Incommensurability in rational choice

When I hear that two options are incommensurable, I imagine things that are very different in value. But incommensurable options could also be very close in value. Suppose an eccentric tyrant tells you that she will spare the lives of ten innocents provided that you either have a slice of delicious cake or listen to a short but beautiful song. You are thus choosing between two goods:

1. The ten lives plus a slice of delicious cake.

2. The ten lives plus a short but beautiful song.

The values of the two options are very close relatively speaking: the cake and song make hardly any difference compared to the ten lives that comprise the bulk of the value. Yet, because the cake and the song are incommensurable, when you add the same ten lives to each, the results are incommensurable.

We can make the differences between the two incommensurables arbitrarily small. Imagine that the tyrant offers you the choice between:

3. The ten lives plus a chance p of a slice of delicious cake.

4. The ten lives plus a chance p of a short but beautiful song.

Making p be as small as we like, we make the difference between the options as small as possible, but the options remain incommensurable.

Well, maybe “noncomparable” is a better term than “incommensurable”, as it is a more neutral term, without that grand sound. Then we can say that (1) and (2) are “noncomparable by a slight amount” (relative to the magnitude of the overall goods involved).

There is a common test for incommensurability. Suppose A and B are options where neither is better than the other, and we want to know if they are equal in value or incommensurable. The test is to vary one of the two options by a slight amount of value, either positive or negative. If after the tweak the two options are still such that neither is better than the other, they must be incommensurable. (Proof: If A′ is slightly better or worse than A, and B is equal to A, then A′ will be slightly better or worse than B. So if A′ is neither better nor worse than B, we couldn’t have had B and A equal.)

But cases of things that are noncomparable by a slight amount show that we need to be careful with the test. The test still offers a sufficient condition for incommensurability: if the fact that neither is better than the other remains after making an option better or worse, we must have incommensurability. But if the two options are noncomparable by a very, very slight amount, a merely very slight variation in one could destroy the noncomparability, and generate a false positive for incommensurability. For instance, suppose that our two options are (3) and (4) with p = 10⁻¹⁰⁰. Now suppose the slight variation on (3) is that we suppose you are given a mint in addition to the goods in (3). A mint beats a 10⁻¹⁰⁰ chance of a song, even if it’s incommensurable with a larger chance of a song. So the variation on (3) beats the original (4). But we still have incommensurability.

(Note: There are two concepts of incommensurability. One is purely value based, and the other is agent-centric and based on rational choice. It is the second one that I am using in this post. I am comparing not pure values, but the reasons for pursuing the values. Even if the values are strictly incommensurable, as in the case of a certainty of a mint and a 10⁻¹⁰⁰ chance of a song, the former is rationally preferable at least for humans.)

Virtue versus painlessness

Suppose we had good empirical data that people who suffer serious physical pain are typically thereby led to significant on-balance gains in virtue (say, compassion or fortitude).

Now, I take it that one of the great discoveries of ethics is the Socratic principle that virtue is a much more significant contributor to our well-being than painlessness. Given this principle and the hypothetical empirical data, it seems that then we should not bother with giving pain-killers to people in pain—and this seems wrong. (One might think a stronger claim is true: We should cause pain to people. But that stronger claim would require consequentialism, and anyway neglects the very likely negative effects on the virtue of the person causing the pain.)

Given the hypothetical empirical data, what should we do about the above reasoning. Here are three possibilities:

1. Take the Socratic principle and our intuitions about the value of pain relief to give us good reason to reject the empirical data.

2. Take the empirical data and the Socratic principle to give us good reason to revise our intuition that we should relieve people’s pain.

3. Take the empirical data and our intuitions about the value of pain relief to give us good reason to reject the Socratic principle.

Option 1 may seem a bit crazy. Admittedly, a structurally similar move is made when philosophers reject certain theodical claims, such as the Marilyn Adams claim that God ensures that all horrendous suffering is defeated, on the grounds that it leads to moral passivity. But it still seems wrong. If Option 1 were the right move, then we should now take ourselves (who do not have the hyptohetical empirical data) to have a priori grounds to hold that serious physical pain does not typically lead to significant on-balance gains in virtue. But even if some armchair psychology is fine, this seems to be an unacceptable piece of it.

Option 2 also seems wrong to me. The intuition that relief of pain is good seems so engrained in our moral life that I expect rejecting it would lead to moral scepticism.

I think some will find Option 3 tempting. But I am quite confident that the Socratic principle is indeed one of the great discoveries of the human race.

So, what are we to do? Well, I think there is one more option:

4. Reject the claim that the empirical data plus the Socratic principle imply that we shouldn’t relieve pain.

In fact, I think that even in the absence of the hypothetical empirical data we should go for (4). The reason is this. If we reject (4), then the above reasoning shows that we have a priori reasons to reject the empirical data, and I don’t think we do.

