## Saturday, December 3, 2022

### A new (but uncertified) world record

## Friday, December 2, 2022

### Moderately pacifist war

I’ve been wondering whether it is possible for a country to count as pacifist and yet wage a defensive war. I think the answer is positive, as long as one has a moderate pacifism that is opposed to lethal violence but not to all violence. I think that a prohibition of all violence is untenable. It seems obvious that if you see someone about to shoot an innocent person, and you can give the shooter a shove to make them miss, you presumptively should.

Here’s what could be done by a moderately pacifist country.

First, we have “officially” non-lethal weapons: tasers, gas, etc. Some of these might violate current international law, but it seems that a pacifist country could modify its commitment to some accords.

Second, “lethal” weapons can be used less than lethally. For instance, with modern medicine, abdominal gunshot wounds are only 10% fatal, yet they are no doubt very effective at stopping an attacker. While it may seem weird to imagine a pacifist shooting someone in the stomach, when the chance of survival is 90%, it does not seem unreasonable to say that the pacifist could be aiming to stop the attacker non-lethally. After all, tasers sometimes kill, too. They do so less than 0.25% of the time, but that’s a difference of degree rather than of principle.

Third, we might subdivide moderate pacifists based on whether they prohibit all violence that foreseeably leads to death or just violence that intentionally leads to death. If it is only intentionally lethal violence that is forbidden, then quite a bit of modern warfare can stand. If the enemy is attacking with tanks or planes, one can intentionally destroy the tank or plane as a weapon, while only foreseeing, without intending, the death of the crew. (I don’t know how far one can take this line without sophistry. Can one drop a bomb on an infantry unit intending to smash up their rifles without intending to kill the soldiers?) Similarly, one can bomb enemy weapons factories.

Whether such a limited way of waging war could be successful probably depends on the case. If one combined the non-lethal (or not intentionally lethal) means with technological and numerical superiority, it wouldn’t be surprising to me if one could win.

## Thursday, December 1, 2022

### Against a moderate pacifism

Imagine a moderate pacifist who rejects lethal self-defense, but allows non-lethal self-defense when appropriate, say by use of tasers.

Now, imagine that one person is attacking you and nine other innocents, with the intent of killing the ten of you, and you can stop them with a taser. Surely you should, and surely the moderate pacifist will say that this is an appropriate use case for the taser.

Very well. Now consider this on a national level. Suppose there are a million enemy soldiers ordered to commit genocide against ten million, and you have two ways to stop them:

Tase the million soldiers.

Kill the general.

If you can tase one person to stop the murder of ten, then (1) should be permissible if it’s the only option. But tasers occasionally kill people. We don’t know how often. Apparently it’s less than 1 in 400 uses. Suppose it’s 1 in 4000. Then option (1) results in 250 enemy deaths.

So maybe our choice is between tasing a million, thereby non-intentionally killing 250 soldiers, and intentionally killing one general. It seems to me that (2) is morally preferable, even though our moderate pacifist has to allow (1) and forbid (2).

Note that a version of this argument goes through even if the moderate pacifist backs up and says that tasers are too lethal. For suppose instead of tasers we have drones that destroy the dominant hand of an enemy soldier while guaranteeing survival (with science fictional medical technology). It’s clearly right to release such a drone on a soldier who is about to kill ten innocents. But now compare:

Destroy the dominant hand of a million soldiers.

Kill the general.

I think (4) is still morally preferable to causing the kind of disruption to the lives of a million people that plan (3) would involve.

These may seem to be consequentialist arguments. I don't think so. I don't have the same intuitions if we replace the general by the general's innocent child in (2) and (4), even if killing the child were to stop the war (e.g., by making the general afraid that their other children would be murdered).

### Normative powers

A normative power is a power to change a normative condition. Raz says the change is not produced “causally” but “normatively”.

Here is a picture on which this is correct. We exercise a normative power by exercising a natural power in such a context that the successful exercise of the natural power is partly constitutive of a normative fact. For instance, we utter a promise, thereby exercising a natural power to engage in a certain kind of speech act, and our exercise of that speech act is partly constitutive of, rather than causal of, the state of affairs of our being obligated to carry out the promised action.

There are two versions of the above model. On one version, there is
an underlying fundamental conditional normative fact *C*, such as that if I have promised
something then I should do it, and my exercise of normative power
supplies the antecedent *A* of
that conditional, and then the normative consequent of *C* comes to be grounded in *C* and *A*. On another version, there there
are some natural acts that are directly constitutive of a normative
state of affairs, not merely by supplying the antecedent of a
conditional normative fact. I think the first version of the model is
the more plausible in paradigmatic cases.

But why not allow for a causal model? Why not suppose that a normative power is a causal power to make an irreducible normative property come to be instantiated in someone? Thus, my power to promise is the power to cause myself to be obligated to do what I have promised.

I think the difficulty with a causal model is the fact that in
paradigm cases of normative power, there is a natural power that
*is* being exercised, and we have the intuition that the exercise
of the natural power is necessary and sufficient for the normative
effect. But on a causal model, why couldn’t I cause a promissory-type
obligation without promising, simply causing the relevant property of
being obligated to come to be instantiated in me? And why couldn’t I
engage in the speech act while yet remaining normatively unbound,
because my normative power wasn’t exercised in parallel with the natural
power?

Maybe the answer to both questions is that I could, but only metaphysically and not causally. In other words, it could be that the laws of nature, or of human nature, make it impossible for me to exercise one of the powers without the other, just as I cannot wiggle my ring finger without wiggling my middle finger as well. On this view, if there is a God, he could cause me to acquire promissory-type obligations without my promising, and he could let me engage in the natural act of promising while blocking the exercise of normative power and leaving me normatively unbound. This doesn’t seem particularly problematic.

Perhaps the real problem for a lot of people with a causal view of normative powers is that it tends to lead to a violation of supervenience. For if it is metaphysically possble to have the exercise of the normative power without the exercise of the natural power, or vice versa, then it seems we don’t have supervenience of the normative on the non-normative. But supervenience does not seem to me to be inescapable.

## Wednesday, November 30, 2022

### Two versions of the guise of the good thesis

According to the guise of the good thesis, one always acts for the sake of an apparent good. There is a weaker and a stronger version of this:

**Weak**: Whenever you act, you act for an end that you perceive is good.**Strong**: Whenever you act, you act for an end, and every end you act for you perceive as good.

For the strong version to have any plausibility, “good” must include cases of purely instrumental goodness.

I think there is still reason to be sceptical of the strong version.

**Case 1:** There is some device which does something
useful when you trigger it. It is triggered by electrical activity. You
strap it on to your arm, and raise your arm, so that the electrical
activity in your muscles triggers the device. Your raising your arm has
the arm going up as an end, but that end is not perceived as good, but
merely neutral. All you care about is the electrical activity in your
muscles.

**Case 2:** Back when they were dating in high school,
Bob promised to try his best to bake a nine-layer chocolate cake for
Alice’s 40th birthday. Since then, Bob and Alice have had a falling out,
and hate each other’s guts. Moreover, Alice and all her guests hate
chocolate. But Alice doesn’t release Bob from his promise. Bob tries his
best to bake the cake in order to fulfill his promise, and happens to
succeed. In trying to bake the cake, Bob acted for the end of producing
a cake. But producing the cake was worthless, since no one would eat it.
The only value was in the trying, since that was the fulfillment of his
promise.

In both cases, it is still true that the agent acts for a good end—the useful triggering of the device and the production of the cake. But in both cases it seems they are also acting for a worthless end. Thus the cases seem to fit with the weak but not the strong guise of the good thesis.

I was going to leave it at this. But then I thought of a way to save
the strong guise of the good thesis. Success is valuable as such. When I
try to do something, succeeding at it has value. So the arm going up or
the cake being produced *are* valuable as necessary parts of the
success of one’s action. So perhaps every end of your action *is*
trivially good, because it is good for your action to succeed, and the
end is a (constitutive, not causal) means to success.

This isn’t quite enough for a defense of the strong thesis. For even if the success is good, it does not follow that you perceive the success as good. You might subscribe to an axiological theory on which success is not good in general, but only success at something good.

