2. Condorcet’s Paradox

Let’s start with a classic case. Alan, Betty and Chico want snacks, but they have different tastes. Their preferences are as follows, where ‘A’ stands for the outcome where everyone gets apples, ‘B’ bananas and ‘C’ cherries:

1. Alan: A > B > C

2. Betty: B > C > A

3. Chico: C > A > BFootnote ⁷

Their preferences, moreover, are transitive. Alan prefers apples to cherries, Betty prefers bananas to apples, and Chico prefers cherries to bananas. All that matters in this case is giving people what they want. We can only choose one snack, however, so there are just three outcomes: apples for all, bananas for all, cherries for all. Which is best?

That depends on how we settle conflicts. Apples are best for Alan, bananas for Betty, and cherries for Chico. We want to know: which is best overall?

The answer is easy if we can just sum up individual desires, measured with a shared cardinal unit, to arrive at the group’s satisfaction level. For example, if Alan’s preferences are vastly stronger, the best outcome is for Alan to get his apples, as in:

Here we resolve the conflict by adding desires and comparing sums, arriving at a tidy transitive ranking: A > B > C.

But consider a non-additive resolution:

A foundational result in social choice theory is Condorcet’s Paradox: Majority Rule can violate Transitivity even when individual preferences are transitive (Arrow Reference Arrow1963; Sen Reference Sen2017).Footnote ⁸ Here is how. Suppose Alan, Betty and Chico vote, pair by pair, on their favourite snacks. Majority Rule will ‘rank’ the outcomes: A > B > C > A. Why? Because two prefer apples to bananas; two prefer bananas to cherries, and two prefer cherries to apples.

The group’s preferences violate not just Transitivity, but even:

Which says that there are no cycles of ‘>’. It is rather shocking that Majority Rule should violate such a basic principle. Hence Condorcet’s Paradox.

I am not, however, here to talk about rules for voting. I am interested in Majority Rule as a principle for how values combine in a single agent’s choice.Footnote ⁹ We have three outcomes, none of which is best in every respect: A is best for Alan, B for Betty and C for Chico. We have three dimensions of value, three ways in which an outcome can be good – one for each individual. We can think of Majority Rule as a method for aggregating dimensions: A > B if A is better than B for most people. This leads to violations of Transitivity and even Acyclicity for betterness: A > B > C > A.

Of course, cycles aside, Majority Rule is a problematic moral principle. One problem is that doesn’t factor in anything besides utility – what about justice, knowledge and other dimensions of value? But this is fixable. We simply generalize Majority Rule to say that A is better than B if it is better along most dimensions, whatever they may be.

The real problem with Majority Rule in morality is that it only cares about whether A beats B along most dimensions, not by how much.Footnote ¹⁰ For example, even in the case where Alan’s preferences are by far the most intense, Majority Rule will not say that it is best overall for Alan to get his apples – not even if we multiply the intensity of his desires by a million, and dull down Betty’s and Chico’s – because the majority prefer cherries. This alone makes Majority Rule fishy as a principle of value.Footnote ¹¹

Thankfully, I am not advocating for this principle, only using it for illustration.Footnote ¹² My point is that Majority Rule can lead to cycles of betterness even given the Internal Aspects View, which means that this view cannot really underwrite Transitivity.

3. Internal aspects and 1D scores

Now we need to show that Majority Rule fits the Internal Aspects View. Let’s start by getting clear on what the view really is.

Temkin (Reference Temkin2012: 229) believes that the debate over Transitivity turns on a deeper conflict between two views of value. The Internal Aspects View says that a thing’s value just depends on what the thing is like in itself, and that we compare values to determine which things are better. The Essentially Comparative View says that some aspects of a thing’s value may depend on what the alternatives are; a certain factor might make A better than B without affecting whether A is better than C. The Internal Aspects View leads to Transitivity; the Essentially Comparative View leads us away.