So, we should go for (4), not just hypothetically but actually.

How should this rejection of the implication be made palatable? This is a difficult question. I think part of the answer is that the link between good consequences and right action is quite complex. It may, for instance, be the case there are types of goods that are primarily the agent’s own task to pursue. These goods may be more important than other goods, but nonetheless third parties should pursue the less important goods. I think the actual story is even more complicated: certain ways of pursuing the more important goods are open to third-parties but others are not. It may even be that certain ways of pursuing the more important goods are not even open to first-parties, but are only open to God.

And I suspect that this complexity is species-relative: agents of a different sort might have rather different moral reasons in the light of similar goods.

Basic goods

Many Natural Law (NL) theorists center their exposition of NL around the concept of a basic good. They give lists of basic goods, such as: health, friendship, knowledge, religion, play, etc. The basic goods are incommensurable: each one provides a different aspect of fulfillment to the possessor.

An NL theorist shouldn't, however, think of the theory as depending on the concept of a basic good. For the concept is a fishy one.

The basic goods are types of goods. Types come at many levels of generality. There does not, however, appear to be a non-arbitrary level of generality at which we get the "basic goods". Let me explain.

Here is a non-arbitrary level of generality: infima species of goods, types of good that there is no way of further subdividing into further subtypes that differ qua goods. Given NL's commitments about incommensurability, one might try to characterize an infima species of good as a type of good such that (a) instances of it are all commensurable and (b) it isn't a proper subtype of another type of good with that property. The basic goods are not infima species. For instance, knowledge can be subdivided into knowledge of necessary truths and knowledge of contingent truths, and we have incommensurability between the types. Knowledge of necessary truths can then be subdivided into mathematical knowledge and non-mathematical knowledge, and again there is incommensurability there. I suspect the infima species are going to be extremely specific, e.g., Smith's intellectual friendship with Kowalska focusing on fundamental political philosophy (and it will probably be more specific than that) or Jones's knowledge of Pythagoras' Theorem on the basis of proof P₁₇ (again, further specificity may be called for).

Here is another non-arbitrary level of generality: the highest genera. There might be just one highest genus, good. Or perhaps the highest genera are good of God and good of a creature. Or perhaps there is an infinite list of highest general but they are all instances of the schema good of N where N is a type of entity.

But the basic goods are neither infima species nor highest genera. They fall at some level of generality in between. And there seems to me to be no non-arbitrary way to delineate them. The best approach might be this: the basic goods (for humans) are the highest genera that fall properly under good of a human. (So if the good of a human is a highest genus, then the basic goods are second-highest genera.) But I doubt that there is a non-arbitrary way to define the highest genera under good of a human. There are many ways of subdividing good of a human, and the traditional subdivisions into basic goods are just one of them. For instance, one might subdivide good of a human into good of a human not in relation to other persons and good of a human in relation to God and good of a human in relation to non-divine persons (and maybe one or more hybrid categories). Or one might subdivide it into intellectual good and non-intellectual good. Etc.

Another option: an epistemic distinction. Perhaps the basic goods are the finest partition of the goods into genera with the property that one cannot fully grasp the distinctive value of any of the goods in any one genus on the basis of a grasp of the values of all the goods in the others. But I suspect that a distinction like this, if it can be made at all, would be liable to point to what is in at least some ways a finer level. Can one really grasp the distinctive value of aesthetic knowledge or friendship with Mother Teresa on the basis of other goods? Moreover, it may be that to grasp friendship one needs to grasp at least one other basic good, since friends promote each other's good not just in respect of friendship.

Fortunately, while the notion of incommensurable goods is important to NL, I do not think the NL theorist really needs a non-arbitrary concept of a basic good. The lists of basic goods are useful as heuristics, and they are a pedagogically valuable way to illustrate incommensurability. Moreover, it may be practically useful for guiding one's decisions and self-examination to have a division of goods that is sufficiently thick but not too fine-grained.

A modest sceptical theism that doesn't lead to moral scepticism

I'll just baldly give the theory without much argument. Axiology is necessary. It's a necessary truth that friendship and knowledge are good, that false beliefs are bad, etc. But the values have many aspects and exhibit much incommensurability. It's also a necessary truth that the good us to be pursued and the bad avoided. This gives some practical guidance, but mainly in cases where the reasons in favor of an action dominate those against. And that's rare. In typical cases agents face competing incommensurable reasons.

There may also be a necessary truth that some goods are fundamental and never to be acted against. The nature of a particular kind of agent then specifies how incommensurability is to be resolved. When the agent should be merciful rather than strictly just, when strictly just, and when the agent is morally free to go either way. The nature of an agent also gives the agent inclinations to act accordingly, inclinations that can be introspective. So we can know how we should resolve cases of incommensurability when they come up for us. We have reliable moral intuitions about us.