But perhaps we can say this. We have a normative power to endow some neutral things with value by making them our ends. And in fact the only way to act for an end that does not have any independent value is by exercising that normative power. And exercising that normative power involves your seeing the thing you’re endowing with value as valuable. And maybe the only way to raise your arm or for Bob to bake the cake in the examples is by exercising the normative power, and doing so involves seeing the end as good. Maybe. This has some phenomenological plausibility and it would be nice if it were true, because the strong guise of the good thesis is pretty plausible to me.

If this story is right, it adds a nuance to the ideas here.

## Tuesday, November 29, 2022

### An odd poker variant

Suppose Alice can read your mind, and you are playing poker against a set of people not including Alice. You don’t care about winning, just about money. Alice has a deal for you that you can’t refuse.

If you win, she takes your winnings away.

If you lose, but you tried to win, she pays you double what you lost.

If you lose, but you didn’t try to win, she does nothing.

Clearly the prudent thing to do is to try to win. For if you don’t try to win, then you are guaranteed not to get any money. But if you do try, you won’t lose anything, and you might gain.

Here is the oddity: you are trying to win in order to get paid, but you only get paid if you don’t win. Thus, you are trying to achieve something, the achievement of which would undercut the end you are pursuing.

Is this possible? I think so. We just need to distinguish between pursuing victory for the sake of something else that follows from victory and pursuing victory for the sake of something that might follow from the pursuit of victory.

### Nonoverriding morality

Some philosophers think that sometimes norms other than moral
norms—e.g., prudential norms or norms of the meaningfulness of life—take
precedence over moral norms and make permissible actions that are
morally impermissible. Let *F*-norms be such norms.

A view where *F*-norms
*always* override moral norms does not seem plausible. In the
case of prudential or meaningfulness, it would point to a fundamental
selfishness in the normative constitution of the human being.

So the view has to be that sometimes *F*-norms take precedence over moral
norms, but not always. There must thus be norms which are neither *F*-norms nor moral norms that decide
whether *F*-norms or moral norms
take precedence. We can call these “overall norms of combination”. And
it is crucial to the view that the norms of combination themselves be
neither *F*-norms nor moral norms.

But here is an oddity. Morality already combines *F*-considerations and first order
paradigmatically moral considerations. Consider two actions:

Sacrifice a slight amount of

*F*-considerations for a great deal of good for one’s children.Sacrifice an enormous amount of

*F*-considerations for a slight good for one’s children.

*Morality* says that (1) is obligatory but (2) is permitted.
Thus, morality already weighs *F* and paradigmatically moral
concerns and provides a combination verdict. In other words, there
already are *moral* norms of combination. So the view would be
that there are moral norms of combination and overall norms of
combination, both of which take into account exactly the same first
order considerations, but sometimes come to different conclusions
because they weigh the very same first order considerations differently
(e.g., in the case where a moderate amount of *F*-considerations needs to be
sacrificed for a moderate amount of good for one’s children).

This view violates Ockham’s razor: Why would we have moral norms of combination if the overall norms of combination always override them anyway?

Moreover, the view has the following difficulty: It seems that the best way to define a type of norm (prudential, meaningfulness, moral, etc.) is in terms of the types of consideration that the norm is based on. But if the overall norms of combination take into account the very same types of consideration as the moral norms of combination, then this way of distinguishing the types of norms is no longer available.

Maybe there is a view on which the overall ones take into account not
the first-order moral and *F*-considerations, but only the
deliverances of the moral and *F*-norms of combination, but that
seems needlessly complex.

## Monday, November 28, 2022

### Oppositional relationships

Here are three symmetric oppositional possibilities:

Competition:

*x*and*y*have shared knowledge that they are pursuing incompatible goals.Moral opposition:

*x*and*y*have shared knowledge that they are pursuing incompatible goals and each takes the other’s pursuit to be morally wrong.Mutual enmity:

*x*and*y*have shared knowledge that they each pursue the other’s ill-being for a reason other than the other’s well-being.

The reason for the qualification on reasons in 3 is that one might say that someone who punishes someone in the hope of their reform is pursuing their ill-being for the sake of their well-being. I don’t know if that is the right way to describe reformative punishment, but it’s safer to include the qualification in (3).

Note that cases of moral opposition are all cases of competition. Cases of mutual enmity are also cases of competition, except in rare cases, such as when a party suffers from depression or acedia which makes them not be opposed to their own ill-being.

I suspect that most cases of mutual enmity are also cases of moral opposition, but I am less clear on this.

Both competition and moral opposition are compatible with mutual love, but mutual enmity is not compatible with either direction of love.

Additionally, there is a whole slew of less symmetric options.

I think loving one’s competitors could be good practice for loving one’s (then necessarily non-mutual) enemies.

### Games and consequentialism

I’ve been thinking about who competitors, opponents and enemies are, and I am not very clear on it. But I think we can start with this:

*x*and*y*are competitors provided that they knowingly pursue incompatible goals.

In the ideal case, competitors both rightly pursue the incompatible goals, and each knows that they are both so doing.

Given externalist consequentialism, where the right action is the one that actually would produce better consequences, ideal competition will be extremely rare, since the only time the pursuit of each of two incompatible goals will be right is if there is an exact tie between the values of the goals, and that is extremely rare.

This has the odd result that on externalist consequentialism, in most sports and other games, at least one side is acting wrongly. For it is extremely rare that there is an exact tie between the values of one side winning and the value of the other side winning. (Some people enjoy victory more than others, or have somewhat more in the way of fans, etc.)

On internalist consequentalism, where the right action is defined by expected utilities, we would expect that if both sides are unbiased investigators, in most of the games, at least one side would at take the expected utility of the other side’s winning to be higher. For if both sides are perfect investigators with the same evidence and perfect priors, then they will assign the same expected utilities, and so at least one side will take the other’s to have higher expected utility, except in the rare case where the two expected utilities are equal. And if both sides assign expected utilities completely at random, but unbiasedly (i.e., are just as likely to assign a higher expected utility to the other side winning as to themselves), then bracketing the rare case where a side assigns equal expected utility to both victory options, any given side will have a probability of about a half of assigning higher expected utility to the other side’s victory, and so there will be about a 3/4 chance that at least one side will take the other side’s victory to be more likely. And other cases of unbiased investigators will likely fall somewhere between the perfect case and the random case, and so we would expect that in most games, at least one side will be playing for an outcome that they think has lower expected utility.

Of course, in practice, the two sides are not unbiased. One might overestimate the value of oneself winning and the underestimate the value of the other winning. But that is likely to involve some epistemic vice.

So, the result is that either on externalist or internalist consequentialism, in most sports and other competitions, at least one side is acting morally wrongly or is acting in the light of an epistemic vice.

I conclude that consequentialism is wrong.

### Precise lengths

As usual, write [*a*,*b*] for the interval of
the real line from *a* to *b* including both *a* and *b*, (*a*,*b*) for the interval of
the real line from *a* to *b* excluding *a* and *b*, and [*a*, *b*) and (*a*, *b*] respectively for the
intervals that include *a* and
exclude *b* and vice versa.

Suppose that you want to measure the size *m*(*I*) of an interval *I*, but you have the conviction that
single points matter, so [*a*,*b*] is bigger than (*a*,*b*), and you want to use
infinitesimals to model that difference. Thus, *m*([*a*,*b*]) will be
infinitesimally bigger than *m*((*a*,*b*)).

Thus at least some intervals will have lengths that aren’t real numbers: their length will be a real number plus or minus a (non-zero) infinitesimal.

At the same time, intuitively, *some* intervals from *a* to *b* should have length
*exactly* *b* − *a*, which is a real
number (assuming *a* and *b* are real). Which ones? The choices
are [*a*,*b*], (*a*,*b*), [*a*, *b*) are (*a*, *b*].

Let *α* be the non-zero
infinitesimal length of a single point. Then [*a*,*a*] is a single point.
Its length thus will be *α*, and
not *a* − *a* = 0. So
[*a*,*b*] can’t
*always* have real-number length *b* − *a*. But maybe at least
it can in the case where *a* < *b*? No. For suppose
that *m*([*a*,*b*]) = *b* − *a*
whenever *a* < *b*.
Then *m*((*a*,*b*]) = *b* − *a* − *α*
whenever *a* < *b*,
since (*a*, *b*] is
missing exactly one point of [*a*,*b*]. But then let *c* = (*a*+*b*)/2 be the
midpoint of [*a*,*b*].
Then:

*m*([*a*,*b*]) =*m*([*a*,*c*]) +*m*((*c*,*b*]) = (*c*−*a*) + (*b*−*c*−*α*) =*b*−*a*−*α*,

rather than *m*([*a*,*b*]) as was
climed.

