Tuesday, May 23, 2017

Natural Law decision theory

One of the things I’ve learned from the St Petersburg Paradox and Pascal’s Wager is that we are rationally required to have attitudes to risk that significantly discount tiny chances of benefits, rather than to maximize expected utility. This requirement is rational because failure to have such attitudes to risk makes one subject to two-person diachronic Dutch Books. But it is also clearly irrational to significantly discount large chances of benefits.

But where are the lines to be drawn? Maybe it’s not worth enduring an hour of sitting on an uncomfortable chair for a 1/101000 chance of any finite length of bliss, but enduring an hour of sitting in such a chair for a 45% chance of 1000 years of bliss is worthwhile. As long as we thought the decisions were to be made on the basis of expected utility, we could have said that the lines are to be non-arbitrarily drawn by multiplying probabilities and utilities. But that fails.

It is possible, I suppose, that there is a metaphysically necessary principle of rationality that says where the line of the negligibility of chances is to be drawn. Perhaps an hour in the uncomfortable chair for a 1/101000 chance of a finite benefit cannot possibly be worthwhile, but for a 1/106 chance of a large enough finite benefit it is worth it, and there is a cut-off precisely at π ⋅ 10−9. But the existence of any such a metaphysically necessary cut-off is just as implausible as it is to think that the constants in the laws of nature are metaphysically necessary.

(Vagueness is of no help. For even if the cut-off is vague, the shape—vague or exact—of the vagueness profile of the cut-off will still look metaphysically contingent.)

One could leave it to the individual. Perhaps rationality requires each individual to have a cut-off but where the cut-off lies is up to the individual. But rationality also places constraints on that cut-off: the person who is unwilling to sit in an uncomfortable chair for an hour for a 45% chance of 1000 years of bliss is irrational. (I deliberately made it 45%. The cut-off isn’t at 1/2, which would be satisfyingly non-arbitrary.) And where the constraints on the cut-off lie is itself something to be explained, and again it is implausible that it is metaphysically necessary.

In morals, we also have similar cut-off phenomena. It is morally wrong to put someone in prison for life for stealing an ordinary book, while a week of community service is morally permissible. Whence the cut-off? The problem in both cases comes from two features of the situation:

  1. We have a parameter that seems to have a normative force independent of our minds.

  2. That parameter appears to be contingent.

Utilitarianism provides an elegant answer, but no analog of that answer seems to apply in the rationality/risk case. Kantianism is out of luck. Divine command theory provides an answer, but one whose analogue in the case of rationality is quite implausible: it is irrational to be unwilling to sit in the uncomfortable chair for the 45% chance of the great benefit, rather than forbidden by God.

Natural Law, on the other hand, provides a framework for both the moral and the rational cases by saying that the parameter necessarily comes from our nature. Our nature is independent of our minds, and hence we do justice to (1). But while it is presumably not a contingent fact that we have the nature we do, it is a contingent fact that the persons that inhabit the world have the natures they do. Humans couldn’t have these normative risk or moral parameters other than they do, but there could easily have existed non-humans somewhat similar to us who did. The explanation is parallel to the Kripkean explanation of the seeming arbitrariness of water having two hydrogen atoms. Water couldn’t have had a different number of hydrogen atoms, but something similar to water could have had.

More and more, I think something like Natural Law is a powerful framework in normative areas outside of what is normally construe to be moral theory: in decision theory and epistemology. (I hedge with the “normally construe”, because I happen to think that both decision theory and epistemology are branches of moral theory.)

Wednesday, May 17, 2017

Could God be divinity?

Here's a plausible thesis:

  1. If it is of x's essence to be F, then Fness is prior to x.
This thesis yields a fairly standard argument against the version of divine simplicity which identifies God with the property of divinity. For if God is divinity, then divinity is prior to divinity by (1), which is absurd.

But (1) is false. For, surely:

  1. It is of a property's essence to be a property.
But propertyhood is a property, so it is of propertyhood's essence to be a property, and so propertyhood is prior to propertyhood if (1) is true, which is absurd. So, given (2), we need to reject (1), and this argument against the God=divinity version of divine simplicity fails.

