Wednesday, March 27, 2024

Knowledge of qualia

Suppose epiphenomenalism is true about qualia, so qualia are nonphysical properties that have no causal impact on anything. Let w0 be the actual world and let w1 be a world which is exactly like the actual world, except that (a) there are no qualia (so it’s a zombie world) and (b) instead of qualia, there are causally inefficacious nonphysical properties that have a logical structure isomorphic to the qualia of our world, and that occur in the corresponding places in the spatiotemporal and causal nexuses. Call these properties “epis”.

The following seems pretty obvious to me:

  1. In w1, nobody knows about the epis.

But the relationship of our beliefs about qualia to the qualia themselves seems to be exactly like the relationship of the denizens of w1 to the epis. In particular, neither are any of their beliefs caused by the obtaining of epis, nor are any of our beliefs caused by the obtaining of qualia, since both are epiphenomenal. So, plausibly:

  1. If in w1, nobody knows about the epis, then in w0, nobody knows about the qualia.


  1. Nobody knows about the qualia.

But of course we do! So epiphenomenalism is false.

Tuesday, March 26, 2024

Today's sunspots

Today I was testing the solar filter I got for the eclipse. (300mm f/5.6, cropped).

Brains, bodies and souls

There are four main families of views of who we are:

  1. Bodies (or organisms)

  2. Brains (or at least cerebra)

  3. Body-soul composites

  4. Souls.

For the sake of filling out logical space, and maybe getting some insight, it’s worth thinking a bit about what other options there might be. Here is one that occurred to me:

  1. Brain-soul (or cerebrum-soul) composites.

I suppose the reason this is not much (if at all) talked about is that if one believes in a soul, the body-soul composite or soul-only views seem more natural. Why might one accept a brain-soul composite view? (For simplicity, I won’t worry about the brain-cerebrum distinction.)

Here is one line of thought. Suppose we accept some of the standard arguments for dualism, such as that matter can’t be conscious or that matter cannot think abstract thoughts. This leads us to think the mind cannot be entirely material. But at the same time, there is some reason to think the mind is at least partly material: the brain’s activity sure seems like an integral part of our discoursive thought. Thus, the dualist might have reason to say that the mind is a brain-soul composite. At the same time, there is a Cartesian line of thought that we should be identified with the minimal entity hosting our thoughts, namely the mind. Putting all these lines of thought together, we conclude that we are minds, and hence brain-soul composites.

Now I don’t endorse (5). The main ethical arguments against (2) and (4), namely that they don’t do justice to the deep ethical significance of the human body, apply against (5) as well. But if one is not impressed by these arguments, there really is some reason to accept (5).

Furthermore, exploring new options, like the brain-soul composite option, sometimes may give new insights into old options. I am now pretty much convinced that the mind is something like the brain plus soul (or maybe cerebrum plus intellectual part of soul or some other similar combination). Since it is extremely plausible that all of my mind is a part of me, this gives me a new reason to reject (4), the view that I am just a soul. At the same time, I do not think it is necessary to hold that I am just a mind, so I can continue to accept view (3).

The view that the mind is the brain plus soul has an interesting consequence for the interim state, the state of the human being between death and the resurrection of the body. I previously thought that the human being in the interim state is in an unfortunately amputated state, having lost all of the body. But if we see the brain as a part of the mind, the amputated nature of the human being in the interim state is even more vivid: a part of the human mind is missing in the interim state. This gives a better explanation of why Paul was right to insist on the importance of the physical resurrection—we cannot be fully in our mind without at least some of our physical components.


Unfortunately, most of the forms of exercise I do are too intense for me to think hard while exercising, though perhaps my subconscious is doing something. The main exception is that when swimming, I can do some thinking, but I am also counting lengths, and I can’t do both at once very well. (I have a project that I keep on putting off where I’d interface a BLE beacon on my person with a phone out of the water and use that to count lengths, but I haven’t done it yet. I suppose I could also get a watch that counts lengths.) Occasionally, I can also do some less deep thinking—say, preparing for class—while biking on flat pavement. But I can’t do serious thinking while rock climbing (however, there is good down time for thinking and writing between climbs), or playing badminton, or kayaking.

Last week, Baylor had a step challenge, so I ended up taking some longer brisk walks, alone. (I walk a fair amount with family.) It reminded me of how it is possible to do a lot of thinking while walking. That’s really nice! Though there is the danger that my achievement-oriented personality will push me to keep on increasing my walking speed, to the point where I won't be able to do deep thinking any more.

Monday, March 25, 2024

Representation and truth

For a while, I’ve been thinking of a teleological/normative account of representation. The basic idea is that:

  1. State S represents reality being such that r if and only if one’s teleology specifies that one should be in state S only if r.