What about (*a*,*b*)?
Can that always have real number length *b* − *a* if *a* < *b*? No. For if we had
that, then we would absurdly have:

*m*((*a*,*b*)) =*m*((*a*,*c*)) +*α*+*m*((*c*,*b*)) =*c*−*a*+*α*+*b*−*c*=*b*−*a*+*α*,

since (*a*,*b*) is
equal to the disjoint union of (*a*,*c*), the point *c* and $(c,b).

That leaves [*a*, *b*)
and (*a*, *b*]. By
symmetry if one has length *b* − *a*, surely so does the
other. And in fact Milovich gave me a
proof that there is no contradiction in supposing that *m*([*a*,*b*)) = *m*((*b*,*a*]) = *b* − *a*.

## Tuesday, November 22, 2022

### Hyperreal expected value

I think I have a hyperreal solution, not entirely satisfactory, to three problems.

The problem of how to value the St Petersburg paradox. The particular version that interests me is one from Russell and Isaacs which says that any finite value is too small, but any infinite value violates strict dominance (since, no matter what, the payoff will be less than infinity).

How to value gambles on a countably infinite fair lottery where the gamble is positive and asymptotically approaches zero at infinity. The problem is that any positive non-infinitesimal value is too big and any infinitesimal value violates strict dominance.

How to evaluate expected utilities of gambles whose values are hyperreal, where the probabilities may be real or hyperreal, which I raise in Section 4.2 of my paper on accuracy in infinite domains.

The apparent solution works as follows. For any gamble with values in
some real or hyperreal field *V*
and any finitely-additive probability *p* with values in *V*, we generate a hyperreal expected
value *E*_{p},
which satisfies these plausible axioms:

Linearity:

*E*_{p}(*a**f*+*b**g*) =*a**E*_{p}*f*+*b**E*_{p}*g*for*a*and*b*in*V*Probability-match:

*E*_{p}1_{A}=*p*(*A*) for any event*A*, where 1_{A}is 1 on*A*and 0 elsewhereDominance: if

*f*≤*g*everywhere, then*E*_{p}*f*≤*E*_{p}*g*, and if*f*<*g*everywhere, then*E*_{p}*f*<*E*_{p}*g*.

How does this get around the arguments I link to in (1) and (2) that
seem to say that this can’t be done? The trick is this: the expected
value has values in a hyperreal field *W* which will be larger than *V*, while (4)–(6) only hold for
gambles with values in *V*. The
idea is that we distinguish between what one might call primary values,
which are particular goods in the world, and what one might call
distribution values, which specify how much a random distribution of
primary values is worth. We do not allow the distribution values
themselves to be the values of a gamble. This has some downsides, but at
least we can have (4)–(6) on *all* gambles.

How is this trick done?

I think like this. First it looks like the Hahn-Banach
dominated extension theorem holds for *V*_{2}-valued *V*_{1}-linear functionals on
*V*_{1}-vector spaces
*V*_{1} ⊆ *V*_{2}
are real or hyperreal field, except that our extending functional may
need to take values in a field of hyperreals even larger than *V*_{2}. The crucial thing to
note is that any subset of a real or hyperreal field has a supremum in a
larger hyperreal field. Then where the proof of the Hahn-Banach theorem
uses infima and suprema, you move to a larger hyperreal field to get
them.

Now, embed *V* in a hyperreal
field *V*_{2} that
contains a supremum for every subset of *V*, and embed *V*_{2} in *V*_{3} which has a supremum
for every subset of *V*_{2}. Let *Ω* be our probability space.

Let *X* be the space of
bounded *V*_{2}-valued
functions on *Ω* and let *M* ⊆ *X* be the subspace of
simple functions (with respect to the algebra of sets that *Ω* is defined on). For *f* ∈ *M*, let *ϕ*(*f*) be the integral of
*f* with respect to *p*, defined in the obvious way. The
supremum on *V*_{2}
(which has values in *V*_{3}) is then a seminorm
dominating *ϕ*. Extend *ϕ* to a *V*-linear function *ϕ* on *X* dominated by *V*_{2}. Note that if *f* > 0 everywhere for *f* with values in *V*, then *f* > *α* > 0 everywhere
for some *α* ∈ *V*_{2}, and
hence *ϕ*(−*f*) ≤ − *α* by
seminorm domination, hence 0 < *α* ≤ *ϕ*(*f*).
Letting *E*_{p}
be *ϕ* restricted to the *V*-valued functions, our construction
is complete.

I should check all the details at some point, but not today.

## Monday, November 21, 2022

### Dominance and countably infinite fair lotteries

Suppose we have a finitely-additive probability assignment *p* (perhaps real, perhaps hyperreal)
for a countably infinite lottery with tickets 1, 2, ... in such a way that each ticket has
infinitesimal probability (where zero counts as an infinitesimal). Now
suppose we want to calculate the expected value or previsio *E*_{p}*U* of
any bounded wager *U* on the
outcome of the lottery, where we think of the wager as assigning a value
to each ticket, and the wager is bounded if there is a finite *M* such that |*U*(*n*)| < *M* for
all *n*.

Here are plausible conditions on the expected value:

Dominance: If

*U*_{1}<*U*_{2}everywhere, then*E*_{p}*U*_{1}<*E*_{p}*U*_{2}.Binary Wagers: If

*U*is 0 outside*A*and*c*on*A*, then*E*_{p}*U*=*c**P*(*A*).Disjoint Additivity: If

*U*_{1}and*U*_{2}are wagers supported on disjoint events (i.e., there is no*n*such

that*U*_{1}(*n*) and*U*_{2}(*n*) are both non-zero), then*E*_{p}(*U*_{1}+*U*_{2}) =*E*_{p}*U*_{1}+*E*_{p}*U*_{2}.

But we can’t. For suppose we have it. Let *U*(*n*) = 1/(2*n*). Fix
a positive integer *m*. Let
*U*_{1}(*n*) be
2 for *n* ≤ *m* + 1 and 0 otherwise. Let *U*_{2}(*n*) be 1/*m* for *n* > *m* + 1 and 0 for *n* ≤ *m* + 1. Then by Binary
Wagers and by the fact that each ticket has infinitesimal probability,
*E*_{p}*U*_{1}
is an infinitesimal *α* (since
the probability of any finite set will be infinitesimal). By Binary
Wagers and Dominance, *E*_{p}*U*_{2} ≤ 1/(*m*+1).
Thus by Disjoint Additivity, *E*_{p}(*U*_{1}+*U*_{2}) ≤ *α* + 1/(*m*+1) < 1/*m*.
But *U* < *U*_{1} + *U*_{2}
everywhere, so by Dominance we have *E*_{p}*U* < 1/*m*.
Since 0 < *U* everywhere, by
Dominance and Binary Wagers we have 0 < *E*_{p}*U*.

Thus, *E*_{p}*U* is
a non-zero infinitesimal *β*.
But then *β* < *U*(*n*) for
all *n*, and so by Binary Wagers
and Dominance, *β* < *E*_{p}*U*,
a contradiction.

I think we should reject Dominance.

### Corruptionism and care about the soul

According to Catholic corruptionists, when I die, my soul will continue to exist, but I won’t; then at the Resurrection, I will come back into existence, receiving my soul back. In the interim, however, it is my soul, not I, who will enjoy heaven, struggle in purgatory or suffer in hell.

Of course, for any thing that enjoys heaven, strugges in purgatory or suffers in hell, I should care that it does so. But should I have that kind of special care that we have about things that happen to ourselves for what happens to the soul? I say not, or at most slightly. For suppose that it turned out on the correct metaphysics that my matter continues to exist after death. Should I care whether it burns, decays, or is dissected, with that special care with which we care about what happens to ourselves? Surely not, or at most slightly. Why not? Because the matter won’t be a part of me when this happens. (The “at most slightly” flags the fact that we can care about “dignitary harms”, such as nobody showing up at our funeral, or us being defamed, etc.)

But clearly heaven, purgatory and hell in the interim state is something we should care about.

## Friday, November 18, 2022

### Social choice principles and invariance under symmetries

A comment by a referee of a recent paper of mine that one of my results in decision theory didn’t actually depend on numerical probabilities and hence could extend to social choice principles made me realize that this may be true for some other things I’ve done.

