3.1. A First Pass

I begin by presenting a schematic first pass of my model of weight. This schematic outline will be fleshed out more formally in Section 3.2. One reason for splitting the presentation of the model into two steps is that I want to emphasize that there are two separate proposals here: (i) a schematic proposal for a theory of weight and (ii) a particular way of filling out the schematic proposal. There may be other ways to fill out the schematic proposal.Footnote ⁶ A second reason is that the detailed approach presented in Section 3.2 depends on a specific way of modelling value, but I want to show that the basic idea does not inherently appeal to this conceptualization of value.Footnote ⁷ Now on to the schematic proposal.

Suppose that I am contemplating the decision between a default action d and an alternative a. Let R ₁, . . ., R _n be a set of propositions, each of which is a potential reason for selecting either the alternative or the default.

In this paper, I will attempt to understand reasons and their weight by considering the effect that coming to know that those reasons obtain have on my decision. This means that I move from a state of uncertainty to a state of knowledge.

Let us consider two types of uncertainty.

• Actual Uncertainty. I don’t know whether any of the potential reasons R ₁, R ₂, . . ., R _n obtain.

• Deliberative Uncertainty. In fact, I do know that R ₁, R ₂, . . ., R _n obtain, but I consider my decision from a hypothetical position in which I do not know whether these reasons obtain.

I will focus primarily on deliberative uncertainty. The reason is that in making decisions we are often already aware of various considerations that bear on our decision, and our task is to assess their bearing and weight. The approach explored here is to consider the effect of coming to know these considerations, that is, of moving from a state of hypothetical ignorance to a state of knowledge. The situation in which we know and want to assess the weight of what we know is more basic than the situation in which we are actually uncertain, and we would like to understand the more basic situation first. I will call the situation in which an agent pretends to be ignorant of the reasons that she possesses for the purpose of assessing the weights of her reasons the deliberative position.

Suppose that in the deliberative position, I view the default d and the alternative a as equally good: I have no more reason to choose one than the other. So I take an attitude of indifference.

If I were to learn only that proposition R _i held, this might break the indifference. Perhaps R _i will shift the decision in favour of a. Let us imagine that we could quantify how much better I should take a to be than d upon learning R _i. Let w _a(R _i) be a real number that represents how much better I should take it to be to select a rather than d overall were I to learn R _i (and only R _i).

Under the evidential interpretation of reasons, we might think of R _i as evidence that a is better than d. Under a justificatory interpretation of reasons, we might think of R _i as a fact that makes a better than d in a certain respect. Thinking about how learning that R _i obtains would affect what I should do helps me to get clear on the force of this justification. Alternatively, the force of R _i on the decision might consist in how I should rationally respond to learning R _i in the deliberative position.

If R _i is a reason for a over d, then R _i will shift the decision toward a, and this is represented by the condition that w _a(R _i) > 0. If R _i is a reason against a, then R _i will shift the decision away from a, and w _a(R _i) < 0. The absolute value |w _a(R _i)| represents the magnitude of the reason, that is, how much better or worse a should be taken to be than d; the sign of w _a(R _i) – that is whether w _a(R _i) is positive or negative – represents the valence of the reason, that is whether the reason makes a better or worse in comparison to d.

Imagine that we start with the function w _a, which assigns to each potential reason R _i a number w _a(R _i). Then we may define R _i to be a reason for a over d if w _a(R _i) > 0, and R _i to be reason against a over d if w _a(R _i) < 0. We may define w _a(R _i) to be the weight of the reason R _i for or against a.

Observe that the notion of weight that we have adopted here is essentially comparative. It reflects how R _i shifts the comparison between a and d, not how R _i changes the evaluation of a in an absolute sense. The reason for taking a comparative approach will be explained in Section 3.2.Footnote ⁸

Let $\skew2\bar{R}_i$ be the negation of R _i. Then $w_a\left(\skew2\bar{R}_i\right)$ represents how the comparison between a and d should change were one to learn that R _i does not hold. Intuitively, w _a(R _i) and $w_a\left(\skew2\bar{R}_i\right)$ are of opposite sign. So if w _a(R _i) > 0, it is intuitive to expect that $w_a\left(\skew2\bar{R}_i\right) < 0$: If R _i is a reason for choosing a over d, then $\skew2\bar{R}_i$ is a reason against choosing a over d. We will return to this.

Suppose that in addition to the default d, there are multiple other actions A = {a, b, c, . . .}. Then we can define w _a(R _i), w _b(R _i), w _c(R _i), . . . as the weight of R _i for/against a over the default, the weight of R _i for/against b over the default, and so on. We can also examine comparisons of pairs of actions not including the default. For example, w _{a, b}(R _i) may be the weight of R _i for/against a over b.

3.2. A More Detailed Approach to Modelling Reasons

The previous section presented a schematic model of the weight of reasons. The schematic model was presented in the service of the following goal.

• Goal: To develop a theory of weight on a par with probability theory that details the structure of weight, and specifically the relations among weights of interrelated propositions.

To achieve this goal, we must put more structure on the weights of reasons than we established in Section 3.1. But what additional structure are we to impose? Recall from Section 3.1 that w _a(R _i) measures the effect on the comparison between an alternative a and the default d of learning that R _i obtains. A natural approach to our task is to first ask what this learning consists in and how it affects the relative values of the alternative and default, and then to try to use the answer to specify the structure of weight. The simplest and most straightforward way of modelling learning is as conditioning on the reason. Here conditioning is understood as the conditioning of expected value as in probability theory. We start by assigning a value to the alternative and the default. These values are expected, or average, values because they depend on various conditions, and we average the value over the relevant conditions. We learn R _i. Then we condition the values of the alternative and default on R _i. This conditioning affects the comparison between the alternative and default, and the effect is the weight w _a(R _i).

As I mentioned, the motivation for this strategy is that it uses simple and straightforward tools that are well understood. An additional advantage is that this model immediately translates the notions of reason and weight into the language of decision theory, and thus promises to provide a link between deciding on the basis of reasons and decision theoretic models that involve the maximization of expected value. This approach also immediately imposes a rich structure on weight; this is the structure of conditional expectations. It was our goal to derive a rich structure. We can then check whether this additional structure is reasonable.

I now proceed to describe this approach more precisely.

