2. TWO FORMS OF CONSEQUENTIALISM

Most people seem to have the intuition that in circumstances like those described in the above example, we should hold a lottery to decide how to distribute the good in question. To justify this intuition from a consequentialist point of view, we need to show that the consequences Footnote ² of holding the lottery are better than the consequences of giving the kidney to either Ann or Bob without holding a lottery. There may be many different consequentialist justifications of the discussed intuition. But the one I will focus on is the following: A consequence (or situation) where Ann has received the kidney as a result of a lottery is (strictly) morally better than a consequence where Ann has received the kidney without ‘winning’ it in a lottery, because in the former case Bob had a chance. I take this (commonly heard) justification to follow from the more general Fair Chance View:

Fair Chance View (FCV). Suppose n individuals are in equal need of an indivisible good m < n of which we are about to distribute, and that the individuals are identical in every other respect that is morally relevant to the decision of who should receive a good. Then a situation (or consequence) where m of these individuals receive the good but all n individuals had an equal chance of receiving the good is (strictly) morally better than a situation where m of these individuals receive the good and it is not true that all n individuals had an equal chance of receiving the good.

Giving people a (or an equal) chance of getting a good, in situations like the one under discussion, is valuable in and of itself, according to the FCV as I understand it, rather than merely instrumentally valuable. (Moreover, I will assume that on this view, a situation where any of the individuals receive the good is fair just in case a lottery was used to determine who was to receive a good.) I will not attempt to make a normative assessment of the view, nor address the many and deep philosophical issues surrounding it. For instance, I will set aside questions about whether the view requires that we distribute equally objective chances of receiving the good, or whether an equal distribution of subjective probabilities suffices (and if so, the subjective probabilities of whom). Instead, I will try to capture this common view a bit more formally.

The main thing to notice, for the present purposes, is the relation between chance and counterfactuals. What does it mean to say that even though Ann actually got the kidney, Bob had a chance of receiving it? It means that things could (in some meaningful sense) have turned out differently, and if they had, Bob would have received the kidney.Footnote ³ In particular, had some random event turned out differently than the way it actually did, Bob would have received the kidney. Using the possible world framework for counterfactuals, we can express this by saying that a situation in world w where Ann has received the kidney is made morally better by the ‘existence’ of a possible world w′ that only differs from w in that, firstly, a random event turns out differently from the way it does in w, and, secondly, Bob receives the kidney.

As already indicated, one claim to be defended in this paper is that unlike modal consequentialism, non-modal consequentialism is incompatible with the Fair Chance View. Before defending this claim, let me define the two views a bit more precisely. Above I said that according to consequentialism, the moral value of an alternative is determined by its consequences. But this description may be somewhat misleading. To be more precise, let us say, using the terminology developed in Broome (Reference Broome1991), that according to consequentialism in its most general form, the value of an alternative is determined by how it distributes consequences across locations. In Broome’s view, there are three dimensions of these locations to consider: different states of the world, different people and different times. To keep things simple, I will assume that each location in both the dimension of time and people is valued equally (as is the case according to most forms of utilitarianism). Hence, for the present purposes, I define consequentialism, in its most general form, as the claim that the value of an alternative is determined by how it distributes consequences across different states of the world.

This characterization is compatible with different forms of consequentialism, for instance depending on how consequences in different states of the world are weighted in the calculation of the overall value of an alternative.Footnote ⁴ But more importantly for the present discussion, consequentialism, thus characterized, comes in different forms depending on, first, whether we allow that modal properties matter for the moral value of the consequence in each (or some) state; and, second, whether we allow for the possibility that, when evaluating the overall value of an alternative, the contributions that consequences in different states make depend on consequences in other states.

What I call ‘non-modal consequentialism’ is a strict version of consequentialism, in that it neither allows that modal properties can be of moral importance nor for value interactions between consequences in different states of the world:

Non-Modal Consequentialism (NMC). The moral value of an alternative is determined by how it distributes non-modal consequences across different states of the world, and consequences in different mutually incompatible states of the world make independent contributions to the overall value of an alternative.

From a general consequentialist theory (as described above) we get non-modal consequentialism when we add the following principles:

First Principle of NMC. The moral value of a consequence in a particular state of the world is fully determined by the non-modal properties of that consequence.

Second Principle of NMC. If alternative A has different consequences depending on whether state of the world s ₁, s ₂, s ₃, etc., turns out to be actual (where s ₁, s ₂, s ₃, etc., are mutually incompatible), then for any of these s_i, the moral value that the consequence in s_i contributes to the overall moral value of A is independent of the consequence in any $s_j\not=s_i$.

I will call a consequentialist theory modal if it violates either the first or the second principle of NMC. Here is what the two principles have in common which justifies calling a theory that violates either of these modal. If a theory does not satisfy the first principle, then the value of a consequence in one state of the world may depend on what occurs in other states of the world, but if a theory does not satisfy the second principle, then the contribution that a consequence in some state of the world makes towards the overall value of an alternative may depend what occurs in other states of the world. So violations of the two principles have in common that there is some sort of value dependency between what occurs in different, mutually incompatible states of the world.

