Having multiple sufficient causes

It would be useful for discussions of causal exclusion arguments for physicalism to have a full taxonomy of the kinds of cases in which one effect E can have two sufficient causes C₁ and C₂.

Here is my tentative list of the cases:

1. Overdetermination: C₁ and C₂ overdetermine E

2. Chaining: C_i sufficiently causes C_j which sufficiently causes E (where i = 1 and j = 2 or i = 2 and j = 1)

3. Constitution: C_i sufficiently causes E by being partly constituted by C_j which sufficiently causes E (where i = 1 and j = 2 or i = 2 and j = 1)

4. Parthood: C_i sufficiently causes E by having the part C_j which sufficiently causes E (where i = 1 and j = 2 or i = 2 and j = 1).

If parthood is a special case of constitution, then (4) is a special case of (3). Moreover (2)–(4) are all species cases of:

5. Instrumentality: C_i sufficiently causes E by means of C_j sufficiently causing E (where i = 1 and j = 2 or i = 2 and j = 1).

Note that the above cases are not mutually exclusive. We can, for instance, imagine a case where we have both chaining and overdetermination. Let’s say I aim a powerful heat gun at a snowball. Just in front of the snowball is a stick of dynamite. The heat melts the snowball. But it also triggers an explosion which blows the snowball apart. Thus, we have overdetermination of the destruction of the snowball by two causes: heat and explosion. However, we also have chaining because the heat causes the explosion.

I wonder if we can come up with an argument that (1)–(4), or maybe (1) and (5), are the only options. That seems right to me.

Arguing for divine simplicity

I want to defend this argument:

1. If God is not simple, then some of God’s parts are creatures.

2. If some of the parts of x are creatures, then x is partly a creature.

3. God is not even partly a creature.

4. So, God is simple.

I think (2) is very plausible. Premise (3) follows from the transcedence of God.

That leaves premise (1) to argue for. Here is one argument:

5. If God is not simple, then God has a part that is not God.

6. Anything that is not God is a creature.

7. So, if God is not simple, then God has a part that is a creature.

Premise (5) is true by definition of “simple”. Premise (6) follows from the doctrine of creation: God creates everything other than God.

But perhaps one doesn’t believe the full doctrine of creation, but only thinks that contingent things are created. I think we can still argue as follows:

8. If God is not simple, then God has contingent parts that are not God.

9. Anything contingent that is not God is a creature.

10. So, if God is not simple, then God has a part that is not God.

Why think (8) is true? Well, let’s think about the motivations for denying divine simplicity. The best reasons to deny divine simplicity are considerations about God’s contingent intentions or God’s contingent knowledge, and the idea that these have to constitute proper parts of God. But that yields contingent parts of God.

Now, what if one rejects even the weaker doctrine of creation in (9)? Then I can argue as follows:

11. If God is not simple, then God’s contingent thoughts are proper parts of God.

12. God is contingently the cause of each of his contingent thoughts.

13. Anything that God is contingently the cause of is a creature of God.

14. So, if God is not simple, then God has a part that is not God.

Again, the idea behind (11) is that it flows from the best motivations for denying divine simplicity.

In search of real parthood

In contemporary mereology, it is usual to have two parthood relations: parthood and proper parthood. On this orthodoxy, it is trivially true that each thing is a part of itself and that nothing can be a proper part of itself.

I feel that this orthodoxy has failed to identify the truly fundamental mereological relation.

If it is trivial that each thing is a part of itself, then that suggests that parthood is a disjunctive relation: x is a part of y if and only if x = y or x is a part^* of y, where parthood^* is a more fundamental relation. But what then is parthood^*? It is attractive to identify it with proper parthood. But if we do that, we can now turn to the trivial claim that nothing can be a proper part of itself. The triviality of this claim suggests that proper parthood is a conjunctive property, namely a conjunction of distinctness with some parthood^† relation. And on pain of circularity, parthood^† is not just parthood.

In other words, I find it attractive to think that there is some more fundamental relation than either of the two relations of contemporary mereology. And once we have that more fundamental relation, we can define contemporary mereological parthood as the disjunction of the more fundamental relation with identity and contemporary mereological proper parthood as the conjunction of the more fundamental relation with distinctness.

But I am open to the possibility that the more fundamental relation just is one of parthood and proper parthood, in which case the claim that everything is a part of itself or the claim that nothing is a part of itself is respectively non-trivial.