But I’ve also been worried that this makes representation much too common in the world. If a bacterium’s nature says that some behavior that should only be triggered under some circumstances, then on this account, the bacterium’s behavior represents the occurrence of these circumstances.

I am kind of willing to bite that bullet. But perhaps I don’t need to.

For a long time I’ve been sensitive to the difference between a proposition p and the second-order proposition that p is true, but this sensitivity has largely been a matter of nitpicking. But today I realized that this distinction may help save the teleological account of normativity with a very small tweak:

  1. State S represents a proposition p if and only if one’s teleology specifies that one should be in state S only if p is true.

It is plausible that only higher organisms have a teleology that makes reference to truth as such.

Remark 1: If we want, we can have both (1) and (2) by distinguishing between “simple representation” and “alethic representation”. Alethic representation is then related to simple representation as follows:

  1. State S alethically represents reality being such that r if and only if S simply represents reality being such that it is true that r.

Remark 2: Given Leon Porter’s argument that truth is not a physical property, it is interesting to note that on the alethic version, representation requires a being that has normative properties that make reference to something nonphysical. In particular, this kind of normativity cannot be grounded in evolution.

Identity and eternity

Suppose that you are an immortal who lived for an infinite amount of time, and each year, your body replaces all its cells with new cells constructed from the matter in your food. Furthermore, you only eat local food, and at the beginning of each year it is randomly chosen by a coin toss whether you will live in Australia or America. Moreover, in the world we are imagining, the food in Australia and America has no matter in common.

Consider these two plausible principles:

  1. If x and y are people living in worlds w1 and w2, respectively, and at no time t in their lives do they have any matter in common, then x ≠ y.

  2. The identity of an already existing person never depends on what will happen to that person in the future.

But now whether you exist in year n does not depends on what happens in year n, since you are immortal and by (2) your identity was already determined in year n − 1. By the same token, whether you exist in year n does not depend on what happens in year n − 1, and so on. In particular, it follows that whether you exist now does not depend on any particular coin toss. However, by (1) whether you exist does depend on the totality of the coin tosses, since if all the coin tosses go differently from how they actually do, the matter in the body would always be different, and hence by (1) the person would be different.

But it is quite paradoxical that your existence depends on the coin tosses collectively and yet each one is irrelevant. This points to the hypothesis that beings that are significantly changeable cannot be eternal (and slightly supports causal finitism).

If you think that your identity also depends on your memories, add that in Australia and America you form different memories. If you think that your identity depends on your soul, then instead of running the argument about a human being, run it against something soulless.

If you think all complex objects have something like soul (as I do), the argument may not impress.

Friday, March 22, 2024

Tables and organisms

A common-sense response to Eddington’s two table problem is that a table just is composed of molecules. This leads to difficult questions of exactly which molecules it is composed of. I assume that at table boundaries, molecules fly off all the time (that’s why one can smell a wooden table!).

But I think we could have an ontology of tables where we deny that tables are composed of molecules. Instead, we simply say that tables are grounded in the global wavefunction of the universe. We then deny precise localization for tables, recognizing that nothing is localized in our quantum universe. There is some approximate shape of the table, but this shape should not be understood as precise—there is no such thing as “the set of spacetime points occupied by the table”, unless perhaps we mean something truly vast (since the tails of wavefunctions spread out very far very fast).

That said, I don’t believe in tables, so I don’t have skin in the game.

But I do believe in organisms. Similar issues come up for organisms as for tables, except that organisms (I think) also have forms or souls. So I wouldn’t want to even initially say that organisms are composed of molecules, but that organisms are partly composed of molecules (and partly of form). That still generates the same problem of which exact molecules they are composed of. And in a quantum universe where there are no sharp facts about particle number, there probably is no hope for a good answer to that question.

So maybe it would be better to say that organisms are not even partly composed of molecules, but are instead partly grounded in the global wavefunction of the universe, and partly in the form. The form delineates which aspects of the global wavefunction are relevant to the organism in question.

Monday, March 18, 2024

Simplicity and Newton's inverse square law

When I give talks about the way modern science is based on beauty, I give the example of how everyone will think Newton’s Law of Gravitation

  1. F = Gm1m2/r2

is more plausible than what one might call “Pruss’s Law of Gravitation”

  1. F = Gm1m2/r2.00000000000000000000000001

even if they fit the observation data equally, and even if (2) fits the data slightly better.

I like the example, but I’ve been pressed on this example at least once, because I think people find the exponent 2 especially plausible in light of the idea of gravity “spreading out” from a source in concentric shells whose surface areas are proportional to r2. Hence, it seems that we have an explanation of the superiority of (1) to (2) in physical terms, rather than in terms of beauty.