What else might properties do?

Suppose that we think of properties as the things that fulfill some functional roles: they are had in common by things that are alike, they correspond to fundamental predicates, etc. Then there is no reason to think that these functional roles are the only things properties do. It is prima facie compatible with fulfilling such functional roles that a property do many other things: it might occupy space, sparkle, eat or think.

Can we produce arguments that the things that fulfill the functional roles that properties are defined by cannot occupy space, sparkle, eat or think? It is difficult to do so. What is it about properties that rules out such activity?

Here's one candidate: necessity. The functional roles properties satisfy require properties to exist necessarily. But all things that occupy space are contingent. And all things that sparkle or eat also occupy space. So no property occupies space, sparkles or eats. (Yes, this has nothing to say about thinking.) Yeah, but first of all it's controversial that all properties are necessary. Many trope theorists think that typical tropes are both contingent and properties. Moreover, it may be that my thisness is a property and yet as contingent as I am. Second, it is unclear that everything that occupies space has to be contingent. One might argue as follows: surely, for any possible entity x, it could be that all space is vacant of x. But it does not follow that everything that occupies space has to be contingent. For we still have the epistemic possibility of a necessary being contingently occupying a region space. Christians, for instance, believe that the Second Person of the Trinity contingently occupied some space in the Holy Land in the first century--admittedly, did not occupy it qua God, but qua human, yet nonetheless did occupy it--and yet the standard view is that God is a necessary being. (Also, God is said to be omnipresent; but we can say that omnipresence isn't "occupation" of space, or that all-space isn't a region of space.)

So the modal argument isn't satisfactory. We still haven't ruled out a property's occupying space, sparkling or eating, much less thinking. In general, I think it's going to be really hard to find an argument to rule that out.

Here's another candidate: abstractness. Properties are abstract, and abstracta can't occupy space, sparkle, eat or think. But the difficulty is giving an account of abstracta that lets us be confident both that properties are abstract and that abstract things can't engage in such activities. That's hard. We could, for instance, define abstract things as those that do not stand in spatiotemporal relations. That would rule out occupying space, sparkling or eating--but the question whether all properties are abstracta would now be as difficult as the question whether a property can occupy space. Likewise, we could define abstract things as those that do not stand in causal relations, which would rule out sparkling, eating and thinking, but of course anybody who is open to the possibility that properties can do these activities will be open to properties standing in causal relations. Or we could define abstractness by ostension: abstract things are things like properties, propositions, numbers, etc. Now it's clear that properties are abstracta, but we are no further ahead on the occupying space, sparkling, eating or thinking front--unless perhaps we can make some kind of an inductive argument that the other kinds of abstracta can't do these things, so neither can properties. But whether propositions or numbers can do these things is, I think, just as problematic a question as whether properties can.

All in all, here's what I think: If we think of the Xs (properties, propositions, numbers, etc.) as things that fulfill some functional roles, it's going to be super-hard to rule out the possibility that some or all Xs do things other than fulfilling these functional roles.

For more related discussion, see this old contest.

Tuesday, May 16, 2017

Pascal's Wager and the bird-in-the-hand principle

My thinking about the St Petersburg Paradox has forced me to reject this Archimedean axiom (not the one in the famous representation theorem):

