More on fungibility and naturalism

In an earlier post, I argued against materialism on the grounds that persons are non-fungible but material objects are fungible or at least persons are non-derivatively non-fungible, while material objects are at best derivatively non-fungible.

Here's a pathway to arguing that if naturalism is true, then at least some persons are fungible. Since no persons are fungible, it follows that naturalism is false.

Start with the thought that:

1. Something wholly composed of fungible parts is fungible.

If this is right, and material objects are composed of elementary particles, then all material objects are fungible since elementary particles are fungible. And that's all we need for our argument.

But (1) may not be quite right. After all, arguably, the Mona Lisa is (derivatively) non-fungible, but all the elementary particles making it up are. There would be no loss if we replaced the particles of the Mona Lisa one by one. The non-fungibility of the Mona Lisa is grounded in the non-fungibility of the arrangement of the parts: If suddenly the Mona Lisa was burnt up, but by coincidence the particles in the ashes and smoke arranged themselves in an exactly similar arrangement, something of value would be lost. There is something special here about the arrangement.

What makes the arrangement of the Mona Lisa's particles special is the specialness of the artistic process that produced that arrangement. This suggests:

2. Something wholly composed of fungible parts arranged by a fungible process is fungible.

A fungible process is one such that it is value-relevant whether it is replaced by an exact copy. The painting of the Mona Lisa is like that. Given (2), it is very plausible that things composed of fungible parts are at best derivatively non-fungible, with their non-fungibility derived from that of the process of generation.

Now you or I perhaps did have our parts get arranged by a non-fungible process: our parents' loving union. But even persons produced by in-vitro fertilization had their parts arranged by our biological parents' bodies through their gametes, and the process of gamete production in a person is arguably non-fungible.

However, at least one person—namely, a first human person—has no person as a biological parent, on pain of an infinite regress. If theism is true, that person may still be the product of a non-fungible process of creation by a (divine) person. But naturalism rules out not only dualism but also theism. A naturalist who does not believe in an infinite past will have to hold that there is a first person who is in no way produced by a person. And there it seems that the process producing that first person is fungible—it plausibly doesn't matter value-wise which of two exactly similar brute animals mated with a brute animal to produce a person. (If it is responded that primates like those we descend from are themselves non-fungible, then just take the argument further back in our evolutionary past.)

Extinction, creationism and presentism

An intuition a lot of people have is:

1. The extinction of a species is a very bad thing.

On the other hand, extinction is central to the evolutionary process. Some species go extinct, making room for others, and thereby producing a greater diachronic biodiversity. Diachronic biodiversity is a very good thing, and the production of this diversity (wholly or at least to a significant degree) by natural evolutionary processes is a very good thing. This suggests:

2. If species at least typically arise by evolutionary processes, (1) is false.

What can we say given (2)? Well, we could argue:

3. Species at least typically arise by evolutionary processes. (Scientifically known fact)
4. Therefore, the extinction of a species is not a very bad thing. (By 2 and 3)

I am inclined to go for this. To soften the counterintuitive consequence, we might note that what is particularly bad is extinctions where no new species are likely to evolve to fill the niche, and the kind of environmental degradation that we want to oppose not only leads to the extinction of species but stands in the way of the evolution of new species that fill the niche. Though that's an empirical question, and I don't know if it's so.

Another move is to argue:

5. Both (1) and (2) are true.
6. Therefore, species do not arise by evolutionary processes, even typically. (By 5)
7. If species do not arise by evolutionary processes, creationism is true. (Since creationism is the best alternative to evolution.)
8. So, creationism is true. (By 6 and 7)

A yet different move is to deny (2). My thinking behind (2) was based on the value of diachronic biodiversity. But perhaps diachronic biodiversity is only as valuable as I think it is if presentism is false. It is only if the past organisms in extinct species really exist, even if pastly so, that they contribute in a valuable way to biodiversity. So one might replace (2) by:

9. If species at least typically arise by evolutionary processes and presentism is false, (1) is false.

And then one might argue:

10. Claims (1) and (3) are true.
11. Therefore, presentism is true. (By 9 and 10)

So I think we have three main options:

• Deny that extinction is a very bad thing.
• Deny evolution and affirm creationism.
• Affirm presentism.

I should note that in (1), I am thinking of on-balance badness rather than just intrinsic badness.