For instance, in the past I’ve proved theorems on qualitative
probabilities. A qualitative probability is a relation ≼ on the subsets of some sample space *Ω* such that:

≼ is transitive and reflexive.

⌀ ≼

*A*if

*A*∩*C*=*B*∩*C*= ⌀, then*A*≼*B*iff*A*∩*C*≼*B*∩*C*(additivity).

But need not think of *Ω* as
a space of possibilities and of ≼ as a
probability comparison. We could instead think of it as a set of people
who are candidates for getting some good thing, with *A* ≼ *B* meaning that it’s at
least as good for the good thing to be distributed to the members of
*B* as to the members of *A*. Axioms (1) and (2) are then
obvious. And axiom (3) is an independence axiom: whether it is at least
as good to give the good thing to the members of *B* as to the members of *A* doesn’t depend on whether we give
it to the members of a disjoint set *C* at the same time.

Of course, for a general social choice principle we need more than
just a decision whether to give one and the same good to the members of
some set. But we can still formalize those questions in terms of
something pretty close to qualitative probabilities. For a general
framework, suppose a population set *X* (a set of people or places in
spacetime or some other sites of value) and a set of values *V* (this could be a set of types of
good, or the set of real numbers representing values). We will suppose
that *V* comes with a transitive
and reflexive (preorder) preference relation ≤. Now let *Ω* = *X* × *V*. A value
distribution is a function *f*
from *X* to *V*, where *f*(*x*) = *v* means
that *x* gets something of value
*v*.

We want to generate a reflexive and transitive preference ordering
≼ on the set *V*^{X} of value
distributions.

Write *f* ≈ *g* when
*f* ≼ *g* and *g* ≼ *f*, and *f* ≺ *g* when *f* ≼ *g* but not *g* ≼ *f*. Similarly for values
*v* and *w*, write *v* < *w* if *v* ≤ *w* but not *w* ≤ *v*.

Here is a plausible axiom on value distributions:

- Sameness independence: if
*f*_{1},*f*_{2},*g*_{1},*g*_{2}are value distributions and*A*⊆*X*is such that (a)*f*_{1}≼*f*_{2}, (b)*f*_{1}(*x*) =*f*_{2}(*x*) and*g*_{1}(*x*) =*g*_{2}(*x*) if*x*∉*A*, (c)*f*_{1}(*x*) =*g*_{1}(*x*) and*f*_{2}(*x*) =*g*_{2}(*x*) if*x*∈*A*.

In other words, the mutual ranking between two value distributions does not depend on what the two distributions do to the people on whom the distributions agree. If it’s better to give $4 to Jones than to give $2 to Smith when Kowalski is getting $7, it’s still better to give $4 to Jones than to give $2 to Smith when Kowalski is getting $3. There is probably some other name in the literature for this property, but I know next to nothing about social choice literature.

Finally, we want to have some sort of symmetries on the population.
The most radical would be that the value distributions don’t care about
permutations of people, but more moderate symmetries may be required.
For this we need a group *G* of
permutations acting on *X*.

- Strong
*G*-invariance: if*g*∈*G*and*f*is a value distribution, then*f*∘*g*≈*f*.

Here, *f* ∘ *g* is the
value distribution where site *x* gets *f*(*g*(*x*)).

Additionally, the following is plausible:

- Pareto: If
*f*(*x*) ≤*g*(*x*) for all*x*with*f*(*x*) <*g*(*x*) for some*x*, then*f*≺*g*.

**Theorem:** Assume the Axiom of Choice. Suppose ≤ on *V* is reflexive, transitive and
non-trivial in the sense that it contains two values *v* and *w* such that *v* < *w*. There exists a
reflexive, transitive preference ordering ≼ on the value distributions satisfying
(4)–(6) if and only if there is such an ordering that is total if and
only if *G* has locally finite
action on *X*.

A group of symmetries *G* has
locally finite action a set *X*
provided that for each finite subset *H* of *G* and each *x* ∈ *X*, applying finite
combinations of members of *G*
to *x* generates only a finite
subset of *X*. (More precisely,
if ⟨*H*⟩ is the subgroup
generated by *G*, then ⟨*H*⟩*x* is finite.)

If *X* is finite, then local
finiteness of action is trivial. If *X* is infinite, then it will be
satisfies in some cases but not others. For instance, it will be
satisfied if *G* is permutations
that only move a finite number of members of *X* at a time. It will on the other
hand fail if *X* is a infinite
bunch of people regularly spaced in a line and *G* is shifts.

The trick to the proof of the Theorem is to reduce preferences
between distributions to comparisons of subsets of *X* × *V* and to reduce
comparisons of subsets of *X* to
preferences between binary distributions.

**Proof of Therem:** Suppose that *G* has locally finite action. Define
*Ω* = *X* × *V*.
By Theorem 2 of my invariance of non-classical probabilities
paper, there is a strongly *G*-invariant regular (i.e., ⌀ ≺ *A* if *A* is non-empty) qualitative
probability ≼ on *Ω*. Given a value distribution *f*, let *f*^{*} = {(*x*,*v*) : *v* ≤ *f*(*x*)}
be a subset of *Ω*. Define *f* ≼ *g* iff *f*^{*} ≼ *g*.

Totality, reflexivity, transitivity and strong *G*-invariance for value distributions
follows from the same conditions for subsets of *Ω*. Regularity of ≼ on the subsets of *Ω* and additivity implies that if
*A* ⊂ *B* then *A* ≺ *B*. The Pareto condition
for ≼ on the value distributions
follows since if *f* and *g* satisfy are such that *f*(*x*) ≤ *g*(*x*)
for all *x* with strict
inequality for some *x*, then
*f*^{*} ⊂ *g*^{*}.
Finally, the complicated sameness independence condition follows from
additivity.

Now suppose there is a (not necessarily total) strongly *G*-invariant reflexive and transitive
preference ordering ≼ on the value
distributions satisfying (4)–(6). Given a subset *A* of *X*, define *A*^{†} to be the value
distribution that gives *w* to
all the members of *A* and *v* to all the non-members, where
*v* < *w*. Define
*A* ≼ *B* iff *A*^{†} ≼ *B*^{†}.
This will be a strongly *G*-invariant reflexive and transitive
relation on the subsets of *X*.
It will be regular by the Pareto condition. Finally, additivity follows
from the sameness independence condition. Local finiteness of action of
*G* then follows from Theorem 2
of my paper. ⋄

Note that while it is natural to think of *X* has just a set of people or of
locations, inspired
by Kenny Easwaran one can also think of it as a set *Q* × *Ω* where *Ω* is a probability space and *Q* is a population, so that *f*(*x*,*ω*) represents
the value *x* gets at location
*ω*. In that case, *G* might be defined by symmetries of
the population and/or symmetries of the probability space. In such a
setting, we might want a weaker Pareto principle that supposes
additionally that *f*(*x*,*ω*) < *g*(*x*,*ω*)
for some *x* and *all*
*ω*. With that weaker Pareto
principle, the proof that the existence of a *G*-invariant preference of the right
sort on the distributions implies local finiteness of action does not
work. However, I think we can still prove local finiteness of action in
that case if the symmetries in *G* act only on the population (i.e.,
for all *x* and *ω* there is an *y* such that *g*(*x*,*ω*) = (*y*,*ω*)).
In that case, given a subset *A*
of the population *Q*, we define
*A*^{†} to be the
distribution that gives *w* to
all the persons in *A* with
certainty (i.e., everywhere on *Ω*) and gives *v* to everyone else, and the rest of
the proof should go through, but I haven’t checked the details.

## Thursday, November 17, 2022

### Cerebrums and rattles

Animalists think humans are animals. Suppose I am an animalist and I think that I go with my cerebrum in cerebrum-transplant cases. That may seem weird. But suppose we make an equal opportunity claim here: all animals that have cerebra go with their cerebra. If your dog Rover’s cerebrum is transplanted into a robotic body, then the cerebrumless thing is not Rover. Rather, Rover inhabits a robotic body or that body comes to be a part of Rover, depending on views about prostheses. And the same is true for any animal that has a cerebrum.

It initially seems weird to say that some animals can survive reduced
to a cerebrum and others cannot. But it’s not that weird when we add
that the ones that can’t survive reduced to a cerebrum are animals that
don’t *have* a cerebrum.