  1. For any finite utility U and non-zero probability ϵ > 0, there is a finite utility V such that a gamble that offers a probability ϵ of getting V is always better than a certainty of U.
Roughly speaking, one must reject (1) on pain of being subject to a two-player Dutch Book. But rejecting (1) is equivalent to affirming:
  1. There is a finite utility U and a non-zero probability ϵ > 0, such that no gamble that offers a probability ϵ of getting some finite benefit is better than certainty of U.
With some plausible additional assumptions (namely, transitivity, and that the same non-zero probability of a greater good is better than a non-zero probability of a lesser one), we get this bird-in-the-hand principle:
  1. There is a finite utility U and a non-zero probability ϵ > 0, such that for all finite utilities V, the certainty of U is better than a probability ϵ of V.
Now, Pascal's Wager, as it is frequently presented, says that:
  1. Any finite price is worth paying for any non-zero probability of any infinite payoff.
By itself, this doesn't directly violate the bird-in-the-hand principle, since in (3), I said that V was finite. But (4) is implausible given (3). Consider, for instance, this argument. By (3), there is a finite utility U and a non-zero probability ϵ > 0 such that U is better than an ϵ chance at N days of bliss for every finite N. A plausible limiting case argument suggests that then U is at least as good as an ϵ chance at an infinite number of days of bliss, contrary to (4)--moreover, then U+1 will be better than an ϵ chance at an infinite number of days of bliss. Furthermore, in light of the fact that standard representation theorem approaches to maximizing expected utility don't apply to infinite payoffs, the natural way to argue for (4) is to work with large finite payoffs and apply domination (Pascal hints at that: he gives the example of a gamble where you can gain "three lifetimes" and says that eternal life is better)--but along the way one will violate the bird-in-the-hand principle.

This doesn't, however, destroy Pascal's Wager. But it does render the situation more messy. If the probability ϵ of the truth of Christianity is too small relative to the utility U lost by becoming a Christian, then the bird-in-the-hand principle will prohibit the Pascalian gamble. But maybe one can argue that little if anything is lost by becoming a Christian even if Christianity is false--the Christian life has great internal rewards--and the evidence for Christianity makes the probability of the truth of Christianity not be so small that the bird-in-the-hand principle would apply. However, people's judgments as to what ϵ and U satisfy (2) will differ.

Pleasantly, too, the bird-in-the-hand principle gives an out from Pascal's Mugger.

Friday, May 12, 2017

More on St Petersburg

I’ve been thinking about what assumptions generate the St Petersburg paradox. As stated, the paradox depends on the assumption that we should maximize expected utility, an assumption that will be rejected by those who think risk aversion is rational.

But one can run the St Petersburg paradox without expected utility maximization, and in a context compatible with risk aversion. Suppose finite utilities can be represented by finite real numbers. Assume also:

  1. Domination: If a betting portfolio B is guaranteed to produce at least as good an outcome as A no matter what, then B is at least as good as A.

  2. Archimedeanism: For any finite utility U and non-zero probability ϵ > 0, there is a finite utility V such that a gamble that offers a probability ϵ of getting V is always better than a certainty of U.

  3. Transitivity: If C is better than B and B is at least as good as A, then C is better than A.

(Note: For theistic reasons, one might worry about Construction when the Vi are very negative, but we can restrict Construction to positive finite utilities if we add the assumption in Archimedeanism that V can always be taken to be positive.)

For, given these assumptions, one can generate a gambling scenario that has only finite utilities but that is better than the certainty of any finite utility. Proceed as follows. For each positive integer n, let Vn be any finite utility such that probability 1/2n of Vn is better than certainty of n units of utility (this uses Archimedeanism; the apparent use of the Axiom of Choice can be eliminated by using the other axioms, I think) and Vn ≥ Vn − 1 if n > 1. Toss a fair coin until you get heads. Let your payoff be Vn if it took n tosses to get to heads.

Fix any finite utility U. Let n be a positive integer such that U < n. Then the gambling scenario offers a probability of 1/2n of getting at least Vn, so by Domination, Transitivity and the choice of Vn, it is better than U.

And the paradoxes in this post apply in this case, too.

If we have expected utility maximization, we can take Vn = 2n and get the classic St Petersburg paradox.

Given the plausibility of Domination and Transitivity, and the paradoxes here, it looks like the thing to reject is Archimedeanism. And that rejection requires holding that there is a probability ϵ so small and finite utility U so large that no finite benefit with that probability can outweigh U.

Wednesday, May 10, 2017

Teleology and the direction of time

It would be depressing to think that one will never swim as fast as one is swimming today. But it would uplifting to think that that one has never swum as fast as one is swimming today.

