Wednesday, September 30, 2020

Privation, false belief and time

According to the privation theorist, when an evil occurs, some entity is lacking such that were that entity to exist, there would have been no evil there.

The challenge for the privation theorist is to find the missing entity. Here is one special case of this challenge: the case of justified false belief.

Suppose yesterday I had a justified belief that today you will not dance a jig. And yet today you did dance a jig. What is the missing entity, such that if it existed, there would have been no evil? I can think of only two candidates for the missing entity: truth and correspondence.

Consider first the truth suggestion. My belief yesterday lacked truth. But is this missing truth an entity? It had better be for it to help the privation theorist. But then by choosing to dance a jig today, you brought it about that an entity did not exist yesterday. But we surely do not have such routine and easy power over what entities existed in the past!

The above argument generalizes: whatever the missing entity is, it had better not be an entity that would have existed yesterday. It needs to be an entity that would have existed today, presumably at the very time when by dancing a jig and thereby ensured its nonexistence. (This is already problematic. For what went wrong was my belief yesterday, not your action today, so it seems that what makes the evil an evil should be located yesterday, because of the world-to-mind direction of fit of belief.)

So, perhaps correspondence is such an entity, and perhaps the correspondence between mind and reality exists precisely at the time at which that reality exists. But this is also problematic. For suppose that today you secretly dance a jig, and tomorrow Alice forms the false belief that you didn’t. If correspondence exists at the time of the reality, then were it to be the case that Alice instead tomorrow formed the belief that you did dance a jig, there would have been a correspondence between mind and reality today, during your jigging. Thus, depending on which belief Alice forms tomorrow, either a correspondence does or does not exist today, and so once again we have a routine and ordinary power over what entities existed in the past, which is absurd.

Could we say that the correspondence exists at both the time of the belief and the time described by the belief? But that doesn’t help: Alice by forming a belief tomorrow still affects whether there is a correspondence today.

I suppose one could have a messy view on which when a true belief is about the past, the correspondence exists at the time of the belief, and when the true belief is about the future, the correspondence exists at the time of the described reality. But now suppose that I have a conjunctive true belief about what you did yesterday and what you will do tomorrow. That belief is about both the past and the future. When does it exist? If we say that it exists today or yesterday, then your action tomorrow can affect what exists today or yesterday, and that’s absurd as before. So that correspondence has to exist tomorrow.

But then we are forced into this even messier view: if a belief is true, the correspondence exists at the time of the belief or at the last of the times the belief is about, whichever comes later. But now we have more problems. For suppose you will exist forever (in heaven, say), and I form the true belief that you will never dance a jig. But there is no last time that the belief is about! Indeed, whatever the time t the correspondence exists at, there will be a later potential jig-dancing time at which by choosing whether to dance a jig you control whether that correspondence exists at t. And it seems implausible that just by living forever you gain the power to create or annihilate past entities.

Note that the problem here afflicts anyone who thinks that tokens of correspondence between belief and reality are genuine entities.

Tuesday, September 29, 2020

More on the privation theory of evil

Back in April, I suggested that there are two possible privation theories concerning evil:

  1. every evil is a privation

  2. for every evil, what makes it be evil is a privation.

Well, Aquinas essentially scooped me, in the first article of the De Malo, by distinguishing two senses of evil in the statement “evil is a privation”. If “evil” means the evil thing, the claim is false. But by “evil” we could mean the evilness of the evil thing, and then Aquinas holds the claim to be true. And it seems to me that the evilness of the evil thing is basically that which makes it evil, so Aquinas’ theory is basically my theory (2).

I think too much of our current literature on the privation theory of evil suffers from a failure to explicitly make the distinction between the evil thing and the evilness of the evil thing. As a result, some of the counterexamples in the literature are only counterexamples to (1). And indeed it’s not hard to find uncontroversial counterexamples to the claim that every evil entity is a privation. Josef Stalin was an evil entity (I hope he has repented since), but he was never a privation; an act of adultery is an evil thing, but it is not a privation.

Consider, for instance, the most discussed example in the literature: pain. It gets pointed out that pain is not a lack of pleasure or any other kind of privation. That is very likely true. But Aquinas’ version of the privation theory does not require him to hold that pain is a privation. He can just say that pain is an evil thing, but evil things don’t have to be privations. Rather, what makes the pain be an evil is a privation. Of course that still requires a privative theory as to what makes pain be an evil. But there are such theories. For instance, one might hold a modification of Mark Murphy’s theory about pain and say that what makes pain in paradigmatic cases bad is a privation of a correspondence between our mental states and our desires, given that in paradigmatic cases we desire not to be in pain (and it’s not much of a bullet to bite to say that pain isn’t bad when it doesn’t go against our desires).

