Two more counterexamples to utilitarianism

It’s an innocent and pleasant pastime to multiply counterexamples to utilitarianism even if they don’t add much to what others have said. Thus, if utilitarianism is true, I have to do so. :-)

Suppose you capture Hitler. Torturing him to death would appal many but, given fallen human nature, likely significantly please hundreds of millions more. This pleasure to hundreds of millions could far outweigh the pain to one. Moreover, even of those appalled by the torture, primarily only Nazis and a handful of moral saints would actually feel significant displeasure at the torture. For being appalled by an immoral action is not always unpleasant except to someone with saintly compassion—indeed there is a kind of pleasure one takes in being appalled. Normally in the case of counterexamples to utilitarianism one worries about making people more callous, the breakdown of law and order, giving a bad example to others, and so on. But the case of Hitler is so exceptional that likely the negative effects from a utilitarian point of view would be minimal if any.

One might think that an even better thing to do from the utilitarian point of view would be to kill Hitler painlessly, and then mark up his body so it looks like he was tortured to death, and publically lie about it.

Yet it is wrong to torture even Hitler, and it is wrong to lie that one has done so (especially if only for public pleasure).

Bailey's Priority Principle

Andrew Bailey formulated and defended the Priority Principle (PP), that we think our thoughts in a primary rather than inherited way. His main argument for PP is a two-thinkers argument: if I think my thoughts in an inherited way, then something else—the thing I inherit the thoughts from—thinks them as well, but there aren’t two thinkers of my thoughts. While this argument is plausible, I think it skirts around the main intuition behind the PP. That intuition is that there is something implausible about us being thinkers in a derivative way. This intuition, however, is quite compatible with there being something that derives its thoughts from us, but not so Bailey’s argument, which (unless I am missing something) equally rules out the hypothesis that we inherit our thoughts and the hypothesis that our thoughts are inherited by something else.

Is there a way to argue for PP in concert with this intuition, namely to argue that whether or not there are two thinkers of my thoughts, I am their primary thinker? Such an argument would also escape the following apparent counterexample. Social organizations can have thoughts, derivative in a complex way from their members’ thoughts. But now suppose I join a club, and everyone else resigns membership. Then the club’s opinion on matters relavant to the club’s subject matter comes to be inherited from me. So now there are two thinkers, the club and me, though I am the primary one. This case (which to be fair I am not completely sure of) is a counterexample to Bailey’s argument but not to its conclusion.

My students came up with two closely related arguments, which we might put something like this. First, among our thoughts are intentions. If these are derivative, we are puppets of the primary intender, contrary to our freedom. Second, some of our thoughts are deliberate. It is a contradiction in terms that we think deliberately and yet our deliberate thought is inherited from a prior deliberate thinker—puppetry is incompatible with deliberativeness.

These arguments do not directly show that we are always primary thinkers, so they immediately imply only a weaker version of the PP (WPP), namely that sometimes we think non-derivatively. WPP is still interesting. For instance, it rules out standard perdurantist theories on which we inherit all our thoughts from our temporal parts. Furthermore, WPP makes PP moderately likely: for it is plausible that if there is any thought-inheritance it always goes in the same direction.

That said, maybe there is some reason to accept WPP without PP. Here is one kind of case. Possessing a concept is, perhaps, a way of thinking. But given some moderate semantic externalism, sometimes we possess a concept—say, of a quark—by inheriting it from an expert. Or suppose that the extended mind thesis is true, so that we count as knowing some things because they recorded on our devices. Maybe electronic devices don’t have knowledge, so this isn’t exactly knowledge inheritance. But imagine that you train a parrot to remember all your credit card numbers (a foolish idea) and you carry the parrot with you always. Now you inherit the knowledge of the numbers (under some description common between you and the parrot, definitely not “credit card number”) from the parrot. I am dubious of the extended mind thesis, but there is no need to stick one’s neck out. WPP does justice to many of our intuitions.

Correction to "Goodman and Quine's nominalism and infinity"

In an old post, I said that Goodman and Quine can’t define the concept of an infinite number of objects using their logical resources. Allen Hazen corrected me in a comment in the specific context of defining infinite sentences. But it turns out that I wasn’t just wrong about the specific context of defining infinite sentences: I was almost entirely wrong.

To see this, let’s restrict ourselves to non-gunky worlds, where all objects are made of simples. Suppose, further, that we have a predicate F(x) that says that an object x is finite. This is nominalistically and physicalistically acceptable by Goodman and Quine’s standards: it states a physical feature of a physical object, namely its size qua made of simples. (If the simples all have some finite amount of energy with some positive minimum, F(x) will be equivalent to saying x has a finite energy.)

Now, this doesn’t solve the problem by itself. To say that an object x is finite is not the same as saying that the number of objects with some property is finite. But I came across a cute little trick to go from one to the other in the proof of Proposition 7 of this paper. The trick transposed to the non-gunky mereological setting is this. Then following two statements are equivalent in non-gunky worlds satisfying appropriate mereological axioms:

1. The number of objects x satisfying G(x) is finite.

2. There is a finite object z such that for any objects x and y with G(x) and $G(y), if x ≠ y, then x and y differ inside z (i.e., there is a part of z that is a part of one object but not of the other).

To see the equivalence, suppose (2) is true. Then if z has n simples, and if x is any object satisfying G(x), then all objects y satisfying G(x) differ from x within these n simples, so there are at most 2ⁿ objects satisfying G(x). Conversely, if there are finitely many satisfiers of G, there will be a finite object z that contains a simple of difference between x and y for every pair of satisfiers x and y of G (where a simple of difference is a simple that is a part of one but not the other), and any two distinct satisfiers of G will differ inside z.

I said initially that I was almost entirely wrong. In thoroughly gunky worlds, all objects are infinite in the sense of having infinitely many parts, so a mereologically-based finiteness predicate won’t help. Nor will a volume or energy-based one, because we can suppose a gunky world with finite total volume and finite total energy. So Goodman and Quine had better hope that the world isn’t thoroughly gunky.

Property inheritance

There seems to be such a thing as property inheritance, where x inherits a property F from y which has F in a non-derivative way. Here are some examples of this phenomenon on various theories:

1. I inherit mass from my molecules.

2. A person inherits some of their thoughts from the animal that constitutes the person.

3. A four-dimensional whole inherits its temporary properties from its temporal parts.

These are all cases of upward inheritance: a thing inheriting a property from parts or constituent. There can, however, be downward inheritance.

4. When a whole has the property of belonging to you, so do its parts, and often the parts inherit the property of being owned from the whole, though not always (you can buy a famous chess set piece by piece).

There may also be cases of sideways inheritance.

5. A layperson possesses the concept of a quark by inheritance from an expert to whom they defer with respect to the concept.

There seems to be some kind of a logical connection between property inheritance and property grounding, but the two concepts are not the same, since x’s possession of a property can be grounded in y’s possession of a different property—say, a president’s being elected is grounded in voters’ electing—while inheritance is always of the same property.

It is tempting to say:

6. An object x inherits a property F from an object y if and only if x’s having F is grounded in y’s having the same property F.

That’s not quite right. For if p grounds q, then p entails q. But this bundle of molecules’ having mass may not not entail my having mass, since it might be a contingent feature of the bundle that they are my molecules, so there is a possible world where the bundle exists and has mass, but I don’t (if only because I don’t exist). It seems that what we need in (6) is something weaker than grounding. But partial grounding seems too weak to plug into an account of property inheritance. Consider my property of knowing something. One of my pieces of knowledge is that you know something. So my knowing something is partially grounded in your knowing something, but I do not think that this counts as property inheritance. (Suppose one bites the bullet and says that my knowing something is inherited from you. Then, oddly, I have the property of knowing something both by inheritance and not by inheritance—inherited and non-inherited property possession are now compatible. I don’t know if that’s right, but at least it’s odd.)

I think we can at least say:

7. An object x inherits a property F from an object y only if x’s having F is grounded in y’s having the same property F.

But I don’t know how to turn this into a necessary and sufficient condition.