A quick argument for tropes

1. Every thought is a trope of a thinker.
2. There are thoughts.
3. So, there are tropes.

Fungibility, persons and materiality

This seems plausible:

1. All purely material objects are fungible.
2. No persons are fungible.
3. So, no persons are purely material objects.

Maybe that's not quite right. One might think that some objects care about (a colleague gave the examples of the Mona Lisa and Grandpa's Bible) are non-fungible. But I think it's plausible that material objects are at most derivatively non-fungible, deriving their non-fungibility from the non-fungibility of people. Thus:

4. No purely material object is non-derivately non-fungible.
5. Every person is non-derivatively non-fungible.
6. So, no person is a purely material object.

Presentism

I explained presentism to my 11-year-old daughter. I told her that according to presentism past and future events aren't real and reality is three-dimensional. She found this view scary. It made her think we live in something like a "flat world", something akin to a "two-dimensional" world. I take it her worry was that this impoverished the world, in much the way that denying a spatial dimension would. Her emotional reaction to presentism is much like mine is: presentism contracts the world to something way too thin.

My daughter then connected this with Zeno's paradox of the arrow. I think her thought might have been something like this: The eternalist can say that movement is what happens in virtue of the arrow being in different places at different times. But according to the presentist, the arrow is where it is, and that's that: that's all of reality. So presentism denies real change (which is ironic since presentists are largely motivated by the idea of saving change).

Now all this isn't entirely fair to the presentist. The presentist does not deny that the arrow was elsewhere and will be elsewhere. Still, "was" and "will be" are operators akin to modal ones like "can be". The fact that the arrow was elsewhere is related to the fact that the arrow is here in something like the way that the fact that the arrow could be elsewhere is related to that fact. Being in different places at different times ends up being akin to one's position being contingent. And that doesn't seem to do justice to the reality of motion. This is all suggestive, though it's probably not much of a knockdown argument.

Some issues with causal decision theory

Let U be utility, and let H_i be all the epistemically open causal hypotheses. To evaluate the value of an option A, causal decision theory says we should calculate the weighted sum:

• V₁(A)=Σ_iE(U|H_i&A)P(H_i).

Problem: But what if one of the causal hypotheses is incompatible with option A? For instance, H₂ might be the hypothesis that one is brainwashed into not choosing A. In that case, P(H_i&A)=0 and E(U|H_i&A) is apt to be undefined.

The natural solution is to modify the sum (1) to include only those hypotheses H_i such that P(H_i&A)>0, and renormalize the sum to compensate. For instance, if P(H_i&A)>0 for 1≤i≤m and P(H_i&)=0 for m<i≤n, then we should calculate the value of option A at:

• V₂(A)=a⁻¹Σ_i=1^mE(U|H_i&A)P(H_i)

where a=Σ_i=1^mP(H_i) is our normalization factor. And now we make decisions in the straightforward way: we choose the option that has the highest V₂-style value.

But this turns out to betray the core intuitions of causal decision theory. Suppose there are two equally likely causal hypotheses: H₁ says you're free to choose between pizza and falafel and H₂ says you're brainwashed into eating falafel. Suppose you prefer falafel to pizza. But of course your future is bleak if you're brainwashed into eating falafel. Wonderful as falafel is, eating falafel at every meal is going to be miserable. Now, V₂(pizza) includes in the sum only the term for hypothesis H₁, since H₂ is incompatible with pizza. Thus, V₂(pizza) is the value of eating pizza. But V₂(falafel) includes in the sum terms for both hypotheses H₁ and H₂. And since half of the weight of that sum will correspond to H₂, and on H₂ we have the misery of being brainwashed into falafel for the rest of our lives, the theory based on V₂ requires us to choose pizza, lest it turn out that we were brainwashed into eating falafel. But that's exactly the sort of silliness that causal decision theory was created to eliminate: your present decision whether to eat falafel or not makes not a whit of difference to whether you've been brainwashed. To go for pizza here is to act like an evidential decision theorist.

So our simple modification of V₁ to V₂ solves the problem of V₁ being undefined, but at the price of betraying causal decision theory intuitions.

There is, however, another modification. When choosing between some options, say pizza and falafel, we should define the value by only considering in our sums the hypotheses on which all options are causally possible. Interestingly, this means that the value of an option depends on which contrast class of options we are considering: the value of option A_j as chosen from between A₁,...,A_k will in general depend on what the options are. Thus, we should write the value as:

• V₃(A_j;A₁,...,A_k)=a⁻¹Σ_i=1^mE(U|H_i&A_j)P(H_i),

where it is assumed that H₁,...,H_m are the hypotheses that are compatible with each of A₁,...,A_k and a=Σ_i=1^mP(H_i).