The story about pain doesn’t end here. One might, and I think should, question whether the correct ontology of the world includes such entities as “matches” between mental states and desires for pain to be a privation of. I think what Aquinas would likely say is that because being is said analogically, “matches” do exist in an analogical sense, and hence we can correctly talk of their privation. I think this is problematic. For once we allow that “matches” exist analogically, we should equally allow for privations and other lacks to exist analogically—and Aquinas indeed does. And then we run into the problem that even positive things can count as lacks: for instance, sight could count as a lack of the lack of sight. And once we have gone this far, the privation theory becomes trivial.

But the point remains: once we have seen Aquinas’s distinction between the evil being a privation and the evilness of the evil being a privation, the critiques of the privation theory are apt to get a lot more complex.

Monday, September 28, 2020

Causal finitism and functionalism

Say that a possible thought content has finite complexity provided that the thought content can be represented by a sentence of finite length in a language whose basic terms are the fundamental concepts in the thought content.

  1. Necessarily, if functionalism is true, then the occurrence of a thought content with infinite complexity requires infinitely many states to cooperate to produce a single effect.

  2. Infinitely many states cannot cooperate to produce a single effect.

  3. It is possible for a thought content with infinite complexity to occur.

  4. So, functionalism is false.

I have two separate ideas to defend (1). First, it seems like a system capable of producing a thought content must go through a number of states proportional to the complexity of that thought content in producing it if functionalism is true. Second, the occurrence of a thought content of infinite complexity requires infinitely many constituent states. Moreover, thoughts have to be unified: to think the conjunction of p and q is not just to think p and to think q but to think them in a unified way. On functionalism, the unification has to be causal in nature. To unify the infinitely many constituent states would require them to have the capability of producing some effect together.

If I were a functionalist, I would deny (3). The cost of that is that then most truths end up unthinkable, which seems implausible.

Friday, September 25, 2020

Intention and Christian ethics

Some ethicists think that the permissibility of an action typically does not depend on the intentions with which the action is done. (We obviously need “typically”. It is possible to make promises as to one’s intentions. And saying “This statement is intended” is permissible only if one intends to say it.)

Here is a deep reason why Christians cannot agree. Perhaps the deepest Christian ethical commitment is to something like this:

  • An action is permissible if and only if it is loving (possible weaker variant: it is not unloving).

But whether an action is loving or unloving typically depends on intentions.

Racism and power structures

Many, but not all, theorists think that racially based discrimination only counts as racism, and its perpetrators as racists, when the discrimination aligns with the power structures in society.

This has the following odd consequence: Simply by defeating Nazis militarily, without in any way changing their hearts or minds, you can turn them into non-racists.

Or imagine a protracted racially-based genocidal civil war between the Xs and the Ys, with the tide of war going back and forth. Sometimes, the Xs are ahead, but then the Ys pull ahead, and then later the Xs regain the lead. On the view in question, whenever the Xs are ahead, their acts of genocide constitute racism, but when they fall behind, it is the Ys that are the racists. (This is like an absurd opposite of the view that history is written by the victors.)

(One might object that genocide doesn’t count as discrimination. But it clearly does.)

(The above argument is only a vivid way of putting a well-known objection, that the alignment view of racism makes for too much dependence of the fact of racism on changing social conditions.)

One can escape from the above arguments by weakening the alignment principle to say:

  • x’s racially based discrimination only counts as racism when the discrimination aligns with the power structures that actually obtain or that x desires.

There will probably still be other counterexample.

As usual, my disclaimer: this is very far from my main areas of philosophy.

Thursday, September 24, 2020

Discrimination and coin tosses

Bob is deciding whom to hire for a job where race is clearly irrelevant to job performance. There are two clear front-runners. Bob hires the white front-runner because that candidate is white.

Bob has done something very wrong. Why was it wrong? A naive thought is that what he did wrong was to take into account something irrelevant to job performance while deciding whom to hire. But that can’t be right. For suppose that all the job-performance related facts were on par as far as Bob could tell. And then suppose that Alice when dealing with a similar case just said to herself “Heads, A, and tails, B”, tossed a coin, got tails, and hired candidate B. Alice didn’t do anything wrong. But Alice also made a decision on the basis of something irrelevant to job performance, namely whether the prior heads/tails assignment to a candidate matched the outcome of the coin toss.

In terms of deciding on irrelevancies, the paradigm of a fair tie-breaking procedure—a coin flip—and the paradigm of an unfair tie-breaking procedure—a racist decision—look very similar.

Here is a standard thing to say about this (cf. Scanlon): When the job-performance related facts are tied, and we still have to choose, we just have to choose on the basis of something not related to job performance. But that something had better not be something that forms the basis for large-scale patterns of dominance in society. Both Alice’s and Bob’s procedures are based on something not related to job performance, but Bob’s procedure is an instance of a large-scale social pattern of dominance.

