## Epistemic counterparts 7: Distributed plural roles

So far, we have looked at cases in which an agent has a descriptive belief (e.g., "the creature approaching through the woods is a bear"), which gets reported as a singular belief ("Mary beliefs Mark is a bear"). But sometimes we attribute singular beliefs even though the subject appears to have only a general (quantified) attitude about the relevant individual.

A murder has been committed. The detective has figured out that the culprit probably comes from a certain mountain village. She knows little about that village, but believes that all its inhabitants are poor peasants. You are one of the villagers. We might say:

(1) The detective thinks you are a poor peasant.
(2) The detective thinks you might be the murderer.
(3) The detective believes that every villager might be the murderer.
(4) Every villager is believed by the detective to be poor.

Similarly, the detective might say to herself:

(5) Everyone who lives in that village might be the murderer.

On the "Quine/Kaplan model" that I defended in earlier posts, 'S believes that x is F' is true iff there is a suitable role R such that (i) in all worlds doxastically accessible for S, whatever plays R is F, and (ii) in the actual world, x plays R. The puzzle raised by the above examples is that there does not seem to be any relevant role R.

Let's get clear about this problem. It is easy to find a function from worlds to individuals (and therefore a role) which singles out (say) you in the actual world and a villager in each of the detective's belief worlds. No such role is contextually salient when we utter a sentence like (1), but that's not a problem, since I haven't assumed that the relevant role is always transparent to speakers and hearers. Moreover, I have already accepted that the relevant roles are often underdetermined (because it really only matters how they track the target individual through the subject's belief worlds).

Still, there is a problem. For one thing, any role that tracks you as a villager across the detective's belief worlds would appear to be rather gerrymandered, and gerrymandered roles are usually unsuitable. For example, we can't assume that you are tracked as the 40th oldest inhabitant of the village (which would not be terribly gerrymandered), because the detective may not know that the village has 40 inhabitants. In general, you can't be tracked as the villager who is F unless the detective knows there is a unique F among the villagers, and given her ignorance about the village it's hard to find a non-gerrymandered candidate for F. (And note that we would need to find a different such property for each villager.)

Another part of the problem is that there are many roles tracking the actual villagers as villagers across the detective's belief worlds, but most of them will yield false predictions if they are chosen as "suitable". For example, there are ways of tracking you as a villager that never identify you as the murderer. Relative to any such role, (2) would come out false. But it is true.

So the "Quine/Kaplan model" can't easily account for our puzzles. But the counterpart-theoretic semantics that I built out of the model can.

Let's start with a slightly simpler case. Suppose the detective learns that there are two bakers in the village, both of whom are old. You are one of the bakers. The detective has a descriptive belief about the bakers: she believes that the two bakers in the village are old. We can report this descriptive believe as a singular (or "plural"?) belief about the pair of bakers: she believes that they are old. And now it seems that we can "distribute" this belief over the members of the plurality: since you are one of the bakers, and the detective believes of the two bakers that they are old, we can say that she believes of you that you are old.

Consider any of the detective's belief worlds, w. In this world, there are two bakers in the village. Call them b1 and b2. If the bakers in the actual world are you and Bob, then b1 and b2 are the w-counterparts of you and Bob. But which of them is your counterpart, and which is Bob's? Any choice would be arbitrary. We could identify you with b1 and Bob with b2, or we could identify you with b2 and Bob with b1.

Instead of making the arbitrary choice, let's keep track of both possibilities. We may think of this in terms of multiple counterpart relations: one relation links you to b1 at w and Bob to b2, another links you to b2 and Bob to b1.

In the semantics I introduced in part 4 of this series, modal operators quantify jointly over worlds and counterparts: □Fx is true iff at every accessible world, every counterpart of x is F. Now we also quantify over counterpart relations: Fx is true iff at all accessible worlds, relative to every counterpart relation, every counterpart of x is F. By duality, ◇Fx is true iff at some accessible world, relative to some counterpart relation, some counterpart of x is F.

This triple quantification may seem odd. But it makes sense if we think of de re possibility as expressing individual possibilities for the relevant objects. (Compare Plurality, p.258f.) w is one possibility for the world, but it is two possibilities for you and Bob: given that the world is w, you might be b1 while Bob is b2, or you might be b2 while Bob is b1. De re modals quantify over these more fine-grained possibilities. So we need to consider w twice over when we evaluate de re statements about you and Bob. With multiple counterpart relations, the two possibilities for you and Bob are represented as the two ways of combining w with a counterpart relation.

Instead of multiple counterpart relations, we could (following Lewis) use counterpart relations between sequences. We would then say that the pair (you, Bob) has two counterparts at w: (b1, b2) and (b2, b1). There are subtle differences between the two approaches (using multiple counterpart relations and counterpart relations between sequences), but for present purposes they are largely interchangeable.

In our formal semantics, we could now simply assume that a model contains multiple counterpart relations, or counterpart relations between arbitrary sequences (or between arbitrary assignment functions). But an informative semantics should explain where these ingredients come from: which aspects of the modeled situation they represent.