The fact that the value of an option in general depends on what other options are in view will be a controversial but not fatal consequence.

Another interesting consequence is that if all the causal hypotheses are deterministic, then the value V₃ will be undefined. Thus, decisions presuppose the possibility of indeterminism. It's plausible that an argument can be made from this for incompatibilism, an argument well worth exploring.

Here, however, is a puzzle. Suppose that hitherto your preferences have been that you liked meatballs a little more than falafel which in turn you liked a little more than pizza (with transitivity). There are now two causal hypotheses each with probability 1/2: according to H₁ you have had no intervention, but according to H₂ you've been brainwashed against eating pizza and your brain response to falafel has been modified so that you will enjoy falafel far more than any food you've ever had. You are now choosing between falafel, meatballs and pizza. The V₃ calculation tells you to consider only those hypotheses compatible with all three options, i.e., to consider only hypothesis H₁. So you will go for meatballs. But surely that's a mistake. Falafel is a better option than meatballs, since if you weren't brainwashed against pizza, meatballs are only slightly better, while if you were brainwashed against pizza, falafel is far better. When evaluating falafel, you do need to take the possibility of H₂ into account.

So maybe the whole approach that fixes causal decision theory by restricting the list of causal hypotheses is wrong? Maybe instead we should restrict outcomes, only considering those outcomes that are caused by one's choice. Thus, in our falafel-pizza story, you simply don't count the disvalue of being brainwashed, since that doesn't causally depend on your decision, when evaluating the utilities. Unfortunately, it's not so simple, since surely we can come up with cases where there are subtle interactions between prior conditions, such as brainwashing, and one's decision. I don't know what to do. But fortunately I don't believe in these kinds of decision theories.

In-vitro fertilization and marriage

Traditional sexual norms say that non-marital sex is impermissible. Those who think in-vitro fertilization (IVF) is permissible but who want to hold on to this norms now need to decide:

• Is (voluntary[note 1]) reproduction outside of marriage permissible?

It would be odd, I think, to say that premarital sex is impermissible but premarital reproduction is permissible. First, the main natural law reasons for thinking premarital sex is impermissible involve the reproductive potential of sex. Second, it seems that children bind a couple together much in the way that sex does.

Suppose, then, that the rule that sex needs to be within marriage extends to reproduction. This leads to interesting questions at the beginning and end of a marriage.

End: Suppose one member of a married couple is about to die. Can that spouse give consent to IVF if the IVF would have to occur after death?

It would be in principle permissible (though often prudentially inadvisable) for a married couple to have sexual relations even if they somehow foresee that (a) fertilization will likely occur in a few days but (b) the man will die before fertilization occurs. By the same token, if IVF is permissible, it might be permissible for the couple to give consent to IVF during the marriage, even though the actual reproduction occurs after death (and hence after the end of the marriage).

Beginning: Suppose the couple's gametes are already available to medical professionals prior to their marriage (say, due to some kind of surgery). Is it permissible for an unmarried couple to give their authorization for the union of the gametes on the understanding that the medical professionals will only unite the gametes after the couple is married?

If this happens, the child will not be in any way a fruit of the marriage, since no marital action—not even the giving of consent—is involved in the reproduction. The connection between the child and the marriage would be merely temporal. This would not do justice to the connection between reproduction and marriage. (One can also come up with an argument for this conclusion by combining the Beginning and End scenarios.)

If this is right, then the crucial thing for the extension of the traditional sexual norm to IVF would be the provision of consent: it is this that must occur during marriage in order that the child be the fruit of the marriage.

But this in turn emphasizes once again the way that in non-coital reproduction the essential involvement by the couple is simply the provision of consent. And that troubles me, even if I do not yet have a fully worked out argument against IVF on this basis (though I do have other arguments against IVF here).

A difference between coital and non-coital reproduction

Non-coital reproduction (at least if it is to satisfy standard medical ethics rules) such as in-vitro fertilization requires the couple's consent, but does not require any specifically reproductive activity by the couple other than the giving of consent. Coital reproduction, on the other hand, requires the couple's consent, and more: it requires that the couple engage in coitus.