Dualism, humans and galaxies

Here is a mildly interesting thing I just noticed: given dualism, we cannot say that we are a part of the Milky Way galaxy. For galaxies, if they exist at all, are material objects that do not have souls as parts.

Hachette v. Internet Archive

I am not a lawyer, but I love constructing counterexamples. I’ve been thinking about the Hachette v. Internet Archive. The Archive scanned a bunch of books they owned, and then lent the scans to users on the Internet, making sure that for each physical book, only one scan was lent at a time. The courts ruled that this was copyright infringement.

I imagine a sequence of cases for a library (the Internet Archive is officially a library in California):

1. A user comes to the library and reads a book in the ordinary way.

2. The book is delicate and valuable, so the library puts the book in a metal box with a window, with delicate robotically controlled page flippers controlled by buttons outside the box. The user reads the book through the window while flipping pages with the buttons.

3. Same as 2, but the user wears glasses.

4. Same as 2, but the user lives across the street from the library, and the librarian has placed the box in the library window so pointed that the user, and the user alone, can read the book via an ordinary refracting telescope, with the user having buttons connected to the page flippers via long wires.

5. Same as 4, but the telescope is digital: it has an optical sensor connected electronically to a screen (basically, a CCTV system).

6. Same as 5, but the optical sensor is inside the library while the digital telescope’s screen is in the user’s home.

7. Same as 6, but the user can be arbitrarily far away, because wires are very long.

8. Same as 7, but instead of custom wiring for the connection between the sensor and the screen and the user’s button’s and the page flippers, a TCP/IP protocol over the Internet is employed.

9. Same as 8, but to reduce wear and tear on the book, the sensor caches the images, so that if the user chooses to jump to a page that has already been viewed, the flippers do not need to operate. The cache is deleted at the end of the session.

10. Same as 9, but the cache is not deleted at the end of the session, but is kept for the next session with the same user.

11. Same as 10, but the cache is kept for the next user. Only one user can access the book at once.

12. Same as 10, but a full cache is generated for all the pages once and for all users. Only one user can operate the system at once.

13. Same as 11, but the whole cache is copied to the user’s device and removed once the loan period expires.

Here, 12 is pretty much what the Archive was doing when it was letting users download books for a period, and 11 was what they were doing when it was letting users view the books via a browser.

If 11 and 12 are not allowed, at what point do things become impermissible?

If the worry is about there being copying, then there is already copying at 5: the electrical data from the sensor is copied to the screen. If the user flips through the book, the copying is of the book as a whole, though the copies are constantly deleted. But it seems to me (who am not a lawyer) that we shouldn’t make a significant distinction between a digital and an analogue telescope. Moreover, even viewing by eye involves copying. As soon as a page of a book is exposed to light, a large stream of copies of the data in the book appear in the air as patterns of light. Whether these copies are in the air or in glass (as in the case where glasses or an analogue telescope is used) does not seem significant. Is it significant if the data is in electrons rather than photons, as in 5?

Perhaps the worry is about non-evanescent copies. That sounds reasonable. When a book is exposed to light, the copies that are immediately made are evanescent (though very large in quantity), and likewise if a sensor and a screen arrangement (e.g., CCTV) is used. Thus, 1-8 might be distinguished from 9-12, and maybe the caching introduced at 9 is the problem.

However, the Internet and other electronic devices already have caching built in at various levels, so there is already some “hidden” caching introduced at 9, so the existence of caching doesn’t seem that significant. It seems like caching is just an efficiency improvement that does not significantly affect the normative issues.

Dignity, ecosystems and artifacts

1. If a part of x has dignity, x has dignity.

2. Only persons have dignity.

3. So, a person cannot be a proper part of a non-person. (1–2)

4. A person cannot be a proper part of a person.

5. So, a person cannot be a proper part of anything. (3–4)

6. If any nation or galaxy or ecosystem exists, some nation, galaxy or ecosystem has a person as a proper part.

7. So, no nation, galaxy or ecosystem exists. (5–6)

Less confidently, I go on.

8. If tables and chairs exist, so do chess sets.

9. If chess sets exist, so do living chess sets.

10. A living chess set has persons as proper parts. (Definition)

11. So, living chess sets do not exist. (4,10)

12. So, tables and chairs don’t exist. (8–9,11)

All that said, I suppose (1) could be denied. But it would be hard to deny if one thought of dignity as a form of trumping value, since a value in a part transfers to the whole, and if it’s a trumping value, it isn’t canceled by the disvalue of other parts. (That said, I myself don’t quite think of dignity as a form of value.)

Pairs

As a warmup to his arguments against the existence of ordinary objects, Trenton Merricks argues against the existence of pairs of gloves.

Here’s another argument against pairs of gloves. I recently bought a pack of 200 nitrile gloves. How many pairs am I buying? Intuitively, there were a hundred pairs in the box. But if so, then we have have an odd question: For which distinct gloves of x and y in the box, do x and y in the box constitute a pair? If they all do, then there are 200⋅199/2 = 19,900 pairs in the box, while sure we would feel ripped off if the box said “19,900 pairs”.

Well, we might say this, starting at the top of the box: the first and second gloves are a pair, the third and fourth are a pair, and so on. But now suppose that something went wrong in the packing, and only 199 gloves went into the box (maybe that actually happened—I didn’t count). Then the box has 49 pairs, plus one more glove. But which of the gloves is the extra? Is it the bottom one, the top one, or some one in the middle? There seems to be no answer here.

Moreover, sometimes I only use one glove at a time. If so, then there is a 50% chance that at this point the next two gloves from the box that I put on aren’t actually a pair, and so when I put them on, I am not actually putting on a pair of gloves.

Perhaps, you say, all these difficulties stem from the fact that nitrile gloves do not have a left and right distinction. But suppose they did, and I got sent a messy box with 100 left gloves and 100 right gloves. Now, if every left glove and every right glove make a pair, there are 100⋅100 = 10,000 pairs, but it would be clearly a rip-off to label the box “10,000 pairs”: clearly, there would be 100 pairs. But now we would once again have the insuperable question of which left glove with which right glove makes a pair.

Maybe the problem disappears if one buys things by the single pair, as the “true pairs” are the ones one buys? I doubt it. If you saw me walking around today, you’d have said I was wearing a pair of black running shoes. But what happened was this: Some years back, I bought a pair of running shoes. The stitching on the right shoe gave out all too soon, and I patched it with a punctured bike inner tube (I save inner tubes that are themselves too far gone to keep patching, as they are useful for various projects), and wore it for another couple of months, but eventually gave in and got a second pair of the same make, model, size and color. After a year or two, I noticed that the left shoe on my newer pair was now more worn than the left shoe on my older pair (I didn’t throw the first pair out). And you can guess what I did: I started wearing the right shoe from the newer pair with the left shoe from the older pair. And that’s what I was wearing today. So, if the true pairs are as purchased, you would have been objectively wrong if you thought you saw me wearing a pair of shoes today: I was wearing two half-pairs. But this is absurd.

One might say: shoes become a pair when customarily worn together. But how many days do I need to wear them together for them to become a pair? And what if I bought two pairs of shoes of the same sort, and every morning randomly chose which left one and which right one to wear?

Perhaps the problems afflicting pairs don’t afflict more tightly bound artifacts. But I suspect it’s largely just a difference in vividness of the problem.

Continuous variation

Some arguments against restricted composition—the view that some but not all pluralities compose a whole—are based on the idea that a feature that cuts reality at the joints, such as composition, cannot be vague, and that if composition is restricted, one can have a continuous series of cases from a case of composition to a case of having lack of composition.

But now suppose I owe you ten dollars. Then there is a continuous series of cases where the amount I pay you ranges from zero to $20. The properties wrong, right and supererogatory cut nature at the joints. But as my payment moves from $9.99 to $10.00, it switches from wrong to right, and as it hits $10.01, it switches from merely right to supererogatory. So, one can have a case where the presence of joint-cutting features depends on something that varies continuously. And there is no vagueness: if I pay less than $10, I definitely wrong you; if I pay $10 or more, I definitely do right.

