Vagueness and Action

Consider the following scenario. Emily decides to have a drink of water. The following things happen: Emily’s elbow forms an angle of 116.3 degrees, her shoulder muscle contracts, and her hand moves a total of 19.7cm towards the glass of water waiting on the table in front of her.

style='font-size:9.0pt;line-height:122%'>The above is a fairly precise description of the sequence of events that transpired after Emily decided to have a drink of water—it could, in principle, have been described in completely precise terms. It might be tempting, then, to think that actions are completely precise things.

An action certainly consists in a completely precise sequence of events—denying this sounds as though it would involve a commitment to vague objects or vague events of some kind. So there is certainly some sense in which actions are guaranteed to be precise. However, there is a way of formulating the question of whether actions are precise in the theoretical framework developed above where the question requires a little thought. The most straightforward way to reflect the idea that actions are precise is to maintain that the agent’s action set—the set of propositions they are in a position to make true—consists only of precise propositions. Perhaps, for example, prior to deciding to have a drink of water, the precise proposition describing the sequence of events outlined in the first paragraph is one of the things Emily was in a position to make true.

The hypothesis that the agent’s action set consists only of precise propositions entails a natural precisification of Practical Irrelevance.

Irrelevance to Action: The proposition a rational agent makes true is deter­mined solely by their preferences (or V -values) among precise propositions.

Maximize states that the action a rational agent makes true is determined solely by their preferences/V -values among action propositions; if the action propositions are always precise, then theabove principlefollows.Insomesense this principlestatesthat some of our attitudes toward the vague are epiphenomenal—you can have preferences among vague propositions, but they won't affect how you behave.

It should be noted that this principle is consistent with other attitudes toward the vague being relevant. For example, the value you assign to a precise proposition might be determined by attitudes, such as credences and desires, you have towards the vague. By Averaging, the value of a precise proposition can be written as a weighted sum of values of vague propositions.^[134] It could turn out that what you really care intrinsically about is vague, and that the values you assign to the precise propositions only represent what you care instrumentally about—things that raise the expected value by making it more likely that the vague matters you care about are true. Even though you only need to know how the agent assigns expected values to the precise to know what she'll do, you still need to know her credences and desires about vague matters to know what those expected values are.

Be that as it may, I want to argue that even this modest form of Practical Irrelevance is false. Vagueness seeps even into our actions, and consequently the action a person chooses cannot be determined purely by looking at their preferences among precise matters.

Vague Actions: The propositions we are in a position to make true (i.e. that are among our action propositions) at any given time are almost always vague propositions.

As we shall see shortly, we must distinguish this thesis sharply from the thesis that the action propositions are sometimes borderline (neither determinately true nor determinately false). Indeed, the action propositions for a determinately rational agent with determinate preferences will usually be determinately true or false, their vagueness notwithstanding.

For Vague Actions to be plausible at all, we must distinguish sharply between what happened after Emily decided to have a drink of water, and what she did. Some of the things that happened after Emily decided to have a drink are relevant to the goodness of the outcome of her decision—if she had bent her elbow at a slightly different angle, for example, she would have spilt her drink. But lots of other things happened too: the crickets in her garden continued chirping, someone somewhere died in a car crash, and so on. This even applies to some of Emily's bodily movements: her heart continued beating, she continued to digest her last meal, and so forth—these are things that merely happened to Emily, not things she did. The difference is that most of these things are beyond her control, and are neither among nor are entailed by the things she was in a position to make true. As a rule of thumb, the things Emily made happen closely reflect the things we hold her responsible for—the kinds of things for which she can be subject to blame or praise.

Another issue in the vicinity is that the notion of making true that is relevant to decision theory is an intentional notion: for example, to have made it true that I ate the apple I must have intended to eat the apple.

So the first distinction we must be clear on is the distinction between the things that happened which Emily (intentionally) made happen, and those which she didn't (intentionally) make happen. Clearly only the former kind of things are the things Emily gets to choose from before she makes her decision, and are the kinds of things taken into account when we evaluate her actions after the decision.

The second thing we need to get clear on is how to think about the things she did do, and their role in principles like Maximize. The issue is that even if we exclude from our attention the things that happened that were beyond Emily's control, there are still a multitude of things that Emily did intentionally make happen in the above scenario. Some of the things Emily did in the above example include: picking up the glass, picking something up, moving her arm, and so on. The proposition that Emily picked up the glass and the proposition that Emily picked something up, for example, have different expected values: if the table had on it both a scalding hot plate and a glass of water, for example, then the value of picking something up is worse than the value of picking the glass up, provided she initially knows she's going to do one of the two things and assigns non-zero credence to picking up the plate.