But these moral intuitions are about humans. Intelligent sharks would have a nature that resolves incommensurables differently, and our moral intuitions wouldn't directly tell us much about how intelligent sharks should act (except in cases of domination and maybe the deontic constraint not to act directly against the most fundamental goods). So we have a reasonable scepticism about for our insight into how a morally upright intelligent shark would act. But this scepticism of course in no way detracts from our knowledge of how we should resolve incommensurables.

For exactly the same reason, we have a reasonable scepticism about how God would act, about what resolutions between incommensurables are necessitated by his nature and which are left to choice. But this scepticism in no way detracts from our moral knowledge.

I don't think the scepticism is total. We can engage in limited analogical speculation. But this needs modesty if the theory is right.

Let me end with a little argument. When we think of particularly outlandish ethics cases, such as actions that affect an infinite number of people, we get stuck or even misled. No surprise on the above theory. We aren't made for such decisions. Those are decisions for more godlike beings than us. Perhaps our nature simply fails to specify the resolutions for these cases, as they aren't relevant to us in our niche. Imagine asking an intelligent amoeba about sexual ethics!

Weak and strong incommensurability

X and Y are weakly incommensurable iff there is a dimension of evaluation where X beats Y and a dimension of evaluation where Y beats X. X and Y are strongly incommensurable iff they are weakly incommensurable and a rational agent doesn't have on balance reason to choose X over Y and doesn't have on balance reason to choose Y over X.

Weak incommensurability is precisely what is needed for the possibility of a rational agent choosing either over the other.

Weak incommensurability is evidence of strong incommensurability. But there are cases where weak incommensurability fails to yield strong incommensurability. One kind of case is extremity. If one is choosing between being a superb nurse and a very mediocre mathematician, there is weak incommensurability, but one may have on balance reason to be a nurse (all other things being equal). But when choosing between being a nurse and a mathematician and one's professional quality would be moderately close, it's also strongly incommensurable.

Another quick argument for the possibility of incommensurability

Suppose Curley is deciding whether to accept a $1,000,000 bribe and lose his soul, or remain honest. It seems quite possible to have situations like this where neither option is preferable to Curley. Of course, prima facie that need not be a case of incommensurability--it might be a case of equal preference. But we cannot say that all similar cases like this are cases of equal preference. For in most cases like this, Curley also wouldn't have a preference between a $1,010,000 bribe and honesty. If that too has to be a case of equal preference rather than incommensurability, then by the transitivity of equal preference, we would have to say in such cases that Curley has equal preference as to a $1,000,000 bribe and a $1,010,000 bribe. But of course that's false: in most cases like this, he prefers the larger bribe.

Dominating reasons

Some things just aren't reasons for a choice. For instance, the fact that a portion of ice cream has an odd number of carbon atoms is by itself not a reason at all for eating the ice cream, and the fact that I find hot chocolate unpleasant is by itself not a reason to choose the hot chocolate. (The "by itself" qualifier is needed. I might have some instrumental reason for consuming an odd number of carbon atoms, and I might be ascetically training myself to consume what is unpleasant.)

Sometimes, however, something can be a reason for A without being a reason for A rather than B. For instance, that I enjoy hot chocolate to degree 100 is a reason to have hot chocolate. But if I enjoy ice cream to degree 150 on the very same scale, then my enjoying hot chocolate to degree 100 is not by itself a reason to have hot chocolate rather than ice cream. In the absence of other reasons, it would then make no rational sense to choose hot chocolate over ice cream, since my reason for hot chocolate is strictly dominated by my reason for ice cream.

At least roughly speaking:

• Reason R (not necessarily strictly) dominates reason S if and only if S is not at all a reason for choosing an action supported by S over an action supported by R.
• Reason R strictly dominates reason S if and only if R dominates S and S does not dominate R.

And of course reasons can be replaced by sets of reasons here. Then, Buridan's Ass cases are ones where the reasons for each action non-strictly dominate the reasons for the other.

Rational choice between A and B occurs only when one has reason to choose A over B and reason to choose B over A. Thus, rational choice between A and B occurs only when the reasons for neither option dominate the reasons for the other.

Definition: Reasons R and S are incommensurable if and only if neither dominates the other.

Thus, rational choice is possible only given sets of reasons that are incommensurable.

So close to a good ordering of all subsets of reals...

Let X be the set of integers, or a circle, or the real line, or Euclidean n dimensional space. Imagine a point is "uniformly" randomly chosen in X. For any two subsets A and B of X, we would like to be able to say if one of the subsets is more probable as the location of the point. Here are some conditions we want to impose on the ≤ comparison:

1. ≤ is a total preorder: for any A, B and C, we have A≤B or B≤A; A≤B and B≤C implies A≤C; and A≤A.
2. For any translation t of X (where we deem rotations on the circle to count as "circular translations"), we have tA≤tB if and only if A≤B. (≤ is translation-invariant)
3. If A is a proper subset of B, then A<B (i.e., A≤B but not B≤A).
4. If m(A)<m(B) for d-dimensional Hausdorff measure (including of course Lebesgue measure), for any d between 0 and the dimension of the space (inclusive), then A<B.