Let V _d be the value of the default and V _a be the value of the alternative. These values depend on the state of the world. Formally, let Ω be the set of states of the world. V _a then maps each state in Ω to a real number. The same applies to V _d. V _a and V _d are random variables. (Random variables are just functions whose values depend on the state of the world.) There is a probability measure P over Ω that assigns probabilities to events (subsets) of Ω.Footnote ⁹ Define $\widetriangle {V}_a$ by $\widetriangle {V}_a = V_a-V_d$. $\widetriangle {V}_a$ is the difference between the value of a and the value of d. Since V _a and V _d depend on the state, in general, $\widetriangle {V}_a$ depends on the state as well. I refer to $\widetriangle {V}_a$ as the comparative value of a, since taking a difference, it measures the value of a relative to the value of d. $\widetriangle {V}_a$ measures the comparison between a and d.

On a substantive level, I wish to remain neutral about what constitutes value. It may amount to or include human welfare, under the various conceptions of welfare, and it may include other values, such as those of fairness, freedom, rights and merit. It may represent moral or prudential value. Similarly, the ‘outcome’ that would result from taking action a is interpreted broadly to potentially include information about rights, merit, intentions and so on.

The expected comparative value of a is $\mathbb {E}\left[\widetriangle {V}_a\right]$. Expectation is defined in the ordinary way. For example, if there are three states, ω₁, ω₂ and ω₃, then

\begin{align*} \mathbb {E}\left[\widetriangle {V}_a\right] = \widetriangle {V}_a\left(\omega _1\right)P\left(\omega _1\right)+\widetriangle {V}_a\left(\omega _2\right)P\left(\omega _2\right) + \widetriangle {V}_a\left(\omega _3\right) P\left(\omega _3\right). \end{align*}

Here P(ω_i) is the probability of state ω_i and $\widetriangle {V}_a\left(\omega _i\right)$ is the comparative value at state ω_i. More generally, $\mathbb {E}\left[\widetriangle {V}_a\right] = \int _\Omega \widetriangle {V}_a dP$. So the expected comparative value is just a weighted average of the comparative value of a at different states, where the weights in the weighted average are just the probabilities of the states.

There is a question of how we interpret the probability measure P. We can give P one of the standard interpretations, as either objective or subjective probability. However, I will focus on a special interpretation, which I call deliberative probability. Deliberative probability corresponds to the situation of deliberative uncertainty discussed in Section 3.1. Deliberative probability is a hypothetical probability constructed for the purpose of deliberation. I will elaborate on deliberative probability at greater length in Section 5.

A potential reason is just a proposition, which is modelled formally as just a set of states (a subset of Ω). If R is a potential reason, I assume that P(R) > 0. This assumption is for simplicity. One may want to posit additional restrictions on which propositions can count as potential reasons. For example, if one does not like the evidential account of reasons, one might want to insist that a reason is not merely evidence that favours taking an action, but stands in some stronger relation to the action such as justifying the action. So there may be some initial filtering operation that eliminates some propositions as not being potential reasons. In the interest of simplicity, I ignore this filtering operation here.

However, I do make one assumption about potential reasons: I assume that the truth of all potential reasons is determined before any action is chosen.Footnote ¹⁰ I also assume that if, in deliberation, the agent adopts an attitude of uncertainty toward a potential reason R, then she does not view her choice of action as influencing the probability of R. Thus, if there are propositions whose truth (or the probability that they obtain) is influenced by the choice of action, these propositions are not potential reasons. For example, the fact that I promised is a potential reason. A (non-material) conditional proposition such as ‘if Ann were to choose a, then Bob would be harmed’ can be a potential reason (not to choose a) since the probability assigned to this proposition is not influenced by whether Ann chooses a or not.Footnote ¹¹ In contrast, the proposition ‘Bob is harmed’ is not a potential reason if whether Bob is harmed depends on whether Ann chooses a.

As the example of ‘Bob is harmed’ shows, it is natural to assume that the probabilities of propositions describing the consequences of actions are influenced by the choice of action. I implicitly model the effects of actions on consequences through the different values that different actions have at various states. For example, it may be that V _a(ω) ≠ V _b(ω) because a and b have different consequences at ω. With a little additional machinery it would be straightforward to explicitly model the influence of actions on the probabilities of consequences, but to minimize notation and space, I choose not to do so.

Having discussed potential reasons, I now spell out what it is for a proposition R to be a reason for action a. In the deliberative position, I have an estimate of the comparative value of a, $\mathbb {E}\left[\widetriangle {V}_a\right]$. We may assume, as in Section 3.1, that in the deliberative position, the alternative and default are considered to be equally good. This would be expressed by the condition $\mathbb {E}\left[\widetriangle {V}_a\right]=0$: the expected difference between the two actions is zero. I call this assumption initial neutrality. It is natural to assume that in the absence of reasons, all actions are equally good. However, this assumption is completely inessential to the analysis. Indeed, sometimes we want consider some specific set of reasons given some fixed background information that favours one action over another so that the alternative and default do not start on even footing.

Now imagine that for some potential reason R, I were to learn R and nothing else. Then I would update my evaluation of action a. The value of action a would become $\mathbb {E}\left[V_a\left|R\right.\right].$ If there are finitely (or countably) many states, $\mathbb {E}\left[V_a\left|R\right.\right] = \sum _{\omega \in R} V_a\left(\omega \right) \frac{P\left(\omega \right)}{P\left(R\right)}$. More generally $\mathbb {E}\left[V_a\left| R\right.\right] = \frac{\int _R V_a dP}{P\left(R\right)}$. Similarly, I update my evaluation of the difference between the action and the default, attaining $\mathbb {E}\left[\widetriangle {V}_a\left|R\right.\right]$.

Definition 1. Let R be a potential reason, and let a be an action. R is a reason for a if

(2) \begin{align} \mathbb {E}\left[\widetriangle {V}_a\left|R\right.\right] > \mathbb {E}\left[\widetriangle {V}_a\right]. \end{align}

R is a reason against a if $\mathbb {E}\left[\widetriangle {V}_a\left|R\right.\right] < \mathbb {E}\left[\widetriangle {V}_a\right].$

It is useful to rewrite (2) more explicitly: R is a reason for a if $\mathbb {E}\left[V_a - V_d\left|R\right.\right] \ge \mathbb {E}\left[V_a-V_d\right].$ In other words, R is a reason for a if the comparison between a and the default d becomes more favourable to a once R is taken into account.

Definition 2. The weight of R for/against a is

(3) \begin{align} w_a\left(R\right)= \mathbb {E}\left[\widetriangle {V}_a\left|R\right.\right] - \mathbb {E}\left[\widetriangle {V}_a\right]. \end{align}

Thus the weight of a reason for/against an action a quantifies the change in the comparison between action a and the default that is brought about by taking R into account. Combining Definitions 1 and 2, we see that w _a(R) > 0 if R is a reason for a and w _a(R) < 0 if R is a reason against a. This is exactly what we assumed in the schematic model of Section 3.1. So the more detailed model presented here is indeed a way of filling in the schematic model of Section 3.1.