The person who thinks survival reduced to a cerebrum is implausible for an animal might, however, say that this is what’s odd about it. An animal reduced to cerebrum lacks internal life support organs (heart, lungs, etc.) It is odd to think that some animals can survive without internal life support and others cannot.

But compare this: Some animals can partly exist in spatial locations where they have no living cells, and others cannot. The outer parts of my hairs are parts of me, but there are no living cells there. If my hair is in a room, then I am partly in that room, even if no living cells of mine are in the room. But on the other hand, there are some animals (at least the unicellular ones, but maybe also some soft invertebrates) that can only exist where they have a living cell.

One might object that the spatial case and the temporal case are different, because in the spatial case we are talking of partial presence and in the temporal case of full presence. But a four-dimensionalist will disagree. To exist at a time is to be partly present at that time. So to a four-dimensionalist the analogy is pretty strict.

Finally, compare this. Suppose Snaky a rattlesnake stretched along a
line in space. Now suppose we simultaneously annihilate everything in
Snaky. Now, “simultaneously” is presumably defined with respect to some
reference frame *F*_{1}.
Let *z* be a point in Snaky’s
rattle located just prior (according to *F*_{1}) to Snaky’s
destruction. Then Snaky is partly present at *z*. But with a bit of thought, we can
see that there is another reference frame *F*_{2} where the only parts
of Snaky simultaneous with *z*
are parts of the rattle: all the non-rattle parts of Snaky have already
been annihilated at *F*_{2}, but the rattle has
not. Then in *F*_{2} the
following is true: there is a time at which Snaky exists but nothing
outside of Snaky’s rattle exists. Hence Snaky can exist as just a
rattle, albeit for a very, very short period of time.

Hence even a snake can exist without its life-support organs, but only for a short period of time.

## Monday, November 14, 2022

### Reducing goods to reasons?

In my previous post I cast doubt on reducing moral reasons to goods.

What about the other direction? Can we reduce goods to reasons?

The simplest story would be that goods reduce to reasons to promote them.

But there seem to be goods that give no one a reason to promote them. Consider the good fact that there exist (in the eternalist sense: existed, exist now, will exist, or exist timelessly) agents. No agent can promote the fact that there exist agents: that good fact is part of the agent’s thrownness, to put it in Heideggerese.

Maybe, though, this isn’t quite right. If Alice is an agent, then
Alice’s existence is a good, but the fact that some agent or other
exists isn’t a good as such. I’m not sure. It seems like a world with
agents is better for the existence of agency, and not just better for
the particular agents it has. Adding *another* agent to the world
seems a lesser value contribution than just ensuring that there is
agency at all. But I could be wrong about that.

Another family of goods, though, are necessary goods. That God exists is good, but it is necessarily true. That various mathematical theorems are beautiful is necessarily true. Yet no one has reason to promote a necessary truth.

But perhaps we could have a subtler story on which goods reduce not just to reasons to promote them, but to reasons to “stand for them” (taken as the opposite of “standing against them”), where promotion is one way of “standing for” a good, but there are others, such as celebration. It does not make sense to promote the existence of God, the existence of agents, or the Pythagorean theorem, but celebrating these goods makes sense.

However, while it might be the case that something is good just in
case an agent should “stand for it”, it does not seem right to think
that it is good *to the extent that* an agent should “stand for
it”. For the degree to which an agent should stand for a good is
determined not just by the magnitude of the good, but the agent’s
relationship to the good. I should celebrate my children’s
accomplishments more than strangers’.

Perhaps, though, we can modify the story in terms of goods-for-*x*, and say that *G* is good-for-*x* to the extent that *x* should stand for *G*. But that doesn’t seem right,
either. I should stand for justice for all, and not merely to the degree
that justice-for-all is good-for-me. Moreover, there goods that are good
for non-agents, while a non-agent does not have a reason to do
anything.

I love reductions. But alas it looks to me like reasons and goods are not reducible in either direction.

### The 2018 Belgium vs Brazil World Cup game

In 2018, the Belgians beat the Brazilians 2-1 in the 2018 World Cup soccer quarterfinals. There are about 18 times as many Brazilians and Belgians in the world. This raises a number of puzzles in value theory, if for simplicity we ignore everyone but Belgians and Brazilians in the world.

An order of magnitude more people *wanted* the Brazilians to
win, and getting what one wants is good. An order of magnitude more
people would have felt significant and appropriate *pleasure* had
the Brazilians won, and an appropriate pleasure is good. And given both
wishful thinking as well as reasonable general presumptions about there
being more talent available in a larger population base, we can suppose
that a lot more people *expected* the Brazilians to win, and it’s
good if what one thinks is the case is in fact the case.

You might think that the good of the many outweighs the good of the few, and Belgians are few. But, clearly, the above facts gave very little moral reason to the Belgian players to lose. One might respond that the above facts gave lots of reason to the Belgians to lose, but these reasons were outweighed by the great value of victory to the Belgian players, or perhaps the significant intrinsic value of playing a sport as well as one can. Maybe, but if so then just multiply both countries’ populations by a factor of ten or a hundred, in which case the difference between the goods (desire satisfaction, pleasure and truth of belief) is equally multiplied, but still makes little or no moral difference to what the Belgian players should do.

Or consider this from the point of view of the Brazilian players. Imagine you are one of them. Should the good of Brazil—around two hundred million people caring about the game—be a crushing weight on your shoulders, imbuing everything you do in practice and in the game with a great significance? No! It’s still “just a game”, even if the value of the good is spread through two hundred million people. It would be weird to think that it is a minor pecadillo for a Belgian to slack off in practice but a grave sin for a Brazilian to do so, because the Brazilian’s slacking hurts an order of magnitude more people.

That said, I do think that the larger population of Brazil imbues the
Brazilians’ games and practices with *some* not insignificant
additional moral weight than the Belgians’. It would be odd if the
pleasure, desire satisfaction and expectations of so many counted for
*nothing*. But on the other hand, it should make no significant
difference to the Belgians whether they are playing Greece or Brazil:
the Belgians shouldn’t practice less against the Greeks on the grounds
that an order of magnitude fewer people will be saddened when the Greeks
lose than when Brazilians do.

However, these considerations seem to me to depend to some degree on
which decisions one is making. If Daniel is on the soccer team and
deciding how hard to work, it makes little difference whether he is on
the Belgian or Brazilian team. But suppose instead that Daniel is has
two talents: he could become an excellent nurse or a top soccer player.
As a nurse, he would help relieve the suffering of a number of patients.
As a soccer player, in addition to the intrinsic goods of the sports, he
would contribute to his fellow citizens’ pleasure and desire
satisfaction. In *this* decision, it seems that the number of
fellow citizens *does* matter. The number of people Daniel can
help as a nurse is not very dependent on the total population, but the
number of people that his soccer skills can delight varies linearly with
the total population, and if the latter number is large enough, it seems
that it would be quite reasonable for Daniel to opt to be a soccer
player. So we could have a case where if Daniel is Belgian he should
become a nurse but if Brazilian then a soccer player (unless Brazil has
a significantly greater need for nurses than Belgium, that is). But once
on the team, it doesn’t seem to matter much.

The map from axiology to moral reasons is quite complex, contextual, and heavily agent-centered. The hope of reducing moral reasons to axiology is very slim indeed.

## Friday, November 11, 2022

### Species flourishing

As an Aristotelian who believes in individual forms, I’m puzzled
about cases of species-level flourishing that don’t seem reducible to
individual flourishing. On a biological level, consider how some species
(e.g., social insects, slime molds) have individuals who do not
reproduce. Nonetheless it is important to the flourishing of the
*species* that the species include some individuals that do
reproduce.

We might handle this kind of a case by attributing to other
individuals their *contribution* to reproduction of the species.
But I think this doesn’t solve the problem. Consider a non-biological
case. There are things that are achievements of the human species, such
as having reached the moon, having achieved a four minute mile, or
having proved the Poincaré conjecture. It seems a stretch to try to
individualize these goods by saying that we all contributed to them.
(After all, many of us weren’t even alive in 1969.)

I think a good move for an Aristotelian who believes in individual
forms is to say that “No man or bee is an island.” There is an external
flourishing in virtue of the species at large: it is a part of
*my* flourishing that humans landed on the moon. Think of how
members of a social group are rightly proud of the achievements of some
famous fellow-members: we Poles are proud of having produced Copernicus,
Russians of having launched humans into space, and Americans of having
landed on the moon.