I want to propose an account of why Bob did wrong and Alice did not that seems to me to differ slightly from the standard story (or maybe it just is a version of it). To that end, consider a third story. Carl runs a graduate program where he has to make lots of hard choices about current students, e.g., about travel-funding, stipend-renewal, lab and office allocation, etc., and these choices often involve ties on the usual academic metrics. (This is not a description of the Baylor philosophy program: we have lots of funding, and rarely if ever had to break ties regarding funding.) Carl is lazy and has decided to simplify things for himself by saving the number of coin tosses he has to make. Instead, whenever a student is admitted, Carl chooses a random number between one and a thousand and assigns that number to the student, re-rolling the random number generator if that number matches the number of a student already in the program. Thereafter, whenever a tie is to be broken, Carl always breaks the tie in favor of the student with the higher pre-assigned number.

Carl’s tie-breaking procedure is like Alice’s in terms of randomness and lack of alignment with larger social patterns of discrimination. But it’s still a terrible procedure. It’s terrible, because it distributes benefits and burdens in a seriously unequal way: if you got randomly assigned a low number at admission, you are stuck with it and keep on missing out on goodies that people assigned a high number got.

One can now explain what goes wrong in Bob’s procedure as something rather like what went wrong in Carl’s procedure: given structural racism, the minority candidate, call him Dave, passed over by Bob has tended to have been on the negative side of many other decisions (some of them perhaps being racist tie-breaking decisions, and many of them being even more unjust than that). Bob’s procedure has contributed to Dave having a life of tending to get the short end of the stick, just as Carl’s procedure has led a number of students having a graduate career with a tendency to getting the short end of the stick. And a tendency to getting the short of the end of the stick is something we should (at least typically) not contribute to.

This is close to the standard account about Bob’s racism. It likewise involves the large-scale patterns of dominance in society. But it seems to me also importantly different: The large-scale patterns of dominance in society are relevant to Bob’s action insofar as they make it likely that Dave has been on the unfavorable side of too many decisions. In the graduate program case, there may be no larger social patterns that match the ones within the program (or at least not pre-existing ones), and even within the program there need not be any significant interpersonal patterns of dominance between the persons assigned high numbers and low numbers, especially if the initial numerical assignments and the tie-breaking procedure are kept secret from the students, who just say things like, “My luck is terrible!” (This is going to be most likely in a program where students are oblivious to their social environment due to a focus on their individual research.)

In the alternate account, the focus is on the individual rather than the group, and the larger social facts are relevant precisely as they have impacted the individual. But this may seem to miss out on a common dimension of invidious discrimination. If I am a member of a group and someone else in the group is unfairly discriminated against, then that is apt to harm me in two ways: first, because I am apt to have a special concern for other members of the group (either because they are members of the group, or because persons more closely related to me tend to be members of the group), and harm to someone I have a special concern for is harm to me, and, second, because seeing someone like me get harmed scares me.

But I think this fits with my individualistic story by just multiplying the number of times that Dave gets the short end of the stick: sometimes he gets the short end of the stick directly and sometimes he gets it indirectly by having someone else in his group get it.

At the same time, I have to say that this is material I know next to nothing about. Take it with a grain of salt.

Tuesday, September 22, 2020

Basing public decisions on controversial comprehensive doctrines

According to Rawlsians, it is wrong to make decisions in the public arena based on controversial comprehensive doctrines.

Here is a counterexample. We meet up with aliens who are incredibly truthful: not only have we never found one to lie, but we have never found one to even accidentally say the false. The aliens inform us that they have figured out whether controversial comprehensive doctrine D is true. We have very good reason to believe the aliens. The aliens inform us of a fanatical plan: Each state has two weeks to submit a statement to the aliens as to whether D is true. Those states that either fail to submit such a statement or who get the answer wrong will be completely destroyed. And then the aliens will leave.

It is obvious that the right thing for the state to do is to do its best to figure out whether D is true, and then on the basis of its best determination make the statement to the aliens. It is clearly wrong to refuse to answer, and it is also clearly wrong to simply guess without basing oneself on one’s best determination of the truth value of the comprehensive doctrine. It is clearly the right thing in this case to make a public decision on the basis of a controversial comprehensive doctrine.

Of course, this is a very special case: a case where what rides on the decision is so important that it trumps most other considerations. But we can lower the amount that rides on the decision and still keep the intuition. Suppose that instead the aliens promise to kill one per thousand people at random if we get the answer wrong. That should, I think, be enough to overcome any qualms we may have about using controversial comprehensive doctrines in public policy. But now there are real-world issues riding on controversial comprehensive doctrines where what is at stake is of a similar or greater magnitude. Abortion is a paradigm case.

Non-shareable reasons in the public arena

There is a lot of discussion about whether it is permissible to use reasons based on controversial comprehensive doctrines, such as religious reasons, in the public arena. But at least such reasons can be formulated in terms most people understand.