I used to think the direction of time was defined by the predominant direction of causation. That may be the case, but if one takes humanistic cases like the above as central, one might think that perhaps the predominant direction of teleology is a better way to define the direction of time. Of course, telê are there to be achieved, and so the direction of teleology needs to fit well with the direction of causation, at least in the case of things that concern us. Moreover, there is some reason to think that teleology is behind all causation—causation aims at an effect.

Certamen machine

My kids are involved in a Classics oriented quiz game called Certamen at school. These involve teams and buttons and a machine that determines the order in which buttons were pressed. Surprisingly, these machines seem to cost a ridiculous $500 and up, despite seeming to be quite a simple thing: 12 buttons, display which order the buttons are pressed in, lock out fellow team members once one member of a team has pressed it.

So I offered my kids' school to design and build one for them as a fun summer project for me and an opportunity for my kids to learn to solder. I ordered about $60 of parts, mostly from Aliexpress, centered on an Arduino Mega (I haven't done any Arduino-based programming, but I've used the Arduino toolchain with an ESP8266 before). The parts have started to come in, including the Mega, so I've started writing some code and prototyping. According to my oscilloscope, the quick and dirty polling code I have gets a worst-case detection speed of 0.1 milliseconds, which should be good enough for a quiz game. (I continue to be grateful to the Austin guy who gave had an oscilloscope for sale for $50 on Craigslist, but when I wanted to buy it, gave it to me for free because he liked the sorts of things I was going to use it for.)

I am a bit nervous about signal problems over the three five-meter CAT6 cables (the most expensive single parts of the project) from the control box to the buttons, but I ordered some capacitors for noise suppression, and once my RJ45 jacks come in, I can do some testing.

Monday, May 8, 2017

Good-bye, (Aristotelian) matter

Of course, there are material things like oaks and people, and it’s distinct from immaterial things like angels. But for a long time I’ve been wondering why my fellow Aristotelians think that there is matter, a component of material things. In the process of reflection, I have given up on matter as a fundamental ontological category. Of course, for theological and common-sense purposes, I need to have the concept of a material substance, but here I hope there is some reduction, such as that a material substance is a substance that has at least one geometric property. My Aristotelianism now inclines to be more like Leibniz’s than like the historical Aristotle’s or Aquinas’s. Material substances, on my view, are much like Leibniz’s monads; they are like Aristotle’s gods or Aquinas’s angels, plus whatever properties or causal powers are needed for them to count as material. I am my own form, and in this form there inhere accidents.

What philosophical work does matter play, particularly in Aristotelian theories?

  1. Many Aristotelians say that something remains through substantial change, namely matter.

The persistence of matter through substantial change is said to do justice to the intuition that the corpse is the remains of the living creature: that there is something in the corpse that was in the living creature. But it is notoriously difficult to remain faithful to the Aristotelian emphasis that identity always comes from form and allow that anything in the corpse is identical to anything in the prior living body. Absent a solution to this, the Aristotelian has to say that there is one bunch of matter prior to death, a bunch of matter informed by the form of the living body, and a different bunch of matter after death, informed by the forms of the substances making up the corpse. But that does not do justice to the common-sense intuition.

In the vicinity, too, there is the question of why it is that the corpse is physically like the living body. But this is not to be accounted for by matter, but by accidents such as shape, mass and color. Accidents are possessed by substances. Either accidents can or cannot survive the destruction of their underlying substance. If they can, then we have an explanation of why the corpse is physically like the living body. If they cannot, then adding that there is matter in both—and even that it is the same matter—does not help: we simply have to bite the bullet and say that the accidents of the living body have the power to cause similar accidents in the corpse.