Again, it is useful to unwind the definition.

\begin{align*} \begin{split} w_a\left(R\right) &= \mathbb {E}\left[V_a - V_d\left|R\right.\right] - \mathbb {E}\left[V_a - V_d\right]\\ &= \underbrace{\left(\underbrace{\mathbb {E}\left[V_a\left|R\right.\right] - \mathbb {E}\left[V_d\left|R\right.\right]}_{\rm {alternative \ vs. \ default}}\right) - \left(\underbrace{\mathbb {E}\left[V_a\right] -\mathbb {E} \left[V_d\right]}_{\rm{alternative \ vs. \ default}}\right)}_{\scriptsize{\rm Taking} \ R \ {\rm Into \ Account \ vs. \ Not \ Taking} \ R \ {\rm Into \ Account}} \end{split} \end{align*}

where the second equality uses the linearity of expectations. This shows that the notion of weight is doubly comparative; it measures a difference of differences. It measures the way that taking R into account changes the comparison between action a and the default.

One might wonder why reasons and weight are defined in this comparative way. Why not use non-comparative value V _a rather than comparative value $\widetriangle {V}_a$? Under this non-comparative approach, R is a reason for a if $\mathbb {E}\left[V_a\left|R\right.\right] > \mathbb {E}\left[V_a\right]$.Footnote ¹² Imagine that I am deciding whether or not to go on holiday. Suppose that I win the lottery. Winning the lottery improves the value of all options. It makes going on holiday better, and it also makes staying home better. If we used the non-comparative definition, then winning the lottery would be a reason both for going on holiday and for not going on holiday, but that cannot be. What matters is how winning the lottery affects the comparison between going on holiday and not going on holiday. If winning the lottery shifts the comparison between going on holiday and not going towards going, then winning the lottery is a reason for going on holiday.Footnote ¹³

According to Definition 1, we decide whether R is a reason for a by comparing the (comparative) value of a when R is taken into account to the value of a when R is not taken into account. An alternative approach would be to compare the value of a when the proposition R is true to the value of a when R is false. That is, we may want to say that R is a reason for a if and only if

(4) \begin{align} \mathbb {E}\left[\widetriangle {V}_a\left|R\right.\right] \ge \mathbb {E}\left[\widetriangle {V}_a\left|\skew2\bar{R}\right.\right]. \end{align}

It turns out that these definitions are equivalent. That is, (4) holds if and only if (2) holds.Footnote ¹⁴ It turns out, however, that $\mathbb {E}\left[\widetriangle {V}_a\left|R\right.\right]-\mathbb {E}\left[\widetriangle {V}_a\left|\skew2\bar{R}\right.\right]$ would not be a good definition of weight.Footnote ¹⁵

Definitions 1–2 are definitions of one’s pro tanto reasons and their weight. A pro tanto reason is a (not necessarily decisive) consideration that counts in favuor of or against some action. In contrast a pro toto reason is a complete and decisive all things considered reason for or against an action.Footnote ¹⁶ Let R ₁, . . ., R _n be the reasons bearing on the decision. As reasons are here formally modelled as sets of states, we represent the conjunction of reasons as their intersection $\bigcap _{i=1}^n R_i$. If the default and alternative are the only options, and we assume initial neutrality, then if $w_a\left(\bigcap _{i=1}^n R_i\right) > 0$, then $\bigcap _{i=1}^n R_i$ can be regarded as a pro toto reason in favour of a.Footnote ¹⁷ This interpretation will be supported by the analysis of Section 4.

Imagine that there are many actions available, a, b, c, and so on. Then we can define V _b and V _c, $\widetriangle {V}_b$ and $\widetriangle {V}_c$ and w _b(R) and w _c(R) analogously to $V_a, \widetriangle {V}_a$ and w _a(R). The comparative values $\widetriangle {V}_a, \widetriangle {V}_b$ and $\widetriangle {V}_c$ and weights w _a(R), w _b(R) and w _c(R) are all defined relative to the same default. We can also focus on comparisons between non-default alternatives. Define V _{a, b} = V _a − V _b and $w_{a,b}\left(R\right) = \mathbb {E}\left[V_{a,b}\left|R\right.\right] - \mathbb {E}\left[V_{a,b}\right]$. We can then say that R is reason for choosing a over b if w _{a, b}(R) > 0 and R is reason against choosing a over b if w _{a, b}(R) < 0. Given these definitions, it follows thatFootnote ¹⁸

(5) \begin{align} w_{a,b}\left(R\right) = w_a\left(R\right) - w_b\left(R\right). \end{align}

While the values w _a(R) and w _b(R) depend on the choice of default, the difference w _a(R) − w _b(R) is independent of which action is chosen as default.Footnote ^19, Footnote ²⁰ Along the lines of (5), we have the general relation w _{a, c}(R) = w _{a, b}(R) + w _{b, c}(R).Footnote ²¹ This shows that the model has specific consequences for the relation of the weights of a reason for comparisons between different pairs of actions.Footnote ²²

5.1. How Can We Interpret Deliberative Probability?

The construction of weights in Section 3.2 depends on deliberative probability. But how do we assign deliberative probabilities? To get a grip on this operation, imagine assessing the value of a large tract of land. You conduct a test to check whether there is oil beneath the land. The test can come back positive or negative. Suppose that the test comes back positive and this greatly increases your estimate of the value of the land. If the test had come back negative, this would not have changed your estimate very much. This means that you must have initially assigned a low probability to there being oil under the land because if you had assigned oil a high probability, then you would already have factored the likelihood of oil into your assessment of the land and it would have been the discovery of no oil that would have moved your assessment. The possibility that there is oil has high significance in the sense that if true, it would move your assessment very much, which corresponds to its having low (initial) probability. The same principle applies to deliberative – as opposed to actual – probability. In general, propositions that are highly significant in the sense that they move us much more than their negations have low deliberative probability.

Let us apply this idea to some examples. First consider the case of promising to go to the movies. I promised Jim I would go to the movies. Promising is a strong reason to go, but not promising is not a strong reason not to go. So taking account of the fact that I promised moves my decision of whether to go very much toward going. Had I not promised, taking account of the fact that I had not promised would not move my decision very much. Not having promised to go may be viewed as a very weak reason not to go because the fact that I have not promised rules out one of the many reasons to go. Since promising moves me very much and not promising moves me very little, promising has high significance, and not promising has low significance. Correspondingly, promising has low deliberative probability and not promising has high deliberative probability.