Proposition 1. Given the Axiom of Choice, there is an ordering ≤ satisfying 1-4.

Proof: Start with an ordering such that A is less than B if and only if A=B, or m(A)<m(B) for any d-dimensional Hausdorff measure, or A is a proper subset of B. This ordering is translation-invariant, and it extends to a preorder satisfying 1-4 by the main theorem of Section 2 here.

That sounds great! We can finally compare probabilities of landing in arbitrary sets, it seems. Well, almost. Given a uniform distribution, we would at least want the invariance also to hold for coordinate reflections (where we reflect the kth coordinate, for any k).

Proposition 2. There is no ordering ≤ satisfying 1-3 and the coordinate reflection condition.

That's a consequence of the final proposition in the paper I linked to above.

What a surprising difference these reflections make! With just translations, we have a lovely invariant order (though presumably not unique) respecting strict inclusions of sets. When we add coordinate reflections, we don't. Technically, the difference is that once we have reflections and translations, our symmetry group is no longer commutative. And of course, in the Euclidean space case if we add rotations, all is lost, too (that, too, is easy to show).

Philosophical corollary. There can be incommensurably probable events, and hence incommensurably valuable events (since two chances at the same good will be incommensurably good if the chances are incommensurably probable).

Paradoxes of comparison

There are three sets, A, B and C, each consisting of the same number of people, whose lives are endangered by the same sort of danger, and whose future prospects as far as you know are on par. There are also three hungry kids, x, y and z, who will survive if you don't give them breakfast, but who would benefit from your giving them breakfast once (you have no opportunity to do anything more for them). Suppose you have a choice between two actions:

1. Save all the people in A and feed nobody.
2. Save all the people in B and feed x.

Now, it sure seems like 2 is the better action than 1. One might even formulate a general principle:

• (*) If an action saves the same number of lives and feeds more hungry children, and all else is on par, then it's better.

But actually this is false. For suppose that B is a proper subset of A, and there are 100 people in A who are not in B. Since we said that A and B have the same number of people, this can only be the case when A and B are infinite sets. In this scenario, if one goes for 2, there will be 100 people whom one won't be saving. So we should modify (*), perhaps to:

• (**) If an action saves the same number of lives and feeds more hungry children, and the sets of lives saved and children fed are disjoint between the two actions, and all else is on par, then the action is better.

But paradox ensues when we specify that A and B have no people in common, but C is a subset of A missing 100 people, and then add the option:

3. Save all the people in C and feed y and z.

For by one application of (**), 2 is better than 1, and by another application of (**), 3 is better than 2. But 3 is not better than 1, because the 100 people in A who aren't in C die if you go for 3, and that's not balanced out by the two children fed.

(There is a literature on infinite utilities, and I am not claiming any originality for this case.)

One could take this as yet another argument against the transitivity of "better than". But that doesn't get us out of paradox, since denying that transitivity is itself paradoxical. Moreover, there is already something paradoxical in having to deny (*)—that principle sure seemed plausible.

We could conclude that one can't have infinite sets of people, and make this be one of the family of arguments against actual infinites. Maybe.

But I want to do something else here. I think this, like a number of other paradoxes (which need not all involve infinity; I have a hunch that White's puzzle, as per Joyce's reply discussed in the link, is in this family), is due to us having two ways of comparing. We have an uncontroversial and unproblematic inclusion or domination comparison. It is uncontroversial that all other things being equal, if you can save all the people in A or all the people in B, and the people in A are a proper subset of those in B, then you should save the people in B. It is uncontroversial that if p entails q, then q is at least as likely as p. And so on.

But we also insist on comparing apples to oranges, comparing where there is no inclusion or domination relation. Typically, five oranges are more valuable than one apple, and five apples are more valuable than one orange. To make such comparisons we often assign numbers—say, cardinalities, utilities, prices or probabilities—to the things we are comparing, but we can also just make ordinal comparisons without assigning numbers (I didn't assign any utilities when I gave the ethical story).

I think a lot of paradoxes have the consequence that comparisons without domination are fishy. They need not satisfy transitivity. They might suffer from some arbitrariness. In the ethical sphere, this can be manifested in incommensurability of options. In probability theory, this surfaces in difficulties surrounding infinite sample spaces or nonmeasurable sets (as in White's puzzle, since nonmeasurable sets and non-exact probabilities are of a piece, I think).

Yet we need comparisons-without-domination.