Modal details in Unger's argument against his existence

Unger famously argues that he doesn’t exist, by claiming a contradiction between three claims (I am quoting (1) and (2) verbatim, but simplifying (3)):

1. I exist.

2. If I exist, then I consist of many cells, but a finite number.

3. If I exist and I consist of many but a finite number of cells, then removal of the least important cell does not affect whether I exist.

Unger then says:

these three propositions form an inconsistent set. They have it that I am still here with no cells at all, even while my existence depends on cells. … One cell, more or less, will not make any difference between my being there and not. So, take one away, and I am still there. Take another away: again, no problem. But after a while there are no cells at all.

But taken literally this is logically invalid. Premise (2) says that I consist of many but a finite number of cells. But to continue applying premise (3), Unger needs that premise (2) would still be true no matter how many cells were taken away. But premise (2) does not say anything about hypothetical situations. It says that either I don’t exist, or I consist of a large but finite number of cells. In particular, there are no modal operators in (2).

Now, no doubt this is an uncharitable objection. Presumably (2) is not just supposed to be true in the actual situation, but in the hypothetical situations that come from repeated cell-removals. At the same time, we don’t want (2) to be ad hoc designed for this argument. So, probably, what is going on is that there is an implied necessity operator in (2), so that we have:

4. Necessarily, if I exist, then I consist of many cells, but a finite number.

The same issue applies to (3), since (3) needs to be applied over and over in hypothetical situations. Another issue with (3) is that to apply it over and over, we need to be told that removal of the cell is possible. So now we should say:

5. Necessarily, if I exist and I consist of many but a finite number of cells, then removal of the least important cell is possible and does not affect whether I exist.

Now, I guess, we can have a valid argument in S4.

Is this a merely technical issue here? I am not sure. I think that once we’ve inserted “Necessarily” into (4) and (5), our intuitions may start to shift. While (2) is very plausible if we grant the implied materialism, (4) makes us wonder whether there couldn’t be weird situations where I exist but don’t consist of many but a finite number of cells. First, it’s not obviously metaphysically impossible for me to grow an infinitely long tail? That, however, is a red herring. The argument can be retooled only to suppose that I necessarily have many cells and I actually have a finite number. But, second, and more seriously, is it really true that there is no possible world where I exist with only a few cells? In fact, perhaps, I once did exist with only a few cells in this world!

Similarly about (5). It’s clear that right now I can survive the loss of my least important cell. But it is far from clear that this is a necessary truth. It could well be metaphysically possible that I be reduced to some state of non-redundancy where every cell is necessary for my existence, where removal of any cell severs an organic pathway essential to life. I would be in a very different state in such a case than I am right now. But it’s far from clear that this is impossible.

Perhaps, though, the modality here isn’t metaphysical modality, but something like nomic modality. Maybe it’s nomically impossible for me to be in a state where every cell is non-redundant. Maybe, but even that’s not clear. And it’s also harder to say that the removal of the least important cell has to (in the nomic necessity sense) be nomically possible. Couldn’t it be that nomically the only way the least important cell could be removed would be by cutting into me in ways that would kill me?

Furthermore, once we’ve made our modal complications to the argument, it becomes clear that of the three contradictory premises (1), (4) and (5), premise (1) is by far the most probable. Premise (1) is a claim about my own existence, which seems pretty evident to me, and is only a claim about how things actually are now. Premises (4) and (5) depend on difficult modal details, on how things are in other worlds, and on metaphysical intuitions that are surely more fraught than those in the cogito.

(One of the things I’ve discovered by teaching metaphysics to undergraduates, with a focus on formulating logically valid arguments, is that sometimes numbered arguments in published work by smart people are actually quite some distance from validity, and it’s hard to see exactly how to make them valid without modal logic.)

Sexual symmetry and asymmetry

I want to think a bit about conservative Christian views of sex and gender, but before that I want to offer two stories to motivate a crucial distinction.

Electrons and Positrons

Electrons and positrons (a positron is a positively charged anti-particle to the electron) are very different in one way but not so much in another. If you take some system of electrons and positrons, and swap in a positron for an electron, the system will behave very differently—it will be attracted to the things that the electron was repelled by and vice versa. But if you replace all the electrons by positrons and all the positrons by electrons, it won’t make a significant difference (technically, there may be some difference due to the weak force, but that’s dominated by electromagnetic interaction). Similarly, a cloud of electrons behaves pretty much like a cloud of positrons, but a mixed cloud of electrons and positrons will behave very differently (electrons and positrons will collide releasing energy).

Electrons and positrons are significantly pairwise non-interchangeable, but globally approximately interchangeable.

We might conclude: electrons and positrons significantly differ relationally but do not differ much intrinsically.

On the other hand, if you have a system made of photons and electrons, and you swap out a photon and replace it by an electron, it will make a significant difference, but likewise typically if you swap out all the photons and electrons, it will also make a significant difference (unless the system was in a rare symmetric configuration). Thus, photons and electrons are significantly pairwise and globally interchangeable, and hence significantly differ both relationally and intrinsically.

Heterothallic Isogamous Organisms

Isogamous sexually-reproducing organisms have equally sized gametes among their sexes, and hence cannot be labeled as “female” and “male” (biologists define “female” and “male” in terms of larger and smaller gametes, respectively). Instead these sexes get arbitrarily labeled as plus and minus (I will assume there are only two mating types for simplicity). In heterothallic organisms, the sexes are located in different individuals, so two are needed for reproduction. Humans are heterothallic but not isogamous. But there are many species (mostly unicellular, I believe) that are heterothallic and isogamous.

We can now suppose a heterothallic and isogamous species with pretty symmetric mating roles. In such a species, again, we have significant individual non-interchangeability in a system. If Alice is a plus and Bob is a minus, they can reproduce, but if you swap out Bob for a plus, you get a non-reproductive pair. But if the mating roles are sufficiently similar, you can have global approximate interchangeability: if in some system you put pluses for the minuses and minuses for the pluses, things could go on much as before. A group of pluses may behave very much like a group of minuses (namely, over time the population will decrease to zero), but a mixed group of pluses and minuses is apt to behave very differently. We thus have pairwise non-interchangeability but approximate global interchangeability.

We might similarly say: pluses and minuses in our heterothallic and isogamous species significantly differ relationally but do not differ much intrinsically. On the other hand, cats and dogs significantly differ both relationally and intrinsically.

The Distinction

We thus have a distinction between two kinds of differences, which we can label as relational and intrinsic. I am not happy with the labels, but when I use them, please think of my two examples: particles and isogamous organisms. These two kinds of differences can be thought of as denying different symmetries: intrinsic differences are opposed to global interchange of the types of all individuals; relational differences are opposed to pairwise interchange of the types of a pair of individuals.

Conservative Christian Views of Sex and Gender

Conservative Christians tend to think that there are significant differences between men and women. In addition to cultural traits, there are two main theological reasons for thinking this:

1. Marriage asymmetry: Men and women can marry, but men cannot marry men and women cannot marry women.

2. Liturgical asymmetry: Only men can serve in certain liturgical “clerical” roles.

Of these, the marriage asymmetry is probably a bit more widely accepted than the liturgical asymmetry. (Some also think there is an authority asymmetry in the family where husbands have a special authority over wives. This is even more controversial among conservative Christians than the liturgical asymmetry, so I won’t say more about it.)

We could suppose an arbitrary divine rule behind both asymmetries. But this is theologically problematic: a really plausible way of reading the difference between the Law of Moses and the Law of the Gospel is at that in the Law of the Gospel, we no longer have arbitrary rules whose primary benefit is obedience, such as the prohibition on eating pork.

If we are to avoid supposing an arbitrary divine rule, we need to suppose differences between men and women to explain the theologically grounded asymmetries. And this is apt to lead conservative Christians to philosophical and theological theorizing about normative differences such as women being called more to “receptivity” and men more to “givingness”, or searching through sociological, psychological and biological data for relevant differences between the behavior and abilities of men and women. The empirical differences tend to lie on continua with wide areas of overlap between the sexes, however, and the normative differences are either implausible or likewise involve continua with wide areas of overlap (men, too, are called to receptivity).