I will call the more fundamental relation “real parthood”. It is a relation that underlies paradigmatic instances of proper parthood. And now genuine metaphysical questions open up about identity, distinctness and real parthood. We have three possibilities:

1. Necessarily, each thing is a real part of itself.

2. Necessarily, nothing is a real part of itself.

3. Possibly something is a real part of itself and possibly something is not a real part of itself.

If (1) is true, then real parthood is necessarily coextensive with contemporary mereological parthood. If (2) is true, then real parthood is necessarily coextensive with contemporary mereological proper parthood.

My own guess is that if there is such a thing as parthood at all, then (3) is true.

For the more fundamental a relation, the more I want to be able to recombine where it holds. Why shouldn’t God be able to induce the relation between two distinct things or refuse to induce it between a thing and itself? And it’s really uncomfortable to think that whatever the real parthood relation is, God has to be in that relation to himself.

Perhaps, though, the real parthood relation is a kind of dependency relation. If so, then since nothing can be dependent on itself, we couldn’t have a thing being a real part of itself, and real parthood would be coextensive with proper parthood.

All this is making me think that either real parthood is necessarily coextensive with proper parthood, or it is not necessarily coextensive with either of the two relations of contemporary mereology.

Inclusive vs. proper parthood

Contemporary analytic philosophers seem to treat the “inclusive” concept of parthood, on which each object counts as an improper part of itself, as if it were more fundamental than the concept of proper parthood.

It seems to me that we should minimize the number of fundamental relations that all objects have to stand in. We are stuck with identity: every object is identical with itself. But anything beyond that we should avoid as much as we can.

Now, it is plausible that whatever parthood relation—inclusive parthood or proper parthood—is the more fundamental of the two is in fact a fundamental relation simpliciter. For it is unlikely that parthood can be defined in terms of something else. But if we should minimize the number of fundamental relations that all objects must stand in, then it is better to hold that proper parthood rather than inclusive parthood is a fundamental relation. For every object has to stand in inclusive parthood to itself. But it is quite possible to have objects that are not proper parts of anything else.

On this view, proper parthood will be a fundamental relation, and improper parthood is just the disjunction of proper parthood with identity.

Sameness without identity

Mike Rea’s numerical-sameness-without-identity solution to the problem of material constitution holds that the statue and the lump have numerical sameness but do not have identity. Rea explicitly says that numerical sameness implies sharing of all parts but not identity.

Does Rea here mean: sharing of all parts, proper or improper? It had better not be so. For improper parthood is transitive.

Proposition. If improper parthood is transitive and x and y share all their parts (proper and improper), then x = y.

Proof: But suppose that x and y share all parts. Then since x is a part of x, x is a part of y, and since y is a part of y, y is a part of x. Moreover, if x ≠ y, then x is a proper part of y and y is a proper part of x. Hence by transitivity, x would be a proper part of x, which is absurd, so we cannot have x ≠ y. □

So let’s assume charitably that Rea means the sharing of all proper parts. This is perhaps coherent, but it doesn’t allow Rea to preserve common sense in Tibbles/Tib cases. Suppose Tibbles the cat loses everything below the neck and becomes reduced to a head in a life support unit. Call the head “Head”. Then Head is a proper part of Tibbles. The two are not identical: the modal properties of heads and cats are different. (Cats can have normal tails; heads can’t.) This is precisely the kind of case where Rea’s sameness without identity mechanism should apply, so that Head and Tibbles are numerically the same without identity. But Tibbles has Head as a proper part and Head does not have Head as a proper part. But that means Tibbles and Head do not share all their proper parts.

Here may be what Rea should say: if x and y are numerically the same, then any part of the one is numerically the same as a part of the other. This does, however, have the cost that the sharing-of-parts condition now cannot be understood by someone who doesn’t already understand sameness without identity.

An Aristotelian account of proper parthood (for integral parts)

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

Shape and parts

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

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

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

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

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

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

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

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

Gunk, etc.

If we think parts are explanatorily prior to wholes, then gunky objects—objects which have parts but no smallest parts—involve a vicious explanatory regress. But if one takes the Aristotelian view that wholes are prior to parts, then the regress involved in gunky objects doesn’t look vicious at all: the whole is prior to some parts, these parts are prior to others, and so on ad infinitum. It’s just like a forward causal regress: today’s state causes tomorrow, tomorrow’s causes the next day’s, and so on ad infinitum.