So what should we do? In the ethical sphere, perhaps what we need is basically what Aquinas says about the order of charity. Aquinas thinks that when choosing between an equal benefit to one's parent or to a stranger, one should bestow the benefit on one's parent. But what if the benefit to the stranger is greater? If only slightly greater, we should still benefit our parent. But if much greater, we should benefit the stranger. But where is the line drawn? Aquinas refuses to answer. There is no rule, it seems. Rather, this is just somehting for the wise and virtuous agent to know. And maybe there is an analogue to this answer in the case of the non-ethical paradoxes.

How can I knowingly and freely do wrong?

I accept the following two claims:

1. Every free action is done for a reason.
2. If an action is obligatory, then I have on balance reason to do it.

Consider cases where I know that an action is obligatory, but I don't do it. How could that be? Well, one option is that I don't realize that obligatory actions are ones I have on balance reason to do. Put that case aside: I do know it sometimes when I do wrong. So I know that I have on balance reason to do an action, but I refrain from it. But then how could I have a reason for my refraining? And without a reason, my action wouldn't be free.

It strikes me that this version of the problem of akrasia may not be particularly difficult. There is no deep puzzle about how someone might choose a game of chess over a jog for a reason. A jog is healthier but a game of chess is more intellectually challenging, and one might choose the game of chess because it is more intellectually challenging. In other words, there is a respect in which the game of chess is better than the jog, and when one freely chooses the game of chess, one does so on the basis of some such respect. The jog, of course, also has something going for it: it is healthier, and one can freely choose it because it is better in respect of health.

Now, suppose that the choice is between playing a game of chess and keeping one's promise to visit a sick friend. Suppose the game of chess is more pleasant and intellectually challenging than visiting the sick friend. One can freely choose the game of chess because there are respects in which it is better than visiting the friend. There are, of course, respects in which the game of chess is worse: it is a breaking of a promise and a neglecting of a sick friend. But that there are respects in which visiting the sick friend is better does not make there be no reason to play chess instead, since there are respects in which the chess game is better.

But isn't visiting the sick friend on balance better? Certainly! But being on balance better is just another respect in which visiting the sick friend is better. It is still in some other respects better to play the game of chess. If one freely chooses to play the game of chess, then one chooses to do so on account of those other respects. That one option is on balance better is compatible with the other option being in some respects better. It is no more mysterious how one can act despite the knowledge that another option is on balance better than how one can act despite the knowledge that another option is more pleasant. The difference is that when one chooses against an action that one takes to be on balance better, one may incur a culpability that one does not incur when one chooses against an action that is merely more pleasant, but the incurring of that culpability is just another reason not to do the action.

But isn't it decisive if an action is on balance better? Isn't it irrational to go against such a decisive reading? Well, one can understand a decisive reason in three ways: (a) a reason that in fact decides one; (b) a reason that cannot but decide one; and (c) a reason that rationality requires one to go with. That an action is on balance better need not be what decides you, even if in fact you do the on balance better action. Now, granted, rationality requires one to go with an on balance better action. But that rationality requires something does not imply you will do it.

But if you don't, aren't you irrational, and hence not responsible? Well, if by irrational one means lack of responsiveness to reasons, then that would indeed imply lack of responsibility, but that is not one's state when one chooses to do the wrong thing for a reason. It need not even be true that one is not responsive to what is on balance better. For to be responsive to a reason does not require that one act on that reason. The person who chooses the chess game over the jog is likely quite responsive to reasons of health. If she were not responsive to reasons of health, it might not be a choice but a shoo-in. Likewise, the person who chooses against what is on balance better is responsive to what is on balance better, but goes against it.

Now, of course, the person who knowingly does what she knows she on balance has reason not to do, does not respond to the reason in the way that she should. In that sense, she is irrational. But that sense of irrationality is quite compatible with responsibility.

Choice, rationality and contrastivity

Suppose I choose A over B. For me to have chosen over B, B must have been a relevant alternative. For instance, I am choosing to write this post over doing dishes, but I am not choosing to write this post over plugging in a soldering iron and grabbing its hot tip. Why? Well, I was impressed by some reasons in favor of doing dishes but not impressed by any reasons in favor of holding the tip of a hot soldering iron.

To choose A over B, I not only needed to have a reason to choose A, but also a reason to choose B. Moreover, plausibly, choices are contrastive and so are the reasons for them. If so, then the reason to choose B would have to have been a contrastive reason, a reason for choosing B over A. If this is right, then to choose A over B, I need a reason for A over B and a reason for B over A. Now when A rationally dominates B for me in the sense that any of my reasons for B is at least as much a reason for A, then I have no reason for B over A. But lacking a reason for B over A, I cannot choose A over B, paradoxical as that sounds. I may have reason to do A rather than B, but this isn't a matter of choice, because B is not a relevant alternative to A, since in the context of a choice between A and B, there are no reasons for B, i.e., no reasons for B over A.