A different sort of challenge seems to be posed by reasons that the agent is unable to share, reasons such as hunches, feelings, etc. Not infrequently, we just see, or at least think we see, that something is the right decision, but cannot say why. And while on the one hand such reasons cannot very satisfactorily enter into discussion, on the other hand without them our public knowledge of what we should do will be deeply impoverished, and our public deliberation will be biased in favor of the opinions of verbally skilled elites. And it seems plausible that it is more important to get to the truth than to have a satisfactory discussion.

Of course, one could say that the fact that one has a hunch that something is to be done is itself a shareable, discussable reason, and hence acceptable in the public sphere. But now, I think, the notion of what is a shareable, discussable reason becomes largely trivialized.

Monday, September 21, 2020

Limits of neuroscience

I think that our best physicalist view right now is a functionalism on which mental states are identified with types of computation in a hardware-agnostic way (i.e., whatever the hardware is, as long as the same type of computation is done, the mental states get tokened).


  1. If functionalism is true, what human discipline, if any, will discover which functional processes (e.g., the execution of what algorithms) constitute consciousness?

There are, I think, three plausible answers:

  1. None. We wouldn’t be able to know the answer.

  2. Philosophy and neuroscience working together.

  3. Neuroscience working alone.

I think the most plausible answer is (2), with (3) being a runner up.

In this post I want to give a quick argument that (4) is not the answer.

Neuroscience is a natural science. The natural sciences do not discover substantive facts about worlds whose laws of nature are radically different from ours. Some possible worlds whose laws of nature are radically different from ours contain beings with functional processes isomorphic to the ones running in us. Thus, if neuroscience discovered which functional processes constitute consciousness, it would discover about those worlds that they contain consciousness. That would be a substantive fact about these worlds, and that would contradict the assumption that neuroscience is a natural science.

Consciousness and intersubjectivity

This argument is logically valid:

  1. Science only explains intersubjective phenomena.

  2. Consciousness is subjective and not intersubjective.

  3. So, science doesn’t explain consciousness.

The obvious thing to attack here is the second conjunct in (2). If physicalism is true, consciousness is some intersubjective phenomenon, say a certain pattern of neuronal firings or some functional state. But note that on a physicalist view like this, every subjective phenomenon is also intersubjective: for the subject’s internal states then count as intersubjective. So on that view, (1) becomes largely trivialized. So if we think (1) is not only true but significant and non-trivial, we should not be physicalists.

Thursday, September 17, 2020

A tale of two universes

Suppose that Mary lives in a universe whose physics is radically different from ours. She loves mathematics and is amazingly good at it. She has very little knowledge of biology, beyond what little superficial information she can glean by turning her eye-stalks on herself while spending all her life suspended in a vortex of Z-force. One day, she is given a complete description of the physical state of the laws of our universe and of the physical arrangement of matter in the Solar System over the past billion ears. She thinks this is really cool mathematically, and thinks through all the mathematico-physical facts involved.

Here is the question: Does Mary have enough information to know that the Solar System contains conscious life?

If not, physicalism seems to be in trouble.

Another variant of the knowledge argument

We don’t have anybody like Mary who knows all of physics and yet has not yet seen color, and as Dennett has pointed out, it is hard to imagine what things look like from the point of view of someone who knows so much more than we do.

But here is a variant that may be easier for us to wrap our minds about: Imagine two people, one a completely colorblind early 21st century neuroscientist specializing in the visual system and the other a completely colorblind ordinary person. Suppose both receive color vision. If physicalism is true, the neuroscientist knows a lot more about what seeing red is like, even though (because we’re still in the early days of neuroscience) they don’t know that much about it overall. Thus, if physicalism is true, the neuroscientist would learn less by seeing red the first time, and should be less surprised. But would we really expect them to learn less and be less surprised than the ordinary person?

This variant is inspired by a remark I heard Brandon Rickabaugh give in a conference talk, that consciousness only seems more mysterious when one knows more neuroscience.

Wednesday, September 16, 2020

Agent causation

I have long identified as having an agent-causal theory of free will. But I have just realized that my Aristotelian take on agent causation is far enough from the most common agent-causal theories—those of Clare and O’Connor—that it may be misleading to talk of myself as accepting an agent-causal theory of free will.

Standard agent-causal theories distinguish between agent causation and event causation as two distinct and real things in the world. For instance, they hold that I agent cause my writing of this post but there is an event causal relation between my being in this armchair and the cushion being squished. Agent causation has the agent as the cause and event causation has an event as the cause. But I think in both cases the cause is the same: it is a substance, namely myself. I cause the writing of this post and I cause the cushion to be squashed. On the standard view on which agent causation is distinguished solely by the fact that the cause is the agent, both are cases of agent causation. But that would be misleading to say.