The second principle of NMC is related to a property called separability that has been much discussed in decision theory. Separability is usually discussed as a property of preferences – or, in a moral context, as a property of what we might call ‘betterness judgements’.Footnote ⁵ To explain separability, let us represent each alternative A by an n-tuple, e.g. A = ⟨a ₁, a ₂, ..., a_n⟩, where the a_is are the possible consequences of A. Now take the alternative A and create two new alternatives: A_b created by replacing a_i in the original alternative with b and A_c created by replacing a_i with c. Do the same for alternative D: create D_b by replacing d_i in the original alternative with b and D_c by replacing d_i with c. A betterness judgement (or preference) is separable just in case for any manoeuvre like the one just described, A_b is better than (or preferred to) A_c if and only if D_b is better than (or preferred to) D_c.

Decision theorists typically start with an ordering of alternatives, or a set of properties of orderings, and then show what kind of value functions can represent such an ordering (or an ordering with those properties). But it can be useful to start with a property of a valuation and see what ordering properties it implies. Let us suppose that when a non-modal consequentialist orders a set of alternatives according to ‘betterness’, she first finds out the moral value of each alternative, in accordance with the two principles of NMC, and then orders the alternatives according to moral value. (Moreover, let us restrict our attention to alternatives where both probabilistic and causal independence holds between the alternatives and the states of the world.) Then since moral value, according to her, satisfies the second principle of NMC, her betterness judgement satisfies separability. In the next section I will explain why some have thought that separability is inconsistent with a view like the Fair Chance View. In section 4 we will see that that thought is mistaken.

But first, let me briefly explain why I think the distinction I have just made is important. Surely everyone accepts that modal properties are morally important, someone might say. So what is the point of discussing this distinction between modal and non-modal consequentialism?Footnote ⁶ Well, the best known version of consequentialism, namely classical utilitarianism, is non-modal. According to classical utilitarianism, one should always choose the act that maximizes the total amount of pleasure over pain: ‘the greatest happiness for the greatest number’, as it is often put. Of course, people might feel (psychological) pleasure and pain due to what could have been. However, after we have accounted for such attitudes, an evaluation of alternatives should, according to classical utilitarians, satisfy both the first and the second principle of NMC.Footnote ⁷

Economists and decision theorists have traditionally also been very reluctant to accept that what could have been matters for the (rational) evaluation of actual outcomes. In a classical defence of the ‘Independence Axiom’, found in some form or other in most decision theories (and implied by separability as previously defined), Nobel laureate Paul Samuelson for instance argues for formal separability as captured by the second principle of NMC, based on an intuition like the one expressed by the first principle of NMC. A simple version of the axiom Samuelson was defending states that if (A)₁ is at least as good as (B)₁ and (A)₂ is at least as good as (B)₂, then an alternative that results in either (A)₁ or (A)₂ depending on whether a coin comes up heads or tails, is at least as good as an alternative that results in either (B)₁ or (B)₂ depending on how the coin lands.Footnote ⁸

Here is Samuelson’s informal justification of the axiom:

In other words, the reason an evaluation or ordering of alternatives should satisfy separability (as the second principle of NMC states), is that there should be no desirabilistic dependencies between mutually incompatible outcomes (as the first principle of NMC states).

Finally, it might be worth mentioning that certain consequentialist theories should be classified as ‘modal’ for reasons that have nothing to do with what they say about the fairness of lotteries. These include John C. Harsanyi’s (Reference Harsanyi, Sen and Williams1977) and Richard Arneson’s (Reference Arneson1990) views. Both authors claim (roughly) that, for example, social policies should maximally satisfy people’s hypothetical preferences; i.e. those preferences that people would have in ideal circumstances. What is true in a counterfactual world therefore makes a difference to the moral value of outcomes and alternatives in the actual world. Although I will focus on the fairness of lotteries in this paper, it should become clear that the framework developed in section 5 also provides a formal model in which to state and explore claims made by theories like Harsanyi’s and Arneson’s.

3. NON-MODAL CONSEQUENTIALISM VS. FAIR CHANCE

I will focus on Leonard Savage’s version of decision theory to show why a non-modal consequentialist theory is incompatible with the Fair Chance View (Savage Reference Savage1972). According to Savage’s theory, the value of an alternative A, denoted by U(A), is given by:

Savage’s equation$U(A)=\displaystyle {\sum _{s_{i}\in \mathcal {S}}}u(s_{i}(A))\cdot Pr(s_{i})$

where $\mathcal {S}$ is a set of mutually exclusive and collectively exhaustive states of the world that determine the consequence of A, Pr a probability measure on $\mathcal {S}$, s_i(A) represents the consequence of A when s_i happens to be the actual state of the world, and u is a value measure on (maximally specific) consequences.

Savage’s equation satisfies the second principle of non-modal consequentialism; i.e. the separability property. In decision theoretic jargon, Savage’s utility function is additively separable: the value of each alternative is a weighted sum of the values of each of its possible consequences. Hence, the value that u(s_i(A)) contributes towards the overall value of A is independent any s_j(A). But, crucially, for Savage’s theory to be an appropriate formalization of non-modal consequentialism, we need to assume that each s_i(A) is a non-modal consequence.