That, anyway, is the breezy version. When we look closer, it is less obvious what Temkin takes to be essential to the Internal Aspects View. Consider this gloss:

There are three claims here. First, betterness is determined by comparing values. Second, an outcome’s value cannot change as we compare it to different alternatives. Third, values can (‘perhaps’) be ranked along a linear scale, like notches on a ruler.Footnote ¹³

Let’s call this combination of claims the 1D Internal Aspects View. Put precisely, the view combines the following:

Let me explain the symbols. V is a function that tells us the value of an outcome, all things considered, in a context of comparison. For example, ‘V _B(A)’ denotes A’s value when compared with B. (When context doesn’t matter, I write ‘V(A)’.) To say one value is higher, we use ‘>>’. For example: ‘V _B(A) >> V _A(B)’ means that A’s value is higher than B’s when comparing the pair. The function f in 1D Scoring maps values (in the set V) to real numbers (in the set R); f(v _i ) > f(v _j ) means that v _i ’s number is greater.Footnote ¹⁴

Scores tells us that A is better than B just if, when comparing the pair, A’s value is higher overall; Internal Scores tells us that A has the same value no matter what it is compared with; and 1D Scoring tells us that values can be ranked like numbers on a line.Footnote ¹⁵

Do these claims entail Transitivity, as Temkin says?Footnote ¹⁶ Yes. Suppose Transitivity fails: A > B > C but not A > C. Given Scores and Internal Scoring, the options’ values must have the same structure: V(A) >> V(B) >> V(C), but not V(A) >> V(C). Given 1D Scoring, there must be a way to plug in numbers for these values and substitute ‘is greater than’ for ‘>>’. But this can’t be done. ‘Is greater than’ is transitive.

So the 1D Internal Aspects View entails Transitivity, and the entailment crucially depends on 1D Scoring. But why should 1D Scoring be essential to the Internal Aspects View? It does not follow from the core idea: betterness depends on internal values. (This may be why Temkin introduces 1D Scoring with a parenthetical ‘perhaps’.) Nor does 1D Scoring appear in Temkin’s most explicit statement of the view, where Scores and Internal Scoring are front and centre.Footnote ¹⁷ It seems to me that these latter two principles alone suffice for an Internal Aspects View. Such a view, at any rate, could not be essentially comparative, since values would stay fixed.Footnote ¹⁸

We need a name for this kind of view, and so, with apologies to Temkin, I will call it the ‘Internal Aspects View’.Footnote ¹⁹ This label will cover any view that rejects essential comparativity by embracing Scores and Internal Scoring. So defined, the Internal Aspects View does not require 1D Scoring, and can be paired with all sorts of other rules – even Majority Rule. Now let’s see how this particular pairing leads to cycles in Condorcet’s Paradox.Footnote ²⁰

4. Nontransitive betterness from internal aspects

It is finally time to ask what value scores are. 1D Scoring suggests that they are like numbers on a line. But when Transitivity and Acyclicity fail, values can’t have that structure; the number line never cycles back. So what, then, is a value?

We can start from the idea that values have multiple dimensions. In the snack example, there are three relevant dimensions: goodness for Alan, for Betty and for Chico. For simplicity, let’s just stick with these three, and let’s assume we can measure them with cardinal numbers – units of utility. We give each outcome a 3D score (a, b, c), representing how good it is for Alan, Betty and Chico, respectively. This is a single score: the outcome’s ‘real’ value. It depends only on ‘internal features’ – namely, utilities – and it measures how good an outcome is, all things considered. It is meant to be the ‘key feature’ in determining which outcomes are better.

We thus have a conception of scores that obeys Scores and Internal Scoring, though not necessarily 1D Scoring. The first key idea is:

This says that values can be ordered triples of real numbers. In our case, the first number measures goodness for Alan, the second for Betty and the third for Chico. (More generally, the idea is to have n-tuple scores given n dimensions of value.)

This does not by itself, however, get us beyond 1D Scoring and Transitivity. For suppose we used the utilitarian’s additive scoring rule:

This squishes our n dimensions into one number: the sum, which then settles which things are better. A > B if A’s sum is higher. This is barely a step beyond 1D Scoring.

Instead of adding the numbers, we could use a (long-foreshadowed) spin on Majority Rule. The rule says:

In other words: one value is higher than another, all things considered, if it is higher along the majority of dimensions.

Now suppose we add 3D Scores and Majority Rule (Scoring) to the Internal Aspects View. The resulting 3D Internal Aspects View leads to nontransitivities in Condorcet’s case. We can use the numbers from before:

Where Alan’s preferences are most intense. The 3D Internal Aspects View yields a cycle: A > B > C > A. For A is better than B in most respects; B is better than C in most respects; and C is better than A in most respects. (This is possible because, thanks to 3D Scores, we have more dimensions to play with.) By Majority Rule (Scoring), we will get a betterness cycle, just as Majority Rule leads to cycling preferences. This cycle persists even if we modulate the intensity of individual preferences, so long as everyone’s ranking remains the same.