But I now think I’ve come to realize why this is not a good response to my example. I am talking of Newtonian gravity here. The “spreading out” intuition is based on the idea of a field of force as something energetic coming out of a source and spreading out into space around it. But that picture makes little sense in the Newtonian context where the theory says we have instantaneous action at a distance. The “spreading out” intuition makes sense when the field of force is emanating at a uniform rate from the source. But there is no sense to the idea of emanation at a uniform rate when we have instantaneous action at a distance.

The instantaneous action at a distance is just that: action at a distance—one thing attracting another at a distance. And the force law can then have any exponent we like.

With General Relativity, we’ve gotten rid of the instantaneous action at a distance of Newton’s theory. But my point is that in the Newtonian context, (1) is very much to be preferred to (2).

Beauty and simplicity in equations

Often, the kind of beauty that scientists, and especially physicists, look for in the equations that describe nature is taken to have simplicity as a primary component.

While simplicity is important, I wonder if we shouldn’t be careful not to overestimate its role. Consider two theories about some fundamental force F between particles with parameters α1 and α2 and distance r between them:

  1. F = 0.8846583561447518148493143571151840833168115852975428057361124296α1α2/r2

  2. F = 0.88465835614475181484931435711518α1α2/r2 + 2−64.

In both theories, the constants up front are meant to be exact and (I suppose) have no significantly more economical expression. By standard measures of simplicity where simplicity is understood in terms of the brevity of expression, (2) is a much simpler theory. But my intuition is that unless there is some special story about the significance of the 2 + 2−64 exponent, (1) is the preferable theory.

Why? I think it’s because of the beauty in the exponent 2 in (1) as opposed to the nasty 2 + 2−64 exponent in (2). And while the constant in (2) is simpler by about 106 bits, that additional simplicity does not make for significantly greater beauty.

Friday, March 15, 2024

A tweak to the Turing test

The Turing test for machine thought has an interrogator communicate (by typing) with a human and a machine both of which try to convince the interrogator that they are human. The interrogator then guesses which is human. We have good evidence of machine thought, Turing claims, if the machine wins this “imitation game” about as often as the human. (The original formulation has some gender complexity: the human is a woman, and the machine is trying to convince the interrogator that it, too, is a woman. I will ignore this complication.)

Turing thought this test would provide a posteriori evidence that a machine can think. But we have a good a priori argument that a machine can pass the test. Suppose Alice is a typical human, so that in competition with other humans she wins the game about half the time. Suppose that for any finite sequence Sn of n questions and n − 1 answers of reasonable length (i.e., of a length not exceeding how long we allow for the game—say, a couple of hours) ending on a question that could be a transcript of the initial part of an interrogation of Alice, there is a fact of the matter as to what answer Alice would make to the last question. Then there is a possible very large , but finite, machine that has a list of all such possible finite sequences and the answers Alice would make, and that at any point in the interrogation answers just as Alice would. That machine would do as well as Alice at the imitation game, so it would pass the Turing test.

Note that we do not need to know what Alice would say in response to the last question of Sn. The point isn’t that we could build the machine—we obviously couldn’t, just because the memory capacity required would be larger than the size of the universe—but that such a machine is possible. We could suppose constructing the database in the machine at random and just getting amazingly lucky and matching Alice’s dispositions.

The machine would not be thinking. Matching the current stage in the interrogation to the database and just giving the item in the line for that is not thinking. The point is obvious. Suppose that S1 consists of the question “What is the most important thing in life?” and the database gives the rote answer “It is living in such a way that you have no regrets.” It’s obvious that the machine doesn’t know what it’s saying.

Compare this to a giant chess playing machine which encodes for each of the 1040 legal chess positions the optimal next move. That machine doesn’t think about playing chess.

If the Turing test is supposed to be an a posteriori test for the possibility of machine intelligence, I propose a simple tweak: We limit the memory capacity of the machine to be within an order of magnitude of human memory capacity. This avoids cases where the Turing test is passed by rote recitation of responses.

Turing himself imagined that doing well in the imitation game would require less memory capacity than the human brain had, because he thought that only “a very small fraction” of that memory capacity was used for “higher types of thinking”. Specifically, Turing surmised that 109 bits of memory would suffice to do well in the game against “a blind man” (presumably because it would save the computer from having to have a lot of data about what the world looks like). So in practice my modification is one that would not decrease Turing’s own confidence in the passability of his test.