To relate the above characterization of non-modal consequentialism back to the example of Ann, Bob and the kidney, let L be a lottery that gives Ann and Bob an equal chance of receiving the kidney (depending for instance on whether a fair coin lands heads up or tails up), and A (B) the alternative of giving the kidney to Ann (Bob) without holding a lottery. Then the Fair Chance View implies the following betterness judgement, which I will refer to as the Fair Chance Judgement (FCJ): A≺L, B≺L (where ‘..≺...’ denotes ‘... is worse than ...’). Then given that Savage’s theory is an appropriate formalization of NMC, the latter is only compatible with the FCV if the following inequalities can simultaneously be satisfied:

(1) \begin{equation} \sum _{s_{i}\in \mathcal {S}}u(s_{i}(A))\cdot Pr(s_{i})<\sum _{s_{i}\in \mathcal {S}}u(s_{i}(L))\cdot Pr(s_{i}) \end{equation}

(2) \begin{equation} \sum _{s_{i}\in \mathcal {S}}u(s_{i}(B))\cdot Pr(s_{i})<\sum _{s_{i}\in \mathcal {S}}u(s_{i}(L))\cdot Pr(s_{i} ) \end{equation}

where each s_i(α) is a non-modal consequence.

Let ANN (BOB) represent the consequence where Ann (Bob) receives the kidney. Then A (B) is certain to have ANN (BOB) as consequence, but L will either result in ANN or BOB. Thus we can represent A (B) by ANN (BOB), and L by the n-tuple ⟨ANN, BOB⟩. There do not seem to be any modal properties built into the description of these consequences, so non-modal consequentialists should be happy with this representation of the three alternatives.

But now we run into trouble. For according to NMC, the value of the three alternatives is then given by: U(A) = u(ANN), U(B) = u(BOB) and U(L) = u(ANN) · 0.5 + u(BOB) · 0.5. But obviously, u(ANN) · 0.5 + u(BOB) · 0.5 can never be greater than u(ANN) and also greater than u(BOB). If one of u(ANN) and u(BOB) is greater than the other, then the value of the lottery L falls strictly between the values of the ‘risk-free’ alternatives A and B, but if the value of ANN and BOB is the same, then the value of the lottery will be the same as that of the risk-free alternatives. Hence, it seems, NMC is not compatible with the Fair Chance Judgement (and since the latter is implied by the FCV, NMC is incompatible with the FCV).

The above tension clearly has something to do with separability; in particular, the additively separable form of Savage’s utility function. If the above is the right description of the alternatives, then if the value of an alternative is a probability weighted sum of the values of its possible consequences, then the value of L can never be greater than the value of both A and B. In the next section I discuss an attempt to make the FCV compatible with consequentialism by dropping separability, thus violating the second principle of NMC. But the above tension can also be put down to the way in which the alternatives have been described. In sections 4 and 5 I discuss two attempts, one old and one novel, to make the FCV compatible with consequentialism by describing the alternatives (and their consequences) in a way that violates the first principle of NMC. (The novel attempt also violates the second principle of NMC.) Perhaps unsurprisingly, I will argue that only my new solution succeeds in making consequentialism compatible with the intuition behind the FCV.

4. CONSEQUENTIALISM WITHOUT SEPARABILITY

Suppose we calculate the values of the three alternatives as follows:

where ⊤ is a tautology and r a risk function of a risk seeking agent – i.e. an agent who prefers a gamble with an expected value of x to a risk-free alternative the value of whose consequence is x – and {S, ¬S} is a partition of the possible states of the world into two equiprobable events.Footnote ⁹ Then we can represent the judgement that L is better than both A and B as maximizing risk-weighted utility; if, for instance, we allow for the possibility that u(ANN) · r(P(S)) + u(BOB) · r(P(¬S)) is greater than both u(ANN) · r(P(⊤)) and u(BOB) · r(P(⊤)). And given how I have characterized consequentialism – i.e. as the view that the value of an alternative is determined by how it distributes consequences across states of the world – risk-weighted utility theory is a consequentialist theory.

This solution satisfies the first principle of NMC. For the consequences of the lottery, thus described (ANN and BOB), do not contain modal properties. But this solution is still incompatible with non-modal consequentialism, since it violates the second principle of NMC. Formally, the risk-weighed utility theory does not satisfy separability. To see this, notice that we can partition the tautology into S and ¬S, and then alternative A can be reformulated as the ‘lottery’ that has ANN as consequence in both the S-states and ¬S-states. Hence, this solution violates the second principle of NMC.

Should modal consequentialists be happy with this solution? There are, in my view, two related reasons for seeking an alternative way of making consequentialism compatible with the FCV. First, it seems to me that the Fair Chance View is an example of a more general phenomenon where what could have been is important for the evaluation of the desirability of what actually occurs. Unlike the above solution, the one I develop in section 5 explicitly models this relationship between what is and what could have been. Second, the above solution suggests that accepting the FCV has something to do with being risk seeking. More precisely, this way of making consequentialism compatible with the FCV suggests that the reason consequentialism as formalized by Savage seems incompatible with the FCV, is that that formalization places certain restrictions on attitudes towards risk. But those who accept my first objection will agree that the problem with Savage’s framework, from the perspective of the FCV, is not so much the theory’s restriction on risk attitudes, but rather its insensitivity to the desirabilistic relationships between what is and what could have been.