This may seem mistaken: After all, doesn't non-coital reproduction require the couple to provide gametes? Indeed, non-coital reproduction requires the couple's gametes. But the provision of the gametes does not require any reproductive action on the part of the couple besides the provision of consent. First, the gametes might be surgically extracted and that requires no activity on the part of the couple beside consent. Second, the gametes might be available to medical professionals due to earlier non-reproductive interventons, say because a cancerous ovary was removed.

And what about pregnancy? Isn't that an activity, even if not per se voluntary? Yes, it is an activity, but it is one that occurs after reproduction has already occurred. Once the gametes have fused into a new organism, reproduction has taken place.

Thus in non-coital reproduction, any reproductive activity by the couple besides the mere provision of consent is inessential. Thus, non-coital reproduction can be entirely outsourced: the couple need only be the source of gametes (some will say even this is not necessary, but I deny that the couple would be reproducing if their gametes are not used), but do not need to do anything to provide them.

This is a difference. Is it a morally significant difference? I think so, but I can't quite put my finger on it right now. A permissible form of reproduction just doesn't seem to me to be the sort of thing that could be outsourced without changing its essential nature.

First cause arguments of a Goedelian sort

We say that a proposition p precludes a proposition q provided that p and q cannot both be true. We say that x is purely good provided that x is good and there is no defect in x.

1. Necessarily, if x causes y, then it is impossible that y exists without x existing. (Follows from essentiality of origins.)
2. S5.
3. Being essentially the cause of every pure good other than perhaps oneself is a perfection.
4. If x is purely good and P is a perfection, then that something has P does not preclude that x exists and x is purely good and something has P.
5. Something is purely good. (For instance, a photon.)

Let x be the purely good thing. By (3) and (4), possibly x exists and x is purely good and something is essentially the cause of everything other than itself that is purely good. Call that something y. Then in some world w, y is the cause of x. By (1) and S5, y is actually the cause of x. By S5, y is also actually essentially the cause of every purely good thing other than itself. So:

6. There is something that is essentially the cause of every purely good thing other than perhaps itself.

We can run the argument without essentiality of origins with a tweak. Simply modify (3) to start "Being essentially the essential cause...", where the essential cause of something is a cause without which it cannot exist.

Whitaker's Words for Kindle

My wife asked me to get a Latin dictionary for my daughter's Kindle, so I dusted off the source files for my PalmOS Whitaker's Words dictionary, modified them a little and generated this dictionary. Enjoy!

Driving Roombas and other IR vehicles over WiFi

I've lately been having fun trying to get lots of the devices in the house to talk to each other. There is something gratifying in using an infrared transmitter from a broken helicopter, an old phone and a laptop to control a Roomba. :-) Here are instructions.

From tropes to the Trinity

Suppose that each cat has its own catness trope, each dog has its own dogness trope, and so on: each thing has its own essence trope. Thus, whenever we have two cats, we have two individuals—understood as bare particulars, bundles, substances, composite wholes, or in some other way—and two catnesses. Now, the question comes: When we say "There are two cats", is the content that

1. There are two individuals each of which is a cat, or that
2. There are two catnesses?

Whenever one is true, so is the other. Moreover, both statements are pretty natural. On a reference magnetism semantics on which the contents of our assertions are determined by charity plus naturalness, the choice between (1) and (2) will be a close one. In fact, it may be so close that it may be indeterminate whether "There are two cats" means (1) or (2). If naturalness favors either reading, it seems to favor (2) slightly, I think. So the choice between (1) and (2) will be a close one, and the answer will be either indeterminate or in favor of (2).

Likewise, does "Felix" refer to the individual or the catness? If the former, then "Felix sits" has the content that this individual has a sitting trope, and if the latter, then "Felix sits" has the content that this catness is coinstantiated with a sitting trope. Again, both candidates are pretty close to as natural. So again, the question whether names refer to individuals or their essence tropes will be a close one on reference magnetism, and maybe the answer will be indeterminate.

Suppose now that by a miracle there came to be two cat individuals that had the very same catness trope. Why not, after all? There doesn't appear to be anything logically contradictory about the supposition. After all, Siamese twins may share a heart. Why can't they share a trope as well?

So now you assign names, Felix and Tiger, and you say: "Felix is not Tiger." While it may have been indeterminate, or at least was close, whether ordinary cat names referred to catnesses or individuals, in the case at hand there is no indeterminacy or closeness. Charity now requires that at least these two names refer to the individuals, not the shared catness, since there is but one shared catness.