The same kind of cycle can arise in other cases, too, not just clashes of utilities. Suppose we are evaluating the work of three philosophers – Dina, Ellie and Fumi – along the dimensions of depth, elegance and funniness. The 3D values might be:

Addition gives a transitive ranking: Dina > Ellie > Fumi. Because the differences in depth are so big, they are decisive. But Majority Rule (Scoring) would turn this into a cycle by ranking Fumi over Dina overall, given that his work is better than hers in most ways.

The 3D Internal Aspects View thus entails cycles in a variety of cases. But I want to emphasize something: it is still clearly an Internal Aspects View. The view obeys Scores and Internal Scoring, though of course the scores are not just single numbers anymore; the value of giving everyone apples, for instance, is (10, 0, 1). We might not be used to thinking of values as internal 3D scores.Footnote ²¹ But it should be immediately clear that these scores, whatever they are, are not essentially comparative. They are not even comparative in the broad sense of having a dimension that counts for more in one comparison than in another. Each vote counts equally in Majority Rule.

And so, the 3D Internal Aspects View retains the core of the Internal Aspects View, without any whiff of comparativity, while still allowing cycles.

We have reached a surprising conclusion. It turns out that there are two distinct sources of nontransitivity: one is the Essentially Comparative View that (potentially) accepts 1D Scoring; the other is an Internal Aspects View that rejects 1D Scoring. Transitivity is assured if values are both reducible to one dimension and invariant across comparisons. But relaxing either assumption can lead to violations.

This result should fundamentally change the way we see the debate over Transitivity, which does not just turn on the Internal Aspects View. We can only get Transitivity by also assuming something like 1D Scoring or Addition.Footnote ²² This assumption deserves more direct discussion, and we should be much more careful to distinguish the question of whether values are internal from our choice of preferred scoring rule.

What scoring rule should we prefer? If we want to resist Transitivity, we have to forgo Addition. We might instead opt for Majority Rule. But this is not very plausible when it comes to dimensions of value. If A is better than B in two ways, but vastly worse in one, it seems ridiculous to insist that A must be better overall.

This raises a question for future work. Is there an aggregation rule far enough away from Addition as to allow for nontransitivity, but not so close to Majority Rule as to be ridiculous? If we find such a rule, we can establish a new kind of nontransitivity, consistent with the Internal Aspects View, that is not just possible but plausible. If we can find a problem inherent in such rules – something more direct than ‘they yield cycles’ – we will have found what Temkin wanted all along: a source for Transitivity.

5. Warm-up: Condorcet in context

I have argued that the 3D Internal Aspects View violates Transitivity in the case of Condorcet’s Paradox. As a warm-up for part two of the paper, let’s now put Condorcet’s case in context, by comparing it with some nearby nonmoral examples in the literature.

Interestingly, at the end of Temkin’s first paper on transitivity (Reference Temkin1987: 187, fn. 51), he gives a case that is almost analogous: the nontransitive dice.

This delightful case does not seem essentially comparative. Each die has a 6D internal score – the numbers – which then determine the chances of victory. Die X will probably beat Die Y if X’s highest four numbers are greater than the highest four numbers of Y. With this rule, the ‘will probably beat’ relation is cyclic: A > B > C > A. This is, roughly, a non-moral spin on Condorcet’s Paradox.Footnote ²⁴ With Majority Rule (Scoring), we can get nontransitive betterness by thinking of each die as an outcome and the numbers (ordered highest to lowest) as six people’s utilities. This case shows that the principles behind Condorcet’s case have been right under our noses.

Temkin’s helpful ‘sports analogy’, by contrast, is only superficially Condorcet-like.Footnote ²⁵ Suppose we rank baseball teams based on their results:

A team’s ‘score’ is its number of wins. So understood, a score isn’t intrinsic to the team; it depends on the whole group.Footnote ²⁶ But the score is still ‘internal’ in that it doesn’t vary across comparisons; it is also 1D, securing Transitivity. The case may seem Condorcet-like if teams beat each other in cycles. But since wins are summed, More Wins is really more like the 1D Internal Aspects View. I mention this case because it nicely illustrates that, pace Temkin, it does not really matter for Transitivity whether values arise from intrinsic properties; what matters is that they not vary across comparisons (see fn. 15, above). Values can be ‘internal’ even if they depend on extrinsic relations.