Current estimates of the memory capacity of the brain are of the order of 1015 bits, at the high end of the estimates in Turing’s time (and Turing himself inclined to the low end of the estimates, around 1010). The model size of GPT-4 has not been released, but it appears to be near but a little below the human brain capacity level. So if something with the model size of GPT-4 were to pass the Turing test, it would also pass the modified Turing test.

Technical comment: The above account assumed there was a fact about what answer Alice would make in a dialogue that started with Sn. There are various technical issues with regard to this. Given Molinism or determinism, these technical issues can presumably be overcome (we may need to fix the exact conditions in which Alice is supposed to be undergoing the interrogation). If (as I think) neither Molinism nor determinism is true, things become more complicated. But there are presumably to be statistical regularities as to what Alice is likely to answer to Sn, and the machine’s database could simply encode an answer that was chosen by the machine’s builders at random in accordance with Alice’s statistical propensities.

Wednesday, March 13, 2024

Do you and I see colors the same way?

Suppose that Mary and Twin Mary live almost exactly duplicate lives in an almost black-and-white environment. The exception to the duplication of the lives and to the black-and-white character of the environment is that on their 18th birthday, each sees a colored square for a minute. Mary sees a green square and Twin Mary sees a blue square.

Intuitively, Mary and Twin Mary have different phenomenal experiences on their 18th birthday. But while I acknowledge that this is intuitive, I think it is also deniable. We might suppose that they simply have a “new color” experience on their 18th birthday, but it is qualitatively the same “new color” experience. Maybe what determines the qualitative character of a color experience is not the physical color that is perceived, but the relationship of this color to the whole body of our experience. Given that green and blue have the same relationship to the other (i.e., monochromatic) color experiences of Mary and Twin-Mary, it may be that they appear the same way.

If this kind of relationalism is correct, then it is very likely that when you and I look at the same blue sky, our experiences are qualitatively different. Your phenomenal experience is defined by its position in the network of your experiences and mine is defined by its position in the network of my experiences. Since these networks are different, the experiences are different. Somehow I find this idea somewhat plausible. It is even more plausible some experiences other than colors. Take tastes and smells. It’s not unlikely that fried cabbage tastes differently to me because in the network of my experiences it has connections to experiences of my grandmother’s cooking that it does not have in your network.

Such a relationalism could help explain the wide variation in sensory preferences. We normally suppose that people disagree on which tastes they like and dislike. But what if they don’t? What if instead the phenomenal tastes are different? What if banana muffins, which I dislike, taste differently to me than they do to most people, because they have a place in a different network of experiences, and if banana muffins tasted to me like they do to you, I would like them just as much?

In his original Mary thought experiment, Jackson says that monochrome Mary upon experiencing red for the first time learns what experience other people were having when they saw a red tomato. If the above hypothesis is right, she doesn’t learn that at all. Other people’s experiences of a red tomato would be very different from Mary’s, because Mary’s monochrome upbringing would place the red tomato in a very different network of experiences from that which it has in other people’s networks of experiences. (I don’t think this does much damage to the thought experiment as an argument against physicalism. Mary still seems to learn something—what it is to have an experience occupying such-and-such a spot in her network of experiences.)

More fun with monochrome Mary

Here’s a fun variant of the black-and-white Mary thought experiment. Mary has been brought up in a black-and-white environment, but knows all the microphysics of the universe from a big book. One day she sees a flash of green light. She gains the phenomenal concept α that applies to the specific look of that flash. But does Mary know what green light looks like?

You might think she knows because her microphysics book will inform her that on such-and-such a day, there was a flash of green light in her room, and so she now knows that a flash of green light has appearance α. But that is not quite right. A microphysics book will not tell Mary that there was a flash of green light in her room. It will tell her that there was a flash of green light in a room with such-and-such physical properties. Whether she can deduce from these properties and her observations that this was her room depends on what the rest of the universe is like. If the universe contains Twin Mary who lives in a room with exactly the same monochromatically observable properties as Mary’s room, but where at the analogous time there is a flash of blue light, then Mary will have no way to resolve the question of whether she is the woman in the room with the green flash or in the room with the blue flash. And so, even though Mary knows all the microphysical facts about the world, Mary doesn’t know whether it is a green flash or a blue flash that has appearance α.

This version of the Mary thought experiment seems to show that there is something very clear, specific and even verbalizable (since Mary can stipulate a term in her language to express the concept α, though if Wittgenstein is right about the private language argument, we might require a community of people living in Mary’s predicament) that can remain unknown even when one knows all the microphysical facts and has all the relevant concepts and has had the relevant experiences: Whether it is green or blue light that has appearance α?

This seems to do quite a bit of damage to physicalism, by showing that the correlation between phenomenal appearances and physical facts is a fact about the world going beyond microphysics.