The proposition that I promised is a descriptive proposition. The notion of a promise may be value-laden, but that I promised is a contingent event, and it is straightforward to assign probabilities to events such as promisings. We are now thinking about deliberative probabilities, which are hypothetical probabilities assigned in the deliberative position, but we can think of the assignments of deliberative probabilities as being analogous to the assignment of subjective and objective probabilities when we are thinking about contingent propositions.

However, what about more purely normative propositions such as the proposition that it is wrong to kill? Ordinarily, it is difficult to think about what it means to assign probabilities to normative propositions of this sort. Let us nevertheless try to use the same technique as above to determine the deliberative probability of its being wrong to kill.

That it is wrong to kill is a strong reason not to kill, whereas if it were not wrong to kill, then that it is not wrong to kill would not be a strong reason to kill. In general the fact that something is not wrong is a very weak reason to do it; it is only a weak reason to do it because it merely rules out one reason not to do it. Since the wrongness of killing would move me very much in my deliberations but, were it not wrong to kill, the lack of wrongness would not move me very much, that it is wrong to kill is highly significant, and correspondingly it has very low deliberative probability. This shows how in principle deliberative probabilities can be assigned to normative propositions.Footnote ²⁹

In the application of deliberative probabilities, it is important to make a distinction between background and active facts. A background fact is a fact that is taken as given in deliberation, and to which we do not assign weight. In contrast, an active fact is a fact explicitly involved in deliberation and to which we do assign weight.

To illustrate this distinction, let R be the proposition that it is wrong to kill. We may treat R as either a background or an active fact. If we treat R as a background fact, then we assign R a deliberative probability of 1; we do not suspend our belief in R, but treat the wrongness of killing as a background condition that is taken as given and not in question. In contrast, if we treat R as an active fact – we want to assess the weight of its being wrong to kill – then because R is highly significant, R receives a deliberative probability very close to zero. In assessing the weights of reasons it is important to distinguish background facts, which are taken as given and not assigned weight, and active facts that are not taken as given and to which we do assign weight.

Finally, why do we need deliberative probability at all? A central idea in this paper is that the weight of a reason for an action a over a default d is the effect that taking the reason into account has on the comparison between a and d. Taking a reason R into account amounts to accepting R as true, and evaluating the comparison between a and d conditional on the truth of R. But what does it mean to not take R into account? Deliberative probabilities are used to model what it is to not take a reason into account.Footnote ³⁰ There are several potential ways of modelling the attitude of not taking R into account. The agent might treat R as false or she might adopt an attitude of uncertainty toward R. There are other possibilities: If I do not take the fact that my action will hurt you into account, it might be that the possibility that my action will hurt you has not occurred to me. Let us refer to this attitude as unawareness of the possibility. This paper models that attitude that an agent takes towards R when she does not take R into account as uncertainty about R and deliberative probability is used to represent that uncertainty. Observe that the alternative attitude of treating R as false can be approximated by assigning R low deliberative probability. Indeed, as explained above, this is what happens when R is a highly significant proposition. While there are formal models of unawareness (Modica and Rustichini Reference Modica and Rustichini1994; Dekel et al. Reference Dekel, Lipman and Rustichini1998; Halpern Reference Halpern2001; Heifetz et al. Reference Heifetz, Meier and Schipper2006), uncertainty is much better understood, and there is a standard and well understood way of modelling it. In addition to modelling the attitude toward R, in my model, deliberative probabilities are directly important for modelling what it is to be a reason and the weight of a reason: The expectations in (2) and (3) would not be defined without deliberative probabilities. Deliberative probabilities are also integral to my characterization of independent reasons as can be seen from an examination of Theorem 1 below. These concepts all depend on the attitude that the agent takes toward reasons in the deliberative position.

5.2. A More Formal Discussion

I now discuss the ideas of the preceding section a little more formally. Let us first define the importance of reason R for/against a by:

(8) \begin{align} i_a\left(R\right) =\left| \mathbb {E}\left[\widetriangle {V}_a\left|R\right.\right] - \mathbb {E}\left[\widetriangle {V}_a\left|\skew2\bar{R}\right.\right]\right|. \end{align}

Importance resembles weight: The importance measures how much difference it makes whether or not R is true for the decision between a and d.

Now consider Figure 1.

Figure 1.

Weight and Importance

The line is a segment of the number line. The expected value $\mathbb {E}[\widetriangle {V}_a]$ is a real number on the line. $\mathbb {E}[\widetriangle {V}_a]$ is our initial estimate of the benefit of choosing alternative a rather than the default d. If the agent were to learn R, this would move the expectation in one direction to $\mathbb {E}[\widetriangle {V}_a\left|R\right.]$. The distance traversed in this movement is the absolute value of the weight of R, |w _a(R)|. If the agent were to learn that R does not hold, or in other words, if the agent were to learn the negation $\skew2\bar{R}$, this would move her expectation in the opposite direction. Whichever proposition R or $\skew2\bar{R}$ was initially viewed as more unlikely would move the agent further from her initial starting point; an unlikely discovery changes one’s view more than a likely discovery. Indeed, it follows from the law of iterated expectations that we can relate the probability of R to the weights of R and $\skew2\bar{R}$ through the equationFootnote ³¹

(9) \begin{align} P\left(R\right) = \frac{1}{1+\left|\frac{w_a\left(R\right)}{w_a\left(\skew2\bar{R}\right)}\right|}. \end{align}

Let us call the ratio $S\left(R\right)=\left|\frac{w_a\left(R\right)}{w_a\left(\skew2\bar{R}\right)}\right|$ the deliberative significance of R. It is not difficult to see that $\left|\frac{w_a\left(R\right)}{w_a\left(\skew2\bar{R}\right)}\right|= \frac{1-P\left(R\right)}{P\left(R\right)}$. So the deliberative significance of R can be interpreted as the odds against R. The greater the odds against R, the more significant it would be to learn that R is true. This accords with the previous informal discussion of significance. Under this interpretation, (9) is a familiar formula for probability in terms of odds.