But I think we are now in a position to see that there is a logical shortcoming behind the focus of this search. For differences between men and women can be relational or intrinsic, and the search has tended to focus on the intrinsic.

However, I submit, purely relational differences are sufficient to explain both the marriage and liturgical asymmetries. One way to see this is to pretend that we are a heterothallic isogamous species (rather than heterothallic anisogamous species that we actually are), consisting of pluses and minuses rather than females and males.

Then, if marriage has an ordering to procreation, that would neatly explain why pluses and minuses can marry each other, but pluses can’t marry pluses and minuses can’t marry minuses. No intrinsic difference between pluses and minuses is needed to explain this. Thus, as soon as we accept that marriage has an ordering to procreation, we have a way to explain the marriage asymmetry without any supposition of intrinsic differences.

Likewise, if there is going to be an incarnation, and only one, and the incarnate God is going to be incarnate as a typical organism of our species, then this incarnation must happen as a plus or a minus. And if married love is a deep and passionate love that is a wonderful symbol for the love between God and God’s people, then if the incarnation is as an individual of one of the sexes, God’s Church would then symbolically have the opposite sex. And then those whose liturgical role it is to stand in for the incarnate God in the marriage-like relationship to the Church would most fittingly have the sex opposite to that of the Church. Thus, if the incarnate God is incarnate as a plus, the Church would be figured as a minus, one can explain why it is fitting that the clergy in the relevant liturgical roles would be pluses; if the incarnate God is incarnate as a minus, we have an explanation of why the clergy in these roles would be minuses as well. (Interestingly, on this story, it’s not that the clergy are directly supposed to be like the incarnate God in respect of sex, but that their sex is supposed to be the opposite to that of the Church, and given that in the species there are only two sexes, this forces them to have the same sex as the incarnate God: the clergy need to have a sex opposed to the sex opposed to that of the incarnate God.)

Now, we are not isogamous, and we have female and male, not plus and minus. But we can still give exactly the same explanations. Even though in an anisogamous species there are significant intrinsic biological differences between the sexes, we need not advert to any of them to explain either the marriage or the liturgical asymmetry. The marriage asymmetry is tied to the pairwise non-interchangeability of the sexes and explained by the procreative role of marriage. The liturgical asymemtry is tied to the marriage asymmetry together with the symmetry-breaking event of God becoming incarnate in one of the sexes.

As far as this story goes, there need not be any morally significant intrinsic differences between male and female to explain the marital and liturgical asymmetries. The relational difference, that you need male and female for a mating pair, is morally significant on this story, but in a way that is entirely symmetric between male and female. And then we have one symmetry-breaking event: God becomes incarnate as a male. We need not think that there is any special reason why God becomes incarnate as a male or a female—it could equally well have been as a female. The decision whether to become incarnate as a male or a female could be as arbitrary as the decision about the exact eye color of the incarnate God (though, of course, eye color does not ground either significant intrinsic or significant relational differences). But if it were an incarnation as a female, other changes would be fitting: the clergy who symbolize the nuptial role of the incarnate God would fitting be female, in the exodus story it would fitting be female lambs and goats that would be sacrificed, and it would be fitting that Sarah be asked to sacrifice her first-born daughter.

I am not saying that there are no morally significant intrinsic differences between male and female. There may be. We are, after all, not only heterothallic but also anisogamous, and so there could turn out to be such intrinsic differences. But we need not suppose any such to explain the two asymmetries, and it is safer to be agnostic on the existence of these intrinsic differences.

Nothing in this post is meant as an argument for either the marriage asymmetry and the liturgical asymmetry. I have argued for the marriage asymmetry elsewhere, but here I am just saying that it could be explained if we grant the procreative ordering of marriage. And my arguments for the liturgical asymmetry are based on fittingness. But fittingness considerations do not constrain God. While we can explain why the clergy are of the same sex as the incarnate God by the nuptial imagery story that I gave above, God could instead have chosen to make the clergy be of the opposite sex as the incarnate God, in order to nuptially signify the people with the clergy, or God could chosen to make the clergy be of both sexes, to emphasize the fact that salvation is tied to the humanity (see St. Athanasius on this) and not the sex of the incarnate God. But when many things are fitting, God can choose one, and we can then cite its fittingness as a non-deterministic explanation.

Though, I suppose, I have at least refuted this argument:

3. The only way to explain the marriage and liturgical asymmetries is by supposing morally significant intrinsic differences between female and male.

4. There are no such intrinsic differences.

5. So, probably, the asymmetries don’t exist either.

I have refuted it by showing that (3) is false.

Against group intentional action

Alice, Bob and Carl are triumvirate that unanimously votes for some legislation, for the following reasons:

1. Alice thinks that hard work and religion are intrinsically bad while entertainment is intrinsically good, and believes the legislation will decrease the prevalence of hard work and religion and increase that of entertainment.

2. Bob thinks that hard work and entertainment are intrinsically bad while religion is intrinsically good, and believes the legislation will decrease the prevalence of hard work and entertainment and increase that of religion.

3. Carl thinks that religion and entertainment are intrinsically bad while hard work is intrinsically good, and believes the legislation will decrease the prevalence of religion and entertainment and increase that of hard work.

If groups engage in intentional actions, it seems that passing legislation is a paradigm of such intentional action. But what is the intention behind the action here?

When I first thought about cases like this, I thought they were a strong argument against group intentional action. But then I became less sure. For we can imagine an intrapersonal version. Suppose Debbie the dictator was given a card by a trustworthy expert that she was informed contains a truth, with the expert departing at that point. Before she could read it, however, she accidentally dropped the card in a garbage can. Reaching into the garbage can, she found three cards in the expert’s handwriting, two of them being mere handwriting exercises and one being the advice card:

1. Hard work and religion are intrinsically bad while entertainment is intrinsically good, and the legislation will decrease the prevalence of hard work and religion and increase that of entertainment.
2. Hard work and entertainment are intrinsically bad while religion is intrinsically good, and the legislation will decrease the prevalence of hard work and entertainment and increase that of religion.
3. Religion and entertainment are intrinsically bad while hard work is intrinsically good, and the legislation will decrease the prevalence of religion and entertainment and increase that of hard work.

Oddly, Debbie’s own prior views are so undecided that she just sets her credence to 1/3 for each of these propositions, and enacts the legislation. What is her intention?

But now I think there is a plausible answer: Debbie’s intention is to increase whichever one of the trio of entertainment, religion and hard work is good and decrease whichever two of them are bad.

Could we thus say that that is what the triumvirate intends? I am not sure. Nobody on the triumvirate has such an abstract intention.

So perhaps we still have an argument against group intentional action, of the form:

1. If there is group intentional action, the triumvirate acts intentionally.

2. Something only acts intentionally if it has an intention.

3. The triumvirate has no intention.

4. So, there is no group intentional action.

Reasons of identity

In paradigm instances of parental action, my reason for action is the objective fact that I am a parent, not because of the subjective fact that I think I'm a parent or identify with being a parent. There are times when it makes sense to act on the subjective fact. If I'm asked by someone (say, a counselor) whether I identify with being a parent, my answer needs to be based on the subjective fact that I so identify. But those are atypical cases. 

I suspect this is generally true: cases when one acts on what one is are primary and cases when one acts on what one identifies as are secondary. It is, thus, problematic to define any feature that is significantly rationally relevant to ordinary action in terms of what one identifies with. 

Goodman and Quine and transitive closure

In the previous post, I showed that Goodman and Quine’s counting method fails for objects that have too much overlap. I think (though the technical parts here are more difficult) that the same is true for their definition of the ancestral or transitive closure of a relation.

GQ showed how to define ancestors in terms of offspring. We can try to extend this definition to the transitive closure of any relation R over any kind of entities:

1. x stands in the transitive closure of R to y iff for every object u that has y as a part and that has as a part anything that stands in R to a part of u, there is a z such that Rxz and both x and z are parts of R.