So, if we take seriously the Aristotelian account of causation as substance causation, we shouldn’t distinguish agent causation from other kinds of causation by whether the cause is an agent or something else. But we can still make the distinction. My writing this post is an actualization of a power of my will (or my practical rationality, if you prefer). My squashing the cushion is an actualization of the power of my weight. Agent causation is distinguished from other kinds of causation not by what does the causing, but how it does the causing. In agent causation, the substance causes by actualizing its will (and any substance with a will is an agent). In other kinds of causation, the substance causes by actualizing a different power.

So, I think the fundamental relation underlying causation is actually at least ternary: agent X causes event E by actualizing power P.

This neatly integrates agent causation with reasons causation. There are more and less proximate powers. I now have a nearly proximate power to speak Polish (some motivation might be needed to make it proximate). When I was an infant, I had a remote power to speak Polish, by having a nearly proximate power to learn Polish. When the power P in the causal relation is specified as the maximally proximate power, in the agential case, it is a maximally proximate power of the will. And a maximally proximate powers of the will are tied to reasons: it is only by having a reason to do something that I am able to will it (one wills under the guise of the good). So, my reasons for action supervene on the maximally proximate power for action. Thus, the ternary description of agent causation neatly includes the reasons.

Tuesday, September 15, 2020

Knowledge of qualia

Consider our old friend Mary, who grew up in a black and white room, learned all of physics, and then saw a red tomato, allegedly learning a new fact about the world, what red looks like. Suppose Mary now went back to her black and white room and returned to contemplating the foundations of quantum mechanics, just as she did before. At that point, clearly Mary knows what red looks like. But unless she is visualizing red stuff, she is not having any red qualia at that point. So:

  1. One can know what red is like without having any red qualia.

Moreover, presumably whatever state her mind has—regardless of whether the mind is physical or not—in her black and white room after seeing the red tomato is a state that could have been induced in her (by a neurosurgeon or a demon) without her having had any red qualia. One might worry whether that induced state would count as knowledge, but if one adds that she gets testimonial evidence that her mental representations of qualia are correct despite based on false memories, it could be knowledge. Thus:

  1. One can know what red is like without having or having had any red qualia.

It is possible to agree to (2) while holding that the knowledge of qualia argument against physicalism is a good argument. For one might hold that the state of mind that allows for (2) is not a state that can simply come from learning all the physical facts. It is a state that might require some kind of neurosurgical or supernatural intervention. But it seems to me that when one accepts (2), it becomes significantly less plausible that one cannot learn what red is like just by learning all the physical facts.

There is another move the defender of the knowledge argument can make. They can deny (1) and (2), holding that when Mary is back to thinking about quantum mechanics, she doesn’t know what red is like, but that we are inclined to incorrectly say that she knows it because she has the skill of coming to know it at a moment’s notice by visualizing something red. This is a good move, but it has a pitfall: it makes knowledge of what red is like significantly disanalogous to ordinary knowledge, such as of multiplication tables, which one has even when it is merely dispositional, when one is not thinking about it. But if knowledge of what is red is like has this significant disanalogy to ordinary knowledge, that makes it less likely that it is factual knowledge—which the argument requires it to be.

Sunday, September 13, 2020

Concrete dumbbells

My son needed some 25 lb dumbbells for home workouts, since we took him out of his school's workouts due to COVID, but our local stores were out. So we made concrete ones, using 3D printed molds. A bonus of concrete: because they're bigger for the same weight, they make one look stronger.

Build instructions are here.

Friday, September 11, 2020

Non-instrumental pursuits and uncaused causes

Here’s a curious fact: It is one thing to pursue something because it is a non-instrumental good and another to pursue it as a non-instrumental good, or to pursue it non-instrumentally. A rich eccentric might offer me $100 for pursuing some non-instrumental good. I might then do a Google Image search for “great art”, and spend a few seconds contemplating some painting. I would then be pursuing the good of contemplation because it is a non-instrumental good, but not as a non-instrumental good. (What if the eccentric offered to double the payment if I pursued the good non-instrumentally? My best bet would then be to just forget all about the offer and hope I end up pursuing some good non-instrumentally anyway.)

Thinking about the above suggests an important thesis: To pursue a good non-instrumentally is something positive, not merely the denial of instrumentality. Simply cutting out of the world the story about the rich eccentric and keeping my contemplation in place does not make the contemplation be pursued as a non-instrumental good. Rather, such world surgery makes the contemplation non-rational. To make the contemplation a non-instrumental pursuit of a good requires that I add something—a focus on that good in itself. We don’t get non-instrumental pursuit by simply scratching out the instrumentality, just as we don’t get an uncaused cause by just deleting its cause—rather, an uncaused cause is a cause of a different sort, and a non-instrumental pursuit is a pursuit of a different sort.

Some fun distinctions

Isn’t it funny how very similar gestures can signal respect and disrespect? Under ordinary circumstances, crossing to the other side of the street to avoid near someone is a form of disrespect. But in a pandemic it signals a respectful desire not to make the other nervous. Though I suppose even apart from a pandemic, one would have moved out of the way of dignitaries.