Finally, notice that Condorcet’s case only features a cycle because there are more than two ‘voters’. Does that show that, whenever there are cycles and only two dimensions, we need an essentially comparative story? No! Consider Handfield’s ‘ertnog’ cycle, which emerges just from mass and height. First a definition:

Next, measurements:

Where ‘>’ means ‘more ertnog’, it follows that A > B > C > A, even though ‘>’ depends only on height and mass, which are intrinsic and invariant. The result is another cycle arising not from comparativity, but multidimensionality, this time with 2D ‘scores’. We can see here that Majority Rule is not the only conceivable rule that can squeeze nontransitive rankings from internal aspects; it is just a striking example because it is not comparative in any way: each voter counts equally in each pairwise comparison. Ertnog, by contrast, is comparative in that mass only counts when height is close. (We will analyse this kind of comparativity in §9, below.)

Handfield’s case is also, obviously, not normative. There is no ethics of ertnog. But I submit that there is a simple normative analogue: Temkin’s ‘fireplace’ example.Footnote ²⁷ Temkin (Reference Temkin2012: 4) reports having had three housewarming options:

The options may appear to form a cycle. Fireplace2 > Fireplace1 because a small comfort boost is worth over $300; Fireplace1 > Nothing because a medium comfort boost is worth over $800; and yet Nothing > Fireplace2 because $1,100 is too much to spend on any amount of fireside comfort. This is like the ertnog structure: a big difference in cost is decisive, but when costs are small, comfort is decisive, or at least can be. (With ertnog, big height differences are decisive, and otherwise mass is decisive.) Contra Temkin, then, we seem to have a moral case better suited to a non-additive, multidimensional treatment than an essentially comparative one.

Once we distinguish the two sources of nontransitivity, we have double the resources with which to interpret Temkin’s example and others like it. Rather than exhibiting comparativity, some paradoxes might instead be exceptions to Addition.

6. Internalizing

Let’s recap.

We began with Temkin’s idea that Transitivity is linked to our views of value. The Internal Aspects View entails Transitivity; the Essentially Comparative View does not. But I have argued that this link only holds given 1D Scoring, or something like it (such as additive 3D scores). If we allow for complex scores, aggregated in the right ways, there can be nontransitivity even on the Internal Aspects View – defined as Scores plus Internal Scoring. The upshot is that Internal Scoring is not the fundamental issue on which Transitivity depends. 1D Scoring, though less familiar, is just as crucial.

Now I want to propose a new question, which will be my topic for the second half of the paper. Once we give up 1D Scoring, is there any need for the Essentially Comparative View? Given enough dimensions of value, aggregated in sufficiently funny ways, it is not just Condorcet’s case and Temkin’s fireplace that seem to fit the ‘internal aspects’ treatment. For any case where an Essentially Comparative View delivers a nontransitivity, we may be able to internalize the case, in the following sense: we could cook up an Internal Aspects View with multiple dimensions and a non-additive aggregation rule that gives rise to the very same nontransitivity. There would then be no need for essential comparativity.

The question, then, is whether every (apparent) instance of essentially comparative value can, and perhaps should, be ‘internalized’, where this involves reinterpreting the case to involve internal scores aggregated in a non-additive way.Footnote ²⁸ We will consider two examples: Spectrum Arguments and person-affecting principles. Temkin sees these as the home turf of the Essentially Comparative View. It would be surprising if even they could be internalized.Footnote ²⁹

7. A spectrum argument

Consider 100 populations. In the first, many people live fabulous lives. In the last, vastly more eke out lives barely worth living. The 98 populations in between fall on a spectrum; each is bigger, and less fabulous on average, than the last. Within each population, lives are equally good, and no member shows up in any other population.Footnote ³⁰

The picture is this, where ‘How Many’ tells us the size of the population, and ‘How Good’ tells us the quality of life for each member. (My presentation draws on Hare (Reference Hare2017).)

Now the nontransitivity. When we compare populations 1 and 2, it seems clear that 2 is better, given that the population is so much bigger – ten times the people! – with only a tiny reduction in average quality of life. For the same reason, each population n > 1 seems better than population n – 1. But when we compare populations 1 and 100, the first seems worse. Given that population 100 is vastly inferior in average quality of life, it seems that population 1 is positively better, all things considered, despite its puny size.