We now have several principles:

• Rationality of Choice: one can only choose between options for which there are reasons in the context of choice
• Contrastivity of Reasons: reasons in the context of choice are always reasons for an option over the alternatives
• Domination Principle: choice between A and B is impossible when every reason for B is at least as much a reason for A
• Incommensurability Principle: choice between A and B is only possible when there is a reason for each of these that isn't, or isn't as much, a reason for the other.

The Domination and Incommensurability Principles are equivalent, and are basically endorsed by Aquinas. The argument at the beginning of the post shows that Rationality of Choice plus Contrastivity of Reasons implies the Domination and Incommensurability Principles.

An interesting consequence of the Incommensurability Principle is that one's moral psychology had better not endorse both of the following theses:

• Total Ordering of Strengths: for any two desires d₁ and d₂, either they are equal in strength, or one is stronger than the other
• Desires are Reasons: the reasons on the basis of which one chooses are desires and their strengths are the strengths of reasons.

For Total Ordering of Strengths, Desires are Reasons and Incommensurability together implies that there are no choices.

Humean compatibilists are committed to Desires are Reasons. Humean determinists are committed to Total Ordering of Strengths given how on Humean grounds we can test the strength of desire by seeing what the agent is determined by her psychological state to choose. If this is right then if Rationality and Contrastivity are true, Humeanism needs to be rejected.

An infinity argument for incommensurability

Consider these plausible claims:

1. If worlds w₁ and w₂ contain the exact same individuals, and each individual in w₂ is better off than she is in w₁, then w₂ is a more valuable world.
2. A world can contain an infinite number of individuals.
3. If worlds w₁ and w₂ differ only in respect of which particular individuals exist in them, and perhaps some further value-insignificant respects, if an identity of indiscernibles principle requires it, then w₁ and w₂ are equal in value.
4. If worlds w₁ and w₂ differ only in respect of w₁ lacking one individual that exists in w₂ and that has a good life in w₂, and perhaps in some value-insignificant way, then w₂ is more valuable than w₁.
5. If a and b are equally valuable, then c more (less) valuable than b if and only if c is more (less) valuable than a.
6. Being more valuable than is transitive.
7. Nothing is more valuable than itself.

Now, making liberal use of (2), imagine this sequence of worlds:

• w₁: God plus an infinite sequence of spatiotemporally disconnected individuals x₁, x₂, ... who are almost exactly alike, differing only in that x_i enjoys on balance flourishing of value i (plus any other insignificant differences needed to avoid violation of the identity of indiscernibles)
• w₂: just like w₁, except that the individuals are all different: y₁, y₂, ...
• w₃: just like w₂, except that y_i now enjoys on balance flourishing of value i+1, for all i
• w₄: just like w₃, except that in the place of y_i we have x_i+1, for all i

Now, notice that w₄ is basically w₁ minus the individual x₁, with perhaps some value-insignificant differenecs. By (4), it follows that w₁ is more valuable than w₄. By (3), it follows that w₂ has the same value as w₁. By (1), it follows that w₃ is more valuable than w₂. By (3), it follows that w₄ has the same value as w₃. By (5) and (6) it follows that w₄ is more valuable than w₄. Which contradicts (7).

I think the controversial assumptions are (2) and (3). It's really hard to deny (1), (4), (5), (6) or (7). So, either there can't be an actual infinity or (3) is false. Now, the falsity of (3) would imply a really radical form of incommensurability: situations that are exactly alike except for the particular identities of the individuals involved (and whatever identity of indiscernibles further requires) can differ in value.

I want to hold on to (2). Plainly a world can have an infinite future containing an infinite number of individuals. So I reject (3), and thus accept the radical incommensurability claim above.

And I think the incommensurability claim is independently plausible.

Choice and incommensurability

Right now, I am making all sorts of choices. For instance, I just chose to write the preceding sentence. When I made that choice, it was a choice between writing that sentence and writing some other sentence. But it was not a choice between writing that sentence and jumping up an down three times. Let A be the writing of the sentence that I wrote; let B be the writing of some alternate sentence; let C be jumping up and down three times. Then, I chose between A and B, but not between A and C. What makes that be the case?

Here is one suggestion. I was capable of A and of B, but I was not capable of C. If this is the right suggestion, compatibilism is false. For on standard compatibilist analyses of "is capable of", I am just as capable of C as of A and B. I was fully physically capable of doing C. Had I wanted to do C, I would have done C. So if the capability of action suggestion is the only plausible one, we have a neat argument against compatibilism. However, there is a decisive objection to the suggestion: I can choose options I am incapable of executing. (I may choose to lift the sofa, without realizing it's too heavy.)