5. BROOME’S REDESCRIPTION STRATEGY

Contrary to my suggestion in section 2, many people will undoubtedly have the intuition that the consequence where Ann (Bob) receives the kidney as a result of the lottery L is not the same consequence as Ann (Bob) receiving the kidney as a result of the risk-free alternative A (B). The former consequence is fair whereas the latter is not, which must mean that these are not the same consequence. Hence, it seems, the consequences of A, B and L were not properly described in last section. Similarly to what Broome (Reference Broome1991: ch. 5) suggests, we should perhaps write the fairness directly into the description of the outcomes of the lottery; such that, for instance, L has $ANN\&Fair$, $BOB\&Fair$ as its two possible consequences, but A (B) has ANN (BOB) as the only possible consequence. And then the trouble we saw in section 2 disappears, since an (additively separable) EU function can, of course, simultaneously satisfy:

(3) \begin{eqnarray} EU(A)&=&u(ANN)<EU(L)\nonumber\\ &=&u(ANN\&Fair)\cdot 0.5+u(BOB\&Fair)\cdot 0.5 \end{eqnarray}

(4) \begin{eqnarray} EU(B)&=&u(BOB)<EU(L)\nonumber\\ &=&u(ANN\&Fair)\cdot 0.5+u(BOB\&Fair)\cdot 0.5 \end{eqnarray}

Broome’s solution thus makes a preference for tossing the coin in the example discussed at that start of the paper compatible with consequentialism without giving up separability (i.e. without violating the second principle of NMC). But the solution clearly violates the first principle of NMC. Assuming that the FCV is part of our conception of fairness, then when we start refining our consequence set to include consequences that have fairness written into their description (to deal with examples like the one under discussion), we are in effect creating dependencies between consequences in different mutually incompatible states of the world. The consequence $ANN\&Fair$ for instance implicitly refers to what could have been, in the sense that a necessary condition for $ANN\&Fair$ to be a possible consequence of some alternative C, is that $BOB\&Fair$ is also (at least considered to be) a possible consequence of C.Footnote ¹⁰ For given the FCV, C can only result in $ANN\&Fair$ if C is some sort of lottery or random choice mechanism that has $BOB \& Fair$ as consequence in some state of the world. Hence, given that the moral value of the consequence of the lottery is, on Broome's suggestion, partly a function of this modal property, his suggestion violates the first principle of NMC.

The implicit reference to what could have been is precisely the reason why Broome’s preferred description of the consequences runs into troubles with (what he calls) the Rectangular Field Assumption (RFA). RFA is a technical assumption of many of the traditional decision theoretic representation theorems, such as Savage’s, needed (given the other assumptions of these theorems) to construct from an agent’s preferences a value function that is unique up to positive affine transformation. Recall that an alternative can be represented by an n-tuple of consequences, e.g. A = ⟨a ₁, a ₂, ..., a_n⟩, where a_i is the consequence of alternative A if state of the world s_i happens to be actual. Given any set of alternatives that an agent’s preferences are defined over, each state has associated with it a set of possible outcomes. Call that set for the i-th state C_i. Then take the product of all of these sets: C ₁ × C ₂ × ... × C_n. Now the RFA states that any n-tuple created by picking consequences from some or all of the sets in the product is an alternative in the agent’s preference ordering. That is, if we go through the sets, from C ₁ to C_n, and pick arbitrary consequences from some or all of these sets, then the resulting n-tuple of consequences is an alternative in the agent’s preference ordering.

Going back to our example, we could for instance pick two consequences: ANN and $ANN\&Fair$. The resulting alternative would then be the ordered pair $\langle ANN,\ ANN\&Fair\rangle$. This is an alternative where Ann gets the kidney in any state of the world; but, in addition, if some state turns out to be actual, then Ann gets the kidney fairly! But that is, of course, conceptually impossible on the Fair Chance View. For given this conception of fairness, either Ann or Bob, who have an equal claim to the good, can only receive the kidney fairly if some random mechanism has been used to determine who is to receive the good. But whenever such a random choice mechanism is used, it will not be the case that the same patient receives the kidney in all states of the world. In other words, an outcome is fair only if it is not true that that same patient receives the kidney in all possible states of the world.Footnote ¹¹ Another way to put this, is that given Broome’s description of consequences, the RFA requires that it be possible that a lottery that is unfair results in a consequence that is fair. It seems clear that this requirement goes against the intuition behind the FCV. Hence, it is not at all clear that Broome’s ‘re-description strategy’ makes the FCV compatible with Savage’s consequentialist framework.Footnote ¹²

I should emphasize that the tension with the Rectangular Field Assumption is not the main reason I think we should seek alternative ways of making the Fair Chance View compatible with consequentialism. Not all decision theoretic representation theorems require the RFA,Footnote ¹³ and there are well known problems with the assumption that have nothing to do with fairness. The main problem I have with Broome’s solutions, is rather that unlike the solution suggested in next section, his fails to make explicit the desirabilistic dependency between actual and counterfactual outcomes that seems at the heart of the FCV. Examining the tension between Broome’s solution and the RFA nevertheless does serve an important role in the present argument, since it illustrates the difficulty in dropping the first principle of non-modal consequentialism while holding on to Savage’s framework. The reason the RFA creates trouble for Broome is that some consequences as Broome describes them refer, as we have seen, to what occurs in states of the world in which they themselves do not occur, and thus cannot be combined with any arbitrary consequence as the RFA requires. In other words, the tension with the RFA stems from Broome’s violation of the first principle of NMC.