On the other hand, on the view that parts are explanatorily prior to wholes, upward compositional regresses are unproblematic: the head is a part of the cow, the cow is a part of the earth, the earth is a part of the solar system, the solar system is a part of the Orion arm, the Orion arm is a part of the Milky Way, the Milky Way is a part of the Local Group, and this could go on forever. The Aristotelian, on the other hand, has to halt upward regresses at substances, say, cows.

This suggests that nobody should accept an ontologically serious version of the Leibniz story on which composition goes infinitely far both downward and upward, and that it is fortunate that Leibniz doesn’t accept an ontologically serious version of that story, because only the monads and their inner states are to be taken ontologically seriously. But that's not quite right. For there is a third view, namely that parthood does not involve either direction of dependence: neither do parts depend on wholes nor do wholes depend on parts. I haven't met this view in practice, though.

Fundamental mereology

It is plausible that genuine relations have to bottom out in fundamental relations. E.g., being a blood relative bottoms out in immediate blood relations, which are parenthood and childhood. It would be very odd indeed to say that a is b’s relative because a is c’s relative and c is b’s relative, and then a is c’s relative because a is d’s relative and d is c’s relative, and so on ad infinitum. Similarly, as I argued in my infinity book, following Rob Koons, causation has to bottom out in immediate causation.

If this is right, then proper parthood has to bottom out in what one might call immediate parthood. And this leads to an interesting question that has, to my knowledge, not been explored much: What is the immediate parthood structure of objects?

For instance, plausibly, the big toe is a part of the body because the big toe is a part of the foot which, in turn, is a part of the body. And the foot is a part of the body because the foot is a part of the leg which, in turn, is a part of the body. But where does it stop? What are the immediate parts of the body? The head, torso and the four limbs? Or perhaps the immediate parts are the skeletal system, the muscular system, the nervous system, the lymphatic system, and so on. If we take the body as a complex whole ontologically seriously, and we think that proper parthood bottoms out in immediate parthood, then there have to be answers to such questions. And similarly, there will then be the question of what the immediate parts of the head or the nervous system are.

There is another, more reductionistic, way of thinking about parthood. The above came from the thought that parthood is generated transitively out of immediate parthood. But maybe there is a more complex grounding structure. Maybe particles are immediately parts of the body and immediately parts of the big toe. And then, say, a big toe is a part of the body not because it is a part of a bigger whole which is more immediately a part of the body, but rather a big toe is a part of the body because its immediate parts are all particles that are immediately parts of the body.

Prescinding from the view that relations need to bottom out somewhere, we should distinguish between fundamental parts and fundamental instances of parthood. One might have one without the other. Thus, one could have a story on which we are composed of immediate parts, which are composed of immediate parts, and so on ad infinitum. Then there would be fundamental instances of the parthoood relation—they obtain between a thing and its immediate parts—but no fundamental parts. Or one could have a view with fundamental parts while denying that there are any fundamental instances of parthood.

In any case, there is clearly a lot of room for research in fundamental mereology here.

Mereological perfection

1. Every part of God is perfect.

2. Only God is perfect.

3. So, every part of God is God.

4. So, God has no proper parts (parts that aren’t himself).

5. So, divine (mereological) simplicity is true.

A reductive account of parthood in terms of causal powers

Analytic philosophers like to reduce. But not much work has been done on reduction of parthood. Here’s an attempt, no doubt a failure as most reductive accounts are. But it’s worth trying.

Suppose that necessarily everything has causal powers. Then we might be able to say:

1. x is a part of y if and only if every (token) causal power of x is a causal power of y.

Some consequences:

2. Transitivity

3. Reflexivity

4. If nothing other than x shares a token causal power with x, then x is mereologically simple and does not enter into composition. Plausibly nothing shares a token causal power with God, so it follows that God is mereologically simple and does not enter into composition.

How does this work for hard cases where parthood is controversial?

Suppose I lose a leg and get a shiny green prosthesis. If the prosthesis is a part of me, then the prosthesis’ power of reflecting green light is a power I have. It seems about as hard to figure out whether the power of reflecting green light is a power that I have as it is to figure out whether the prosthesis is a part of me. So here it is of little help.