To get around the last objection, maybe we should talk of capability of choosing A and capability of choosing B. Again, it does not seem that the compatibilist can go for this option. For if determinism holds, then in one sense neither choosing B nor choosing C are available to me—either choice would require a violation of the laws of nature or a different past. And it seems plausible that in that compatibilist sense in which choosing B is available to me—maybe the lack of brainwashing or other psychological compulsion away from B—choosing C is also available to me. Again, if this capability-of-choosing suggestion turns out to be the right one, compatibilism is in trouble.

Here is another suggestion, one friendly to compatibilism. When I wrote the first sentence in this post, I didn't even think of jumping up and down three times. But I did, let us suppose, think of some alternate formulations. So the difference between B and C is that I thought about B but did not think about C. However, this suggestion is unsatisfactory. Not all thinking about action has anything to do with choosing. I can think about each of A, B and C without making a choice. And we are capable of some limited parallel processing—and more is certainly imaginable—and so I could be choosing between D and E while still thinking, "purely theoretically" as we say, about A, B and C. There is a difference between "choosing between" and "theorizing about", but both involving "thinking about".

It seems that the crucial thing to do is to distinguish between the ways that the action-types one is choosing between and those action-types that one is merely theorizing about enter into one's thoughts. A tempting suggestion is that in the choice case, the actions enter one's mind under the description "doable". But that's mistaken, because I can idly theorize about a doable without at all thinking about whether to do it. (Kierkegaard is really good at describing these sorts of cases.) The difference is not in the description under which the action-types enter into the mind, as that would still be a difference within theoretical thought.

I think the beginning of the right thing to say is that those action-types one is choosing between are in one's mind with reasons-for-choosing behind them. And these reasons-for-choosing are reasons that one is impressed by—that actively inform one's deliberation. They are internalist reasons.

But now consider this case. Suppose I could save someone's life by having a kidney removed from me, or I could keep my kidneys and let the person die. While thinking about what to do, it occurs to me that I could also save the other person's life by having both kidneys removed. (Suppose that the other person's life wouldn't be any better for getting both kidneys. Maybe only one of my kidneys is capable of being implanted in her.) So, now, consider three options: have no kidneys removed (K0), have one kidney removed (K1), and have two kidneys removed (K2). If I am sane, I only deliberate between K0 and K1. But there is, in a sense, a reason for K2, namely that K2 will also save the other's life, and it is the kind of reason that I do take seriously, given that it is the kind of reason that I have for K1. But, nonetheless, in normal cases I do not choose between K0, K1 and K2. The reasons for K2 do not count to make K2 be among the options. Why? They do not count because I have no reason to commit suicide (if I had a [subjective] reason to commit suicide, K2 would presumably be among the options if I thought of K2), and hence the reasons for K2 are completely dominated by the reasons for K1.

If this is right, then a consequence of the reasons-for-choice view of what one chooses between is that one never has domination between the reasons for the alternatives. This supports (but does not prove, since there is also the equal-reason option to rule out) the view that choice is always between incommensurables.

A corollary of the lack-of-domination consequence is that each of the options one is choosing between is subjectively minimally rational, and hence that it would be minimally rational to choose any one of them. I think this is in tension with the compatibilist idea that we act on the strongest (subjective) reasons. For then if we choose between A and B, and opt for A because the reasons for A were the strongest, it does not appear that B would have been even minimally rational.

Maybe, though, the compatibilist can insist on two orderings of reasons. One ordering is domination. And there the compatibilist can grant that the dominated option is not among the alternatives chosen between. But there is another ordering, which is denoted in the literature with phrases like "on balance better" or "on balance more (subjectively) reasonable". And something that is on balance worse can still be among the alternatives chosen between, as long as it isn't dominated by some other alternative.

But what is it for an option to be on balance better? One obvious sense the phrase can have is that an action is on balance better if and only if it is subjectively morally better. But the view then contradicts the fact that I routinely make choices of what is by my own lights morally worse (may God have mercy on my soul). Another sense is that an action is on balance better if and only if it is prudentially better. But just as there can be moral akrasia, there can be prudential akrasia.

Here is another possibility. Maybe the compatibilist can say that reasons have two kinds of strength. One kind of strength is on the side of their content. Thus, the strength of reason that I have to save someone's life is greater than the strength of reason that I have to protect my own property. Call this "content strength". The other kind of strength is, basically, how impressed I am with the reason, how much I am moved by it. If I am greedy, I am more impressed with the reasons for the protection of my property than with the reasons for saving others' lives. Call this "motivational strength". We can rank reasons in terms of the content strength, and then we run into the domination and incommensurability stuff. But we can also rank reasons in terms of motivational strength. And the compatibilist now says that I always choose on the basis of the motivationally strongest reasons.