This works fine if no relatum of R overlaps any other relatum of R. But if there is overlap, it can fail. For instance, suppose we have three atoms a, b and c, and a relation R that holds between a + b and a + b + c and between a and a + b. Then any object u that has a + b + c as a part has c as a part, and so (1) would imply that c stands in the transitive closure of R to a + b + c, which is false.

Can we find some other definition of transitive closure using the same theoretical resources (namely, mereology) that works for overlapping objects? No. Nor even if we add the “bigger than” predicate of GQ’s attempt to define “more”. We can say that x and y are equinumerous provided that neither is bigger than the other.

Let’s work in models made of an infinite number of mereological atoms. Write u ∧ v for the fusion of the common parts of both u and v (assuming u and v overlap), u ∨ v for the fusion of objects that are parts of one or the other, and u − v for the fusion of all the parts of u that do not overlap v (assuming u is not a part of v). Write |x| for the number of atomic parts of x when x is finite. Now make these definitions:

2. x is finite iff an atom is related to x by the transitive closure (with respect to the kind object) of the relation that relates an object to that object plus one atom.

3. Axyw iff x and y are finite and whenever x′ is equinumerous with x and does not overlap y, then x′ ∨ y is equinumerous with w. (This says |x| + |y| = |w|.)

4. Say that D_yuv iff A(u−y,u−y,v−y) (i.e., |v−y| = 2|u−y|) and either v does not overlap y or and u ∧ y is an atom or v and y overlap and u ∧ y consists of v ∧ y plus one atom. (This treats u and v as basically ordered pairs (u−y,u∧y) and (v−y,v∧y), and it makes sure that from the first pair to the second, the first component is doubled in size and the second component is decreased by one.)

5. Say that Q⁰yx iff y is finite and for some atom z not overlapping y we have y ∧ z related to something not overlapping x by the transitive closure of D_y. (This takes the pair (z,y), and applies the double first component and decrease second component relation described in (4) until the second component goes to zero. Thus, it is guaranteed that |x| = 2^|y|.)

6. Say that Qyx iff y is finite and Q⁰yx′ for some non-overlapping x′ that does not overlap y and that is equinumerous with x.

If I got all the details right, then Qyx basically says that |x| = 2^|y|.

Thus, we can define use transitive closure to define binary powers of finite cardinalities. But the results about the expressive power of monadic second-order logic with cardinality comparison say that we can only define semi-linear relations between finite cardinalities, which doesn’t allow defining binary powers.

Remark: We don’t need equinumerosity to be defined in terms of a primitive “bigger”. We can define equinumerosity for non-overlapping finite sets by using transitive closure (and we only need it for finite sets). First let T_yuv iff v − y exists and consists of u − y minus one atom and v ∧ y exists and consists of v ∧ y minus one atom. Then finite x and y are equinumerous₀ iff they are non-overlapping and x ∨ y has exactly two atoms or is related to an object with exactly two atoms by the transitive closure of T_yuv. We now say that x and y are equinumerous provided that they are finite and either x = y (i.e., they have the same atoms) or both x − y and y − x are defined and equinumerous₀.

No fix for Goodman and Quine's counting

In yesterday’s post, I noted that Goodman and Quine’s nominalist mereological definition of what it is to say that there are more cats than dogs fails if there are cats that are conjoint twins. This raises the question whether there is some other way of using the same ontological resources to generate a definition of “more” that works for overlapping objects as well.

I think the answer is negative. First, note that GQ’s project is explicitly meant to be compatible with there being a finite number of individuals. In particular, thus, it needs to be compatible with the existence of mereological atoms, individuals with no proper parts, which every individual is a fusion of. (Otherwise, there would have to be no individuals or infinitely many. For every individual has an atom as a part, since otherwise it has an infinite regress of parts. Furthermore, every individual must be a fusion of the atoms it has as parts, otherwise the supplementation axiom will be violated.) Second, GQ’s avail themselves of one non-mereological tool: size comparisons (which I think must be something like volumes). And then it is surely a condition of adequacy on their theory that it be compatible with the logical possibility that there are finitely many individuals, every individual is a fusion of its atoms and the atoms are all the same size. I will call worlds like that “admissible”.

So, here are GQ’s theoretical resources for admissible worlds. There are individuals, made of atoms, and there is a size comparison. The size comparison between two individuals is equivalent to comparing the cardinalities of the sets of atoms the individuals are made of, since all the atoms are the same size. In terms of expressive power, their theory, in the case of admissible worlds, is essentially that of monadic second order logic with counting, MSO(#), restricted to finite models. (I am grateful to Allan Hazen for putting me on to the correspondence between GQ and MSO.) The atoms in GQ correspond to objects in MSO(#) and the individuals correspond to (extensions of) monadic predicates. The differences are that MSO(#) will have empty predicates and will distinguish objects from monadic predicates that have exactly one object in their extension, while in GQ the atoms are just a special (and definable) kind of individual.

Suppose now that GQ have some way of using their resources to define “more”, i.e., find a way of saying “There are more individuals satisfying F than those satisfying G.” This will be equivalent to MSO(#) defining a second-order counting predicate, one that essentially says “The set of sets of satisfiers of F is bigger than the set of sets of satisfiers of G”, for second-order predicates F and G.

But it is known that the definitional power of MSO(#) over finite models is precisely such as to define semi-linear sets of numbers. However, if we had a second-order counting predicate in MSO(#), it would be easy to define binary exponentiation. For the number of objects satisfying predicate F is equal to two raised to the power of the number of objects satisfying G just in case the number of singleton subsets of F is equal to the number of subsets of G. (Compare in the GQ context: the number of atoms of type F is equal to two the power of the number of atoms of type G provided that the number of atoms of type F is one plus the number of individuals made of the atoms of type G.) And of course equinumerosity can be defined (over finite models) in terms of “more”, while the set of pairs (n,2ⁿ) is clearly not semi-linear.

One now wants to ask a more general question. Could GQ define counting of individuals using some other predicates on individuals besides size comparison? I don’t know. My guess would be no, but my confidence level is not that high, because this deals in logic stuff I know little about.

Goodman and Quine and shared bits

Goodman and Quine have a clever way of saying that there are more cats than dogs without invoking sets, numbers or other abstracta. The trick is to say that x is a bit of y if x is a part of y and x is the same size as the smallest of the dogs and cats. Then you’re supposed to say:

1. Every object that has a bit of every cat is bigger than some object that has a bit of every dog.

This doesn’t work if there is overlap between cats. Imagine there are three cats, one of them a tiny embryonic cat independent of the other two cats, and the other two are full-grown twins sharing a chunk larger than the embryonic cat, while there are two full-grown dogs that are not conjoined. Then a bit is a part the size of the embryonic cat. But (assuming mereological universalism along with Goodman and Quine) there is an object that has a bit of every cat that is no bigger than any object has a bit of every dog. For imagine an object that is made out of the embryonic cat together with a bit that the other two cats have in common. This object is no bigger than any object that has a bit of each of the dogs.

It’s easy to fix this:

2. Every object that has an unshared bit of every cat is bigger than some object that has an unshared bit of every dog,

where an unshared bit is a bit x not shared between distinct cats or distinct dogs.

But this fix doesn’t work in general. Suppose the following atomistic thesis is true: all material objects are made of equally-sized individisible particles. And suppose I have two cubes on my desk, A and B, with B having double the number of particles as A. Consider this fact:

4. There are more pairs of particles in A than particles in B.

(Again, Goodman and Quine have to allow for objects that are pairs of particles by their mereological universalism.) But how do we make sense of this? The trick behind (1) and (2) was to divide up our objects into equally-sized pieces, and compare the sizes. But any object made of the parts of all the particles in B will be the same size as B, since it will be made of the same particles as B, and hence will be bigger than any object made of parts of A.

Trope theory and merely numerical differences in pleasures

Suppose I eat a chocolate bar and this causes me to have a trope of pleasure. Given assentiality of origins, if I had eaten a numerically different chocolate bar that caused the same pleasure, I would have had had a numerically different trope of pleasure.

