One of the big questions in formal ethics over the last few decades has concerned the structure of value. The simplest formal models of value map values onto numbers. This raises an obvious question, one that can be asked of all models, about which aspects of the model reflect reality, and which are mere artefacts of the model. This note concerns one such feature: the fact that numbers are ordered.
For any two numbers x and y, either x > y, or y > x, or x = y. Is this also true for value? For any two values, is it always true that one is greater, or they are equal? The terminology around here gets confusing, because the language that some writers use for a negative answer to this question gets used by others for theories that offer a particular theory of how and why this disjunction fails. I’ll say that some values are complete if for any two of them, one is greater or both are equal, and the claim that they are incomplete is the negation of this disjunctive claim.
Incompleteness of value was long discussed, though not under that name, in the economics literature. An important early paper in this tradition is by Amartya Sen (1969), who shows that the famous impossibility results due to Kenneth J. Arrow (1951) don’t hold if we drop completeness, and works out what conditions on choice functions or on ‘weak preference’ suffice for completeness.1 This debate moves into philosophy largely through works by Joseph Raz (1986), Isaac Levi (1986), and, most influentially in recent years, Ruth Chang (2002).
1 I say ‘early’ because this is early relative to the philosophical discussions, but the citations in Sen’s paper are sufficient evidence that this was a thriving literature in the 1960s.
There have been several recent defences of completeness against these views. John Broome (1997) argues against incompleteness using considerations about vagueness. Johan Gustafsson (2022) deploys a moneypump argument against incompleteness. And Cian Dorr, Jacob M. Nebel, and Jake Zuell (2023) argue against any kind of incompleteness on semantic grounds.
Recently Christian Tarsney, Harvey Lederman, and Dean Spears (forthcoming) have presented an aggregation argument against completeness for the value of lives and worlds. They argue that if the values of lives are incomplete, some plausible aggregation principles linking the values of lives to the values of worlds will fail. This is a really fascinating addition to the literature for several reasons.
One notable feature of their argument is that it isn’t an argument for completeness in general. Here’s one reason for believing that some things have incomplete values. John Maynard Keynes (1921) and James M. Joyce (2010) have argued that probabilities are imprecise. If they are right, then the class of gambles on propositions with imprecise probabilities should have incomplete values.2 Nothing in Tarsney et al’s argument tells against this. They argue for completeness for certain kinds of final value, not for all things that can be chosen. Indeed, they don’t even argue for it for all kinds of final value. As they are careful to note, their view is consistent with the view that the values of worlds with different populations are incomplete. This is, to my knowledge, the first argument for this kind of restricted completeness claim in the literature. That makes it interesting, and also more plausible than the unrestricted completeness claims.
2 I’m going to use two distinctions here, roughly following Tarsney et al, which I think are helpful. I’ll say an outcome is a full specification of how things go for an agent or a world, while an option is anything that can be chosen. A particular life of an artist is an outcome; going to art school is an option. A lottery is a probability distribution over outcomes. A gamble is any function from a partition of possibility space to options. So a 50/50 chance of getting this artist’s life or that banker’s life is a lottery; a ticket that pays $100 if this horse wins the cup is a gamble. All lotteries are gambles, and all outcomes are options, but the converse fails in both cases.
This matters for some of the applications of incompleteness. Take the example, tracing back to Raz, about the idea that the life of a (fairly successful) artist might be incomparable in value with the life of a (fairly successful) banker. Tarsney et al reject that idea; the point of their argument is to show it has untenable consequences when it comes to the value of aggregates of lives. But they can accept that the value of going to art school might be incomparable with the value of going to business school. Once uncertainty is involved, and particularly once hard to quantify uncertainty is involved, they allow that incompleteness might come back. This is, I think, a novel position in the debate, and it would be worth discussing for that reason alone.
The other striking feature of their argument concerns its direction. It’s easy to assume, in any area of philosophy really, that epistemology follows metaphysics. That is, if F is grounded in G, it’s easy to assume that our knowledge of features of F should be based in knowledge of features of G, and not vice versa. But this isn’t how epistemology works; anything can be evidence for anything. This is an important point in broader decision theory. Even if the value of a single choice isn’t a function of how it interacts with later choices, sometimes we get evidence for rational constraints on single choices by thinking about rational constraints on dynamic choice. It’s a good idea to check whether we can learn about the value of lives by thinking about the value of worlds, even if the value of worlds is ultimately grounded in the value of lives. The easy assumption I mentioned above means that it’s easy to miss the possibility of learning this way.