But now suppose Joan lives on Earth in a universe which contains both Earth and Twin Earth. The denizens of both planets are prescientific, and at their prescientific level of observation, everything is exactly alike between Earth and Twin Earth. Finer-grained observation, however, would reveal that Earth’s predominant surface liquid is H2O while Twin Earth’s is XYZ, but currently there is no difference. Now, Joan reads a book that tells her in full detail all the microphysical structure of the universe.

Having read the book, Joan wonders: Is water H2O or is it XYZ? Just by reading the book, she can’t know! The reason she doesn’t know it is because her prescientific observations combined with the contents of the book are insufficient to inform her whether she lives on Earth or on Twin Earth, whether she is Joan or Twin Joan, and hence are insufficient to inform her whether the liquid she refers to as “water” is H2O or XYZ.

But surely this shouldn’t make us abandon physicalism about water!

Now Joan and Twin Joan both have concepts that they verbalize as “water”. The difference between these concepts is entirely external to Joan and Twin Joan—the difference comes entirely from the identity of the liquid interaction with which gave rise to the respective concepts. The concepts are essentially ostensive in their differences. In other words, Joan’s ignorance of whether water is H2O or XYZ is basically an ignorance of self-locating fact: is she in the vicinity of H2O or in the vicinity of XYZ.

Is this true for Mary and Twin Mary? Can we say that Mary’s ignorance of whether it is a green or a blue flash that has appearance α is essentially an ignorance of self-locating facts? Can we say that the difference between Mary’s phenomenal concept formed from the green flash and Twin Mary’s phenomenal concept formed from the blue flash is an external difference?

Intuitively, the answer to both questions is negative. But the point is not all that clear to me. It could turn out that both Mary and Twin Mary have a purely comparative recognitive concept of “the same phenomenal appearance as that flash”, together with an ability to recognize that similarity, and with the two concepts being internally exactly alike. If so, then the argument is unconvincing as an argument against physicalism.

Tuesday, March 12, 2024

The epistemic gap and causal closure

In the philosophical literature, the main objection to physicalism about consciousness is the epistemic gap: the alleged fact that full knowledge of the physical does not yield full knowledge of the mental. And one of the main objections to nonphysicalism about consciousness is causal closure: the alleged fact that physical events, like our actions, have causes that are entirely physical.

There is a simple way to craft a theory that avoids both objections. Simply suppose that mental states have two parts: a physical and a non-physical part. The physical part of the mental state is responsible for the mental state’s causal influence on physical reality. The non-physical part explains the epistemic gap: full knowledge of the physical world yields full knowledge of the physical part of the mental state, but not full knowledge of the mental state.

Monday, March 11, 2024

Trust versus prediction

What is the difference between trusting that someone will ϕ and merely predicting their ϕing?

Here are two suggestions that don’t quite pan out.

1. In trusting, you have to have a pro-attitude towards ϕing. But this is false. One can trust a referee will make a fair decision even when one hopes they will make a decision that favors one instead. And you can trust that someone who promised you a punishment will mete it out if you deserve it even if you would rather they didn’t.

2. In trusting, you rely on the person’s ϕing. But this is not always true. A promised benefit might be such that it doesn’t affect any of your actions, but you can still trust you will receive it.

But here is an idea I like. In trusting, you believe that the person will intentionally ϕ as part of her proper functioning, and you believe this on account of the person’s possessing the relevant proper functional disposition. In central cases, “proper functioning” can be replaced with “expression of virtue”, but trust can include non-moral proper function.

A consequence of this account is that it is impossible to trust someone to do wrong, since wrongdoing is never a part of a person’s proper functioning. For trust-based theories of promises, this makes it easy to see why promises to do wrong are null and void: for it makes no sense to solicit trust where trust is impossible.

This account of trust gives a nice extended sense of trust in things other than people. Just drop “intentionally” and “person”. In an extended sense, you can trust a dog, a carabiner, a book, or anything else that has a proper function. This seems right: we certainly do talk of trust in this extended sense.

Consent, desire and promises

I have long argued that desire is not the same as consent: the fact that I want you to do something does not constitute consent to your doing it.

Here is a neat little case that has occurred to me that seems to show this conclusively. Alice borrowed a small sum of money from me, and the return is due today. However, I know that I have failed Alice on a number of occasions, and I have an unpleasant feeling of moral envy as to how she has always kept to her moral commitments. I find myself fantasizing about how nice it would feel to have Alice fail me on this occasion! It would be well worth the loss of the loan not to “have to” feel guilt about the times I failed Alice.

But now suppose that Alice knows my psychology really well. Her knowing that I want her to fail to return the money is no excuse to renege on her promise.