We have another neat little thing here. There is a difference between going out of one’s way to ensure that one isn’t in another’s personal space and going out of one’s way to ensure that the other isn’t in one’s personal space, even though in an egalitarian society, x is in y’s space if and only if y is in x’s space.

And notice how hard it is to formulate that point without reifying “personal space”, just by using distance. I can hear a difference between avoiding my being within a certain distance of another and avoiding the other being within a certain distance of me, but I can’t tell which is which! Maybe, though, we can distinguish (a) avoiding imposing on another the bad-for-them of us being within a certain distance and (b) avoiding imposing on me the bad-for-me of us being within that distance. In other words, the reasons for the two actions are grounded in the same state of affairs but considered as bad for different individuals.

I suppose similar things can happen entirely in third person contexts. I can work for a friendship between x and y considered as a good for x, considered as a good for y, or considered as a good for both. And these are all three different actions.

Wednesday, September 9, 2020

Minor inconveniences and numerical asymmetries

As a teacher, I have many opportunities to cause minor inconveniences in the lives of my students. And subjectively it often feels like when it’s a choice between a moderate inconvenience to me and a minor inconvenience to my students, there is nothing morally wrong with the minor inconvenience to the students. Think, for example, of making online information easily accessible to students. But this neglects the asymmetry in numbers: there is one of me and many of them. The inconvenience to them needs to be multiplied by the number of students, and that can make a big difference.

I suspect that we didn’t evolve to be sensitive to such numerical asymmetries. Rather, I expect we evolved to be sensitive to more numerically balanced relationships, which may have led to a tendency to just compare the degree of inconvenience, in ways that are quite unfortunate when the asymmetry in numbers becomes very large. If I make an app that is used just once by each of 100,000 people, and my app’s takes a second longer than it could, then it should be worth spending about two working days to eliminate that delay. (Or imagine—horrors!—that I deliberately put in that delay, say in the form of a splashscreen!) If I give a talk to a hundred people and I spend a minute on an unnecessary digression, it’s rather like the case of a bore talking my ears off for an hour and a half. In fact, I rather like the idea that at the back of rooms where compulsory meetings are held there should be an electronic display calculating for each speaker the total dollar-time-value of the listeners’ time, counting up continuously. (That said, some pleasantries are necessary, in order to show respect, to relax, etc.)

Sadly, I rarely think this way except when I am the victim of the inconvenience. But it seems to me that in an era where more and more of us have numerically asymmetric relationships, sometimes with massive asymmetries introduced by large-scale electronic content distribution, we should think a lot more about this. We should write and talk in ways that don’t waste others’ time in numerically asymmetric situations. We should make our websites easier to navigate and our apps less frustrating. And so on. The strength of the moral reasons may be fairly small when our contributions are uncompensated and others’ participation is voluntary, but rises quite a bit when we are being paid and/or others are in some way compelled to participate.

One of my happy moments when I actually did think somewhat in this way was some years back when, after multiple speeches, I was asked to say a few words of welcome to our prospective graduate students. There were multiple speeches. I stood up, said “Welcome!”, and sat down. I am not criticizing the other speeches. But as for me, I had nothing to add to them but just a welcome from me, so I added nothing but a welcome from me. I should do this sort of thing more often.

Tuesday, September 8, 2020

The fairness of infinite lotteries and qualitative probabilities

Suppose that we wish to model an infinite fair lottery with tickets numbered by integers by means of qualitative probabilities, i.e., a reflexive and transitive relation ≲ between sets of tickets that satisfies the non-negativity constraint that ∅ ≲ A for all A and the additivity constraint that A ≲ B iff A − B ≲ B − A. Suppose, further, that we want to have the regularity constraint that ∅ < A if A is not empty.

At this point, we want to ask what “fairness” is. One proposal is that fairness is strong translation invariance: if A is a set of integers and n + A is the set {n + m : m ∈ A} of all the members of A shifted over by m, then A and n + A are equally probable. Unfortunately, if we require strong translation invariance, then we violate the regularity constraint, since we will have to assign the same probability to the winning ticket being in {1, 2, 3, ...} as to the winning ticket being in {2, ...}, which (given additivity) violates the constraint that ∅ < {1}.

One possible option that I’ve been thinking about is is to require weak translation invariance. Weak translation invariance says that A ≲ B iff n + A ≲ n + B. Thus, a set might not have the same probability as a shift of itself, but comparisons between sets are not changed by shifts. I’ve spent a good chunk of the last week or two trying to figure out whether (given the other constraints) it is coherent to require weak translation invariance. Last night, Harry West gave an elegant affirmative proof on MathOverflow. So, yes, one can require weak translation invariance.

However, weak translation invariance does not capture the concept of fairness. Here is one reason why.

Say that a set B of integers is right-to-left (RTL) bigger than a set A of integers provided that there is an integer n such that:

  1. n ∈ B but not n ∈ A, and

  2. for every m > n, if m ∈ A, then m ∈ B.