  1. Matter may play a role in diachronic identity.

But since immaterial substances like angels can persist over time, matter isn’t needed to solve the problem of diachronic identity. Moreover, the problem of diachronic identity seems to me, as a four-dimensionalist, to be a pseudoproblem (see also this]). It is no more a problem how the same thing can exist in 2017 and in 2018 than it is a problem how someone can exist in the room and in the hall—just put a leg in each, and you’ll see how. Matter does nothing to help with the latter problem, since presumably it isn’t the same chunk of matter that’s in the room as in the hall. So, why should matter help with the former?

  1. Matter may play a role in problems of material composition.

Matter may also play a role in some specific solutions to the problem of material composition. One might, for instance, identify the lump with the matter and the statue with the substance composed of it, or the lump with one thing made of the matter and the statue with another thing made of the same matter, and then explain away the commonality of many properties, like mass, by the identity of matter. But either the statue and the lump have numerically the same accident of mass or they do not. If they do, then since accidents inhere in substances, not in matter, the commonality of matter doesn’t do any work. If they do not, then the commonality of matter doesn’t seem to have done much—we still have to explain why the two have an exactly similar accident of mass, given that they have numerically distinct ones.

What matter does do, I think, is help differentiate the classic statue–lump case from the horse–ghost case where Bucephalus’s ghost happens to walk right through the living Seabiscuit, in such a way that the ghost horse and the living horse happen to occupy exactly the same space. For we can say that the ghost case is a case of merely spatial colocation, while the statue–lump case is a case of having the same matter. And intuitively there is a difference between the two cases. Interestingly, though, this isn’t the material composition problem that matter usually gets invoked to solve. And since I don’t believe in statues, or in any other entities that could plausibly be thought to make there be two entities of one chunk of matter, this does little for me.

  1. Isn’t hylo-morphism the distinctively Aristotelian solution to the mind-body problem?

Sure. But, even more than the classic Aristotelian solution, my view is a dissolution to the mind-body problem rather than a solution. The form of course affects the accidents that constitute and shape our embodiment. All of this is due to the nexus—ontological, teleological and causal—that exists between the substance and its accidents (both substance–accident and accident–accident). It’s not a case of one thing moving another: it is just the common story of the form affecting the accidents and the accidents affecting one another.

And, yes, of course I agree with the Council of Vienne that the soul is the form of the body. On my view, talk of the soul is talk of the substance qua form and apart from the accidents constituting its materiality, and the substance qua form is a base for all the accidents which constitute us as having bodies. So, the soul is the form of the body.

  1. Physics talks of matter.

Sure, but physics probably doesn’t have a fundamental distinction between matter and energy, I think.

Anyway, I don’t deny that there is matter in the sense of substances that are so configured as to count as material. Quite possibly, where you have a heap of sand, you have a heap of material substances, and hence matter. (But perhaps not: perhaps fundamental physical reality is just a handful of fields.)

All in all, I just see little if any benefit to matter. And there is much mystery about it. Ockham’s razor cuts it away.

Unless, of course, we come to some philosophical problem that can’t be solved without matter, or can’t be solved as well without it…

A way to argue against Strong AI

  1. Strong AIs are finite persons who are implemented by software. (Definition.)

  2. The correct theory of personal identity for Strong AIs would be a version of the psychological theory.

  3. Necessarily, the same theory of personal identity applies to all possible finite persons.

  4. We are finite persons.

  5. So, if Strong AIs are possible, a version of the psychological theory of personal identity applies to us.

  6. But the psychological theory of personal identity is false.

  7. So, Strong AIs are impossible.

Of course, the hard part is to argue for (6), since (6) is so widely accepted.

Friday, May 5, 2017

How not to defend penal substitution

Consider the standard problem for penal substitution views:
  • How is it that an innocent person's suffering harsh treatment removes the guilt of this guilty?
This is just a quick remark. Here is how not to solve the problem: Don't invoke God's sovereignty or power to claim that God can transfer guilt and punishment at will. For if God can transfer guilt and punishment at will, then God could transfer the guilt and punishment to a tree. But wouldn't it be better that a tree should be harshly punished for eternity (say, constantly have its bark ripped off as it grows back) than that Christ suffer?