One awkward feature of the Jeffrey-style decision theory we've been using thus far is that the things that get assigned values are propositions, whereas phrases that denote actions, like ‘picked up a glass', are not propositional. That said, actions clearly have some kind of logical structure—for example, picking up a glass entails picking up something but not vice versa, running to catch the bus entails running, and so on. It is therefore not entirely unnatural to regiment these actions as propositions: we can talk of the proposition that Emily is picking up a glass, that she is picking up something, that she is running to catch the bus, and so on. The presence of this logical structure gives us a straightforward answer to the question we just raised. In the above example, Emily made a number of things happen: it is surely the most specific thing that she did that we should evaluate her actions by, and the thing the value of which she should be attempting to maximize. It doesn't matter if the expected value of picking something up is low (because there's a scalding hot plate on the table) if Emily only picked up something in virtue of doing the more specific act of picking up the glass. In this framework, this can be represented by the conjunction of all of propositions she made true.

To summarize: in any given scenario, we can divide the propositions into those that Emily made true and those which she didn't.^[135] To calculate the expected value of what would be made true in that scenario, we simply conjoin the propositions that Emily made true and calculate the conjunction's expected value. It is important to note, then, that an action proposition is not merely a proposition which it is possible (i.e.

With the action propositions thus delineated, Maximize tells us to make the action proposition with the highest expected value true.^[136] With a clearer question in sight, we are now in position to return to our discussion of Vague Actions.

Let us begin by noting that the precise proposition describing the events following Emily's decision, presented at the beginning of this section, does not primarily consist of things that Emily intentionally made true. While these are certainly things that happened, it is quite implausible to suppose that Emily made it true that her elbow contracted to an angle of exactly 116.3 degrees; this is simply too specific a thing to have been under her control, and she certainly didn't intend anything that particular.

On the other hand, she no doubt made some weaker propositions true. For ex­ample, she made it true that she contracted her elbow by some amount. But recall that an action proposition is not merely a proposition Emily could have made true: it's a proposition that could have been the strongest thing she made true. Presumably, then, the proposition that Emily contracted her elbow by some amount is not the strongest thing she made true—she has more control over her elbow than that. Nonetheless, might the strongest thing she made true still be a precise proposition? Perhaps the strongest thing she made true is the proposition that she contracted her elbow to an angle between 102.4 and 128.6 degrees? Again it seems as though this is just too preciseathingtohavebeenthe strongestthingshemadetrue; what Emilymadetrueis at least partly a matter of her intentions, and she certainly did not have intentions that precise. But more importantly, Emily's ability to control her movements isn't perfect. Someone who does not have perfect motor control cannot hope to bend their elbow to exactly 115 degrees, say, or to some precise interval around 115 degrees—the best they can do is aim to bend elbow at roughly 115 degrees.^[137]

In a similar way, when I decide to extend or flex my elbow, but I don't have perfect motor control, I am doing something that I think will result in my elbow making certain precise angles, although in a way that will leave me uncertain which precise angle I will end up making. If I'm trying to make a right angle, presumably I should be more confident that I will make an angle in the 85-95 degree range than the 95-105 degree range. Indeed, it is natural to think that the probability of each precise angle, conditional on my chosen course of action, will smoothly drop off either side of 90 degree.

However, if the strongest things we make true are always precise, then it seems as though we wouldn't get the smooth curve that seems to be predicted. The difference between the smooth curve and the sharp curve isn't a mere curiosity either—these probabilities are essential to applying decision theory. Averaging says that in order to calculate the value of this action we can look at the value of each precise angle and multiply it by the probability of it conditional on that action and sum them; yet these sums can change between the smooth and sharp curves even if our utilities remain constant.

In our discussion of vague evidence, we considered the idea that one's evidence isn't propositional at all, and that one should update by Jeffrey conditioning over a collection of precise propositions instead. There is an analogous move to be made here as well. Rather than treating actions as a choice between making a single, potentially vague, proposition true, we could imagine an action being a collection of precise propositions paired with probabilities representing the probability that we will make that precise thing true. Thus, perhaps, choosing to make certain elbow movements is a bit like choosing to enter a kind of lottery: the possible results are that I flex my elbow to an angle of exactly x degrees, and each of these results happen with probability q, where the result of graphing q against x conforms to the kind of smooth curve you would expect. Call such things ‘mixed strategies’. Mixed strategies are thus the analogue of Jeffrey conditioning for the case of action: a choice, on this view, is a choice about which mixed strategy over precise eventualities to adopt rather than a choice about which proposition to make true.

Mixed strategies on this account play a fundamental role in thought and action. Indeed, much like the kinds of Jeffrey conditionings that arise through obtaining inexact evidence, there is a straightforward correspondence between mixed strategies and the things I called evidential roles—functions from maximally strong consistent precise propositions to [0,1]. Since, on this picture, it is mixed strategies/evidential roles that play the proposition role—the things we know, believe, make true, desire, and so forth (see chapter 4)—there is little reason not to call these entities propos­itions. The result is a theory of vague propositions much like that outlined in chapter 6, or the expressivist account of vague propositions discussed in chapter 8.