RTL comparison of sets of integers thus always favors sets with larger integers. Thus, the set {2, 3} is RTL bigger than the infinite set {..., − 3, −2, −1, 0, 1, 3}, because the former set has 2 in it while the latter does not.

It looks to me that West’s proof straightforwardly adapts to show that that there is a weakly translation invariant qualitative probability that coheres with RTL ordering: if B is RTL bigger than A, then B is strictly more likely than A. But a probability comparison that coheres with RTL ordering is about as far from fairness as we can imagine: a bigger ticket number is always more likely than a smaller one, and indeed each ticket number is more likely to be the winner than the disjunction of all the smaller ticket numbers!

So, weak translation invariance doesn’t capture the concept of fairness.

Here is a natural suggestion. Let’s add to weak translation invariance the following constraint: any two tickets are equally likely.

I think—but here I need to check more details—that a variant of West’s proof again shows that this won’t do. Say that a set B of integers is right-skewed (RS) at least as big as a set A of integers provided that one or more of the following holds:

  1. A is finite and B has at least as many members than A, or

  2. B has infinitely many positive integers and A does not, or

  3. A is a subset of B.

Intuitively, a probability ordering that coheres with RS ordering fails to be fair, because, for instance, it makes it more likely that the winning ticket will be, say, a power of two than that it be a negative number. But at the same time, a probability ordering that coheres with RS ordering makes all individual tickets be equally likely by (1).

To make this work with West’s proof, replace his C0 with the set of bounded functions that have a well-defined and non-negative sum or whose positive part has an infinite sum.

Monday, September 7, 2020

Two beauties

In a number of cases of beauty, beauty is doubled up: there is the beauty in an abstract state of affairs and there is the beauty in that state of affairs being real, or at least real to an approximation. For instance, the mathematics of Relativity Theory is beautiful in itself. But that it is true (or even approximately true) is also beautiful.

This shows an interesting aspect of superiority that painting and sculpture have over the writing of novels. The novelist discovers a beautiful (in a very broad sense of the word, far broader than the “pretty”) abstract state of affairs, and then conveys it to us. But the painter and sculptor additionally doubles the beauty by making something real an instantiation of it, and it is by making that instantiation real that they convey it to us. The playwright is somewhere in between: the beautiful state of affairs is made approximately real by a play.

The above sounds really Platonic. But we can also read it in an Aristotelian way, if we understand the abstract states of affairs as potentialities. The painter, sculptor and novelist all discover a beautiful potentiality. The painter and sculptor brings that potentiality to actuality. The novelist merely points it out to us.

Half tickets in an infinite lottery

Consider a fair infinite lottery with tickets numbered ..., −3, −2, −1, 0, 1, 2, 3, ..... Consider these events:

  • E: winner is even

  • O: winner is odd

  • E*: winner is even but not zero

  • E+: winner is even and positive

  • O+: winner is odd and positive

  • E: winner is even and negative

  • O: winner is odd and negative.


  1. O+ is equally likely as O

  2. E+ is equally likely as E

  3. E is equally likely as O

  4. all tickets are equally likely to win.

Now then E* is less likely than E by one ticket, and hence also less likely than O by one ticket according to (3). And E* is the same event as the disjunction E+ or E, while O is the same event as the disjunction O+ or O. Therefore, the disjunction O+ or O is one ticket more likely than the disjunction E+ or E. Since O+ and O are equally likely and E+ and E are equally likely, it follows that:

  1. E+ is half a ticket less likely than O+.

But how could one lottery outcome be less likely than another by half a ticket in a lottery where all tickets are equally likely to win? The only option seems to be that the probability of any particular ticket winning is zero. And that seems paradoxical, too.

Sunday, September 6, 2020

Heaven, the goods of others and the defeat of evil

There is a delight in competing athletically with one’s child: if they win, it feels good, and if one wins, it feels good, too. (The hedonic ideal is achieved when the child wins about 60% of the time; then one feels proud of their superiority, but not rarely one has the pleasure of beating a stronger opponent.)

Parental love makes it easy to love another as oneself (to paraphrase what C. S. Lewis says about Eros). It thus gives us an image of what it is like to be in heaven: we will greatly enjoy the goods had by others. This gives us an attractive picture of how the joy of heaven could fit with enduring differences in personal characteristics. Perhaps being an extrovert would not be true to my self and to God’s vocation for me, and so maybe even over an eternity in heaven I won’t be extroverted. But if so, I will still be fully happy for the joy of the heavenly extroverts, without any regret that I am not one of them, while they will be fully happy for me introverted joys, also without any regret that they are not like me.