Observe that S(R) is not indexed by a. More precisely, for all a and b, if w _a(R) ≠ 0 and w _b(R) ≠ 0, then $S\left(R\right) = \left|\frac{w_a\left(R\right)}{w_a\left(\skew2\bar{R}\right)}\right| =\left|\frac{w_b\left(R\right)}{w_b\left(\skew2\bar{R}\right)}\right|$. If R is irrelevant to the choice between a and the default – that is, w _a(R) = 0 – then equation (9) does not apply.Footnote ³²

The significance of R measures how much R moves comparative value relative to its negation. A significant proposition moves comparative value a great deal relative to its negation. Observe that the probability P(R) and the significance S(R) are inversely related: The larger is the significance of R, the smaller is the probability of R.

One can imagine that rather than having intuitions about the deliberative probability of various potential reasons, one has intuitions about the deliberative significance of potential reasons. The examples that I have given above concerning the significance of promising and the significance of its being wrong to kill support this idea. It is then intuitions about deliberative significance that determine deliberative probabilities via (9).

As I said above, the significance of a proposition does not depend on the decision at hand, provided that the proposition is relevant at all to the decision at hand (i.e. w _a(R) ≠ 0). So if R is relevant to the choice between a and the default d, to the choice between b and d, and to the choice between a and b, it is equally significant to all. However, it is not equally important to all (where importance is defined by (8)). Significance concerns a relation between R and its negation as affecting decisions generally, whereas importance relates to the particular decision at hand rather than the relation between R and $\skew2\bar{R}$. Indeed for all actions a, $i_a\left(R\right)= i_a\left(\skew2\bar{R}\right)$. The importance measures how much difference it makes one way or another whether the R is true to the decision at hand. Weight, significance and importance, are related through the following equation:

(10) \begin{align} \left|w_a\left(R\right)\right| = \frac{i_a\left(R\right)}{1+\frac{1}{S\left(R\right)}}. \end{align}

The more significant is R in general and the more important is R to the decision at hand, the greater (the absolute value of) the weight of R.

If we are to understand deliberative probability, we must also understand conditional deliberative probabilities, such as P(R ₂|R ₁): Once reason R ₁ is taken into account, a deliberative probability must be assigned to R ₂. How are we to interpret conditional deliberative probabilities? First observe that conditional probabilities are inherent in the underlying deliberative probability measure because $P\left(R_2 \left|R_1\right.\right) = \frac{P\left(R_1 \cap R_2\right)}{P\left(R_1\right)}$. Since deliberative probability is inherent in deliberative significance via equation (9), and conditional probabilities are inherent in probabilities, conditional probabilities are also inherent in deliberative significance – indeed $P\left(R_2 \left|R_1\right.\right) = \frac{1+S\left(R_1\right)}{1+S\left(R_1 \cap R_2\right)}$.Footnote ³³

To summarize, one makes judgements about the significance of various propositions and these judgements determine the deliberative probabilities in the deliberative position. These judgements are weaker than judgements of weight. As is made evident by (10), to actually derive judgements of weight, in addition to judgements of general significance, one must also make judgements of the importance of potential reasons for specific decisions.

6.1. Independent Reasons

Let $\mathscr {R}= \left\lbrace R_1, \ldots , R_n\right\rbrace$ be a collection of reasons. Below, I will assume that your reasons are mutually consistent. More precisely, say that $\mathscr {R}$ is consistent if $P\left(\bigcap _{i=1}^n R_i\right) >0$. Throughout this section, whenever discussing a collection of reasons $\mathscr {R}$, I will assume that $\mathscr {R}$ is consistent. A reason R ₀ is consistent with $\mathscr {R}$ if $\mathscr {R} \cup \lbrace R_0\rbrace$ is consistent.

We can extend the notion of the weight of a reason to collections of reasons. In particular, define the weight of a consistent collection of reasons $\mathscr {R}=\lbrace R_1,\ldots ,R_n\rbrace$ for/against a as:

(11) \begin{align} \hat{w}_a(\mathscr {R})= w_a\left( \bigcap _{i=1}^n R_i \right). \end{align}

In other words, the weight of a collection of reasons is just the weight of their conjunction (or, more formally, their intersection). The hat $\hat{}$ in $\hat{w}_a\left(\mathscr {R}\right)$ differentiates weight applied to a collection of reasons from weight applied to a single reason w _a(R). Using the convention $\bigcap \emptyset = \Omega$, observe that $\hat{w}_a\left(\emptyset \right)=w_a\left(\Omega \right)=0$.Footnote ³⁴In other words, the weight of a collection of reasons containing nothing is zero; there are no reasons in the collection to move our assessment of the decision.

We can also define conditional weights. That is, we can define the weight of a reason conditional on having accepted some other reasons. This is the weight of the reason not from the initial deliberative position in which knowledge of all relevant reasons has been suspended, but from an intermediate deliberative position in which some of these reasons have been accepted again. The weight of reason R ₀ for/against a conditional on $\mathscr {R}$ (where $\mathscr {R}= \left\lbrace R_1,\ldots ,R_n\right\rbrace$) is:

(12) \begin{align} w_a\left(R_0\left|\mathscr {R}\right.\right) = \mathbb {E}\left[\widetriangle {V}_a\left| R_0 \cap \left(\bigcap _{i=1}^n R_i\right)\right.\right]- \mathbb {E}\left[\widetriangle {V}_a\left| \bigcap _{i=1}^n R_i \right.\right]. \end{align}

That is to say, $w_a\left(R_0\left|\mathscr {R}\right.\right)$ is the difference in the expected comparative value of a that conditioning on R ₀ makes when one has already conditioned on the reasons in $\mathscr {R}$. It follows from our definitions that

(13) \begin{align} w_a\left(R_0\left|\mathscr {R}\right.\right) = \hat{w}_a\left(\lbrace R_0\rbrace \cup \mathscr {R}\right) - \hat{w}_a\left(\mathscr {R}\right). \end{align}

In Section 7, I discuss the possibility of taking weight as fundamental, rather than as derived from values and deliberative probabilities. If this is done, then (13) can be taken as the definition of conditional weights.Footnote ³⁵ This shows that in the same way that conditional probabilities can be derived from unconditional probabilities, conditional weights can be derived from unconditional weights.

As a matter of notation, for any potential reasons R ₀ and R ₁, I write w _a(R ₀|R ₁) instead of w _a(R ₀|{R ₁}), so that, in particular, $w_a\left(R_0\left|\mathscr {R}\right.\right)=w_a\left(R_0 \left|\bigcap _{i=1}^n R_i\right.\right).$ This notational convention only has potential for ambiguity if the reason R ₁ that one conditions on is R ₁ = ∅; however since I exclude inconsistent reasons w _a(R|∅) means that weight of R is conditional on an empty set of reasons, not conditional on the inconsistent reason. So w _a(R|∅) = w _a(R|Ω) = w _a(R): If the set of reasons that one conditions on is empty, one remains with one’s initial assessment. Finally $w_{a,b}\left(R_0\left|\mathscr {R}\right.\right)$ is defined analogously to $w_a\left(R_0\left|\mathscr {R}\right.\right)$, with w _{a, b} playing the role of w _a.