6. MULTIDIMENSIONAL DECISION THEORY

I now finally turn to formulating a version of modal consequentialism that better satisfies the intuition behind the Fair Chance View than the theories already discussed. The framework I develop is a multidimensional extension of Richard Jeffrey’s (Reference Jeffrey1983) decision theoretic framework to counterfactuals. The resulting theory allows us to represent various different judgements according to which counterfactual outcomes influence the desirability of actual outcomes. A general discussion of such judgements would take us too far from the topic of this paper, so I will focus on showing that the multidimensional theory can represent the Fair Chance Judgement as maximizing desirability.Footnote ¹⁴

According to Jeffrey’s theory, both probabilities and desirabilities are measures on a Boolean algebra Ω of propositions (that is, a set of propositions closed under negation and conjunction) from which impossible (zero probability) propositions have been removed.Footnote ¹⁵ The desirability of any particular proposition, p, is according to Jeffrey’s measure a weighted average of the desirabilities of the different (mutually exclusive) ways in which the proposition can be true, where the weight on each way p_i that proposition p can be true is given by the conditional probability of p_i given p. More formally:

Jeffrey’s equation (a)$Des(p)=\displaystyle {\sum _{p_{i}\in p}}Pr(p_{i}\mid p)\cdot Des(p_{i})$

A proposition, in Jeffrey’s framework, is a set of possible worlds. So probabilities and desirabilities are measures on a set of sets of worlds. We can thus interpret the p_is as different worlds compatible with the proposition p; i.e. think of {p ₁, p ₂...} as a world partition that is equivalent to p. So, if we like, we can formulate Jeffrey’s equation as:

Jeffrey’s equation (b)$Des(p)=\displaystyle {\sum _{w_{i}\in p}}Pr(w_{i}\mid p)\cdot Des(w_{i})$

Jeffrey’s evaluation of propositions is clearly consequentialist, given how I have characterized consequentialism. The possible ways in which a proposition can come true can be understood as the possible consequences of the proposition coming true, and we can interpret that which determines the way in which a proposition comes true (if it comes true) as a ‘state’ of our world. Representing the Fair Chance Judgement as maximizing the value of a Jeffrey-desirability function is therefore one possible way of making the Fair Chance View compatible with consequentialism. However, someone who evaluates propositions according to Jeffrey’s equation will not always satisfy separability, since the contingencies that determine how a proposition becomes true are not, in Jeffrey’s theory, probabilistically independent of the proposition that is being evaluated.Footnote ¹⁶

Although Jeffrey’s theory violates separability, the theory as Jeffrey himself interpreted it – i.e. as a version of non-modal consequentialism (where Ω contains only factual propositions) – nevertheless runs into the same problem we have seen with Savage’s: there is no pair of desirability and probability functions relative to which the Fair Chance Judgement can be represented as maximizing desirability. To see this, now let ANN (BOB) be the proposition that Ann (Bob) receives the kidney, and L a proposition that can come true in one of two ways, ANN or BOB, and does so with equal probability. Then for the Fair Chance View to be compatible with Jeffrey’s original theory, there has to be a function Des such that:

(5) \begin{equation} \ \ \ \ Des(ANN)<Des(ANN)\cdot Pr(ANN\mid L)+Des(BOB)\cdot Pr(BOB\mid L) \end{equation}

(6) \begin{equation} \ \ \ \ Des(BOB)<Des(ANN)\cdot Pr(ANN\mid L)+Des(BOB)\cdot Pr(BOB\mid L) \end{equation}

which implies that:

(7) \begin{equation} Des(ANN)<0.5Des(ANN)+0.5Des(BOB) \end{equation}

(8) \begin{equation} Des(BOB)<0.5Des(ANN)+0.5Des(BOB) \end{equation}

But again, a probability mixture of the desirabilities of ANN and BOB can of course never exceed the desirability of both ANN and BOB.Footnote ¹⁷

It could be objected that the consequences of the lottery should be formulated as ANN∧L, in which case there will be pairs of desirability and probability functions relative to which the Fair Chance Judgement can be represented as maximizing desirability. But ANN∧L is not a non-modal consequence: in effect, this description of the consequence has built into it that the consequence in question had 0.5 chance of occurring (since L is the proposition that either ANN or BOB will occur with equal chance). Making Jeffrey’s theory compatible with the Fair Chance Judgement by including the lottery in the description of the consequences moreover suffers from the same problem as Broome’s suggestion, namely that it does not make explicit the importance of counterfactuals for the Fair Chance View.

I will base the extension of Jeffrey’s theory to counterfactuals on Richard Bradley’s (Reference Bradley2012) Multidimensional Possible World Semantics for Conditionals. The basic ingredients in the multidimensional semantics are n -tuples of worlds, ⟨w ₁, w ₂, w ₃, w ₄, ...⟩ (i.e. ordered sets of worlds), where the first world representa a potential actual world and the rest represent potential counter-actual worlds under different suppositions. To keep things simple, let’s assume that what is being supposed true are sentences. Then w ₂ might e.g. represent a counter-actual world under the supposition that sentence P is true, w ₃ a counter-actual world under the supposition that ¬P is true, w ₄ a counter-actual world under the supposition that Q, etc. In what follows I will refer to such an n-tuple as a possible situation: ⟨w ₁, w ₂, w ₃, ...⟩ is the situation that w ₁ is actual, w ₂ would be actual if P, w ₃ would be actual if ¬P, etc.