There are milder and nastier versions of this. A particularly nasty version is when the promisee wants you to break a promise so that you get severely punished: one thinks here of Shylock in the Merchant of Venice. A mildish (I hope) version is where I am glad when people come late to meetings with me because it makes me feel better about my record of unpunctuality.

Or for a very mild version, suppose that I typically come about a minute late to appointments with you. You inductively form the belief that I will do so this time, too. And it is a pleasure to have one’s predictions verified, so you want me to be late.

The above examples also support the claim that we cannot account for the wrong of promise-breaking in terms of overall harm to the promisee. For we can tweak some of these cases to result in an overall benefit to the promisee. Let’s say that I feel pathologically and excessively guilty about all the times I’ve been late to appointments, and your breaking your promise to show up at noon will make me feel a lot better. It might be that overall there is a benefit from your breaking the promise. But surely that does not justify your breaking the promise.

Or suppose that in the inductive case, the value of your pleasure in having your predictions verified exceeds the inconvenience of waiting a minute.

Objection: Promises get canceled in the light of a sufficiently large benefit to the promisee.

Response: The above cases are not like that. For the benefit of relief of my guilt requires that you break the promise, not that the promise be canceled in light of a good to me. And the pleasure of verification of predictions surely is insufficient to cancel a promise.

Promising punishment

I have long found promises to punish puzzling. The problem with such promises is that normally a promisee can release the promisor from a promise. But what’s the point of me promising you a punishment should you do something if you can just release me from the promise when the time for the promise comes?

Scanlon’s account of promising also faces another problem with promises to punish: Scanlon requires that the promisee wants to be assured of the promised action. But of course in many cases of promising a punishment, the promisee does not want any such assurance! (There are some cases when they do, say when they recognize the benefit of being held to account for something.)

Additionally, it seems that breaking a promise is wrong because of the harm to the promisee. But it is commonly thought that escaping punishment is not a harm. Here I am inclined to follow Boethius, however, who insisted that a just punishment is intrinsically good for one. But suppose we follow common sense rather than Boethius, or perhaps we are dealing with a case where the norm whose violation gains a punishment is not a moral norm.

Then there is still something interesting we can say. Let’s say that I promise you a punishment for some action, and you perform that action, but I omit the punishment. Even if the omission of the punishment is not a harm, you might feel a resentment that in your choice of activity you had to take my prospective punishment into account but I wasn’t going to follow-through on the punishment. There is something unfair about this. Perhaps the point is clearest in a case like this: I promise you a punishment each time you do something. Several times you hold yourself back due to fear of punishment, and then finally you do it, and out of laziness I don’t to punish. You then feel: “Why did I even bother to keep to the rule earlier?”

But note that even in a case like this, it seems better to locate the harm in my making of the promise if I wasn’t going to keep it than in the non-keeping of it. So, let’s suppose that the Boethius line of thought doesn’t apply, and suppose that I am now deciding whether to perform the onerous task of punishing you as per promise. What moral reason do I have to punish you now in light of the promise? Well, there are considerations having to do with future cases: if I don’t do it now, you won’t trust me in the future, etc. But we can suppose all such future considerations are irrelevant—maybe this is the last hour of my life. So why is it that I should punish you?

I think there are two mutually-compatible stories one can tell. One story is an Aristotelian one: it’s simply bad for my will that I not keep my promise. The other story is a trust-based one: I solicited your trust, and even if you want me to break trust with you, I have no right to betray your trust. Having one’s trust betrayed is in itself a harm, regardless of whether one is trusting someone to do something that is otherwise good or bad for one.

The Laws of Promising

On a conventionalist theory of promises, there is a social institution of promising, somewhat akin to a game, and a promise is a kind of communicative action that falls under the rules of that institution. But what makes a communicative action fall under the rules of the promissory institution? Well, one of the generally agreed on necessary conditions is that it must be intentional. So now it seems that a part of what makes something a promise is that it be intended to fall under the rules of the promissory institution. And this itself is a rule of the promissory institution.

Thus, the promissory institution needs to make reference to itself in its rules. Is this a vicious circularity?

Maybe not. The Laws of Badminton govern players of badminton. Indeed, the Definitions in the Laws start with: “Player: Any person playing Badminton”. Badminton is nothing but the game governed by these rules, and yet the rules constantly make reference to badminton via the concept of a player (and occasionally make explicit self-reference, as in law 17.6.1 that an umpire shall “uphold and enforce the Laws of Badminton”). Is this a vicious circularity? Here is a reason to think it is not. People can coherently decide to play the game defined by a set of rules referred to under some description such as “The rules posted on” or “The rules customarily in use in this club” or “The Laws of Badminton” or “The rules adopted by the Badminton World Federation” that in fact refers to the same set of rules. The rules can refer to themselves under some of these descriptions as well. We can then suppose that a player is someone who is achieving some measure of minimal success in intentionally following the rules under some such description.