For Temkin, there is only one way to explain this apparent nontransitivity: his Essentially Comparative View. The ‘relevant factors’ (‘or the significance of those factors’) that determine betterness must change, depending on whether we are comparing P1 with P2 or P1 with P100.Footnote ³¹ What does this mean exactly? Temkin is not always clear.Footnote ³² But I think the basic idea is clear enough. Higher quality of life for each is decisive only when we are comparing populations with big differences in quality (see Temkin Reference Temkin1996: 193–196; 2012: 266). When quality is close – as with P1 vs. P2 – total quantity of wellbeing can be decisive. But when the gap in quality is a whopping 99 points – as with P1 to P100 – quantity is moot.

This is a breeze to model on the Essentially Comparative View with 1D Scoring. We could say that the value of a population P in a context is equal to the number of people times the average quality of life (its quantity) – unless P is being compared with another population P* whose average quality is lower by at least 99, in which case P is given infinite value.Footnote ³³ In this way, we get that each population n in the series is better than its predecessor n – 1, but population 1 is better than population 100.Footnote ³⁴

That is one option. But we can also treat the case using an Internal Aspects View, much like Handfield’s treatment of ‘ertnog’ and my take on Temkin’s fireplace (see §6).

Here’s the trick. We give each population P a two-dimensional score: <A_P, T_P>, representing the average utility and total utility, respectively. (Where ‘utility’ measures quality of life, whatever that may be.)Footnote ³⁵ On this 2D Internal Aspects View we then aggregate as follows:

These rules, though simplistic, are vastly more plausible than Majority Rule (Scoring), and they, too, can fit the Internal Aspects View.Footnote ³⁶ The two dimensions really do measure how good a population is all things considered; the overall score is the key factor in determining betterness all things considered; and scores never change even as we swap out alternatives.Footnote ³⁷ And yet we get a cycle: P1 > P2 > … > P100 > P1.

(Notice, by the way, that the nontransitivity here does not depend in any way on there being vagueness. Our simple model has a sharp cut-off for when total quantity starts trumping average quality, and the cycle does not need to include many tiny steps between P1 and P100. There is a cycle between P1, P100 and any intermediate population – for example, P100 > P50 > P1. Now, I am not saying that there cannot be vagueness in this case. Maybe it’s independently plausible that the cut-off should be vague. My point is just that vagueness is not essential to there being a nontransitivity here – or in Temkin’s fireplace case and Handfield’s ertnog case (§6). Multidimensionality can lead to cycles all on its own.)

We have just ‘internalized’ the Spectrum Argument, in the following sense. We have taken a putative nontransitivity that is supposed to be representable only on the Essentially Comparative View and shown how to model it on a multidimensional Internal Aspects View. Indeed, this doesn’t just seem like one way to model the cycle; it seems like the right way to do it. Who needs essential comparativity?

8. A person-affecting principle

Let’s internalize one more classic example.

Suppose that you are making a choice that will affect who exists. If you do A, the world will include a very happy Alan (utility: 10) and a content Chico (utility: 5). If you do B, the world will again feature Alan, merely content this time (utility: 5), alongside an ebullient Betty (utility: 20).

What should you do?

The utilitarian will say: make the world with the happiest people! Do B! But some say morality is not so concerned with making happy people; it is really about ‘making people happy’ (in Narveson’s (1967) slogan). This view is typified by principles like:

Where, crucially, we assume that doing A cannot be ‘better for’ or ‘worse for’ someone if the alternative for them is nonexistence. This principle is all about trying to do what is best for the people who will exist no matter what is done. It recommends, in a choice from A and B, that you do A.

But while the Person-Affecting Principle may be appealing, it violates Transitivity and even Acyclicity. Compare doing B with doing C, where these give rise to the following worlds:

Clearly, on the Person-Affecting Principle, B > C, since B is better for Betty and worse for no one. But now compare:

Here, the principle says C > A, since C is better for Chico and worse for no one. Put the pairs together, we have yet another cycle: A > B > C > A.

What kind of view would allow this? Temkin’s answer is, again, the Essentially Comparative View (Reference Temkin2012: 422–423). The importance of Alan’s welfare changes depending on whether we are comparing two options that both lead to his existence. That A is good for Alan is immaterial when the alternative C, but it greatly increases the value of A relative to B (since Alan will also exist, though less happily, if you do B).