Suppose I am plugged into a room-size heart-lung machine. Is the machine a part of me? Well, the machine has the power of crushing people by its weight. It seems intuitively right to say that by being plugged into that machine, I have not acquired the power of crushing people. So it seems that it’s not a part of me.

Is a fetus a part of the mother? Here, maybe the story is some help. The fetus eventually acquires certain powers of consciousness. These do not seem to be powers of the mother—she can be conscious while the fetus is awake. So, once consciousness is acquired, the fetus is not a part of the mother. But earlier, the fetus as the power to acquire these instances of consciousness, and the mother does not seem to, so earlier, too, the fetus does not seem to be a part of the mother. Here the story is of some help, maybe.

However, one doesn’t need all of (1) for some of the applications. The “only if” part of in (1) is sufficient for the heart-lung machine and pregnancy cases.

Eliminating or reducing parthood

Parthood is a mysterious relation. It would really simplify our picture of the world if we could get rid of it.

There are two standard ways of doing this. The microscopic mereological nihilist says that only the fundamental “small” bits—particles, fields, etc.—exist, and that there are no complex objects like tables, trees and people that are made of such bits. (Though one could be a microscopic mereological nihilist dualist, and hold that people are simple souls.)

The macroscopic mereological nihilist says that big things like organisms do exist, but their commonly supposed constituents, such as particles, do not exist, except in a manner of speaking. We can talk as if there were electrons in us, but there are no electrons in us. The typical macroscopic mereological nihilist is a Thomist who talks of “virtual existence” of electrons in us.

Both the microscopic and macroscopic nihilist get rid of parthood at the cost of ridding themselves of large swathes of objects that common sense accepts. The microscopic nihilist gets rid of the things that are commonly thought to be wholes. The macroscopic nihilist gets rid of the things that are commonly thought to be parts.

But there is a third way of getting rid of parthood that has not been sufficiently explored. The third kind of mereological nihilist would neither deny the existence of things commonly thought to be wholes nor of things commonly thought to be parts. Instead, she would deny the parthood relation that is commonly thought to hold between the micro and the macro things. Parts of the space occupied by me are also occupied by my arms, my legs, my heart, the electrons in these, etc. But these things are not parts of me: they are just substances that happen to be colocated with me. I’ll call this “parthood nihilism”.

This is compatible with a neat picture of organ transplants. If my kidney becomes your kidney, nothing changes with respect to parthood. All that changes is the causal interactions: the kidney that previously was causing certain distributional properties in me starts to cause certain distributional properties in you.

An obvious question is what about property inheritance? Whenever my hand is stained purple, I am partly purple. We don’t want this to be just a coincidence. The common-sense parts theorist has a nice explanation: I inherit being partly purple from my hand being partly purple (note that they’re only properly partly purple—they aren’t purple inside the bones, say). My partial purpleness derives from the partial purpleness of a part of me.

But the parthood nihilist can accept accept this kind of property inheritance and give an account of it: the inheritance is causal. My hand’s being partly purple causes me to be partly purple, which is a distributional property of an extended simple). I guess on the standard view, property inheritance is going to be a kind of grounding: my being partly purple occurs in virtue of my hand’s being a part of me and its being partly purple. On the present nihilism, we have simultaneous causation instead of grounding.

Here’s another difficulty: what about gravity (and relevantly similar forces). I have a mass of 77kg. If my mass is m₁ and yours is m₂ and the distance between us is r, there is a force pulling you towards me of magnitude Gm₁m₂/r². But why isn’t that force equal in magnitude to (m₁ + m₁₁ + m₁₂ + m₁₃ + ...)m₂/r², where m₁₁, m₁₂, m₁₃, ... are the masses of what common sense calls “my parts” (about five kilograms for my head, four for my left arm, four for my right arm, and so on)? After all, wouldn’t all these objects be expected to exert gravitational force?

The first two kinds of nihilists have easy answers to the problem. The microscopic nihilist says that only particles have mass as only particles exist. The macroscopic one says that I am all there is here—the head, arms, etc. don’t exist. The standard common-sense view has a slightly more complicated answer available: gravitational forces only take into account non-inherited mass. But parthood nihilist can give a variant of this: it’s a law of nature that only fundamental particles produce gravitational forces.

There is a fourth kind of view. This fourth kind of view is no longer a mereological nihilism, but mereological causal reductivism. On the fourth kind of view, for x to be a part of y just is for x to be identical with y or for x to be a proper part of y. And for x to be a proper part of y just is for a certain causal relation to hold between x’s properties and y’s properties.