The way to avoid vicious circularity here is that one needs some way of gaining reference to the rules from within the rules, and one can do so by means of an appropriate expression typically having to do with a physical embodiment of the rules, say in an inscription or in a customary practice.

Can make the same move with regard to promises? We could image a group of early humans sitting around and making up “the Laws of Promising” prior to any promises being made, with the Laws of Promising referencing themselves under some description like “The Laws promulgated in the Cave of the Lone Bear on the third full moon since the melting of the snow in the fourth year of the chiefdom of Jas the Bald.” And then the laws could cover communicative actions intended to fall under the Laws of Promising under some relevant description or other. But while we can imagine this, it is highly implausible as a historical claim.

I want to offer a weird alternative to the institutional theory of promises. Let’s first imagine that in your head there is a literal “book of promises” (made of waterproof paper, etc.), and that you can inscribe text in that book using a little pen that moves around in your head. But suppose that moving the pen is not a basic action. The only way to write p in the book of promises is to intentionally communicate to another person that you are inscribing p in the book. Such intentional communication causes, by some weird law of nature, the inscription of p into the book of promises. And then we suppose that it is a fundamental moral law that anything inscribed in the book of promises is to be done, subject to various nuances.

On this account, promising p is inscribing p into the book by intentionally communicating that you are inscribing p into the book. But note that you are not intending to promise: you are intending to inscribe into the book, which is different. So there is no circularity. (Compare here a mind-reading machine which serves you lunch if you press a button with the intention of getting lunch from the machine. There is no circularity.)

Is there such a book? A tempting simple thought is that there is: it is our memory. But that’s not right. Promises are normatively binding even if they are not remembered, though if they innocently forgotten one is typically not culpable for breaking them.

A dualist can suppose that the soul really does contain something like a book of promises, which is not directly available to introspection. When you make a promise, the content is “inscribed” into the “promise book”, and remembered as being inscribed. There is no other way to put things into the soul’s “promise book”, though if there is a God, he could miraculously inscribe things in the book. (Would we then be required to fulfill them? Well, it depends on what the moral rule is. If it says that one must do everything in the book, then we would be required to fulfill what God wrote in the book. But if it only says that one must do everything that one inscribed in the book, then what God inscribed in it may not need to be done.)

Tuesday, March 5, 2024


It is commonly thought that to engage in a speech of a particular sort—assertion, request, etc.—one needs to intend to do so.

But suppose you ask me a question, and I unintentionally blurt out an answer, even though the matter is confidential. Can you correctly tell people that I answered your question, that I asserted whatever it was that I blurted out?

If yes, then one does not need to intend to engage in a speech act of a particular sort in order for that speech act to occur.

But I suspect the that in unintentionally blurting one does not answer or assert. One reason is that if one was answering or asserting, then it seems that one could also unintentionally blurt out a lie. (Imagine that you have a habit of answering a certain question with a falsehood, and you blurt out a falsehood purely out of habit.) But I don’t think a lie can be unintentional.

Moreover, if someone asserts, then what they say is presented for trust. But what is said unintentionally is not presented for trust.

I am not very confident of the above.

Friday, March 1, 2024

Comparing sizes of infinite sets

Some people want to be able to compare the sizes of infinite sets while preserving the proper subset principle that holds for finite sets:

  1. If A is a proper subset of B, then A < B.

We also want to make sure that our comparison agrees with how we compare finite sets:

  1. If A and B are finite, then A ≤ B if and only if A has no more elements than B.

For simplicity, let’s just work with sets of natural numbers. Then there is a total preorder (total, reflexive and transitive relation) ≤ on the sets of natural numbers (or on subsets of any other set) that satisfies (1) and (2). Moreover, we can require the following plausible weak translation invariance principle in addition to (1) and (2):

  1. A ≤ B if and only if 1 + A ≤ 1 + B,

where 1 + C is the set C translated one unit to the right: 1 + C = {1 + n : n ∈ C}. (See the Appendix for the existence proofs.) So far things are sounding pretty good.

But here is another plausible principle, which we can call discreteness:

  1. If A and C differ by a single element, then there is no B such that A < B < C.

(I write A < B provided that A ≤ B but not B ≤ A.) When two sets differ by a single element, intuitively their sizes should differ by one, and sizes should be multiples of one.