Now, imagine that I eat a chocolate bar in my right hand and it causes me to have a trope of pleasure R, and immediately as I have finished eating that one chocolate bar, I switch to eating the chocolate bar in my left hand, which gives me an exactly similar trope of pleasure, L, with no temporal gap. Nonetheless, by essentiality of origins, trope L is numerically distinct from trope R.

To some (perhaps Armstrong) this will seem absurd. But I think it’s exactly right. In fact, I think it may even an argument for trope theory. For it seems pretty plausible that as I switch chocolate bars, something changes in me: I go from one pleasure to another exactly like it. But on heavy-weight Platonism, there is no change: I instantiated pleasure and now I instantiate pleasure. On non-trope nominalism, likewise there is no change. It’s trope theory that gives us the change here.

Does one's vote make a difference?

Suppose that there is a simple majority election, with two candidates, and there is a large odd number of voters. Suppose polling data makes the election too close to call. How likely is it that you can decide which candidate wins?

I could look up this stuff, but it’s more fun to figure it out.

A quick and dirty model is this. We have N people other than you voting, each choosing between candidates A and B with probabilities p and 1 − p respectively. You don’t know what p and 1 − p are, but polling data tells you that p is between 1/2 − a and 1/2 + b for some positive numbers a and b. Your vote decides the election provided that exactly N/2 people vote for candidate A. This requires that N be even (if N is odd, at best you can decide between a candidate winning and the election being undecided, so you can’t decide which candidate wins), which has probability 1/2. Given that N = 2n is even, the probability that the other votes are exactly balanced is (a+b)⁻¹ C(2n,n)∫_1/2−a^1/2+bpⁿ(1−p)^n − 1dp, where C(m,n) is the binomial coefficient. Assuming n is large as compared to a and b, the integral can be approximated by replacing its bounds by 0 and 1 respectively, and some work with Mathematica shows that for large n the probability is approximately 1/(N(a+b)).

So what? Well, suppose you think that candidate A will on average make a person in the jurisdiction be u units of flourishing better off than candidate B will, and there are K persons, where K ≥ N + 1 (there are at least as many persons as candidates). So, the expected amount of difference that your voting for A will make is at least Ku/(2N(a+b)). This is at least u/(a+b). Thus, if the polling data gives you a range between 0.48 and 0.52 for the probability of a person’s preferring candidate A, and half of the people in the jurisdiction vote, the expected amount of difference that your vote makes is 25u. This is quite a lot if you think that which candidate wins makes a significant difference u per governed person.

Interestingly, some numerical work with Mathematica also shows that as number of people increases, then the expected amount of difference your vote makes also increases asymptotically, up to the limit of Ku/(2N(a+b)). So for larger jurisdictions, even though the probability of your vote making a difference is smaller, the expected difference from your vote is a bit bigger.

My quick and dirty model is not quite right. Of course, people don’t come to the polls and randomly choose whom to vote for. A more likely source of randomness has to do with who actually makes it to the polls (who gets sick, who has something come up, who decides it’s pointless to vote, etc.). A better model might be this. We have M people eligible to vote, of whom pM want to vote for A and (1−p)M want to vote for B. Some random subset of the M people then votes. My probabilist intuitions say that this is not that different from my model if the number of actual voters is, say, half of the eligible voters. If I had an election that I was eligible to vote in coming, I might try to figure our the more complex model, but I don’t.

Theology and source critical analysis

There is reason to think that a number of biblical texts—paradigmatically, the Pentateuch—were redacted from multiple sources that scholars have worked to tease apart and separately analyze. This is very interesting from a scholarly point of view. But I do not know that it is that interesting from the theological point of view.

Vatican II, in Dei Verbum, famously teaches:

since everything asserted by the inspired authors or sacred writers must be held to be asserted by the Holy Spirit, it follows that the books of Scripture must be acknowledged as teaching solidly, faithfully and without error that truth which God wanted put into sacred writings for the sake of salvation. … However, since God speaks in Sacred Scripture through men in human fashion, the interpreter of Sacred Scripture, in order to see clearly what God wanted to communicate to us, should carefully investigate what meaning the sacred writers really intended, and what God wanted to manifest by means of their words.

Presumably many other Christian groups hold something similar.

Now, in the case of a text put together from multiple sources, the question is who the “sacred writers” are. I want to suggest that in the case of such a text, the relevant “sacred writers” are the editors who put the texts together, and especially the ones responsible for a final (though this is a somewhat difficult to apply concept) version, and the intentions relevant to figuring out “What God wanted to communicate to us” are the intentions of the final layer of editing. The books in question, such as Genesis, are not anthologies. In an anthology, an editor has some purposes in mind for the anthologized texts, but the texts belong, often in a more or less acknowledged fashion, to the individual authors. The editorial work in putting the Biblical works together from source material is much more creative—it is genuine form of authorship—which is obvious from how much back-and-forth movement there is. Like in an anthology, we should not take the editor’s intentions to align with the intentions of the source material authors, but unlike in an anthology, the final work comes with the editor’s authority, and counts as the assertion of the editor, with the editor’s intentions being the ones that determine the meaning of the work.

If this is right, then I think we can only be fully confident of dealing with inspired teaching in the case of what the editors intend to assert through the final works. Writers typically draw on a multiplicity of sources, and need not be asserting what these sources meant in their original context—think of the ways in which a writer often repurposes a quote from another. Think here of how Homer draws upon a rich variety of fictional and nonfictional source material, but when he adapts them for inclusion in his work, the intentions relevant to “What the Iliad and Odyssey say” are Homer’s intentions.

If what we want to be sure of is “what God wanted to communicate to us”, then we should focus on the redactors’ intentions. In particular, when there is a tension in text between two pieces of source material, exegetically we should focus on what the editor meant to communicate to us by the choice to include material from both sources. (In a text without divine inspiration, we might in the end attribute a tension to editorial carelessness, but in fact scholars rarely make use of “carelessness” as an explanation for phenomena in great works of secular literature.) I think we should be open even to the logical possibility that the editor misunderstood what the source material meant to communicate, but it is the editor’s understanding that is normative for the interpretation of what the text as a whole is saying.

From a scholarly point of view, earlier layers in the composition process are more interesting. But I think that from a theological point of view, it is what the editor wanted to communicate that matters.

I don’t want to be too dogmatic about this, for three reasons. First, it is possible that the source material is an inspired text in its own right. But, I think, we typically don’t know that it is (though in a Christian context, an obvious exception is where the New Testament quotes Jesus’ inspired teaching). Second, it is possible for a writer or editor who has a deep respect for a piece of source material to include the text with the intention that the text be understood in the sense in which the original authors intended it to be understood, in which case the intentions of the authors of the source material may well be relevant. Third, this is not my field—I could be really badly confused.

An impartiality premise

In an argument that David Lewis’s account of possible worlds leads to inductive skepticism, I used this premise:

1. If knowing that x is F (where F is purely non-indexical and x is a definite description or proper name) does not epistemically justify inferring that x is G (where G is purely non-indexical), then neither does knowing x is F and that x is I (now, here, etc.: any pure indexical will do) justify inferring that x is G.

This is less clear to me now than it was then. Self-locating evidence might be a counterexample to this principle. I know that the tallest person in the world is the tallest person in the world. But suppose I now learn that I am the tallest person in the world. It doesn’t seem entirely implausible to think that at this point it becomes reasonable (or at least more reasonable) to infer that the number of people in the world is small. For on the hypothesis that the number of people is small, it seems more likely that I am the tallest than on the hypothesis that the number of people is large. (Compare: That I won some competition is evidence that the number of competitors was small.)

But I think I can fix my argument by using this premise:

2. If knowing that x is F (where F is purely non-indexical and x is a definite description or proper name) and that a uniformly randomly chosen person (or other occupied location) is x would not epistemically justify inferring that x is G (where G is purely non-indexical), then neither does knowing x is F and that x is I (now, here, etc.: any pure indexical will do) justify inferring that x is G.