All that said, I don’t think their argument ultimately works, and the point of this note is to say why. But I want to stress from the top that if my objections work at all, they at most target the details of their argument. The broader methodology, of looking for whether completeness might hold in some particular domain, and of using aggregation as a means of learning about completeness, seems both correct and valuable.
It’s time to move onto details. Assume we have some function v that takes as input, individuals, worlds, or lotteries over these things. Even though we won’t assume the range of v is the real numbers, we will assume that the outputs of v can stand in comparisons. So if a and b are individuals, we’ll assume that things like v(a) ⩾ v(b) make sense. For simplicity, we’ll only assume that such comparisons make sense if the inputs to v on either side of the comparison are either (i) both individuals, (ii) both worlds with the same populations, or (iii) lotteries defined over worlds with the same population.3 Tarsney et al’s restricted completeness thesis is that when a and b fall into one of these three categories, exactly one of v(a) > v(b), v(a) = v(b), and v(a) < v(b) is true. We’ll say that incompleteness is true when none of these three are true.4
3 Recall from the previous footnote that a lottery isn’t a mapping from propositions to outcomes, but from probabilities to outcomes. Tarsney et al allow for a somewhat more liberal set of lotteries under (iii), and this matters to some subsequent arguments they’ll make, but this restricted domain is enough for the purposes of this note.
4 Another terminological note. I’ll write v(a) ⩾ v(b) to mean the disjunction of v(a) > v(b) and v(a) = v(b). This means it isn’t the same as the weak preference relation in Sen (1969), which holds whenever v(a) < v(b) is false. I think my usage is common in contemporary philosophy, and tracks how we normally pronounce ⩾, but there is a possibility of confusion here.
Before getting to Tarsney et al’s argument, I’m going to run through another argument from aggregation principles for completeness as applied to values of lives and worlds. This one won’t involve lotteries, but the result would generalise easily to lotteries. I don’t think this is a good argument, but seeing why it fails makes it easier to see where one might object to Tarsney et al’s more interesting argument.
For any world w, let P(w) be the population of w. Assume that we have two worlds w1 and w2 such that P(w1) = P(w2) = C. (Intuitively, C is our community.) Assume that we have three lives a, b and a+ such that v(a) ~ v(b), v(a+) ~ v(b), and v(a+) > v(a).5 For now we’re just going to be interested in the very simple case where C = {p1, p2}. In those cases, write ⟨x, y⟩ for the world where p1 has life x, and p2 has life y. Now consider the following three somewhat plausible principles.
5 Tarsney et al offer a careful argument for why the believer in incompleteness should be committed to these, but I’ll skip that because I suspect it’s common ground. Indeed, a common argument for incompleteness, the so-called small improvement argument, takes the possibility of such cases to be a motivation for incompleteness.
- Anonymity
- ∀x,y: v(⟨x, y⟩) = v(⟨y, x⟩)
The more general form of this says that if for any person x in C, if the life they have in w1 is exactly as good as the life that π(x) has in w2, where π is an arbitrary permutation of C, then v(w1) = then v(w2). But we’ll just use the binary case for now. Intuitively, permuting who has which life doesn’t change the value of the world as a whole.
- Negative Pareto
- ∀x1,x2,y1,y2: [¬(v(x1) > v(x2)) ∧ ¬(v(y1) > v(y2))] → ¬(v(⟨x1, y1⟩) > v(⟨x2, y2⟩))
The more general form of this says that if it’s true for all people in C that their life in w1 is not better than their life in w2, then w1 is not better than w2. Intuitively, you only make worlds better by making someone’s life better. As Tarsney et al note, you might worry about this because you care about distributional features of value, or about features of value, like natural beauty, that don’t affect welfare. I’m going to set those worries aside, and eventually argue that Negative Pareto fails even when we just care about the welfare of individuals. More generally, in what follows assume w1 and w2 are identical in all aspects of value other than the value of the lives in them, without assuming that there are any such aspects.
- Positive Pareto
- ∀x,y,z: (v(x) > v(z)) → (v(⟨x, y⟩) > v(⟨z, y⟩))
The more general form of this says that if everyone is at least as well off in w1 as w2, and at least one person is better off, then w1 is better than w2. Intuitively, you make things better if you make someone better off without harming anyone else.