Let us focus on a model with only four worlds, w ₁ to w ₄, and suppose Table 1 represents how worlds and sentence match up. In other words, the sentence P is true at worlds w ₁ and w ₂, Q at worlds w ₁ and w ₃. To explain the semantics it is useful to focus on only one supposition: the supposition that P. Table 2 represents the eight possible situations; for each ⟨w_i, w_j⟩ the possibility that w_i is the case and w_j would be if P.Footnote ¹⁸

Table 1. Worlds-sentences

Table 2. Space of possibilities

Now suppose the sentence P expresses the proposition p. Thus the sentence P is true if and only if p. So p is true in the situations in the first and second row of the table, but false in the situations in the third and fourth row. And given that we identify a proposition with the situations where it holds true, we have p = {⟨w ₁, w ₁⟩, ⟨w ₁, w ₂⟩, ⟨w ₂, w ₁⟩⟨w ₂, w ₂⟩}. So a proposition, on the multidimensional semantics, is a set of n-tuples of worlds.

Now let $\square \hspace{-4.0pt}\rightarrow$ stand for the counterfactual conditional connective.Footnote ¹⁹ The conditional sentence $P\square \hspace{-4.0pt}\rightarrow Q$ is true, on the multidimensional semantics, if and only if Q is true in the world that is counter-actual under the supposition that P. In other words, $P\square \hspace{-4.0pt}\rightarrow Q$ is true if Q is true in the second world in the n-tuple that correctly represents what is the case and what would be the case under the supposition that P. So $P\square \hspace{-4.0pt}\rightarrow Q$ is made true by the situations in the first column of the above table, but made false by situations in the second column. And the proposition $p\square \hspace{-4.0pt}\rightarrow q$ is thus the set {⟨w ₁, w ₁⟩, ⟨w ₂, w ₁⟩, ⟨w ₃, w ₁⟩, ⟨w ₄, w ₁⟩}.

Let us then return to the project of representing the Fair Chance Judgement.Footnote ²⁰ Now let P express the proposition (p) that the coin comes up heads and ¬P the proposition that the coin comes up tails. Let Q express the proposition (q) that Ann receives the kidney and ¬Q the proposition that Bob receives the kidney. We have thus made two simplifying assumptions already. Firstly, it might seem more natural to let P (¬P) express the conditional that the coin comes heads (tails) up if tossed. But nothing is lost if we think of this just as a factual sentence. Secondly, we have limited our attentions to situations where either Ann or Bob receives the kidney. But what is distinctive about the FCV is what it has to say about situations where a number of individuals have an equal claim to an indivisible good that some but not all of them get. (Any kind of welfarism for instance condemns a situation where none of the needing patients receives the kidney.) Hence, since we want to focus on what is special about the view, it seems justifiable to limit our attention to situations where one of Anna and Bob does receive the kidney.

To represent the Fair Chance Judgement in a multidimensional model we need to work with two suppositions: the supposition that P and the supposition that ¬P. But actually, we only need to consider two dimensions at a time, since the FCJ only orders alternatives according to what is true and what would be true under a contrary-to-factual supposition. That is, the judgement for instance is that a situation where the coin comes up heads and Ann receives the kidney, is made better or worse depending on whether Ann also receives the kidney under the (contrary-to-factual) supposition that the coin comes up tails. But it says nothing about whether the desirability of this situation depends on what is true under the (matter-of-factual) ‘supposition’ that the coin comes up heads. Hence, for a situation where P is true at the actual world, we need not represent the ‘counter’-world under the supposition that P; and similarly for the situation where ¬P is true. So although semantically we are now working with a three-dimensional model, we only need to worry about two dimensions at a time.

Let θ denote the world we need not consider at each time. If P is true, then θ is the ‘counter’-world under the supposition that P (which is the second world as I am setting up the n-tuples) but if ¬P is true, then θ is the ‘counter’-world under the supposition that ¬P (which is the third world as I am setting up the n-tuples). Table 3 thus represents the space of possible situations we need (call the whole space $\mathcal {W}$).

Table 3. Space of possibilities 2

I will assume that it makes no difference, according to the FCV, whether Ann or Bob actually receives the kidney, since by assumption their situation is symmetrical in all ways that are relevant for the decision of who should receive the kidney. Hence, the fair situations are equally good when Ann receives the kidney as when Bob receives the kidney and similarly for the unfair situations. The FCJ thus orders the situations into two equivalence classes, represented in Table 4, where every situation from the ‘Good’ class is better than every situation from the ‘Bad’ class, but any two situations within the same class are equally good.

Table 4. The good, the bad

What is common to the (‘Bad’) situations in the left column is that the person who actually receives the kidney would also have received it had the coin come up differently. In the situations in the right column, however, whoever actually receives the kidney would not have received it had the coin landed differently.

The Fair Chance Judgement can now be formulated as follows:Footnote ²¹

(10) \begin{eqnarray} &&\langle w_{1}, \theta , w_{3}\rangle \sim \langle w_{2}, \theta , w_{4}\rangle \sim \langle w_{3}, w_{1}, \theta \rangle \sim \langle w_{4}, w_{2}, \theta \rangle \nonumber\\ && \prec \ \ \langle w_{1}, \theta , w_{4}\rangle \sim \langle w_{2}, \theta , w_{3}\rangle \sim \langle w_{3}, w_{2}, \theta \rangle \sim \langle w_{4}, w_{1}, \theta \rangle \end{eqnarray}

and the FCJ can be represented by a function V that satisfies:Footnote ²²

(11) \begin{eqnarray} &&V(\langle w_{1}, \theta , w_{3}\rangle )=V(\langle w_{2}, \theta , w_{4}\rangle )=V(\langle w_{3}, w_{1}, \theta \rangle )=V(\langle w_{4}, w_{2}, \theta \rangle )\nonumber\\ && < \ \ V(\langle w_{1}, \theta , w_{4}\rangle )=V(\langle w_{2}, \theta , w_{3}\rangle )=V(\langle w_{3}, w_{2}, \theta \rangle )=V(\langle w_{4}, w_{1}, \theta \rangle ) \end{eqnarray}