Spelling out the details of this causal relation is difficult. Roughly, it just says that all of x’s properties and relations cause corresponding properties and relations of y. Thus, x’s being properly partly located in Pittsburgh causes y to be properly partly located in Pittsburgh, while x’s being wholly located in Pittsburgh causes y to be at least partly located in Pittsburgh; x’s being green on its left half causes y’s being green in the left half of the locational property that x causes y to have; and so on.

As I said, it’s difficult to spell out the details of this causal relation. But it is no more difficult than the common-sense parts theorist’s difficulty in spelling out the details of property inheritance. Wherever the common-sense parts theorist says that there is a part-to-whole inheritance between properties, our reductionist requires a causal relation.

The reductionism changes the order of explanation. Suppose my hand is the only green part of me and it gets amputated. According to the common-sense parts theorist, I am no longer partly green because the green hand has stopped being a part of me. According to the reductionist, on the other hand, the hand’s no longer contributing to my greenness makes it no longer a part of me.

The reductionist and parthood nihilist, however, have an extra explanatory burden. Why do all these causal relations cease together? Why is it that when my right hand stops causing me to be partially green, my right hand also stops causing me to have five right fingers? The common-sense parts theorist has a nice story: when the part stops being a part, all the relevant grounding relations stop because a portion of the ground is the fact that the part is a part.

But there is also a causal solution. The common-sense parts theorist has to give a story as to when it is that certain kinds of causal interaction—say, a surgeon using a scalpel—cause a part to stop being a part. For each such kind of causal interaction, the reductionist and parthood nihilist can say that there is a cessation of all the causal relations that the common-sense parts theorist would say go with inheritance.

All in all, I think the reductionist has a simpler fundamental ideology than the standard common-sense inheritance view: the reductionist can reduce parthood to patterns of causation. Her theory is overall not significantly more complicated than the common-sense inheritance theory, but it is more complicated than either microscopic or macroscopic nihilism. But she gets to keep a lot more of common-sense than the nihilists do. In fact, maybe she gets to keep all of common-sense, except for pretty theoretical claims about the direction of explanation, etc.

The parthood nihilist has most of the advantages of reductionism, but there is some common-sense stuff that she denies—she denies that my arm is a part of me, etc. Overall parthood nihilism is not significantly simpler than reductionism, I think, because the parthood nihilist’s account of how all the relevant causal relations cease together will include all the complications that the reduction includes. So I think reductionism is superior to parthood nihilism.

But I still like macroscopic nihilism more than reductionism.

Immaterial body parts

Here’s a difficult question: Does an artificial heart literally become a body part of the patient?

And here’s a line of thought suggestive of a negative answer.

1. Necessarily, all our body parts are material.

2. If one could have an artificial heart as a body part, one could have an immaterial artificial heart as a body part.

3. So, one cannot have an artificial heart as a body part.

Why accept 2? Because presumably what makes an artificial heart suitable for being a body part is that it does the job of a heart. But we could imagine an immaterial being which does the job of a heart. For instance, an angel could move blood around the body, and do so in response to electrical activity in the brain stem. Perhaps one could say that an angel couldn't be a body part, because it is already an intelligent being. But we could then imagine something that moves blood around like the angel but doesn’t have a mind.

I am not so confident of premise 1, however. One could, I suppose, turn the argument around: An artificial heart could be a body part, so possibly some of our body parts are immaterial. And if that’s right, then given a view on which body parts are informed by the form of the person, we would have the further interesting conclusion that a form can inform something that isn’t matter.

Accretion, excretion and four-dimensionalism

Suppose we are four-dimensional. Parthood simpliciter then is an eternal relation between, typically, four-dimensional entities. My heart is a four-dimensional object that is eternally a part of me, who am another four-dimensional object.

But there is surely also such a thing as having a part at a time t. Thus, in utero my umbilical cord was a part of me, but it no longer is. What does it mean to have a part at a time? Here is the simplest thing to say:

1. x is a part of y at t if and only if x is a part of y and both x and y exist at t.

But (1) then has a very interesting metaphysical consequence that only a few Aristotelian philosophers endorse: parts cannot survive being accreted by or excreted from the whole. For if, say, my finger survived its removal from the whole (and not just because I became a scattered object), there would be a time at which my finger would exist but wouldn’t be a part of me. And that violates (1) together with the eternality of parthood simpliciter.