Thursday, May 4, 2017


Somehow, I find writing parsers and interpreters one of the most satisfying computer programming activities I've done. I've done this a couple of times in my life, sometimes from scratch and sometimes using a tool like bison. Maybe it's because the resulting linguistic adeptness that the computer shows--even in the case of a very simple language--is somehow impressive. It's fun, for instance, to write a parser that translates a formula like "x^3*y-y^3*x" into a LISP-like representation ["-",["*",["^","x",3],"y"],["*",["^","y",3],"x"]], and that can then interpret the representation given values for x and y. Most recently, I had the fun of doing this in the OpenSCAD 3D design language, to enable passing formulas to functions/modules. This was kind of challenging for me as I'm not very comfortable with functional languages.

What Galileo should have said

The big theological problem that Galileo's opponents had for Galileo wasn't the (not very convincing) biblical arguments that the sun moves and the earth stands still, but a theological objection to Galileo's inference from (a) the greater simplicity of the Copernican hypothesis over its competitors and (b) the fact that the hypothesis fits the data to (c) the truth of the Copernican hypothesis. The theological objection, as I understand it, was that Galileo was endangering the doctrine of divine omnipotence, since if there is an omnipotent God, he can just as easily have made true one of the less simple hypotheses that fit the data. (And, indeed, an earth-centered system can be made to fit the data just as well as a sun-centered one if one has enough epicycles.)

What Galileo should have said is that his argument does not, of course, establish the Copernican hypothesis with certainty, but only as highly probable, and that his argument had the form of the well-established theological argument ex convenientia, or from fittingness: "It was fitting for God to do it, God was able to do it, so (likely) God did it." Such arguments were widely given in the Middle Ages for theological views such as the immaculate conception of Mary. The application is that it is fitting for God to do things in the more elegant Copernican fashion, an omnipotent God was able to do things in such wise, and so (likely) God did it. Not only would the argument form have been one that Galileo's interlocutors would have been familiar with and friendly towards, but Galileo would have the dialectical advantage that he could not be reasonably said to be challenging divine omnipotence if his own argument depended on it. (Maybe Galileo did say something like this. I've seen the use of the argumentum ex convenientia in astronomy attributed to Kepler. Maybe Kepler got it from Galileo.)

And, to be honest, I think that all science is essentially founded on arguments ex convenientia. Which are good arguments.

Tuesday, May 2, 2017

Grounding accidents in substances

Consider this plausible principle:

  1. x partially grounds y if and only if there are cs that fully ground y and x is one of the cs.

But now consider this plausible-sounding Aristotelian claim:

  1. The substance (or its form or its essence—the details won’t matter) partially grounds each of its accidents.

Note that the grounding here is not full. For if my substance fully grounded my accident of sleepiness, then my substance would be metaphysically sufficient for my sleepiness, and I would be always sleepy, which is fortunately not the case.

So, by 2, my sleepiness is partly grounded by my substance (i.e., me?), and merely partly. By 1, then, it follows there are other things, beside my substance, such that my sleepiness is fully grounded by my substance and those other things. What are those other things? Is it other accidents of me? If so, then the problem repeats for them. Or is it something beyond my substance or accidents? But what would that be?

I am inclined to think that the solution to this problem is to reject 1. Somehow, 1 is reminiscent to me of the false view that:

  1. x indeterministically causes y only if there are cs that deterministically cause y and x is one of the cs.

Compositional and non-compositional trope theories

There are two kinds of trope theories: Those on which the tropes are parts of the particular object—call these “compositional” trope theories—and those on which the relation between the object and its tropes is not a whole-to-part relation. Compositional trope theories have an initial advantage over non-compositional ones: they have no need to introduce a new relation to join objects to their tropes.

But this is only an apparent advantage. Consider this old argument. Assume compositional trope theory. Suppose my toe is blue. Then its blueness trope is a part of the toe, which is in turn a part of me, and so the blueness trope is a part of me. Hence I am blue.