There will certainly be many functions satisfying 11 (and thus representing 10): any ordinal utility function, defined over a set of world-triples, can represent this ordering.Footnote ²³ So to some extent we have reached the aim of making the FCV compatible with (modal) consequentialism. For we have found a way of showing that there is a function whose assignment of values to situations corresponds to whether the FCV deems the situation fair. And we do not have to worry about clashes with the Rectangular Field Assumption, since the assumption is not needed for the existence of such a function.

I have however not yet shown that there is a Jeffrey desirability function that represents 10. But we can do so by construction. Let V and Pr assign values to the basic situations (i.e. the ordered triples) in $\mathcal {W}$. The functions extend to any proposition r (i.e. to any set of situations) according to the following rules:

(12) \begin{equation} V(r)=\sum _{\langle w_i, w_j, w_k\rangle \in r}V(\langle w_i, w_j, w_k\rangle )\cdot Pr(\langle w_i, w_j, w_k\rangle \mid r) \end{equation}

(13) \begin{equation} Pr(r)=\sum _{\langle w_i, w_j, w_k\rangle \in r}Pr(\langle w_i, w_j, w_k\rangle ) \end{equation}

Then by construction V is a Jeffrey desirability function: the desirability of a proposition, according to this function, is a weighted average of the desirabilities of the different ways in which the propositions can come true (i.e. the different situations compatible with the proposition), where the weights are given by the appropriate conditional probabilities.

Now let’s see whether this function can represent the Fair Chance Judgement, as formulated in 10. Recall the two equivalence classes of propositions (sets of n-tuples) induced by the FCJ. Let us call the ‘good’ equivalence class G and the ‘bad’ equivalence class ¬G. We can stipulate that:

1. 1. ∀⟨w_i, w_j, w_k⟩ ∈ G: V(⟨w_i, w_j, w_k⟩) = 1

2. 2. ∀⟨w_l, w_m, w_n⟩ ∈ ¬G: V(⟨w_l, w_m, w_n⟩) = −1

Then it is clear that V represents the FCJ, as formulated in 10: for any two basic situations, ⟨w_i, w_j, w_k⟩ and ⟨w_l, w_m, w_n⟩, we have: if ⟨w_i, w_j, w_k⟩ ~ ⟨w_l, w_m, w_n⟩ according to FCJ, then V(⟨w_i, w_j, w_k⟩) and V(⟨w_l, w_m, w_n⟩) are both either -1 or 1; but if ⟨w_i, w_j, w_k⟩ ≺ ⟨w_l, w_m, w_n⟩ according to FCJ, then V(⟨w_i, w_j, w_k⟩) = −1, V(⟨w_l, w_m, w_n⟩) = 1. So we have constructed a Jeffrey desirability function that represents the FCJ.

G and ¬G are also propositions (sets of n-tuples of worlds), and by the above stipulation: V(G) = 1 and V(¬G) = −1. But for any arbitrary proposition r:

(14) \begin{eqnarray} V(r)&=&V(G)\cdot Pr(G\mid r)+V(\lnot G)\cdot Pr(\lnot G\mid r)\nonumber\\ &=&Pr(G\mid r)- Pr(\lnot G\mid r) \end{eqnarray}

Up to now I have been focusing only on propositions that are either completely fair or completely unfair; i.e. propositions that are either subsets of G or ¬G but do not overlap the two sets. And I have formulated the FCJ as only having something to say about propositions that are either completely fair or unfair. But we can easily construct propositions that overlap the two sets. Let’s call such propositions ‘mixed’. m = {⟨w ₁, θ, w ₃⟩, ⟨w ₁, θ, w ₄⟩} is a mixed proposition, for instance, since the first of its elements is an unfair situation but the second is fair. Intuitively, a proposition like m is a biased lottery. In ⟨w ₁, θ, w ₃⟩ Ann gets the kidney no matter how the coin lands. So since this situation is possible given m, but no situation that gives Bob a greater chance than Ann is possible given m, m is biased towards Ann.

Now take any two mixed propositions m ₁ and m ₂: suppose each is a set of two triples, one of which is an element of G but the other of ¬G. According to V, as I have constructed it, V(m ₁) ⩽ V(m ₂) just in case Pr(G∣m ₁) ⩽ Pr(G∣m ₂). It is natural to think of Pr(G∣m_i) as measuring how unbiased m_i is: if Pr(G∣m_i) = 1 then m_i is not at all biased but if Pr(G∣m_i) = 0 then m_i is maximally biased. So the more (less) biased a proposition is, the lower (higher) value V assigns to it. Although I have said nothing about how the Fair Chance View judges biased lotteries, this is exactly what we should want.Footnote ²⁴ For any mixed propositions m ₁ and m ₂, the former should be better than the latter, on this view, if and only if it is less biased; but if they are equally biased, then they should be equally good (or bad). So the function I have constructed does not only capture the intuition that a situation where Ann (Bob) receives the kidney is fair only if Bob (Ann) had a chance; it also captures the intuition that situations where they both had an equal chance at receiving the kidney are more fair than situations where their chances were unequal.

