, and hence by (1) it would be half-true that he is bald. And that seems too much: if Bob is an alligator, he is closer to being non-bald than bald.

A better approach seems to me to be this. A language assigns to each sentence s a set of precisifications and a measure m_s on this set with total measure 1 (i.e., technically a probability measure, but it does not represent chances or credences). The degree of truth of a sentence, then, is the measure of the subset of precisifications that are actually true.

Suppose now that we add to our story a probability measure P representing credences. Then we can form the interesting quantity E_P(m_s) where E_P is the expected value with respect to P. If s is non-vague, then E_P(m_s) is just our credence for s. Then E_P(m_s) is an interesting kind of “sincerity measure” (though it may not be a measure in the mathematical sense) that combines both how true a statement is and how sure we are of it. When E_P(m_s) is close to 1, then it is likely that s is nearly true, and when it is close to 0, then it is likely that s is nearly false. But when it is close to 1/2, there are lots of possibilities. Perhaps, s is nearly certain to be half-true, or maybe s is either nearly true or nearly false with probabilities close to 1/2, and so on.

This is not unlikely worked out, or refuted, in the literature. But it was fun to think about while procrastinating grading. Now time to grade.