This may seem to be a reductio of (1). But if we reject (1), what do we put in its place, assuming four-dimensionalism? I suspect we will have to posit a second relation of parthood, parthood-at-a-time, which is not reducible to parthood simpliciter. And that seems to be unduly complex.

So I propose that the four-dimensionalist embrace (1) and conclude to the thesis that parts cannot survive their accretion or excretion.

Dualist survivalism

According to dualist survivalism, at death our bodies perish but we continue to exist with nothing but a soul (until, Christians believe, the resurrection of the dead, when we regain our bodies).

A lot of the arguments against dualist survivalism focus on how we could exist as mere souls. First, such existence seems to violate weak supplementation: my souls is proper part of me, but if the body perished, my soul would be my only part—and yet it would still be a proper part (since identity is necessary). Second, it seems to be an essential property of animals that they are embodied, an essential property of humans that they are animals, and an essential property of us that we are humans.

There are answers to these kinds of worries in the literature, but I want to note that things become much simpler for the dualist survivalist if she accepts a four-dimensionalism that says that we are four-dimensional beings (this won't be endurantist, but it might not be perdurantist either).

First, there will be a time t after my death (and before the resurrection) such that the only proper part of mine that is located at t is my soul. However, the soul won’t be my only part. My arms, legs and brain are eternally my parts. It’s just that they aren’t located at t, as the only proper part of me that is located at t is my soul. There is no violation of weak supplementation. (We still get a violation of weak supplementation for the derived relation of parthood-at-t, where x is a part-at-t of y provided that x is a part of y and both x and y exist at t. But why think there is weak supplementation for parthood-at-t? We certainly wouldn’t expect weak supplementation to hold for parthood-at-z, where z is a spatial location and x is a part-at-z of y provided that x is a part of y and both x and y are located at z.)

Second, it need not follow from its being an essential property of animals that they are embodied that they have bodies at every time at which they exist. Compare: It may be an essential property of a cell that it is nucleated. But the cell is bigger spatially than the nucleus, so it had better not follow that the nucleus exists at every spatial location at which the cell does. So why think that the body needs to exist at every temporal location at which the animal does? Why can’t the animal be bigger temporally than its body?

Of course, those given to three-dimensional thinking will say that I am missing crucial differences between space and time.

Parthood and composition

I had a very long conversation yesterday with one of our graduate students regarding Weak Supplementation: he finds Weak Supplementation plausible while I find it implausible. Anyway, I felt this was one of those really good philosophical conversations where you get to something at the root of the issue. So, here's where I felt we got to: It is crucial to how one thinks about parthood whether one sees a close connection between parthood and composition. In particular, it is crucial whether one accepts this thesis:

1. Necessarily, if an object has proper parts, it is composed of them.

There are, I think, two kinds of reasons for accepting (1). You could think that proper parthood is defined by composition, say because you think:

2. x is a proper part of y if and only if (and if so, because) there are zs that compose y such that x is one of the zs.

Or you might think that proper parthood is more fundamental than composition and think that there is a way of defining composition in terms of proper parthood, e.g.:

3. The zs compose y if and only if (and if so, because) every one of the zs is a part of y and every part of y overlaps at least one of the zs.

I think Weak Supplementation is pretty plausible if you accept (1). I also think that acceptance of (1) neatly goes along with the kind of bottom-up thinking that we get in van Inwagen's "special composition question": When do things compose a thing?

On the other hand, I am quite sceptical of (1). I think of parts as derivative from the whole. Instead of wanting to know when things compose a thing, I want to know when a thing has proper parts. One way to put the question, in the case of material substances, is this: Given a thing and a region of space (or spacetime), under what circumstances is there a part that exactly fills that region? And a tempting answer, somewhat reminiscent of van Inwagen's "life" answer, is that this happens when the thing has a function fulfilled precisely in that region.