There are multiple versions of (b) depending on how the random choice works, e.g., whether it is a random choice from among actual persons or from among possible persons (cf. self-sampling vs. self-indication).

It takes a bit of work to convince oneself that the rest of the argument still works.

A new kind of project

I did something new and fun this fall: I wrote a computer science paper. It's an analysis of the conditions under which a device equipped with a camera and an accelerometer can identify its position relative to two observed landmarks with known positions. Except for a measure zero set of singular cases with infinitely many solutions, there are always at most two solutions for device positions (this was previously known), and I found necessary and sufficient conditions for there to be a single solution. In particular, if the two landmarks are at the same altitude, there is always a single solution, unless the device is at the same altitude as the landmarks.

I implemented the algorithm on a phone (code here). In the screenshot, the markers 1 and 2 are landmarks, identified and outlined in green with OpenCV library code, and then the phone uses their positions and the accelerometer data to predict where the control markers 3 and 4 are on the screen, outlining them in red.

For someone like me who does some philosophy of science, it was an interesting experience to actually do a real experiment and collect data from it.

I am planning at some point to try to implement the algorithm using infrared LEDs under a TV and the accelerometer and infrared camera inside a right Nintendo Switch joycon. To that end, over the last couple of days I've reverse-engineered two of the joycon infrared camera blob identification modes.

Aristotelian sciences

There is an Aristotelian picture of knowledge on which all knowable things are divided exhaustively and exclusively into sciences by subject matter. This picture appears wrong. Suppose, after all, that p is a fact from one science—say, the natural science fact that water is wet—and q is a fact from another science—say, the anthropological fact that people pursue pleasure. Then the conjunction p and q does not belong to either of these science, or any other science.

One might cavil that a conjunction isn’t another fact over and beyond the conjuncts, that to say p and q is to say p and to say q. I am sceptical, but it’s easy to fix. Just replace my counterexample with something that isn’t a conjunction but is logically equivalent to it, say the claim that it’s not the case that either p or q is false.

Actual result utilitarianism implies a version of total depravity

Assume actual result utilitarianism on which there are facts of the matter about what would transpire given any possible action of mine, and an action is right just in case it has the best consequences.

Here is an interesting conclusion. Do something specific, anything. Maybe wiggle your right thumb a certain way. There are many—perhaps even infinitely many—other things you could have done (e.g., you could have wiggled the thumb slightly differently) instead of that action whose known consequences are no different from the known consequences of what you did. We live in a chaotic world where the butterfly principle very likely holds: even minor events have significant consequences down the road. It is very unlikely that of all the minor variants of what you did, all of which have the same known consequences, the variant you chose has the best overall consequences down the road. Quite likely, the variant action you chose is middle of the road among the variants.

So, typically, whatever we do, we do wrong on actual result utilitarianism.

There is no canonical way to define a regular comparative probability in terms of a full conditional probability

I claim that there is no general, straightforward and satisfactory way to define a total comparative probability with the standard axioms using full conditional probabilities. By a “straightforward” way, I mean something like:

1. A ≲ B iff P(A−B|AΔB) ≤ P(B−A|AΔB)

or:

2. A ≲ B iff P(A|A∪B) ≤ P(A|A∪B) (Pruss).

The standard axioms of comparative probability are:

3. Transitivity, reflexivity and totality.

4. Non-negativity: ⌀ ≤ A for all A

5. Additivity: If A ∪ B is disjoint from C, then A ≲ B iff A ∪ C ≲ B ∪ C.

A “straightforward” definition is one where the right-hand-side is some expression involving conditional probabilities of events definable in a boolean way in terms of A and B.

To be “satisfactory”, I mean that it satisfies some plausible assumptions, and the one that I will specifically want is:

6. If P(A|C) < P(B|C) where A ∪ B ⊆ C, then A < B.

Definitions (1) and (2) are straightforward and satisfactory in the above-defined senses, but (1) does not satisfy transitivity while (2) does not satisfy the right-to-left direction of additivity.

Here is a proof of my claim. If the definition is straightforward, then if A ≲ B, and A′ and B′ are events such that there is a boolean algebra isomorphism ψ from the algebra of events generated by A and B to the algebra of events generated by A′ and B′ such that ψ(A) = A′, ψ(B) = B′ and P(C|D) = P(ψ(C)|ψ(D)) for all C and D in the algebra generated by A and B, then A′ ≲ B′.

Now consider a full conditional probability P on the interval [0,1] such that P(A|[0,1]) is equal to the Lebesgue measure of A when A is an interval. Let A = (0,1/4) and suppose B is either (1/4,1/2) or (1/4, 1/2]. Then there is an isomorphism ψ from the algebra generated by A and B to the same algebra that swaps A and B around and preserves all conditional probabilities. For the algebra consists of the eight possible unions of sets taken from among A, B and [0,1] − (A∪B), and it is easy to define a natural map between these eight sets that swaps A and B, and this will preserve all conditional probabilities. It follows from my definition of straightforwardness that we have A ≲ B if and only if we have B ≲ B. Since the totality axiom for comparative probabilities implies that either A ≲ B or B ≲ A, so we must have both A ≲ B and B ≲ A. Thus A ∼ B. Since this is true for both choices of B, we have

7. (0,1/4) ∼ (1/4,1/2) ∼ (1/4, 1/2].

But now note that ⌀ < {1/2} by (3) (just let A = ⌀, B = {1/2} and C = {1/2}). The additivity axiom then implies that (1/4,1/2) < (1/4, 1/2], a contradiction.

I think that if we want to define a probability comparison in terms of conditional probabilities, what we need to do is to weaken the axioms of comparative probabilities. My current best suggestion is to replace Additivity with this pair of axioms:

8. One-Sided Additivity: If A ∪ B is disjoint from C and A ≲ B, then A ∪ C ≲ B ∪ C.

9. Weak Parthood Principle: If A and B are disjoint, then A < A ∪ B or B < A ∪ B.

Definition (2) satisfies the axioms of comparable probabilities with this replacement.

Here is something else going for this. In this paper, I studied the possibility of defining non-classical probabilities (full conditional, hyperreal or comparative) that are invariant under a group G of transformations. Theorem 1 in the paper characterizes when there are full conditional probabilities that are strongly invariant. Interesting, we can now extend Theorem 1 to include this additional clause:

6. There is a transitive, reflexive and total relation ≲ satisfying (4), (8) and (9) as well as the regularity assumption that ⌀ < A whenever A is non-empty and that is invariant under G in the sense that gA ∼ A whenever both A and gA are subsets of Ω.

To see this, note that if there is are strongly invariant full conditional probabilities, then (2) will define ≲ in a way that satisfies (vi). For the converse, suppose (vi) is true. We show that condition (ii) of the original theorem is true, namely that there is no nonempty paradoxical subset. For to obtain a contradiction suppose there is a non-empty paradoxical subset E. Then E can be written as the disjoint union of A₁, ..., A_n, and there are g₁, ..., g_n in G and 1 ≤ m < n such that g₁A₁, ..., g_mA_m and g_m + 1A_m + 1, ..., g_nA_n are each a partition of E.

A standard result for additive comparative probabilities in Krantz et al.’s measurement book is that if B₁, ..., B_n are disjoint, and C₁, ..., C_n are disjoint, with B_i ≲ C_i for all i, then B₁ ∪ ... ∪ B_n ≲ C₁ ∪ ... ∪ C_n. One can check that the proof only uses One-Sided Additivity, so it holds in our case. It follows from G-invariance that A₁ ∪ ... ∪ A_m ∼ E ∼ A_m + 1 ∪ ... ∪ A_n. Since E is the disjoint union of A₁ ∪ ... ∪ A_m with A_m + 1 ∪ ... ∪ A_n, this violates the Weak Parthood Principle.

Restricted composition and laws of nature

Ted Sider famously argues for the universality of composition on the grounds that:

1. If composition is not universal, then one can find a continuous series of cases from a case of no composition to a case of composition.