The importance of the notion of a conditional weight stems from the fact that in general the weight of a reason changes when we take other reasons into account. That is to say, in general w _a(R ₁) ≠ w _a(R ₁|R ₂). Indeed, it is possible that w _a(R ₁) > 0, so that R ₁ is a reason for action a, but w _a(R ₁|R ₂) < 0, which means that conditional on R ₂, R ₁ is a reason against a. So reasons can change their valence depending on the context, which is here modelled by the set of other reasons that have already been taken into account. Thus, when asking whether R ₁ is a reason for a, it is important to specify the context. Independence of reasons occurs precisely when the weight of a reason is unaffected by other reasons. This is captured formally by the following definition.

Definition 3. $\mathscr {R}$ is a collection of independent reasons for/against a if and only if for all reasons R ₀ in $\mathscr {R}$ and for all subsets $\mathscr {S}$ of $\mathscr {R}$ with R ₀ not in $\mathscr {S}$,

\begin{align*} w_a\left(R_0\left|\mathscr {S}\right.\right) = w_a(R_0). \end{align*}

To abbreviate, we also say that $\mathscr {R}$ is a-independent.

In other words, $\mathscr {R}$ is a collection of independent reasons if conditioning on any subset of reasons $\mathscr {S}$ does not alter the weight of any reason in $\mathscr {R}$ outside the set $\mathscr {S}$. The following proposition is immediate from definitions.

In other words, for a collection of independent reasons the weight of any subcollection is equal to the sum of the weights of the individual reasons. Thus when weights are independent, they can be added.

Define a collection of reasons $\mathscr {R} =\left\lbrace R_1,\ldots ,R_n\right\rbrace$ as (a, b)-independent if $w_{a,b}\left(R_0\left|\mathscr {S}\right.\right)=w_{a,b}\left(R_0\right)$ for all $R_0 \in \mathscr {R}$ and all non-empty $\mathscr {S} \subseteq \mathscr {R}$ with $R_0 \not\in \mathscr {S}$. This is identical to the definition of a-independence, except that w _{a, b} has been substituted for w _a. Observe that the conjunction of a-independence and b-independence is a sufficient but not necessary condition for (a, b)-independence. Let us now interpret $\mathscr {R} =\left\lbrace R_1,\ldots , R_n\right\rbrace$ as all of the agent’s actual reasons. Recall the notion of preference on the basis of one’s reasons ≿ defined by (7) in Section 4. Recall also that initial neutrality is the assumption that for all actions a and b, $\mathbb {E}\left[V_a\right]=\mathbb {E}\left[V_b\right]$. Combining the analogue of Proposition 3 for (a, b)-independence with Proposition 1 and (7) yields the following result.

In words, a is preferred to b if and only if the expected value of a is greater than the expected value of b conditional on one’s reasons if and only if the sum of weights of reasons in favour of a over b is greater than the sum of weights of reasons in favour of b over a; to understand the last condition, observe that if R ₁, . . .R _k support a over b (that is, if R _i is among these reasons, w _{a, b}(R _i) > 0) and R _{k + 1}, . . ., R _n support b over a (so that if R _j is among these reasons, w _{a, b}(R _j) < 0, so that w _{b, a}(R _j) = −w _{a, b}(R _j) > 0), then (3) is equivalent to ∑^k_{i = 1}w _{a, b}(R_i) ⩾ ∑ⁿ_{j = k + 1}w _{b, a}(R_j).

In Section 6.4, I will characterize independent reasons in terms of the structure of underlying values and deliberative probabilities. This will give greater insight into when reasons are independent. However, first I will compare independent to non-independent reasons with regard to the aggregation of reasons in practical decision-making.

6.2. Non-independent Reasons and Conditional Weights

If reasons are not independent, we cannot add the weights of reasons. However, we can still add conditional weights. Suppose for example that there are three reasons R ₁, R ₂ and R ₃. If the reasons are a-independent, then, as we have seen in the previous section, the weight of the conjunction is the sum of the unconditional weights of the individual reasons:

(14) \begin{align} w_a\left(R_1 \cap R_2\cap R_3\right) = w_a\left(R_1\right)+w_a\left(R_2\right)+ w_a\left(R_3\right). \end{align}

Equation (14) is not valid for non-independent reasons. However something similar is valid. For non-independent reasons, the weight of the conjunction is the sum of the conditional weights of the individual reasons:

\begin{align*} w_a\left(R_1 \cap R_2\cap R_3\right) = w_a\left(R_1\right)+ w_a\left(R_2\left|R_1\right.\right) + w_a\left(R_3\left|R_1 \cap R_2\right.\right). \end{align*}

In general, to form the weight $w_a\left(\bigcap _{i=1}^nR_i\right)$ of the conjunction of all of our reasons, we can perform the following procedure.

1. 1. Start with the unconditional weight w _a(R ₁),

2. 2. then add the weight w _a(R ₂|R ₁) of R ₂ conditional on R ₁,

3. 3. then add the weight of w _a(R ₃|R ₁∩R ₂) of R ₃ conditional on R ₁ and R ₂,

4. ⋅⋅⋅ ,

5. n. and finally add the weight $w_a\left(R_n\left|\bigcap _{i=1}^{n-1} R_i\right.\right)$ of R _n conditional on all preceding reasons to arrive at the aggregate weight of all reasons for or against a.

We terminate with the sum:

$$\begin{equation*} w_a\left(R_1\right) + w_a\left(R_2\left|R_1\right.\right) + w_a\left(R_3\left|R_2 \cap R_2\right.\right) + \cdots + w_a\left(R_n\left|\bigcap _{i=1}^{n-1} R_i\right.\right). \end{equation*}$$

When adding non-independent reasons, we must condition on increasingly more information as we add more reasons. The reasons accumulate as we go. We must assess the weight of each new reason in light of the weights of the reasons that we have already taken into account. The discussion is summarized in Proposition 5, which is the analogue of Proposition 3 for independent reasons.

\begin{align*} w_a\left(\bigcap _{i=1}^n R_i\right)=\sum _{i=1}^n w_a\left(R_i\left|\bigcap _{j=1}^{i-1} R_j\right.\right). \end{align*}

The analogue of Proposition 4 for non-independent reasons is the following.