7. SOME IMPLICATIONS

The framework developed in the last section has the advantage over the other modal consequentialist theories I have been discussing in that it explicitly models the desirabilistic relationship between what is and what could have been, which seems at the heart of the Fair Chance View. In addition, it has the following advantage over Broome’s suggestion for how to make consequentialism compatible with the preference for tossing the coin. Broome’s solution consists in adding a primitive fairness property to the outcome space. My suggested solution however consists in enriching decision theory to include counterfactual prospects in such a way that the fairness property emerges as a relationship between actual and counterfactual outcomes. Thus, I contend, my solution better explains what orthodox decision theory lacks when it comes to capturing common intuitions about fair distribution of chances, such as the intuition underlying the FCV.

Broome might of course complain that I have myself added some primitive variable to decision theory, namely the (counterfactual) supposition operator. But those who are already motivated by the Fair Chance View, or more generally recognize that what could have been is often important for the evaluation of actual outcomes, hopefully agree that this extra complexity is more than offset by the benefit of being able to formally represent desirabilistic dependencies between facts and counterfactuals.

Using the multidimensional framework to represent the Fair Chance Judgement has the interesting implication that the extra value generated by the truth of the relevant counterfactual does not supervene on the non-modal facts. The table representing the ‘goodness partition’, for instance, has each actual world in both the ‘Good’ and the ‘Bad’ column. The situations ⟨w ₁, θ, w ₃⟩ and ⟨w ₁, θ, w ₄⟩ for instance share all non-modal facts and differ only in what would be true if ¬P were. But the latter is fair whereas the former is not, which suggests that fairness does not supervene on non-modal facts. More generally, if the multidimensional semantics, formulated as Bradley suggests, is the right semantics for counterfactuals, then counterfactuals don’t supervene on non-modal facts.Footnote ²⁵ But then if the moral value of a situation partly depends on what counterfactuals are true, as the FCV states, then the moral value of a situation does not supervene on its non-modal facts, contrary to what Broome (Reference Broome1991: 114–115) claims.

The non-modal consequentialist will without a doubt point out that the failure of fairness to supervene on non-modal facts is merely an artefact of our model. And she might argue as follows. This failure of counterfactuals to supervene on non-modal facts, according to the version of the multidimensional semantics I have been using here, is a reason for looking for a different semantics for counterfactuals, if we want to insist that fairness is partly determined by what counterfactuals are true. According to the best known semantics for counterfactuals – i.e. the Stalnaker–Lewis semantics (see e.g. Stalnaker Reference Stalnaker and Rescher1968; Lewis Reference Lewis1986) – counterfactuals do supervene on (and are implied by) factual propositions.Footnote ²⁶ Let f ₁ be the set of factual propositions that (on this semantics) imply the counterfactual $\lnot \text{A}\square \hspace{-4.0pt}\rightarrow \text{B}$ and f ₂ the set of factual propositions that imply the counterfactual $\lnot \text{A}\square \hspace{-4.0pt}\rightarrow \lnot \text{B}$. What really explains our judgement that $V(\text{A}\wedge \text{B} \wedge \lnot \text{A}\square \hspace{-4.0pt}\rightarrow \lnot \text{B})$ can be different from $V(\text{A}\wedge \text{B} \wedge \lnot \text{A}\square \hspace{-4.0pt}\rightarrow \text{B})$, the non-modal consequentialist might argue, is the fact that for us, $V(\text{A}\wedge \text{B} \wedge f_2)$ might be different from $V(\text{A}\wedge \text{B} \wedge f_1)$. To put the point in a less abstract manner, although the truth of a particular counterfactual is one difference between a situation where Ann receives the kidney fairly and one where Ann receives the kidney unfairly, what really makes the moral difference is the set of factual propositions that implies the relevant counterfactual.

I do not want to argue against the view that counterfactuals and other modal facts supervene on non-modal facts. However, there are various arrangements of non-modal facts that can make true the particular modal facts we are interested in. For instance, there are various ways of making it true that Ann and Bob have an equal chance of receiving the kidney. Many of these arrangements of non-modal facts are equally good, from a moral perspective. And what makes them morally good, is the fact that they all entail the relevant modal fact; in the case we are considering, the fact that Ann and Bob have an equal chance. In other words, the reason all these different arrangements of non-modal facts are morally good, if the supervenience thesis is true, is that they entail this particular modal fact.

Moreover, and more generally, we should make a distinction between facts that carry value and facts on which the carriers of value supervene.Footnote ²⁷ Even if it is true, as Humeans claim, that all facts supervene on the non-modal facts, that does not, by itself, mean that these non-modal facts are the carriers of value. Perhaps the following analogy will help. Every intrinsic (as opposed to relational) property of a painting is determined by how the atoms that make up the painting are arranged. Hence, the aesthetic qualities of the painting supervene on this arrangement of atoms. These aesthetic qualities, most people think, carry some value, that is not carried by the atoms that make up the painting. Similarly, whether or not Bob could have received the kidney may be determined by the non-modal facts of our world. But that does not mean that this particular counterfactual carries no value over and above these non-modal facts. A plausible consequentialist theory must take that into account.