Fun fact: There is no total preorder on the subsets of the natural numbers that satisfies the proper subset principle (1), the weak translation invariance principle (3) and the discreteness principle (4).

The proof will be given in a bit.

One way to try to compare sets that respects the subset principle (1) would be to use hypernatural numbers (which are the extension of the natural numbers to the context of hyperreals).

Corollary 1: There is no way to assign a hypernatural number s(A) to every set A of natural numbers such that (a) s(A) < s(B) whenever A ⊂ B, (b) s(A) − s(B) = s(1+A) − s(1+B), and (c) if A and B differ by one element, then |s(A)−s(B)| = 1.

For if we had such an assignment, we could define A ≤ B if and only if s(A) ≤ s(B), and we would have (1), (3) and (4).

Corollary 2: There is no way to assign a hyperreal probability P for a lottery with tickets labeled with the natural numbers such that (a) each individual ticket has equal non-zero probability of winning α, (b) P(A) − P(B) and P(1+A) − P(1+B) are always either both negative, both zero, or both positive, and (c) no two distinct probabilities of events differ by less than α.

Again, if we had such an assignment, we could define A ≤ B if and only if P(A) ≤ P(B), and we would have (1), (3) and (4).

I will now prove the fun fact. The proof won’t be the simplest possible one, but is designed to highlight how wacky a total preorder that satisfies (1) and (4) must be. Suppose we have such a total preorder ≤. Let An be the set {n, 100 + n, 200 + n, 300 + n, ...}. Observe that A100 = {100, 200, 300, 400, ...} $ is a proper subset of A0 = {0, 100, 200, 300, ...}, and differs from it by a single element. Now let’s consider how the elegant sequence of shifted sets A0, A1, ..., A100 behaves with respect to the preorder ≤. Because An + 1 = 1 + An, if we had (3), the order relationship between successive sets in the series would always be the same. Thus we would have exactly one of these three options:

  1. A0 ≈ A1 ≈ ... ≈ A100

  2. A0 < A1 < ... < A100

  3. A0 > A1 > ... > A100,

where A ≈ B means that A ≤ B and B ≤ A. But (i) and (ii) each contradict (1), since A100 is a proper subset of A0, while (iii) contradicts (4) since A0 and A100 differ by one element.

This completes the proof. But we can now think a little about what the ordering would look like if we didn’t require (3). The argument in the previous paragraph would still show that (i), (ii) and (iii) are impossible. Similarly, A0 ≥ A1 ≥ ... ≥ A100 is impossible, since A100 < A0 by (1). That means we have two possibilities.

First, we might have A0 ≤ A1 ≤ ... ≤ A100. But because A0 and A100 differ by one element, by (4) it follows that exactly one of these is actually strict. Thus, in the sequence A0, A1, ..., A100 suddenly there is exactly one point at which the size of the set goes up by one. This is really counterintuitive. We are generating our sequence of sets by starting with A0 and then shifting the set over to the right by one (since $A_{n+1}=1+A_n), and suddenly the size jumps.

The second option is we don’t have monotonicity at all. This means that at some point in the sequence we go up and at some other point we go down: there are m and n between 0 and 99 such that Am < Am + 1 and An > An + 1. This again is really counterintuitive. All these sets look alike: they consist in an infinite sequence of points 100 units apart, just with a different starting point. But yet the sizes wobble up and and down. This is weird!

This suggests to me that the problem lies with the subset principle (1) or possibly with discreteness (4), not with the details of how to formulate the translation invariance principle (3). If we have (1) and (4) things are just too weird. I think discreteness is hard to give up on: counting should be discrete—two sets can’t differ in size by, say, 1/100 or 1/2. And so we are pressed to give up the subset principle (1).

Appendix: Existence proofs

Let U be any set. Let be the equivalence relation on subsets of U defined by A ∼ B if and only if either A = B or A and B are finite and of the same cardinality. The subset relation yields a partial order on the -equivalence classes, and by the Szpilrajn extension theorem extends to a total order. We can use this total order on the equivalence classes of subsets to define a total preorder on the subsets, and this will satisfy (1) and (2).

If we want (3), let U be the integers, and instead of the Szpilrajn extension theorem, use Theorem 2 of this paper.

The proof of the “Fun Fact” is really easy. Suppose we have such a total preorder ≤. Let A = {2, 4, 6, ...}, B = {1, 2, 3, 4, ...} and C = {0, 2, 4, 6, 8, ...}. By (1), we have A < C. Suppose first that B ≤ A. Then C = 1 + B ≤ 1 + A = B by (3). Hence C ≤ A by transitivity, contradicting A < C. So A < B by totality. Thus B < C by (3). Since A and C differ by one element, this contradicts (4).