That is, to select an action when reasons are not independent, we add conditional weights.

A final note is in order. Section 4 discussed the possibility that not all reasons are relevant to all decisions. Let $\mathscr {R}$ be the set of all your reasons that are relevant to any decision between any pair of actions. Say that a subset $\mathscr {S}$ of $\mathscr {R}$ is (a, b)-complete if for all $\{R_{i_1},\ldots,R_{i_h}\} \subseteq \mathscr{R}$, $w_{a,b}(\bigcap_{j=1}^h R_{i_j}|\mathscr{S})=0$. This means that once $\mathscr {S}$ is taken into account, no further reasons have any weight. Say that $\mathscr {S}$ is minimal (a, b)-complete if no proper subset of $\mathscr {S}$ is (a, b)-complete. Once a minimal (a, b)-complete set reasons has been taken into account, no other reasons remain relevant to the decision between a and b. Suppose for example that {R ₁, . . ., R _k} with k < n is a minimal (a, b)-complete set. Then $\sum _{i=1}^n w_{a,b}(R_i\left|\bigcap _{j=1}^{i-1} R_j\right.)=\sum _{i=1}^k w_{a,b}(R_i\left|\bigcap _{j=1}^{i-1} R_j\right.)$, so that (under initial neutrality) we should take action a over b if and only if $\sum _{i=1}^k w_{a,b}(R_i\left|\bigcap _{j=1}^{i-1} R_j\right.) \ge 0$; in other words, we need not consider reasons relevant only to other decisions.

6.3. Examples from the Introduction

I now revisit some of the examples from the Introduction to show how they would be dealt with in the framework that I have developed. Recall the example in which I am trying to decide whether to take a treatment for a disease. Two relevant reasons to take the treatment may be:

• • R ₁ : The treatment would prolong my life by at least 1 year.

• • R ₂ : The treatment would prolong my life by at least 2 years.

Observe that R ₂ = R ₁∩R ₂. Clearly these reasons overlap. Handling overlapping reasons such as this was one of the main motivations for developing the framework of this paper.

Let a be the action corresponding to taking the treatment, and let the default d be the action of not taking the treatment. Clearly, the weight of R ₂ in favour of a is greater than the weight of R ₁ in favour of a. To be concrete, suppose that w _a(R ₁) = 3 and w _a(R ₂) = 5. Then observe that

\begin{align*} w_a\left(R_2\left|R_1\right.\right) & = w_a\left(R_1 \cap R_2\right) - w_a\left(R_1\right) = 5-3 =2,\\ w_a\left(R_1\left|R_2\right.\right)&= w_a\left(R_1 \cap R_2\right) - w_a\left(R_2\right) = 5 - 5 =0. \end{align*}

As a special case of Proposition 5, it now follows that

\begin{align*} w_a(R_1 \cap R_2) &= w_a(R_1) + w_a(R_2|R_1)= 2 + 3 = 5 \\ &= w_a(R_2)+ w_a(R_1|R_2) = 5+0 =5. \end{align*}

To arrive at the total weight, I add appropriate conditional weights. The order in which the reasons are considered does not matter. If first I consider that the treatment will prolong my life by at least one year, I add the weight w _a(R ₁) = 3, but having taken this into account, considering that the treatment will actually prolong my life by at least two years, I add a weight of w _a(R ₂|R ₁) = 2, arriving at a total weight of 5. This second reason was partially – but not fully – taken into account when I considered the first reason. On the other hand, if first I consider that the treatment will prolong my life by at least two years, then this has a weight of w _a(R ₂) = 5, and then considering that the treatment will prolong my life by at least one year does not amount to adding a new significant reason – that reason has already been taken into account: We have w _a(R ₁|R ₂) = 0. In either order of considering the reasons, the total weight of their conjunction is 5. In sum, the framework naturally handles the overlap between reasons, and is not subject to any problem of double-counting when reasons are considered in conjunction.

Let us now consider a more conceptually challenging example, which was also raised in the introduction. Suppose that action a is the action of going to the movies, and the default d is the action of staying at home. Let us assume for simplicity that V _d ≡ 0. That is the default of staying at home always (i.e. in every state) yields a value of zero.

Let us suppose that before taking reasons into account, it would be better to stay at home. This violates the initial neutrality assumption but (as explained in footnotes 25–26) the initial neutrality assumption is inessential and nothing of import changes if it is eliminated. Our assumption about the value of the default – that in the absence of reasons, it would be better to stay at home – amounts to the assumption that $\mathbb {E}\left[V_a\right] < 0$. Now consider the following reasons (from the Introduction):

1. R ₁ : I promised Jim I would go.

2. R ₂ : It is important to keep one’s promises.

The reason R ₂ has high deliberative significance conditional on R ₁; similarly, R ₁ has high deliberative significance conditional on R ₂. Using the inverse relation between deliberative significance and deliberative probability, this is equivalent to assuming that P(R ₂|R ₁) and P(R ₁|R ₂) are small. Assume in fact that the deliberative significance is high enough that these two probabilities are approximately zero. Suppose that V _a is positive when and only when R ₁ and R ₂ are both true simultaneously; that is, it would be better to go to the movies than to stay home if it is true both that I promised that I would go to the movies and that it is important to keep one’s promises. Suppose moreover that V _a has a roughly constant negative value when R ₁∩R ₂ is false; that is, given that one of the propositions R ₁ and R ₂ is false, it does not matter very much whether the other is. For example, it may be that Jim would be equally happy if I were to go to the movies (and equally unhappy if I were not to go) regardless of whether I promised to go, so that promising by itself does alter the comparison between going and not going from the standpoint of Jim’s happiness.Footnote ³⁷ Then if P(R ₁|R ₂) and P(R ₂|R ₁) are small enough, it follows that $\mathbb {E}\left[V_a\left|R_1\right.\right]\approx \mathbb {E}\left[V_a\left|R_2\right.\right] \approx \mathbb {E}\left[V_a\right] < 0$. On the other hand our assumptions also imply that $\mathbb {E}\left[V_a\left| R_1 \cap R_2\right.\right] > 0$. It then follows that w _a(R ₁) ≈ w _a(R ₂) ≈ 0, but w _a(R ₁|R ₂) ≈ w _a(R ₂|R ₁) ≈ w _a(R ₁∩R ₂) > 0. That is, if it were not important to keep one’s promises, then the fact that I made a promise to Jim would not be a reason to go, and if I had not made a promise to Jim, then the fact that it is important to keep one’s promises would not be a reason to go. Conditional on either reason, the other reason carries the full weight of their conjunction. The two reasons are, in a sense, ‘perfect complements’. In this case, it is not possible to divide the weight of the conjunction among the two reasons in any meaningful way; it is only in conjunction that the two reasons have weight. This seems true in this particular example. There are many examples in which the weight can only be assigned jointly to a set of reasons; if any subset is put into the background, the remaining active reasons carry all of the weight (see Section 5.1 for the distinction between background and active reasons). Thus in different contexts, in which different background conditions are taken to be assumed, we may assign the weights to different reasons. There are other less extreme cases in which some subsets of reasons have weight on their own, but the whole is still more than the sum of its parts. That is, w _a(R ₁) > 0, w _a(R ₂) > 0 and w _a(R ₁∩R ₂) > w _a(R ₁) + w _a(R ₂). The model can easily represent such possibilities.