However, there is still a puzzle. If it is a part of every human’s good that “I am a member of a species that landed on the moon”, does that mean the good is multiplied the more humans there are, because there are more instances of this external flourishing? I think not. External flourishing is tricky this way. The goods don’t always aggregate summatively between people in the case of external flourishing. If external flourishing were aggregated summatively, then it would have been better if Russia rather than Poland produced Copernicus, because there are more Russians than Poles, and so there would have been more people with the external good of “being a citizen of a country that produced Copernicus.” But that’s a mistake: it is a good that each Pole has, but the good doesn’t multiply with the number of Poles. Similarly, if Belgium is facing off Brazil for the World Cup, it is not the case that it would be way better if the Brazilians won, just because there are a lot more Brazilians who would have the external good of “being a fellow citizen with the winners of the World Cup.”

### More on the interpersonal Satan's Apple

Let me take another look at the interpersonal moral Satan’s Apple, but start with a finite case.

Consider a situation where a *finite* number *N* of people independently make a
choice between *A* and *B* and some disastrous outcome
happens if the number of people choosing *B* hits a threshold *M*. Suppose further that if you fix
whether the disaster happens, then it is better you to choose *A* than *B*, but the disastrous outcome
outweighs all the benefits from all the possible choices of *B*.

For instance, maybe *B* is
feeding an apple to a hungry child, and *A* is refraining from doing so, but
there is an evil dictator who likes children to be miserable, and once
enough children are not hungry, he will throw all the children in
jail.

Intuitively, you should do some sort of expected utility calculation
based on your best estimate of the probability *p* that among the *N* − 1 people other than you, *M* − 1 will choose *B*. For if fewer or more than *M* − 1 of them choose *B*, your choice will make no
difference, and you should choose *B*. If *F* is the difference between the
utilities of *B* and *A*, e.g., the utility of feeding the
apple to the hungry child (assumed to be fairly positive), and *D* is the utility of the disaster
(very negative), then you need to see if *p**D* + *F* is positive
or negative or zero. Modulo some concerns about attitudes to risk, if
*p**D* + *F* is
positive, you should choose *B*
(feed the child) and if its negative, you shouldn’t.

If you have a uniform distribution over the possible number of people
other than you choosing *B*, the
probability that this number is *M* − 1 will be 1/*N* (since the number of people
other than you choosing *B* is
one of 0, 1, ..., *N* − 1). Now,
we assumed that the benefits of *B* are such that they don’t outweigh
the disaster even if everyone chooses *B*, so *D* + *N**F* < 0.
Therefore (1/*N*)*D* + *F* < 0,
and so in the uniform distribution case you shouldn’t choose *B*.

But you might not have a uniform distribution. You might, for
instance, have a reasonable estimate that a proportion *p* of other people will choose *B* while the threshold is *M* ≈ *q**N* for some
fixed ratio *q* between 0 and 1. If
*q* is not close to *p*, then facts about the binomial
distribution show that the probability that *M* − 1 other people choose *B* goes approximately exponentially
to zero as *N* increases.
Assuming that the badness of the disaster is linear or at most
polynomial in the number of agents, if the number of agents is large
enough, choosing *B* will be a
good thing. Of course, you might have the unlucky situation that *q* (the ratio of threshold to number
of people) and *p* (the
probability of an agent choosing *B*) are approximately equal, in which
case even for large *N*, the
risk that you’re near the threshold will be too high to allow you to
choose *B*.

But now back to infinity. In the interpersonal moral Satan’s Apple,
we have infinitely many agents choosing between *A* and *B*. But now instead of the threshold
being a finite number, the threshold is an infinite cardinality (one can
also make a version where it’s a co-cardinality). And this threshold has
the property that other people’s choices can *never* be such that
your choice will put things above the threshold—either the threshold has
already been met without your choice, or your choice can’t make it hit
the threshold. In the finite case, it depended on the numbers involved
whether you should choose *A* or
*B*. But the exact same
reasoning as in the finite case, but now without *any*
statistical inputs being needed, shows that you should choose *B*. For it literally cannot make any
difference to whether a disaster happens, no matter what other people
choose.

In my previous post, I suggested that the interpersonal moral Satan’s
Apple was a reason to embrace causal finitism: to deny that an outcome
(say, the disaster) can causally depend on infinitely many inputs (the
agents’ choices). But the finite cases make me less confident. In the
case where *N* is large, and our
best estimate of the probability of another agent choosing *B* is a value *p* not close to the threshold ratio
*q*, it still seems
counterintuitive that you should morally choose *B*, and so should everyone else, even
though that yields the disaster.

But I think in the finite case one can remove the
counterintuitiveness. For there are mixed strategies that if adopted by
everyone are better than everyone choosing *A* or everyone choosing *B*. The mixed strategy will involve
choosing some number 0 < *p*_{best} < *q*
(where *q* is the threshold
ratio at which the disaster happens) and everyone choosing *B* with probability *p*_{best} and *A* with probability 1 − *p*_{best}, where *p*_{best} is carefully
optimized allow as many people to feed hungry children without a
significant risk of disaster. The exact value of *p*_{best} will depend on the
exact utilities involved, but will be close to *q* if the number of agents is large,
as long as the disaster doesn’t scale exponentially. Now our statistical
reasoning shows that when your best estimate of the probability of other
people choosing *B* is
*not* close to the threshold ratio *q*, you should just straight out
choose *B*. And the worry I had
is that everyone doing that results in the disaster. But it does not
seem problematic that in a case where your data shows that people’s
behavior is not close to optimal, i.e., their behavior propensities do
not match *p*_{best},
you need to act in a way that doesn’t universalize very nicely. This is
no more paradoxical than the fact that when there are criminals, we need
to have a police force, even though ideally we wouldn’t have one.

But in the infinite case, no matter what strategy other people adopt,
whether pure or mixed, choosing *B* is better.

## Thursday, November 10, 2022

### The interpersonal Satan's Apple

Consider a moral interpersonal version of Satan’s
Apple: infinitely many people independently choose whether to give a
yummy apple to a (different) hungry child, and if infinitely many choose
to do so, some calamity happens to everyone, a calamity outweighing the
hunger the child suffers. You’re one of the potential apple-givers and
you’re not hungry yourself. The disaster strikes if and only if
infinitely many people *other than you* give an apple. Your
giving an apple makes no difference whatsoever. So it seems like you
*should* give the apple to the child. After all, you relieve one
child’s hunger, and that’s good whether or not the calamity happens.

Now, we deontologists are used to situations where a disaster happens because one did the right thing. That’s because consequences are not the only thing that counts morally, we say. But in the moral interpersonal Satan’s Apple, there seems to be no deontology in play. It seems weird to imagine that disaster could strike because everyone did what was consequentialistically right.

One way out is causal finitism: Satan’s Apple is impossible, because the disaster would have infinitely many causes.

### More on discounting small probabilities

In yesterday’s
post, I argued that there is something problematic about the idea of
discounting small probabilities, given that in a large enough lottery
*every* possibility with has a small probability. I then offered
a way of making sense of the idea by “trimming” the utility function at
the top and bottom.

This morning, however, I noticed that one can also take the idea of
discounting small probabilities more literally and still get the exact
same results as by trimming utility functions. Specifically, given a
probability function *P* and a
probability discount threshold *ϵ*, we form a credence function *P*_{ϵ} by letting
*P*_{ϵ}(*A*) = *P*(*A*)
if *ϵ* ≤ *P*(*A*) ≤ 1 − *ϵ*,
*P*_{ϵ}(*A*) = 0
if *P*(*A*) < *ϵ* and
*P*_{ϵ}(*A*) = 1
if *P*(*A*) > 1 − *ϵ*.
This discounts close-to-zero probabilities to zero and raises close-to-one
probabilities to one. (We shouldn’t forget the second or things won't work well.)

Of course, *P*_{ϵ} is not in
general a probability, but it does satisfy the Zero, Non-Negativity,
Normalization and Monotonicity axioms, and we can now use LSI^{↑} level-set
integral to calculate utilities with *P*_{ϵ}.

If *U*_{ϵ}
is the “trimmed” utility function from my previous post, then LSI^{↑}_{Pϵ}(*U*) = *E*(*U*_{2ϵ}),
so the two approaches are equivalent.