But while it is clear that we should say that Felix is not Tiger, it is less clear whether we should say that there are two cats. It is tempting to argue that Felix is a cat, Tiger is a cat, Felix and Tiger are two, so there are two cats. Certainly, there are two each of which is a cat. But are there two cats? Remember that it was indeterminate or close whether in general "There are two cats" referred to a duality of individuals or a duality of catnesses. If it should turn out to have in general referred to a duality of catnesses, then we should say "There is one cat". But even if it was indeterminate whether "There are two cats" refers to a duality of individuals or catnesses, we might reasonably in this very exceptional case settle on duality of catnesses. There is also a significant consideration in favor of describing the case by saying "There is one cat": it lets one say that there is exactly one catness trope per cat.

But now suppose we have three individuals, the Father, the Son and the Holy Spirit, who have one divinity trope. Then charity will make us say that the Father, the Son and the Holy Spirit are three individuals. Should we say that they are three Gods or that they are one God? If in ordinary cases we take counting to go with essence tropes, then we unambiguously have to go for the "one God" reading. If in ordinary cases, counting is indeterminate between individuals and their essence tropes, then we might in the case at hand simply reasonably resolve the ambiguity by speaking in the "one God" way, which then lets us say, as appears right, that there is exactly one divinity trope per God.

I am not defending this exact theory of the Trinity. One needs to be very cautious talking of individuals that have divinity due to divine simplicity. But I think something like the above is Aquinas' story.

God, love and creation

Consider this argument (inspired by a comment Dan Johnson made on this post):

1. God does everything out of love.
2. God creates Francis.
3. If God creates Francis out of love, he creates Francis out of self-love, out of love for Francis, or out of love for someone or something other than God or Francis.
4. God does not create Francis out of self-love.
5. God does not create Francis out of love for Francis.
6. God does not create Francis out of love for anyone or anything other than himself or Francis.
7. So, God does not create Francis out of love.
8. Contradiction!

Premise (4) is justified by the idea that our existence doesn't benefit the infinite, self-sufficient, transcendent and triune God. Premise (6), and to some degree (4) as well, is justified by Kantian thoughts about how Francis is an end in himself.

Premise (5), on the other hand, comes from the thought that if God creates Francis out of love for Francis, then God's love for Francis is explanatorily prior to his decision to create Francis. But insofar as God is loving Francis, God must have "already" (in the explanatory order) decided to create him, and so he can't be basing his decision to create on his love for Francis.

I am not confident about (4). God's extended, non-intrinsic, well-being (the kind of well-being that is constituted by the flourishing of our friends or the success of our projects) may be promoted by creating Francis, and because God is God, Kantian worries about Francis existing for God's sake are inappropriate.

But one might also question (1). When I bestow goods on my kids, that they are my kids, that they are in the image of God and that the goods are good yield one or two sufficient reasons to bestow the good. That I love my kids, while true, is a fact about me. That fact may give me additional reason to bestow goods on them, but that additional reason doesn't seem to me to be something that should be in the forefront of my mind. Love is more focused on the beloved than on one's own status as a love. Moreover, when I bestow the goods on my kids because they are my kids, because they are in the image of God, and because the goods are good, I act lovingly. My act of will is partly constitutive of my love for them.

Likewise, then, we can say that when God creates Francis, he does not have to do that because he loves Francis. Instead, God's creation of Francis is partly constitutive of that love. Thus, we have reason to revise (1) to:

9. Everything that God does, God does either out of love or it is partly constitutive of love.

We can now ask the interesting question: Can we procreate not out of love for the child, but in a way partly constitutive of love for that child? Notice a difference between us and God. While God's creation is essentially efficacious, our procreation is very chancy—indeed, most cases where people intend to procreate, they do not actually succeed (then). Moreover, our procreation is not tied to one person: it might be Francis who will result, but it might be someone else. So there will be worlds where Francis's parents do the same as they did in our world, but someone other than Francis comes about or even no one comes about. In those other worlds, they don't love Francis, since there is no Francis to love, and it doesn't even seem right to say that their act is partly consitutive of love for Francis (not just because it is not clear whether one can partly constitute something that doesn't obtain).

I suppose it could be the case, however, that it is a contingent matter that this particular procreative act is partly constitutive of love for Francis—in some worlds it's not partly constitutive of love for any child (maybe it's partly constitutive of love of spouse and love of God), and in others it's partly constitutive of love for another child. So, yes, perhaps the same thing can be said in our case as in God's case, with the difference that the divine creative act is essentially constitutive of love.