In part 4, I suggested that if there is a unique "suitable" role R for a given individual x (by something like the criteria from part 2), then an individual y at another world is a counterpart of x (at the actual world) iff y satisfies R (at the relevant world). This is what we need to generalise. There are different ways to do this. Here is one natural approach, following the above intuitive considerations.

First, we allow for "plural roles" capturing how a plurality of individuals might be known to an epistemic subject. These roles are not functions from worlds to individuals, but (let's say) functions from worlds to sets of individuals. Otherwise they behave much like individual roles.

To illustrate, consider a variant of the Mary Beth Harshbarger case.

A group of rangers is moving towards Mary through the woods. Thinking they are bears, Mary shoots them all with her machine gun. In her defense, she says that she thought the rangers were bears. Now suppose there were dozens of rangers, and Mary only vaguely noticed a big commotion in the woods as they approached. Then it would be hard to find suitable roles for every individual ranger. Instead, we can apply the Quine/Kaplan model directly to the plurality: 'Mary thought the rangers were bears' is true because the rangers played a certain role R – approaching through the woods, causing the commotion – of which Mary thought it was played by a group of bears. This plural role R maps each world w to the set of individuals approaching through the woods at w.

Like singular roles, plural roles determine a counterpart relation between pluralities: the approaching bears at each of Mary's belief worlds are counterparts of the rangers at our world.

Next, we specify how such a counterpart relation between pluralities induces a set of counterpart relations between the individuals that make up the pluralities.

More precisely, consider some belief world w, and the actual world @. We know that a certain plurality Pw in w is a counterpart of a certain plurality P@ in @. For each such pair of worlds, we need to define a set of relations linking members of P@ with members of Pw.

In normal cases, these relations should arguably be "total", so that their domain is all of P@, and their range all of Pw. If Pw and P@ differ in cardinality, this means that the relations can't be one-one. However, we should arguably require them to be as close to one-one as possible, minimizing branching in both directions.

To motivate these assumptions, imagine we are talking to one of the injured rangers, of which there were (say) 32. We might tell him that Mary thought he was a bear (even though there's no definite role by which Mary had identified him). So he must be tracked as one of the approaching bears at every belief world. However, we don't want to say that Mary thought he might be 50 bears, even if in some of Mary's belief worlds 50 bears were approaching. So we don't want to include a counterpart relation that links a single member of P@ to every member of Pw.

I won't settle how exactly the "minimize branching" requirement should be spelled out.

The current recipe for determining counterpart relations assumes that none of the relevant individuals (in P@) fall under any more specific suitable role. If one of the rangers was clearly visible to Mary – although she mistook him for a bear –, then this ranger will be tracked not as an anonymous member of the approaching group, but by a more definite role.

This model nicely handles all of the detective cases. In the case of the two bakers, we can explain why (6) is true (in a suitable context):

(6) The detective thinks you are old.

In short, the explanation is that the detective believes that the two village bakers are old, and this plural belief is distributed over the individual bakers. Along similar lines, we can explain the initial examples (1)–(5).

(The proposal I have outlined resembles a proposal in Ninan 2018. Ninan doesn't invoke multiple counterpart relations or counterparts of sequences. Instead, he suggests that every member of P@ has every member of Pw as a counterpart. This falsely predicts that Mary thought the ranger might be 50 bears, and that the detective could truly say 'both bakers might be murderers', even if she is sure there's only one murderer in the village (because at some belief world, the murderer is a counterpart of both you and Bob).)

The present account makes the somewhat surprising prediction that 'some F might be G' is equivalent to 'every F might be G', assuming the Fs are simply tracked as the Fs. But maybe that's correct. If the detective has no further information about the villagers, and thinks one of them might be the murderer, then we can arguably report her as believing of each villager that they might be the murderer: every villager is a suspect.

To conclude, here are a few variations of the detective scenario that raise some further questions.

First, suppose the detective thinks all the villagers are adults, but in fact one of them is a recently born baby. Does the detective think the baby might be the murderer? Arguably not. Why not? Presumably because here we give up the totality assumption, that every actual villager is tracked as a villager in the detective's belief worlds. The baby is too unlike the villagers in the belief worlds, so it has none of them as a counterpart.

Next, why does 'every villager might be the murderer' sound false if it is known that the baker is innocent? (Let's say there's only one baker now.) Perhaps because we are now tracking the innocent baker by the more specific role of being the baker (or by the coincident role of being the innocent baker). Remember that the anonymous distribution of plural roles only covers members of the plurality that are not tracked by a more specific suitable role.

Next, why does 'every villager might be the murderer' sounds bad if we know the murderer's ordinary name, but OK if we merely know the murderer's pseudonym from a note left behind at the crime scene? Perhaps because if we know that one of the villagers has the ordinary name N, then we track them as the person called N; we don't do this with the pseudonym, because that would trivialise the identification of the murderer.

Next, why does 'every villager might be the baker' sound worse than 'every villager might be the murderer'? Perhaps because the modal quantifies not over the worlds compatible with easily available information. (The baker sentence sounds better if the baker's identity is hard to determine.) Or perhaps because we again track the baker by the more specific baker role, and for some reason this role is more suitable than the murderer role.