Here are two controversial (for very different reasons) applications of this. First, there is a genuine and distinctive good in being a woman and there is a genuine and distinctive good in being a man, and it seems to make sense for a person to desire the goods of the other sex, regardless of whether it is possible to have them oneself. In heaven, however, Joseph can enjoy Mary’s good in being a woman and Mary can enjoy Joseph’s good in being a man, without Joseph regretting that he personally “only” has the good of manhood and Mary regretting that she personally “only” has the good of womanhood. That is what total love is like.

Second, given an eternalist or moving block theory of time, the past will always be fully real. This in turn gives us a solution to the problem that various important goods, such as marriage and self-sacrifice, will not be available in heaven. For we will be able to rejoice in others’ past possession of these goods, without regret for the fact that they aren’t ours and now.

The second point, however, raises the following problem: Won’t we also grieve for others’ past—and even present, if hell is a reality (as I think it is)—subjection to great evils? Maybe, but in God’s plan there is a crucial asymmetry between good and evil. Evils are defeated. How this defeat happens is deeply mysterious. But because of this defeat, I suspect the grief for a defeated evil will not hurt, precisely because of the evil’s being defeated, while goods remain undefeated and hence the joy for them will always delight.

In fact, the last point suggests something to me. A lot of philosophers of religion have said that it’s not enough for theodicy if evils are justly compensated for or their permission is in some way justified. We need these evils to be defeated. I think this is mistaken if all we are after is a response to the problem of evil. But we also need a response to the problem of why the past and present suffering of others doesn’t cause the saints pain in heaven. And it is here that we need the defeat of evil.

Wednesday, September 2, 2020

Against anti-interactionist intuitions

  1. Fundamental and non-fundamental things are at least as deeply metaphysically different from each other as physical and non-physical things.

  2. Fundamental things can cause non-fundamental things and non-fundamental things can cause fundamental things. (E.g., particles can cause a heap, and lab equipment can cause a particle emission.)

  3. Therefore, being at least as deeply metaphysically different as physical and non-physical things are cannot be sufficient for making causal relationships impossible.

"In the right way"

When I was in grad school, we were taught that one should abandon all hope of solving “in the right way” problems, such as the problem of how exactly an intention has to result in an action in order for the action to be done from that intention.

I think that with a robust metaphysics of causation, the problem is soluble.

Solution 1: Causal powers have a teleology: to produce a certain effect in a certain way. That teleology is metaphysically written into the causes. The “in the right way” condition may be infinitely complex, but it has a metaphysical home: it is found in the causal power. What makes it be the case that William James’ mountaineer who intended to kill his buddy by dropping the rope, and then dropped the rope because of the nervousness resulting from the intention did not intentionally drop the rope is because the outcome of events is a mismatch to the description in the teleology of the causal powers constituting the intention.

Solution 2: It is a Thomistic maxim that the effect is the actuality of the cause qua cause. This maxim I suspect needs a qualification: the proper effect—the one that happens in the right way—is the actuality of the cause qua cause. So now we have a neat and simple criterion for when a cause C has caused an effect E in the right way: this happens precisely when E is the actuality of C qua cause. (See here for more discussion.)

I think the reason we were taught to eschew the problem of “in the right way” conditions was because of an implicit reductionistic metaphysics. If we think that the cause is just an arrangement of particles, it is hopeless to have a distinction between proper and improper effects.

An argument that there are more positive odd numbers than positive even ones

Suppose that we embrace these intuitive theses about sets of natural numbers:

  1. If A is a proper subset of B, then B has more members than A.

  2. If A has at least as many members as B, then 1 + A has at least as many members as 1 + B, where 1 + C = {1 + c : c ∈ C} is C shifted over by one to the right.

  3. Either there are more odd positive numbers than even positive numbers or there are at least as many even positive numbers as odd positive numbers.

Let O be the positive odd numbers and E be the positive even numbers. Write A ≲ B to mean that B has at least as many members as A, and write A ∼ B to mean that they have the same number of members. For a reductio, suppose that O ≲ E. Then 1 + O ≲ 1 + E by (2). But 1 + O = E. Thus, O ≲ E ≲ 1 + E. But 1 + E is all the odd numbers starting with three, which is a proper subset of O, which contradicts (1). So we do not have O ≲ E, and hence by (3):

  1. There are more odd positive numbers than even positive numbers!

Of course, the usual Cantorian way of comparing sizes of sets rejects (1).

I think the non-Cantorian’s best bet is either to embrace the conclusion (4) or to deny the special case of totality of comparison in (3). In either case, the non-Cantorian needs to deny the intuitive claim that:

  1. There are equally many odd positive numbers and even positive numbers.

Note that (2) does not say or imply that 1 + A has the same number of members as A. Since that would imply that {1, 2, 3, ...} and {2, 3, 4, ...} have the same number of members, that would beg the question against the typical non-Cantorian for whom (1) is a central intuition. One might also wonder whether there is a way of comparing sets of natural numbers that satisfies (1)–(3): the answer is yes (even with (3) generalized to all pairs of sets of naturals).