From these principles and our assumptions about a, b and a+ we can quickly derive a contradiction.
- v(⟨a+, b⟩) > v(⟨a, b⟩) (Positive Pareto)
- v(⟨a, b⟩) = v(⟨b, a⟩) (Anonymity)
- v(⟨a+, b⟩) > v(⟨b, a⟩) (1, 2)
- ¬[v(⟨a+, b⟩) > v(⟨b, a⟩)] (Negative Pareto)
If v(a) ~ v(b), then it is really plausible that some such a+ exists. For any life, it’s easy to imagine a life that is just like it, but things go just a little better. So really what this shows is that the three principles are incompatible with a and b being incomparable. Since the three principles are very plausible, does that mean that incompleteness is ruled out?
Not so fast! Let’s look more closely at the argument for Negative Pareto. What it says is that you can only make a world better by making some particular person’s life go better. That might seem like it is motivated by the thought that w1 and w2 are alike in all other features, so only the value of the lives could make a difference. But actually Negative Pareto is a bit stronger than the claim that only the values of lives matters. It says that something can only make a decisive difference to the comparison between worlds if it makes a decisive difference to at least one person’s life. That need not be true, and indeed fails in some simple models.
Start with the model developed by Harvey Lederman (forthcoming). When the input to v is an individual, the output in a vector. For simplicity, we’ll take it to be a vector in 2-space, so ⟨x, y⟩, where x and y are numbers. We’ll say that these values are equal iff they are equal in both dimensions. And we’ll say that one value is greater than another iff it is greater in one dimension, and at least as great in the other dimension. Now take the simplest possible theory of value for worlds. The value of a world is just the sum of the values of the individual lives. So if the value of p1’s life is ⟨4, 0⟩, and the value of p2’s life is ⟨0, 4⟩, then the value of the world is ⟨4, 4⟩. (Don’t think too hard about what this means for comparisons between worlds and lives; we’re ignore such comparisons.) Then we compare worlds just the same way we compared lives.
This model looks like only the value of lives matters; the value of a world is just the sum of the value of the lives in it. But it violates Negative Pareto. A simple counterexample is in Table 1.
| w1 | w2 | |
|---|---|---|
| p1 | ⟨4, 0⟩ | ⟨1, 4⟩ |
| p2 | ⟨0, 4⟩ | ⟨4, 0⟩ |
| Total | ⟨4, 4⟩ | ⟨5, 4⟩ |
Each cell in Table 1 gives the value of the life that the person on the row has in the world in the column. It’s easy enough to see that each person has incomparable lives in w1 and w2. But the bottom line shows that w2 is better than w1. Note also that Positive Pareto and Anonymity hold in this model; they are more plausibly consequences of the view that only the values of lives matter.6
6 Though one might worry about the tension Sen (1970) notes between Positive Pareto and liberalism. It would be an idiosyncratic axiology that gave up Positive Pareto for that reason, but it wouldn’t necessarily reject the idea that values of worlds are grounded in the values of lives.
What’s crucial is this. The value of a world is solely a function of the values of the lives of the people in it. But that doesn’t mean that comparisons between worlds are a function of comparisons of lives in those worlds. Moving from facts about the absolute value of lives to facts about the comparative value of lives involves throwing away information. And that information might be crucial to determining how valuable the world as a whole is.
After all that, it’s time to return to Tarsney et al. Their argument doesn’t involve these principles. Instead, they are interested in principles about lotteries. The core principle, or at least what I think is the false principle in their argument, is Negative Dominance.
- Negative Dominance
-
If none of the possible outcomes of lottery L1 are better than any of the possible outcomes of lottery L2, then L1 is not better than L2. (Tarsney, Lederman, and Spears forthcoming, 7)
The ‘outcomes’ here are possible worlds. Their core example uses more or less the four worlds in Table 2. I say ‘more or less’ because while the numbers I use in the first two rows are just instances of the model they give on page 19, they are definitely not committed to the aggregation function I’m using in the row labeled ‘Total’. So really their model is (a more general version of) the first two rows of Table 2.
| w3 | w4 | w5 | w6 | |
|---|---|---|---|---|
| p1 | ⟨4, 1⟩ | ⟨0, 4⟩ | ⟨4, 0⟩ | ⟨0, 4⟩ |
| p2 | ⟨0, 4⟩ | ⟨4, 0⟩ | ⟨4, 0⟩ | ⟨0, 4⟩ |
| Total | ⟨4, 5⟩ | ⟨4, 4⟩ | ⟨8, 0⟩ | ⟨0, 8⟩ |
Here’s the core argument. Let L1 be a lottery that has a 50/50 chance of getting w3 or w4, and L2 a lottery that has a 50/50 chance of getting w5 or w6. All the outcomes of L1 are incomparable with all the outcomes of L2. Their argument for this conclusion is more subtle than my crude aggregation model, but since I agree with that conclusion I’ll skip over those subtleties. So no outcome of L1 is better than any outcome of L2. So, by Negative Dominance, L1 is not better than L2.