In an earlier post, I argued for a paradox in regard to human reproduction. Maybe the above considerations help? Maybe a couple can intentionally procreate in order to perform an act constitutive of love for a child? But I don't think that helps with the argument in that post, since performing an act constitutive of love for a child is either seen as good for the child, in which case premise 2 of that argument covers the case, or it does not (probably, it's seen as good for the couple), in which case premise 1 of that argument covers the case.

By the same token, saying that God's creative act is partly constitutive of God's love for the creature doesn't answer the question of why God creates the creature. I still think the answer that he does so for the good of the creature, which I give in the aforementioned post, can be defended. As the Catechism says, we are created to know and love God. And that's good for us.

Explanation modulo a fact

A notion that I find rather natural is the idea of explanation modulo a fact. This notion can do some of the work that contrastive explanation can.

1. Why did you eat a banana?

can be precisified contrastively in ways like:

2. Why did you eat a banana rather than something else?
3. Why did you eat a banana rather than doing something else with it?
4. Why did you eat a banana rather than not eating anything?

But one can also accomplish the same thing this with requests for explanation modulo a fact. Doing the work of 2 and 3 is easy:

5. Given that you ate, why did you eat a banana?
6. Given that you did something with a banana, why did you eat it?

Doing the work of 4 is a bit harder, but maybe this works:

7. Given that you ate a banana if you ate anything, why did you eat a banana?

When we ask for explanation why p given q, we are asking for something that isn't just an explanation why q, and that gives us a further explanation why p when added to the fact taht q. The question presupposes there is such an additional truth. If the fact that q is the one and only complete explanation why p, then the question has a false presupposition.

It seems, however, that any request for explanation modulo a fact can be rephrased as a request for contrastive explanation. For instead of asking:

8. Given q, why p?

one can ask:

9. Why p and q rather than not-p and q?

This makes explanation modulo a fact not very useful, it seems. However, one can also formulate contrastive questions as requests for explanation modulo a fact. Thus the contrastive question:

10. Why p₁ rather than p₂, p₃, ... or p_n?

can be replaced with:

11. Given p₁, p₂, p₃, ... or p_n, why p₁?

So which is more fundamental? Explanation-modulo or contrastive explanation? I don't know. The literature prefers contrastive explanation, but that may be a historical accident.

In any case, I think all this makes an important fact about contrastive explanation clearer, a fact that I learned from conversation with Dan Johnson: When you make a request for a contrastive explanation, you are also indicating a refusal to accept certain kinds of answers. In the explanation-modulo case, these are answers that merely go through the given fact. But then the defender of the Principle of Sufficient Reason should feel no embarrassment about being unable to give answers to all requests for contrastive explanation—after all, that every fact has an explanation does not make for any kind of guarantee that the explanation will be of the sort we want. Once we start ruling out certain kinds of explanations, we might well be left with none.

Thomson's lamp and the Axiom of Choice

Consider Thomson's lamp: a lamp with a pushbutton switch that toggles it on and off. The lamp starts in the off position, and then in the next half minute the button is pressed, and in the next quarter it is pressed again, and then in the neight eighth again, and so on. Then at the end of the supertask, the lamp is either on or off.

Now keep the lamp but change the story. During each of the ever shorter intervals, a coin is flipped and the switch is pressed if it lands heads, and not pressed if it lands tails. Moreover, the final state of the lamp depends on the results of the coin flips in the following ways:

1. The results of the coin flips determine the final state of the lamp.
2. For any sequence of coin flip results, if any one (and only one) coin flip had a different, the lamp's final state would have been different, too.

Surprisingly, the existence of a lamp that would work in this way implies a version of the Axiom of Choice. To see this, notice that if the coin flips are independent and fair, then the subset of the probability space where the lamp's final state is on is nonmeasurable.[note 1]

But of course, on some technical assumptions, the existence of a nonmeasurable set requires a version of the Axiom of Choice. So if we read the Thomson's lamp story in such a way that the final outcome is determined by which presses are made and which aren't, in such a way that changing a single press changes the final outcome, that story seems to commit us to a version of the Axiom of Choice.

Conversely, it is easy to use the Axiom of Choice for pairs to prove the existence of a function such as would be implemented by the lamp.[note 2]