But we can just internalize again, with 3D Scores as in the snacks case. The only difference is that now we need to add gaps (‘ø’) to represent the missing utilities of the nonexistent, and we need to stipulate that gaps disable dimensions: if a _i = ø or b _j = ø, then neither a _i > b _j nor b _j > a _i . We then aggregate using a rule known as (strong) Pareto:

Which says that a score is higher if it is higher along one dimension and lower along none. And this leads to the cycle above, all safely within the Internal Aspects View, without any need for essential comparativity.

9. Comparative effects from internal aspects

Now we are really pushing the Internal Aspects View to its limits.

In this paper’s first half, I argued that the Internal Aspects View could allow some cases of nontransitive betterness. Now it seems that the view might underwrite all nontransitive betterness, or at least far more than expected. The Person-Affecting Principle and Spectrum Arguments are the home turf of the Essentially Comparative View. But we were able to ‘internalize’ Temkin’s take on them in a fairly natural way, reconstruing each to fit an Internal Aspects View with multiple dimensions.

What, then, do we make of the Essentially Comparative View? Should we give up on comparative values? One reason to say ‘no’ – the boring reason – is that we haven’t looked at all of the examples. Perhaps we will eventually find some cases where the Essentially Comparative View resists internalizing. I’m open to this. It would be rash to chuck out the Essentially Comparative View too quickly.

There is, however, an interesting moral we can already draw from the two cases above. Some kinds of comparativity may be compatible with, or even derivable from, an Internal Aspects View. This is not ‘essential comparativity’, because scores aren’t shifting. But it is comparativity in a broader sense: there is a value dimension that can count more or less depending on the options we are comparing.

Consider the 2D Internal Aspects View we used in the Spectrum Argument. The view has two principles, Quality and Quantity, which tell us how to combine our 2D scores. When quality is close, quantity is decisive; when quality is distant, quality is decisive. This is lexicographic aggregation. To determine whether P > P*, we compare them one dimension of value at a time, in a specific order, and the better option overall is the one that is the first to beat the other along a dimension.Footnote ³⁹ In this case, quality comes first, and quantity comes second. P ‘beats’ P* in quality if the average quality of life in P is 99 higher than life in P*. P ‘beats’ P* in quantity if the total quantity of good things in life is higher in P than in P*.

This allows for a kind of comparative effect – screening off. Whenever P beats P* in quality, quantity doesn’t matter to whether P > P*, even though it might be decisive in determining whether P* > P**. In our example above, P1 > P100, despite the massive quantity of good in P100, because quality screens off quantity. This is why P100’s high total goodness is irrelevant in this comparison even though it is relevant, indeed decisive, when we compare P100 with P99, whose quality does not beat P100’s.Footnote ⁴⁰

It is tempting to confuse effects such as screening-off, which are certainly comparative in some sense, with Temkin’s Essentially Comparative View, which is basically the denial of the Internal Aspects View.Footnote ⁴¹ But there is a difference. Screening-off can fit the Internal Aspects View; indeed, it can be the effect of aggregating internal scores in a lexicographic way. Average quality and total quantity are internal to P1, but how quantity affects betterness depends on how close the alternative is in quality.

Now, what about the Person-Affecting Principle? It’s not lexicographic, but it also seems comparative, in some sense. We have three dimensions – one per person – and we are ignoring any dimension that has a gap in it. For example, when we compare A with B, we have the following scores: (10, ø, 5) and (5, 20, ø). Only the first dimension, goodness for Alan, makes a difference; so, A > B even though B is great for Betty. But when we compare A with C, goodness for Alan is irrelevant, since Alan does not exist if C is chosen; its score is (ø, 5, 20), with a gap in Alan’s dimension.

This is not quite ‘screening off’. It isn’t as though Alan’s wellbeing is being trumped by some inherently superior value.Footnote ⁴² Instead we have gappy aggregation.Footnote ⁴³ If there is a gap in one of A’s dimensions, that dimension does not affect whether A is better or worse than B. This seems to be compatible with the Internal Aspects View, since how good A is for Betty – and whether its goodness for Betty is defined at all – is fully intrinsic to A. The role of the gap in A’s score, we might say, is to ‘disable’ the dimension, to make B’s goodness for Betty irrelevant to whether A > B.