Of course, the compositionalist has an answer to this argument: there are two different kinds of parthood here. The toe is, as the medievals would say, an integral part of me. And the blueness trope is a non-integral part of the toe. Transitivity holds for integral parts. It may or may not hold for non-integral parts, but it certainly doesn’t hold across types of parthood: if y is an integral part of x and z is a non-integral part of y, it does not follow that y is any kind of part of x.

But notice now that the compositionalist has lost the main advantage over the non-compositionalist. The compositionalist’s initial advantage was not having to introduce a new kind of relation over and beyond the familiar composition relation. But the familiar composition relation was the one between wholes and integral parts, and our compositionalist now has to introduce a new relation over and beyond that. Granted, it is a new relation of the same type as the familiar one. But this actually makes the compositionalist’s theory more complicated. For now the compositionalist has two relations, integral composition and non-integral composition, plus a new relation type, composition. But the non-compositionalist need only have two relations, integral composition and the object-to-trope relation. These two relations don’t need to have a new relation type to fall under. In other words, the non-compositionalist has only one mystery in her theory—what is the object-to-trope relation—while the compositionalist has two mysteries—what is the object-to-trope relation and what is the type composition.

The same point applies more generally to compositional ontologies versus relational ontologies.

Rapid cell replacement: A failed argument against materialism

I thought I had a nice argument against materialism, but it didn’t work out. Still, it’s fun to think about the argument and why it doesn’t work.

Start with this plausible thesis, which seems at least naturally necessary:

  1. If any cell in a human body blinks out of existence and a new cell, exactly like the one before, blinks into existence sufficiently quickly in the same orientation, then the result would not interrupt the human’s life or any train of consciousness.

Now imagine that very, very quickly one-by-one every cell in my body blinks out of existence and is replaced by a new cell formed by a coincidental quantum fluctuation. Moreover, suppose each replacement happens sufficiently quickly in the sense of (1), and indeed so quickly that all of the replacements are done in less than the blink of an eye. Applying claim (1) billions of times, I conclude that neither my existence nor my train of consciousness would be interrupted by this process.

But if materialism is true, the resulting entity would have insufficient causal connection to me to be me. Thus, if materialism is true, I would have to cease to exist as a result of these rapid replacements. But it seems this would violate (1) at some point. (Moreover, the resulting being would not be the product of natural selection, so on evolutionary functionalist theories, the being would not have mental states. Furthermore, in any case, its brain states would not have the kinds of connections with the external world that give rise to content according to the best materialist theories, so its thoughts would be largely contentless.)

But the argument I just gave doesn’t work. First, (1) is false in the case of a human zygote, since the destruction of one’s only cell would kill one. What made (1) plausible was the thought that we had many cells, and the replacement of any one of them with a randomly produced cell would make no difference. So, (1) needs to be modified to remain plausible:

  1. If any cell in a human body consisting of many cells blinks out of existence and a new cell, exactly like the one before, blinks into existence sufficiently quickly in the same orientation, then the result would not interrupt the human’s life or any train of consciousness.

But now it no longer follows that a quick cell-by-cell replacement would have to keep me alive. For here is a possible hypothesis: For a replacement cell to come to be a part of the body, it has to come to be sufficiently causally intertwined with the rest of the body. This takes some time. It could well be that if the cells are replaced one by one in less than the blink of an eye, the new cells don’t have time to become intertwined with the rest of the body. Thus, the body comes to have fewer and fewer cells as the gradual replacement process continues. If the replacement process were to stop, pretty quickly the replacement cells would come to be causally intertwined with the veteran cells, and would come to be a part of the body. But it doesn’t stop. As a result, eventually the process leads to a state where I don’t have “many” cells in my body, and hence (2) becomes inapplicable.

What if, on the other hand, the replacement is done more slowly, so that there is time for cells to causally intertwine and become a part of the body? Then there need be no problem for materialism, because now the resulting entity does have a sufficient causal connection to me to be me.

There is, of course, a vagueness problem for the materialist: When do I cease to exist in the process? But that's another argument. I think typical materialists who think that they exist cannot escape vague existence.