Although I maintain that an agent's action propositions are often vague, it is important to distinguish this claim from the thought that the action propositions are often borderline (i.e. neither determinately true nor determinately false). Indeed, it is natural to think that determinately rational agents with determinate prefer­ences always have determinately true or determinately false action propositions. The thought, to put it roughly, is that if A has the optimal value out of all of my action propositions (and this is a determinate truth) and A is borderline, then it's borderline whether I've satisfied Maximize, and, therefore, it's at best borderline whether I'm being rational.^[138] We can turn this into an argument that the action set of any determinately rational agent with determinate preferences consists only of propositions which are determinately true or determinately false on the assumption that it's determinate that there are no ties among her preferences on actions. Let S be a determinately rational agent and V her value function.

1.        For any pair of propositions A and B: either it's determinate that V (A) < V (B) or it's determinate that V (B) < V (A). (By the assumption that it's determinate what Ss preferences are, and the fact that there are no ties.)

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2.        For any proposition A, either it's determinate that A is an action proposition or it's determinate that A is not an action proposition. (It's determinate what Ss action propositions are.)

3.        For any action proposition A, determinately: A if and only if V (B) < V (A) for every other action proposition B. (The agent determinately satisfies Maximize and the action propositions are pairwise incompatible.)

4.        Therefore, for any action proposition A, either it's determinate that A (if A is optimal) or it's determinate that —A (if A is not optimal).

The first and second premises are debatable, but are probably true in many relevant cases; for example, the first premise is entailed by the thought that it is a precise matter what credence and utility the agent assigns to a proposition. However, even though this is usually false—ascriptions of desire and belief are often slightly vague—the vagueness usually isn't enough to make a difference to the ordering of propositions. (3) follows from the claim that the agent is determinately rational and that it's determinate that there are no ties in A. If the agent is determinately rational and it's determinate that there are no ties, then she determinately makes the best proposition in her action set true: this makes the right to left direction true. If we assume that the action propositions are incompatible with one another, then the other action propositions must be false giving us the other direction.^[139] Finally, from the precision of the right hand side of (3) we may infer that for every action proposition A, determinately: A iff [something determinate]. This ensures (4).

Summarizing, it follows that if A is an action proposition of a given agent, and it's borderline whether A, then either: (i) the agent is not determinately rational, (ii) the agent doesn't have determinate preferences, (iii) it's not determinate which propositions are the agent's action propositions, or (iv) it's not determinate that there are no ties. If none of the caveats (i)-(iv) obtain, then the action propositions will always consist of determinately true or false propositions.

Note, however, that this conclusion is completely compatible with Vague Actions. As we have noted several times already, being vague is not the same as being neither determinately true nor false—the proposition that Bruce Willis is bald, for example, is a vague proposition despite the fact that it's determinately true. Vague Actions just says that the action propositions are usually vague, and this is completely compatible with their usually being determinately true or false. Moreover, the mere fact that they consist of determinately true or false propositions does not mean that our attitudes toward the vague are irrelevant, because even if an action proposition turns out to be determinatelytruewewon't Fypicallyknowwhich it is priortodeliberation,and it's our ignorance, and in particular our ignorance about the vague, that changes how we act.

Let us summarize the points we have made above and relate them to the practical irrelevance thesis. The thought we started off with was the idea that the behaviour of a rational person will always be a function of their rational desires and beliefs. Moreover, if the behaviour of an agent is always a precise matter, then the desires and beliefs which matter for the agent's behaviour will be ones with precise contents. Cer­tainly, there are conceptions ofbehaviour, popular among early kinds ofbehaviourists, that identify behaviour fairly narrowly with precise bodily movements. However, it is quite clear that this conception of behaviour will neither play the role in decision theory we need it to play nor will it track the kinds of thing we care about when we make evaluative judgements about a person's behaviour. Here I have argued that behaviour, as it figures in decision theory, is best described in terms of a fundamentally intentional notion: that of making something true. Making true is a propositional attitude in the sense that it requires you to be in a certain state of mind, although like other externalist attitudes such as knowledge, it cannot be held unless the world around you complies.

On the alternative account of behaviour, our attitudes towards the vague are practically important. For example, it's possible that on a given occasion the only thing that you've made true is a vague proposition. In this case you simply cannot describe your behaviour purely in terms of the precise things you have made true. More importantly, for a rational person, which proposition is made true is determined by which proposition they assign the highest value to, so one's beliefs and desires in the vague determine which action they perform.

Thus, on this picture it is possible for two people to assign the exact same values to all precise propositions and be in a position to make the same propositions true, while rationally behaving in different ways.^[140] If Alice and Bob, say, agree about the precise in the sense that they assign the same values to all the precise propositions, they might still end up behaving differently if they assign different values to the vague. The best proposition in Alice's action set might be different from the best proposition in Bob's action set—for Alice's best proposition might be one of the vague propositions Alice and Bob disagree about. This can happen even if Alice and Bob have the same action set, since all that agreement about the precise ensures is that they will both make the same proposition true in the special case where their action sets are the same and consist only of precise propositions.^[141] Moreover, this will result in a concrete difference in behaviour: since Alice and Bob will choose the highest value proposition to make true, they will end up making different vague propositions true.

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