I then see no reason why the whole would have to be composed of its proper parts. I think I can imagine a stories like the following being true. There is something that looks like a normal human hand. It has five fingers as proper parts. And that's all. There is no such part as the fingerless palm, since the fingerless palm just doesn't have the right kind of functional unity, and I shall suppose that in this scenario there are no cells or particles. The hand then has six parts: the hand itself (an improper part) and the five fingers. Since the hand isn't composed of the fingers, the hand isn't composed of its proper parts. More interestingly, perhaps, I might be persuaded to think of the tropes of an immaterial substance as its proper parts, but I wouldn't want to say that the substance is composed of its tropes--that would be a bundle theory--nor would I want to say that the substance is composed of its tropes and a bare particular. So, then, an immaterial substance might have proper parts, but it wouldn't be composed of them.

A consideration against Weak Supplementation

The Axiom of Weak Supplementation (WS) says that if y is a proper part of x, then there is a part of x that doesn't overlap y. Standard arguments against WS adduce possible counterexamples. But I want to take a different tack. Proper parthood seems to be a primitive relation or a case of a primitive relation (the proper metaphysical component relation seems a good candidate; cf. here). Moreover, this relation does not involve any entities besides the two relata--it's not like the relationship of siblinghood, which holds between people who have a parent in common.

But if R is a primitive binary relation that does not involve any entities besides the two relata, then it is unlikely that the obtaining of R between two entities should non-trivially entail the existence of a third entity. (By "non-trivially", I want to rule out cases like this: everything trivially entails the existence of any necessary being; if mereological universalism is true, then the existence of any two entities trivially entails the existence of their sum.) But if WS is true, then existence of two entities in a proper parthood relationship non-trivially entails the existence of another part. Hence, WS is unlikely to be true.

Elegance and stipulation

Depending on metaphysics, wholes depend on their parts or the parts depend on the wholes. But nothing depends on itself: that would be a vicious circularity. So nothing is a part of itself. On my own preferred story about parts, they are modes of wholes. But perhaps apart from God, nothing is a mode of itself. So, again, nothing is a part of itself (we shouldn't say that God is a part of himself, except trivially if everything is a part of itself).

Yet contemporary usage in mereology makes each thing a part of itself. One is free to stipulate how one wishes. If "part" is the ordinary notion, the contemporary mereologist can stipulate that parthood* is a disjunction of parthood and identity, i.e.,

1. x is a part* of y if and only if x is a part of y or x=y.

However, while one can stipulate how one wishes, one wouldn't expect a disjunctive stipulation to cut reality at its joints.

Why does this matter? Well, one of the interesting questions about parts is what axioms of mereology are true. We have several criteria for what makes a plausible axiom. It's supposed to be intuitive in itself, it's supposed to not lead to paradox, but it's also supposed to be elegant. It seems, however, that one can always ensure the elegance of any axioms with stipulation (just stipulate a zero-place predicate that says that the conjunction of the axioms is true). So it seems we want axioms to be elegant when expressed in terms that cut nature at the joints. In mereology, this would mean that we want axioms to be elegant when expressed in terms of proper parthood* (since proper parthood* is just parthood, the joint-carving natural concept) rather than in terms of parthood*.

This is a bit problematic. For it seems that the standard axioms of mereology get some of their prettiness by using the overlap relation:

2. Oxy iff x and y have a part* in common.

But the overlap relation is a nasty disjunctive mess when expressed in terms of proper parthood*:

3. Oxy iff x=y or x is a proper part* of y or y is a proper part* of x or x and y have a proper part* in common.

This suggests that much of the apparent elegance of the axioms of classical mereology may be spurious. They end up being a mess when you rewrite them in terms of parthood rather than parthood*.

I think the above negative conclusion about the elegance of the axioms of classical mereology is premature, and buys into a mistaken way to measure the elegance of the axioms of a theory. The mistake is to think that one rewrites all the axioms in what Lewis calls "perfectly natural" terms, and then looks at how brief the result is. Mathematicians frequently think that some set of axioms--say, group axioms--are quite "elegant and natural" even when rewriting the axioms in terms of the set membership relation ∈ produces a mess, as it generally does. (Just think of what a mess is produced when anything using the ordered pair (x,y) is rewritten using the set {{x},{x,y}}, and how just about everything in mathematics uses functions and hence ordered pairs.)

One can indeed make any set of axioms brief by careful choice of stipulations. But in some cases the stipulation will itself be very messy (the extreme case is where one replaces all the axioms with a single zero-place predicate) and in other cases there will be many stipulations. But if one can make a set of axioms brief by making a small number of relatively simple stipulations, that is impressive.