However, there are several good arguments that L1 is indeed better than L2. Again, they have a careful argument for this, but since I agree I’ll just note a much cruder argument. Take the value of the lotteries to be the weighted average of the vectors. So the value of L1 will be ⟨4, 4½⟩, while the value of L2 will be ⟨4, 4⟩. Hence L1 is more valuable than L2, contradicting the conclusion of the last paragraph that it is not more valuable.
They conclude that there is something wrong with the model, and while this is not obvious from my simple model, they show that similar problems will arise however one treats incompleteness between lives and/or worlds. That is, any model that allows for incompleteness in the value of lives will, if one endorses Negative Dominance, lead to a contradiction. So we must give up one of incompleteness (for lives) and Negative Dominance. I say we should keep incompleteness, but why should you believe me?
Well, let’s look at their argument for Negative Dominance, and compare it to the way that in the model I just sketched, L1 turned out to be more valuable than L2. They write,
The central idea of this story is that the evaluative ranking of outcomes is more fundamental than, and integral to explaining, the ranking of lotteries. The value of outcomes is actual value … The value of lotteries is something different in kind… [T]he thing that ultimately matters in evaluating lotteries, and grounds any comparisons between them, is their potential for actual value—that is, the values and probabilities of their potential outcomes … Nonfinal value is derivative of final value . So the comparative nonfinal value of a pair of lotteries must be explicable in terms of the comparative final value of their outcomes. In particular, if L1 is (nonfinally) better than L2, then the explanation of this fact must involve some possible outcome of L1 being (finally) better than some possible outcome of L2. (Tarsney, Lederman, and Spears forthcoming, 8, italics in original, bolding added)
The arguments about final and nonfinal value here don’t justify the two words I’ve bolded in the text. What’s true is that the value of lotteries must be grounded in the values of their outcomes. But this doesn’t mean that comparisons between lotteries are grounded in comparisons between outcomes. Comparisons between outcomes throw out lots of information that the full story about the values of those outcomes includes, and the information thrown out might be relevant to the comparison between lotteries.
For comparison, consider the view (defended by Gallow (2015)) that even though all the causal facts supervene on microphysical facts, they need not supervene on microphysical causal facts. Even if we just care about causation in the macro-world, we might need to look at non-causal facts in the micro-world to determine what those facts are. My claim here is that even if we want to look at comparisons in things made of outcomes, i.e., lotteries, we might need to look at non-comparative facts about those outcomes.
Again, it helps to look at the crude model. My argument that L1 is more valuable than L2 just relied on two principles. First, the value of a world is the sum of the values of the lives in that world. Second, the value of a lottery over worlds is the weighted average of the values of those worlds, where the weights are given by the probabilities in the lottery. That’s a model the value of worlds is solely dependent on the value of lives, and where the nonfinal value of lotteries is completely dependent on the final values of the outcomes of those lotteries. But it’s not a world where Negative Dominance holds. That’s because Negative Dominance requires more than the dependence of nonfinal value on final value. It requires the dependence of comparative nonfinal value on comparative final value. That’s a further assumption, and one that the defender of incomparable value should be happy to reject.
So ultimately my argument is that Negative Dominance is false for the same reason that Negative Pareto is false. The argument for it only supports 1 below, but what’s needed is an argument for 2.
- Facts about the absolute value of worlds/lotteries are grounded in facts about the absolute values of lives/outcomes.
- Facts about the comparative value of worlds/lotteries are grounded in facts about the comparative values of lives/outcomes.
If incompleteness holds, there are simple models, like the ones I’ve discussed, where 1 is true, and all of 2, Negative Pareto and Negative Dominance are false.7 So if we antecedently accept incompleteness, we shouldn’t think these arguments about how the value of complex objects like worlds or lotteries are grounded is a reason to reject it.