6.4. Characterization of Independent Reasons in terms of Values and Probabilities

We can characterize independent reasons in terms of properties of the corresponding value functions and deliberative probabilities. To do so, we actually need a stronger definition of independence. For any reason R, let R ¹ = R; that is, R ¹ is just the reason R itself, and let $R^0=\skew2\bar{R}$; so R ₀ is the negation (formally, the complement) of R. We can refer to R ¹ as the ‘affirmation’ of R and R ⁰ as the ‘disaffirmation’ of R. Let $\mathscr {R}=\left\lbrace R_1, \ldots ,R_n\right\rbrace$, and define a truth assignment for $\mathscr {R}$ to be a function t : {1, . . ., n} → {0, 1}. Then define $\mathscr {R}^t= \left\lbrace R^{t\left(1\right)}_1,\ldots , R^{t\left(i\right)}_i, \ldots , R^{t\left(n\right)}_n\right\rbrace$ to be such that for each potential reason R _i, R ^t(i) affirms R _i if t(i) = 1 and disaffirms R _i if t(i) = 0. For example if $\mathscr {R}=\left\lbrace R_1,\ldots , R_5\right\rbrace$ and t(i) = 1 if i is odd and t(i) = 0 if i is even, then $\mathscr {R}^t=\left\lbrace R_1,\skew2\bar{R}_2,R_3,\skew2\bar{R}_4,R_5\right\rbrace$. Say that $\mathscr {R}$ is logically independent if for all truth assignments t, $\mathscr {R}^t$ is consistent.Footnote ³⁸ We now have the following strengthened definition of independence.

Thus, a collection of reasons is strongly a-independent if the collection is a-independent and negating any subset to the reasons, the collection remains a-independent.Footnote ³⁹ I need one more definition: $\mathscr {R}$contains an a-empty reason if there exists R in $\mathscr {R}$ such that for all subsets $\mathscr {S}$ of $\mathscr {R}$ with R not in $\mathscr {S}$, $w_a\left(R\left|\mathscr {S}\right.\right)=0$. An a-empty reason is a reason that has no weight, no matter which other reasons we condition on. I now state a theorem that characterizes strong a-independence in terms of properties of the comparative value of a, $\widetriangle {V}_a$, and the deliberative probability P.

Let us interpret the result. What is needed in order for reasons to be strongly independent? First, the reasons R ₁, . . ., R _n must be probabilistically independent with respect to deliberative probability. That is, in deliberation, learning that any one of the reasons obtains provides no information about any of the other reasons. Second, the (comparative) value of a must be decomposable into a sum of factors U _i (condition 2(b)i), and each factor U _i is influenced (in expectation) by only one reason R _i (condition 2(b)ii).

Let me make this concrete by means of an example. This is an important example of independent reasons. Suppose that I am a utilitarian and the various actions a correspond to different policies. U^a_i(ω) is the utility of agent i in state ω if action a is taken. There is a default action d such that U^d_i ≡ 0 for all i; in other words, the default action produces a zero utility at any state of the world. Perhaps the action a is some policy contemplated by the government and the default d is doing nothing or sticking with the status quo. For a utilitarian, $\widetriangle {V}_a = V_a = \sum _i U^a_i$. That is, since the default delivers zero utility, the benefit (or cost) that results from choosing a is the sum of utilities this choice would lead to for the people affected. Let R^a_i be the proposition that action a is good (or bad) for agent i. (This proposition can be made more precise in various ways.) Suppose the fact that the policy is good for one agent is completely uninformative about its effects on any other agent. This is understood from the standpoint of the deliberative position: When I deliberate about whether the action is taken, I do not take the proposition that it is good for any given agent to be informative with regard to the question of whether it is good for any other agent. Then according to Theorem 1, the reasons R^a_i are independent reasons for or against taking the action. Thus utilitarianism is a paradigmatic case of independent reasons, where the reasons are the net utility effects that the action under consideration has on various people.

Finally, I mention one variant of Theorem 1. Let $\mathscr {F}_{\mathscr {R}}$ be the field generated by $\mathscr {R}$.Footnote ⁴⁰ For any event R, let 1_R be the indicator function that takes value 1 at states ω in R and takes value 0 in states ω not in R. Then in Theorem 1, condition 2b can be replaced byFootnote ⁴¹

(15) \begin{align} \mathbb {E}\left[\widetriangle {V}_a\left|\mathscr {F}_{\mathscr {R}}\right.\right] = \sum _{i=1}^n w_a\left(R_i\right)1_{R_i}+ \sum _{i=1}^n w_a\left(\bar{R_i}\right) 1_{\bar{R_i}} + \mathbb {E}\left[\widetriangle {V}_a\right]. \end{align}

Note that this is still a characterization in terms of V and P, because w _a(R _i) and $w_a\left(\skew2\bar{R}_i\right)$ are defined in terms of V and P (see Definition 2). This has the advantage of eliminating the existential quantifier in condition 2(b)ii by effectively identifying U _i: that is, $U_i = w_a\left(R_i\right) 1_{R_i} + w_a\left(\skew2\bar{R}_i\right) 1_{\skew2\bar{R}_i}$ for i = 1, . . ., n−1 and $U_n = \widetriangle {V}_a - \sum _{i=1}^{n-1} U_i$.Footnote ⁴² The condition (15) says that we start with $\mathbb {E}\left[\widetriangle {V}_a\right]$ and for each reason R _i and each state ω, we add w _a(R _i) or $w_a\left(\skew2\bar{R}_i\right)$, depending on whether R _i or $\skew2\bar{R}_i$ obtains. Observe that the probabilistic independence of R ₁, . . ., R _n (condition 2a) is still required.