A theory can, thus, be elegant even if it is messy and long when all the axioms are written out in perfectly natural terms to the extent that the theory can be elegantly generated from an elegantly small set of elegant stipulations. Classical mereology can satisfy this elegance condition on theories even if I am right that the natural concept of parthood does not allow for proper parts. One just makes the fairly elegant (it's just a disjunction of two natural conditions) disjunctive stipulation (1), and then uses this stipulated notion of parthood* to elegantly stipulate a notion of overlap by means of (2), and then elegantly formulates the rest of the theory in terms of these.

The suggestion I am making is that we measure the complexity of a theory in terms of the brevity of expression in a language that has significant higher-order generative resources that, nonetheless, start with perfectly natural terms. These generative resources allow, in particular, for multiple levels of stipulation. We philosophers have a tendency to simply ignore stipulative definitions. But they do matter. If one takes classical mereology and rewrites the axioms in terms of (proper) parthood, one gets a mess; but the hierarchical stipulative structure of the classical theory is a part of the theory. Furthermore, the generative resources should also allow one to see an axiom schema as simpler and better unified than the sum total of the individual axioms falling under the schema. An axiom schema is not just the sum of the axioms falling under it.

This approach would also let one compare the complexity two different higher-level scientific theories, say in geology or organic chemistry, and say that one is simpler than the other even if both are equally intractable messes when fully expanded out in the vocabulary of fundamental physics. And one can do this even if one does not know how to make the needed stipulations--nobody knows how to define "tectonic plate" in the terminology of fundamental physics, but we can suppose the stipulation to have been made and proceed onward. All this makes it easier to be a reductionist about higher-level theories (I'm not happy about this, mais c'est la vie).

None of this should be news at all to those who are enamoured of computational notions of complexity.

One deep question here is just what generative resources the language should have.

And another deep question should be asked. When we formulate axioms by careful use of stipulation or axiom schemata, what we are really doing is describing the axioms in higher level terms: we are describing a set of sentences formulated in lower level terms. Patterns in reality are sometimes most aptly described not by first-order sentences in fundamental terms, but by describing how to generate those first-order sentences (say, as instances of a schema, or as the result of filling out a sequence of stipulations). We should then ask: How can such patterns be explanatory? I think that if such patterns are explanatory, if they are not mere coincidence, then in an important way reality is suffused with logos, in both of the main sense of the word (language and rationality).

There are, I think, three main options here. One is that we create this reality with our language. Realism forbids that. The second is that we are living in a computer simulation. But this explains the linguistic-type patterns only in contingent reality. But the axioms of mereology or of set theory are not merely contingent. The third is a supernaturalist story like theism, panentheism or pantheism.

A way of thinking about parthood

Take as primitive a "metaphysical component of" relation. Thus, accidents and essences are metaphysical components of their substances. I am interested in a family of accounts of parthood on which

1. x is a part of y if and only x is a metaphysical component of y and F(x) (and maybe G(x)).

In other words, to be a part just is to be a metaphysical component and to be the right sort of thing (and, maybe, for the thing you're a component of to be the right sort of thing). In other words, the relation of parthood would be defined in terms of the relation of being a metaphysical component (which, I suspect, is just the "mode of" or "accident or essence of" relationship) and something that isn't a relation between the part and whole.

Accretion of particles

There is good Aristotelian reason to think that when a particle accretes to a substance--say, when I eat it--the particle ceases to exist. For prior to being accreted, it seems the particle is an independent substance. But a substance can't have substances as parts. And it seems absurd to think that the particle would change from being a substance to a non-substance.

But the view that post-accretion the object that is now a part of the substance is distinct from the object that was accreted is counterintuitive. Here I want to run a partners in crime response to this by showing how a very different and yet fairly mainstream set of assumptions leads to the same counterintuitive conclusion. This should help make that conclusion a little less counterintuitive.

To that end, assume reality is four-dimensional, that objects have lots of temporal parts and that parthood is transitive. This is a fairly commonly accepted set of assumptions. Now suppose I ate a particle x. I will argue that the particle perished in the process. For suppose that x is still a part of me. Let u be a temporal part of x prior to the accretion. Then x is a part of me. But u is a part of x. So by transitivity of parthood, u is a part of me. But that's absurd. So we must deny that x is a part of me on this set of assumptions, too. Hence, this set of assumptions leads to the same conclusion that it is impossible for an object to exist as not a part of something and then continue to exist as a part of that object.