This is problematic. First, it suggests a picture that just does not seem to be that of freedom—we are at the mercy of the non-rational strengths of reasons. For it is the content strength rather than the the motivational strength of a reason that is a rational strength. Thus, the choices we make are only accidentally rational. The causes of the choices are reasons, but what determines which of the reasons we act on is something that is not rational. Rationality only determines which of the options we choose between, and then the choice itself is made on the non-rational strengths. This is in fact recognizable as a version of the randomness objection to libertarian views of free will. I actually think it is worse than the randomness objection. (1) Agent-causation is a more appealing solution for incompatibilists than for compatibilists, though Markosian has recently been trying to change that. (2) The compatibilist story really looks like a story on which we are in bondage to the motivational strengths of reasons. (3) The content strength and content of the outweighed reasons ends up being explanatorily irrelevant to the choice. And (4) the specific content of the reasons that carried the day is also explanatorily irrelevant—all that matters is (a) what action-type they are reasons for and (b) what their motivational strength is.

In light of the above, I think the compatibilist should consider giving up on the language of choice, or at least on taking choice, as a selection between alternatives, seriously. Instead, she should think that there is only the figuring out of what is to be done, and hold with Socrates that there is no akrasia in cases where we genuinely act—whenever we genuinely act (as opposed to being subject to, say, a tic) we do what we on balance think should be done. I think this view would give us an epistemic-kind of responsibility for our actions, but not a moral kind. Punishment would not be seen in retributivist terms, then.

St. Anselm on the magnitude of sin

St. Anselm believes that the least of our sins puts us in an infinite debt to God and is infinitely bad. Anselm's own argument for this thesis is uses some implicit premises. Here is my best reconstruction:

1. (Premise) To sin is to oppose the will of God.
2. (Premise) If it is not permissible to do A in order to preserve a good G, then A is at least as bad as the loss of G.
3. (Premise) It is not permissible to oppose the will of God "even to preserve the whole of creation", even "if there were more worlds as full of beings as this", and even if "they increased to an infinite extent".
4. (Premise) The badness of the loss of the whole of creation if the whole of creation consisted of planets as full of beings as Earth and increased to an infinite extent would be infinite.
5. (Premise) If something is at least as bad as an infinite bad, then it's infinitely bad.
6. To oppose the will of God is at least as bad as something infinitely bad. (2, 3 and 4)
7. To oppose the will of God is infinitely bad. (5 and 6)
8. Every sin is infinitely bad. (1 and 7)
9. (Premise) To do something infinitely bad puts us in infinite debt.
10. Every sin sin is infinitely bad and puts us in infinite debt. (8 and 9)

Probably the most obvious thing to worry about is the conjunction of (2) and (3). Premise (2) appears to be a consequentialist claim. Premise (3), on the other hand, is a premise that the Christian tradition does indeed endorse, but the Christian tradition is not a consequentialist tradition (witness St Augustine on lying and St Paul's insistence that we do not do evil that good might come of it). Thus the argument brings together premises that originate in two competing moral theories, and it is unsurprising that when you do that, you can derive surprising things from their conjunction.

But this is too quick. For there is a lot of plausibility to the consequentialist intuition that Anselm invokes in (2). When one thinks about Augustine and Kant on lying, it "feels right" to say that they value honesty over life. On the side of Kant at least, this is mistaken. Kant's objection to lying isn't that honesty is a greater good than life, because Kant's ethics is not centered on the good (at least not in a way that would make this statement work). So perhaps it is right to affirm the consequentialist intuition in (2) as well as the deontological intuition in (3). And if one does that, then one gets the view that certain actions are infinitely bad.

However, this approach is mistaken. For one does not do justice to both consequentialist and deontological intuitions by supposing that wrong actions are infinitely bad. For consider this situation: If you don't kill an innocent person, a hundred other people will each kill one innocent person. (How can you set this up? One way is statistical. You know that if a certain temptation T is given to people, one percent of people will commit a murder. So you're told that if you don't kill one person, temptation T will be offered to 10,000 people.) Clearly the hundred murders are worse than one murder, and yet it's wrong, according to the deontologist, to kill one innocent person. Moreover, this case is directly a counterexample to (2).

One might try to get out of this counterexample by denying that infinities can be compared. But if one does that, then one can no longer say things that mortal sin is worse than venial sin, and the like. Besides, one does want to be able to compare infinities. If I had a choice between preventing one stranger from committing one murder of a stranger and a hundred strangers from committing one murder apiece, obviously I should prevent the latter.

A different kind of objection to the argument would be that in (2), "at least as bad" should be read as "not less bad" (I am worried about cases of incommensurability). But if one makes the same replacement in (5), it is far from clear whether (5) remains true. For it may be that something is infinitely bad in an aesthetic way (say, an infinitely long bad novel) and something else is finitely bad in an incommensurable way (say, a toothache of finite length); the latter is not less bad than the former, but it does not follow that the latter is infinite.

Nonetheless, even though St Anselm's argument for the infinite badness of sin fails, I think there are alternate ways to make the thesis plausible. But that's a subject for another post (hopefully over the next couple of days).