One can also do the same thing within Buchak’s
REU theory, since that theory is equivalent to applying LSI^{↑} with a probability transformed
by a monotonic map of [0,1] to [0,1] keeping endpoints fixed, which is
exactly what I did when moving from *P* to *P*_{ϵ}.

## Wednesday, November 9, 2022

### How to discount small probabilities

A very intuitive solution to a variety of problems in infinite decision theory is that “for possibilities that have very small probabilities of occurring, we should discount those probabilities down to zero” when making decisions (Monton).

Suppose throughout this post that *ϵ* > 0 counts as our threshold of
“very small probabilities”. No doubt *ϵ* < 1/100.

In this post I want to offer a precise and friendly amendment to the
solution of neglecting small probabilities. But first why we need an
amendment. Consider a game where an integer *K* is randomly chosen between − 1 and *N* for some large fixed positive
*N*, so large that 1/(2+*N*) < *ϵ*, and you get
*K* dollars. The game is clearly
worth playing. But if you discount “possibilities that have very small
probabilities”, you are left with *nothing*: every possibility
has a very small probability!

Perhaps this is uncharitable. Maybe the idea is not that we discount
to zero *all* possibilities with small probabilities, but that we
discount such possibilities until the total discount hits the threshold
*ϵ*. But while this sounds like
a charitable interpretation of the suggestion, it leaves the theory
radically underdetermined. For *which* possibilities do we
discount? In my lottery case, do we start by discounting the
possibilities at the low end ( − 1, 0, 1, ...) until we have hit the
threshold? Or do we start at the high end (*N*, *N* − 1, *N* − 2, ...)
or somewhere in the middle?

Here is my friendly proposal. Let *U* be the utility function we want to
evaluate the value of. Let *T*
be the smallest value such that *P*(*U*>*T*) ≤ *ϵ*/2.
(This exists: *T* = inf {*λ* : *P*(*U*>*λ*) ≤ *ϵ*/2}.)
Let *t* be the largest value
such that *P*(*U*<*t*) ≤ *ϵ*/2
(i.e., *t* = sup {*λ* : *P*(*U*<*λ*) ≤ *ϵ*/2}).
Take *U* and replace any values
bigger than *T* with *T* and any values smaller than *t* with *t*, and call the resulting utility
function *U*_{ϵ}. We now
replace *U* with *U*_{ϵ} in our
expected value calculations. (In the lottery example, we will be
trimming from both ends at the same time.)

The result is a precise theory (given the mysterious threshold *ϵ*). It doesn’t neglect all
possibilities with small probabilities, but rather it trims
low-probability outliers. The trimming procedure respects the fact that
often utility functions are defined up to positive affine
transformations.

Moreover, the trimming procedure can yield an answer to what I think is the biggest objection to small-probability discounting, namely that in a long enough run—and everyone should think there is a non-negligible chance of eternal life—even small probabilities can add up. If you are regularly offered the same small chance of a gigantic benefit during an eternal future, and you turn it down each time because the chance is negligible, you’re almost surely missing out on an infinite amount of value. But we can apply the trimming procedure at the level of choice of policies rather than of individual decisions. Then if small chances are offered often enough, they won’t all be trimmed away.

## Tuesday, November 8, 2022

### A principle about infinite sequences of decisions

There are many paradoxes of infinite sequences of decisions where the sequence of individual decisions that maximize expected utility is unfortunate. Perhaps the most vivid is Satan’s Apple, where a delicious apple is sliced into infinitely many pieces, and Eve chooses which pieces to eat. But if she greedily takes infinitely many, she is kicked out of paradise, an outcome so bad that the whole apple does not outweigh it. For any set of pieces Eve eats, another piece is only a plus. So she eats them all, and is damned.

Here is a plausible principle:

- If at each time you are choosing between a finite number of betting portfolios fixed in advance, with the betting portfolio in each decision being tied to a set of events wholly independent of all the later or earlier events or decisions, with the overall outcome being just the sum or aggregation of the outcomes of the betting portfolios, and with the utility of each portfolio well-defined given your information, then you should at each time maximize utility.

In Satan’s Apple, for instance, the overall outcome is not just the sum of the outcomes of the individual decisions to eat or not to eat, and so Satan’s Apple is not a counterexample to (1). In fact, few of the paradoxes of infinite sequences of decisions are counterexamples to (1).

However, my unbounded expected utility maximization paradox is.

I don’t know if there is something particularly significant about
a paradox violating (1). I think there is, but I can’t quite put my finger
on it. On the other hand, (1) is such a complex principle that it may just seem *ad hoc*.

## Wednesday, November 2, 2022

### Must we accept free stuff?

Suppose someone offers you, at no cost whatsoever, something of specified positive value. However small that value, it seems irrational to refuse it.

But what if someone offers you a random amount of positive value for free. Strict dominance principles say it’s irrational to refuse it. But I am not completely sure.

Imagine a lottery where some positive integer *n* is picked at random, with all
numbers equally likely, and if *n* is picked, then you get 1/*n* units of value. Should you play
this lottery for free?

The expected value of the lottery is zero with respect to any
finitely-additive real-valued probability measure that fits the
description (i.e., assign equal probablity to each number). And for any
positive number *x*, the
probability that you will get less than *x* is one. It’s not clear to me that
it’s worth going for this.

If you like infinitesimals, you might say that the expected value of
the lottery is infinitesimal and the probability of getting less than
some positive number *x* is
1 − *α* for an infinitesimal
*α*. That makes it sound like a
better deal, but it’s not all that clear.

Of course, infinite fair lotteries are dubious. So I don’t set much store by this example.

### Two different ways of non-instrumentally pursuing a good

Suppose Alice is blind to the intrinsic value of friendship and Bob can see the intrinsic value of friendship. Bob then told Alice that friendship is intrinsically valuable. Alice justifiedly trusts Bob in moral matters, and so Alice concludes that friendship has intrinsic value, even though she can’t “see” it. Alice and Bob then both pursue friendship for its own sake.

But there is a difference: Bob pursues friendship because of the particular ineffable “thick” kind of value that friendship has. Alice doesn’t know what “thick” kind of value friendship has, but on the basis of Bob’s testimony, she knows that it has some such value or other, and that it is a great and significant value. As long as Alice knows what kinds of actions friendship requires, she can pursue friendship without that knowledge, though it’s probably more difficult for her, perhaps in the way that it is more difficult for a tone-deaf person to play the piano, though in practice the tone-deaf person could learn what kinds of finger movements result in aesthetically valuable music without grasping that aesthetic value.

The Aristotelian tradition makes the grasp of the particular thick kind of value involved in a virtuous activity be a part of the full possession of that virtue. On that view, Alice cannot have the full virtue of friendship. There is something she is missing out on, just as the tone-deaf pianist is missing out on something. But she is not, I think, less praiseworthy than Bob. In fact Alice’s pursuit of friendship involves the exercise of a virtue which Bob’s does not: the virtue of faith, as exhibited in Alice’s trust in Bob’s testimony about the value of friendship.

## Tuesday, November 1, 2022

### Pursuing a thing for its own sake

Suppose you pursue truth for its own sake. As we learn from Aristotle, it does not follow that you don’t pursue truth for the sake of something else. For the most valuable things are both intrinsically and instrumentally valuable, and so they are typically pursued both for their own sake and for the sake of something else.

What if you pursue something, but not for the sake of something else. Does it follow that you pursue the thing for its own sake? Maybe, but it’s not as clear as it might seem. Imagine that you eat fiber for the sake of preventing colon cancer. Then you hear a study that says that fiber doesn’t prevent colon cancer. But you continue to eat fiber, out of a kind of volitional inertia, without any reason to do so. Then you are pursuing the consumption of fiber not for the sake of anything else. But merely losing the instrumental reason for eating fiber doesn’t give you a non-instrumentally reason. Rather, you are now eating fiber irrationally, for no reason.

Perhaps it is impossible to do something for no reason. But even if
it is impossible to do something for no reason, it is incorrect to
*define* pursuing something for its own sake as pursuing it not
for the sake of something else. For that you *pursue something for
its own sake* states something positive about your pursuit, while
that you *don’t pursue it for the sake of anything else* states
something negative about your pursuit. There is a kind of valuing of the
thing for its own sake that is needed to pursue the thing for its own
sake.