2. Given such a continuous series, there won’t be any abrupt cut-off in composition.

3. But composition is never vague, so there would have to be an abrupt cut-off.

Consider this argument that every velocity is an escape velocity:

4. If it’s not the case that every velocity is an escape velocity from a spherically symmetric body of some fixed size and mass, then one can find a continuous series of cases from a case of insufficiency to escape to a case of sufficiency to escape.

5. Given such a continuous series, there won’t be any abrupt cut-off in escape velocity.

6. But escape velocity is never vague, so there would have to be an abrupt cut-off.

It’s obvious that we should deny (5). There is an abrupt cut-off in escape velocity, and there is a precise formula for what it is: (2GM/r)^1/2 where G is the gravitational constant, M is the mass of the spherical body, and r is its radius. As the velocity of a projectile gets closer and closer to the (2GM/r)^1/2, the projectile goes further and further before turning back. When the velocity reaches (2GM/r)^1/2, the projectile goes out forever. There is no paradox here.

Why think that composition is different from escape velocity? Why not think that just as the laws of nature precisely specify when the projectile can escape gravity, they also precisely specify when a bunch of objects compose a whole?

My suspicion is that the reason for thinking the two are different is thinking that composition is something like a “logical” or maybe “metaphysical” matter, while escape is a “causal” matter. Now, universalists like David Lewis do tend to think that the whole is a free lunch, nothing but the “sum of the parts”, in which case it makes sense to think that composition is not something for the laws of nature to specify. But if we are not universalists, then it seems to me that it is very natural to think of composition in a causal way: when a proper plurality of xs are arranged a certain way, they cause the existence of a new entity y that stands in a composed-by relation to the xs, just as when a projectile has a certain velocity, that causes the projectile to escape to infinity.

Some may be bothered by the fact that laws of nature are often taken to be contingent, and so there would be a world with the same parts as ours but different wholes. That would bother one if one thinks that wholes are a free lunch. But if we take wholes seriously, it should no more bother us than a world where particles behave the same way up to time t₁, and then behave differently after t₁ because the laws are different.

Humeans have good reason to reject the above view, though. If the laws of composition are to match our intuitions about composition, they are likely to be extremely complex, and perhaps too complex to be part of the best system defining the laws on a Humean account of laws. But if we are not Humeans about laws, and think the simplicity of laws is merely an epistemic virtue, the explanatory power of laws of composition might make it reasonable to accept very complex such laws.

That said, we all have reject the simple causal version of the above view, where a proper plurality composing a whole causes the whole’s existence. For instance, I am composed by a plurality of parts that includes my hair, but my hair is not a cause of my existence: I would have just as much existed had I never developed hair. So a more complex version of the causal view is needed: initial parts (maybe the DNA in the zygote that I started as) causally contribute to the existence of the whole, but the causal relation runs in a different direction with respect to later parts, like teeth: perhaps I and my teeth together cause the teeth to be parts of me.

(I don’t endorse the more complex causal view either. I prefer, but still do not endorse, an Aristotelian alternative: when y is in a certain condition, it causes the existence of all of the parts. This is much neater because the causation always runs in the same direction.)

More on full conditional probabilities and comparative probabilities

I claim that there is no general, straightforward and satisfactory way to define a total comparative probability with the standard axioms using full conditional probabilities. By a “straightforward” way, I mean something like:

1. A ≲ B iff P(A−B|AΔB) ≤ P(B−A|AΔB)

or:

2. A ≲ B iff P(A|A∪B) ≤ P(B|A∪B).

The standard axioms of comparative probability are:

3. Transitivity, reflexivity and totality.

4. Non-negativity: ⌀ ≤ A for all A

5. Additivity: If A ∪ B is disjoint from C, then A ≲ B iff A ∪ C ≲ B ∪ C.

A “straightforward” definition is one where the right-hand-side is some expression involving conditional probabilities of events definable in a boolean way in terms of A and B.

To be “satisfactory”, I mean that it satisfies some plausible assumptions, and the one that I will specifically want is:

6. If P(A|C) < P(B|C) where A ∪ B ⊆ C, then A < B.

Definitions (1) and (2) are straightforward and satisfactory in the above-defined senses, but (1) does not satisfy transitivity while (2) does not satisfy the right-to-left direction of additivity.

Here is a proof of my claim. If the definition is straightforward, then if A ≲ B, and A′ and B′ are events such that there is a boolean algebra isomorphism ψ from the algebra of events generated by A and B to the algebra of events generated by A′ and B′ such that ψ(A) = A′, ψ(B) = B′ and P(C|D) = P(ψ(C)|ψ(D)) for all C and D in the algebra generated by A and B, then A′ ≲ B′.

Now consider a full conditional probability P on the interval [0,1] such that P(A|[0,1]) is equal to the Lebesgue measure of A when A is an interval. Let A = (0,1/4) and suppose B is either (1/4,1/2) or (1/4, 1/2]. Then there is an isomorphism ψ from the algebra generated by A and B to the same algebra that swaps A and B around and preserves all conditional probabilities. For the algebra consists of the eight possible unions of sets taken from among A, B and [0,1] − (A∪B), and it is easy to define a natural map between these eight sets that swaps A and B, and this will preserve all conditional probabilities. It follows from my definition of straightforwardness that we have A ≲ B if and only if we have B ≲ B. Since the totality axiom for comparative probabilities implies that either A ≲ B or B ≲ A, so we must have both A ≲ B and B ≲ A. Thus A ∼ B. Since this is true for both choices of B, we have

7. (0,1/4) ∼ (1/4,1/2) ∼ (1/4, 1/2].

But now note that ⌀ < {1/2} by (3) (just let A = ⌀, B = {1/2} and C = {1/2}). The additivity axiom then implies that (1/4,1/2) < (1/4, 1/2], a contradiction.

I think that if we want to define a probability comparison in terms of conditional probabilities, what we need to do is to weaken the axioms of comparative probabilities. My current best suggestion is to replace Additivity with this pair of axioms:

8. One-Sided Additivity: If A ∪ B is disjoint from C and A ≲ B, then A ∪ C ≲ B ∪ C.

9. Weak Parthood Principle: If A and B are disjoint, then A < A ∪ B or B < A ∪ B.

Definition (2) satisfies the axioms of comparable probabilities with this replacement.

Here is something else going for this. In this paper, I studied the possibility of defining non-classical probabilities (full conditional, hyperreal or comparative) that are invariant under a group G of transformations. Theorem 1 in the paper characterizes when there are full conditional probabilities that are strongly invariant. Interesting, we can now extend Theorem 1 to include this additional clause:

6. There is a transitive, reflexive and total relation ≲ satisfying (4), (8) and (9) as well as the regularity assumption that ⌀ < A whenever A is non-empty and that is invariant under G in the sense that gA ∼ A whenever both A and gA are subsets of Ω.

To see this, note that if there is are strongly invariant full conditional probabilities, then (2) will define ≲ in a way that satisfies (vi). For the converse, suppose (vi) is true. We show that condition (ii) of the original theorem is true, namely that there is no nonempty paradoxical subset. For to obtain a contradiction suppose there is a non-empty paradoxical subset E. Then E can be written as the disjoint union of A₁, ..., A_n, and there are g₁, ..., g_n in G and 1 ≤ m < n such that g₁A₁, ..., g_mA_m and g_m + 1A_m + 1, ..., g_nA_n are each a partition of E.

A standard result for additive comparative probabilities in Krantz et al.’s measurement book is that if B₁, ..., B_n are disjoint, and C₁, ..., C_n are disjoint, with B_i ≲ C_i for all i, then B₁ ∪ ... ∪ B_n ≲ C₁ ∪ ... ∪ C_n. One can check that the proof only uses One-Sided Additivity, so it holds in our case. It follows from G-invariance that A₁ ∪ ... ∪ A_m ∼ E ∼ A_m + 1 ∪ ... ∪ A_n. Since E is the disjoint union of A₁ ∪ ... ∪ A_m with  A_m + 1 ∪ ... ∪ A_n, this violates the Weak Parthood Principle.