But there is a problem. We can get the same effect as gappy aggregation with the Essentially Comparative View. Recall our three options:

Now, when we compare A with B, we might think of their values simply as the two gappy triples: (10, ø, 5) and (5, 20, ø). But notice that only the first dimension matters here, since it is the only one without a gap. So why not prune the other dimensions? We would then get svelte new scores: V _B(A) = (10) and V _A(B) = (5). We can then get the same result as before with Pareto (Scoring): A > B. Ditto for the other pairs. Here, the betterness relations are exactly the same as before, but the triples are no longer values. They are instead ‘proto-values’ that determine an option’s value depending on context. For example, A’s proto-value is (10, ø, 5). This means (i) A’s value is (10) when compared with something whose value has a gap in the third but not the first dimension; (ii) A’s value is (5) when compared with something whose value has a gap in the first but not the third dimension; and (iii) A’s value is (10, 5) when compared with something whose value has no gap in the first or third dimensions. The triple is not a value itself, but it tells us how to find the 2D or 1D value.Footnote ⁴⁴

Here we have two views that seem like notational variants. One says that values are internal, with gappy dimensions that can be ignored during aggregation; the other view says that proto-values are internal, with gappy dimensions that are pruned to arrive at an option’s true context-relative value. Both views generate cycles, and both involve a kind of multidimensionality. To decide between them, we would need some kind of a priori constraint on value theories to serve as a tiebreaker.

Is there any such tiebreaker? I think so. One constraint on a value theory is that there should be exactly one dimension of value for each genuine and distinct way in which things can be good or bad. (The ‘dimensionality constraint’, if you like.) For example, if A’s value is (10, ø, 5) – where the dimensions reflect the utilities of Alan, Betty and Chico – it follows that A is good to degree 10 with respect to Alan’s utility, that A is good to degree 5 with respect to Chico’s utility, and that these are not the same way of being good: (10, ø, 5) is not the same value as (5, ø, 10). The two utilities are not fungible in ethical value as dollar bills are in their value as cash.Footnote ⁴⁵

Now, to my mind, one respect in which B is good is that it is good for Betty. I cannot say that B is better for Betty than A is, since she does not exist given A, nor can I say that A is good with respect to Betty’s happiness (there it is undefined). But I can still say that B is good in virtue of Betty’s happiness. That suggests that B’s overall value score has to include a ‘Betty’ dimension with a score of 20. B’s value cannot just be, say, (5). And yet, (5) is B’s value when compared with A, on the Essentially Comparative View that prunes gappy dimensions. That’s why I think we should prefer the view on which B has the internal value of (5, 20, ø).

I conclude that the Person-Affecting Principle is probably better with gappy internal values than with gapless comparative values. For similar reasons, I think the Spectrum Argument is best understood with 2D internal values; average utility and total utility seem like genuine and distinct dimensions of value.Footnote ⁴⁶

But I might be wrong. In fact, I might be very wrong. On some views, there cannot in principle be any tiebreaker between two value theories so long as they agree about which value relations obtain between which things; any further difference is merely notational. I call this the:

If this is true, there is nothing more to a theory of value than what it says about overall value relations. If it’s false, there must be more to values than the relations they underwrite. I myself suspect that the thesis is false. There should be many dimensions of value that do not change relative to the alternative, since there are many ways, independent of the alternatives, in which a thing can be good.

Let’s sum up. The Spectrum Argument can be internalized as a case of lexicographic aggregation; this leads to a kind of comparativity – namely, screening – that isn’t ‘essentially’ comparative. The Person-Affecting Principle can be internalized as a case of gappy aggregation; this, too, leads to a kind of non-essential comparativity – namely, the disabling of dimensions. In both cases, I think the internalized view is an improvement, since there ends up being one value dimension for each genuine and distinct way in which things can be good. My assumption here is that values, and value dimensions, have some independent reality and can be assessed in themselves; they are not just ‘whatever determines betterness’. That said, I have left open the possibility that I am wrong about this, and I have not leapt to the conclusion that we should always internalize, though I think we probably should do it sometimes.

The moral is that we do not want a sharp divide between, on the one side, the Internal Aspects View, and on the other, all comparative phenomena. Complex scores make the relationship more nuanced and interlinked. To see a simple dichotomy here is restrict one’s view of some of the most dazzling ideas in axiology – lofty hypotheses about the nature of value – to the limited lens of 1D Scoring. Why look at the Milky Way through a magnifying glass?Footnote ⁴⁸