It is tempting to say that you pursue a thing for its own sake
provided that you pursue it because of the intrinsic value you take it
to have. But that, too, is incorrect. For suppose that a rich benefactor
tells you that they will give you a ton of money if you gain something
of intrinsic value today. You know that truth is valuable for its own
sake, so you find out something. In doing so, you find out the truth
*because* the truth is intrinsically valuable. But your pursuit
of that truth is entirely instrumental, despite your reason being the
intrinsic value.

Hence, to pursue a thing for its own sake is not the same as to pursue it because it has intrinsic value. Nor is it to pursue it not for the sake of something else.

I suspect that pursuing a thing for its own sake is a primitive concept.

### Human worth and materialism

A typical human being has much more intrinsic value than any 80 kg arrangement of atoms.

If materialism is true, a typical human being is an 80 kg arrangement of atoms.

So, materialism is not true.

## Monday, October 31, 2022

### Transsubstantiation and magnets

On Thomistic accounts of transsubstantiation, the accidents of bread and wine continue to exist even when the substance no longer does (having been turned into the substance of Christ’s body and blood). This seems problematic.

Here is an analogy that occurred to me. Consider a magnet. It’s not crazy to think of the magnet’s magnetic field as an accident of the magnet. But the magnetic field extends spatially beyond the magnet. Thus, it exists in places where the magnet does not.

Now, according to four-dimensionalism, time is rather like space. If
so, then an accident existing *when its substance does not* is
rather like an accident existing *where its substance does not*.
Hence to the four-dimensionalist, the magnet analogy should be quite
helpful.

Actually, if we throw relativity into the mix, then we can get an even closer analogy, assuming still that a magnet’s field is an accident of the magnet. Imagine that the magnet is annihilated. The magnetic field disappears, but gradually, starting near the magnet, because all effects propagate at most at the speed of light. Thus, even when the magnet is destroyed, for a short period its magnetic field still exists.

That said, I don’t know if the magnet’s field is an accident of it. (Rob Koons in conversation suggested it might be.) But it’s comprehensible to think of it as such, and hence the analogy makes Thomistic transsubtantiaton comprehensible, I think.

## Friday, October 28, 2022

### Does our ignorance always grow when we learn?

Here is an odd thesis:

- Whenever you gain a true belief, you gain a false belief.

This follows from:

- Whenever you gain a belief, you gain a false belief.

The argument for (2) is:

You always have at least one false belief.

You believe a conjunction if and only if you believe the conjuncts.

Suppose you just gained a belief

*p*.There is now some false belief

*q*that you have. (By (3))Before you gained the belief

*p*you didn’t believe the conjunction of*p*and*q*. (By (4))So, you just gained the belief in the conjunction of

*p*and*q*. (By (5) and (7))The conjunction of

*p*and*q*is false. (By (6))So, you just gained a false belief. (By (8) and (9))

I am not sure I accept (4), though.

### “Accuracy, probabilism and Bayesian update in inﬁnite domains”

The paper has just come out online in Synthese.

Abstract: Scoring rules measure the accuracy or epistemic utility of a credence assignment. A significant literature uses plausible conditions on scoring rules on finite sample spaces to argue for both probabilism—the doctrine that credences ought to satisfy the axioms of probabilism—and for the optimality of Bayesian update as a response to evidence. I prove a number of formal results regarding scoring rules on infinite sample spaces that impact the extension of these arguments to infinite sample spaces. A common condition in the arguments for probabilism and Bayesian update is strict propriety: that according to each probabilistic credence, the expected accuracy of any other credence is worse. Much of the discussion needs to divide depending on whether we require finite or countable additivity of our probabilities. I show that in a number of natural infinite finitely additive cases, there simply do not exist strictly proper scoring rules, and the prospects for arguments for probabilism and Bayesian update are limited. In many natural infinite countably additive cases, on the other hand, there do exist strictly proper scoring rules that are continuous on the probabilities, and which support arguments for Bayesian update, but which do not support arguments for probabilism. There may be more hope for accuracy-based arguments if we drop the assumption that scores are extended-real-valued. I sketch a framework for scoring rules whose values are nets of extended reals, and show the existence of a strictly proper net-valued scoring rules in all infinite cases, both for f.a. and c.a. probabilities. These can be used in an argument for Bayesian update, but it is not at present known what is to be said about probabilism in this case.

### Choices on a spectrum

My usual story about how to reconcile libertarianism with the Principle of Sufficient Reason is that when we choose, we choose on the basis of incommensurable reasons, some of which favor the choice we made and others favor other choices. Moreover, this is a kind of constrastive explanation.

This story, though it has some difficulties, is designed for choices between options that promote significantly different goods—say, whether to read a book or go for a walk or write a paper.

But a different kind of situation comes up for choices of a point on a spectrum. For instance, suppose I am deciding how much homework to assign, how hard a question to ask on an exam, or how long a walk to go for. What is going on there?

Well, here is a model that applies to a number of cases. There are
two incommensurable goods one better served as one goes in one direction
in the spectrum and the other better served as one goes in the other
direction in the spectrum. Let’s say that we can quantify the spectrum
as one from less to more with respect to some quantity *Q* (amount of homework, difficulty of
a question or length of a walk), and good *A* is promoted by less of *Q* and incommensurable good *B* is promoted by more of *Q*. For instance, with homework,
*A* is the student’s having time
for other classes and for non-academic pursuits and *B* is the student’s learning more
about the subject at hand. With exam difficulty, *A* may be avoiding frustration and
*B* is giving a worthy
challenge. With a walk, *A* is
reducing fatigue and *B* is
increasing health benefits. (Note that the claim that *A* is promoted by less *Q* and *B* is promoted by more *Q* may only be correct within a
certain range of *Q*. A walk
that is too long leads to injury rather than health.)

So, now, suppose we choose *Q* = *Q*_{1}. Why did
one choose that? It is odd to say that one chose *Q* on account of reasons *A* and *B* that are opposed to each
other—that sounds inconsistent.

Here is one suggestion. Take the choice to make *Q* equal to *Q*_{1} to be the conjunction
of two (implicit?) choices:

Make

*Q*at most*Q*_{1}Make

*Q*at least*Q*_{1}.

Now, we can explain choice (a) in terms of (a) serving good *A* better than the alternative, which
would be to make *Q* be bigger
than *Q*_{1}. And we can
explain (b) in terms of (b) serving good *B* better than the alternative of
making *Q* be smaller.

Here is a variant suggestion. Partition the set of options into two
ranges *R*_{1},
consisting of options where *Q* < *Q*_{1} and
*R*_{2}, where *Q* > *Q*_{1}. Why
did I choose *Q* = *Q*_{1}? Well, I
chose *Q* over all the choices
in *R*_{1} because *Q* better promotes *B* than anything in *R*_{1}, and I chose *Q* over all the choices in *R*_{2} because *Q* better promotes *A* than anything in *R*_{1}.

On both approaches, the apparent inconsistency of citing opposed goods disappears because they are cited to explain different contrasts.

Note that nothing in the above explanatory stories requires any
commitment to there being some sort of third good, a good of balance or
compromise between *A* and *B*. There is no commitment to *Q*_{1} being the best way to
position *Q*.

### Simplicity and gravity

I like to illustrate the evidential force of simplicity by noting
that for about two hundred years people justifiably believed that the
force of gravity was *G**m*_{1}*m*_{2}/*r*^{2}
even though *G**m*_{1}*m*_{2}/*r*^{2 + ϵ}
fit the observational data better if a small enough but non-zero *ϵ*. A minor point about this struck
me yesterday. There is doubtless some *p* ≠ 2 such that *G**m*_{1}*m*_{2}/*r*^{p}
would have fit the observational data *better*. For in general
when you make sufficiently high precision measurements, you never find
*exactly* the correct value. So if someone bothered to collate
all the observational data and figure out exactly which *p* is the best fit (e.g., which one
is exactly in the middle of the normal distribution that best fits all
the observations), the chance that that number would be 2 up to the requisite number of significant
figures would be vanishingly small, even if *in fact* the true
value is *p* = 2. So simplicity
is not merely a tie-breaker.

Note that our preference for simplicity here is actually infinite.
For if we were to collate the data, there would not just be *one*
real number that fits the data better than 2 does, but a *range* *J* of real numbers that fits the data
better than 2. And *J* contains uncountably many real
numbers. Yet we rightly think that 2 is
more likely than the claim that the true exponent is in *J*, so 2 must be infinitely more likely than most of
the numbers in *J*.