# Identifying the subjects of common knowledge

Suppose that it’s public information/common belief/common ground among a group G that the government has fallen (p). What does this require about what members of G know about each other?

Here are three possible situations:

1. Each knows who each of the other group members is, attributing to (de re) to each whatever beliefs (etc) are required for it to be public information that p.
2. Each has a conception corresponding to each member of the group. One attributes, under that conception, whatever beliefs (etc) are required for it to be public information that p.
3. Each has a concept of the group as a whole. Each generalizes about the members of the group, to the effect that every one of them has the beliefs (etc) required for it to be public information that p.

Standard formal models of common belief suggest the a type 1 situation (though, as with all formal models, they can be reinterpreted in many ways). The models index accessibility relations by group members. One advantage of this is that once we fix which world is actual, we’re in a position to unambiguously read off the model what the beliefs of any given group member is—one looks at the set of worlds accessible according to their accessibility relation. What it takes in these models for A to believe that B believes that p is for all the A-accessible worlds to be such that all worlds B-accessible from them are ones where p is true. So also: once we as theorists have picked our person (A), it’s determined what B believes about A’s beliefs—there’s no further room in the model for further qualifications or caveats about the “mode of presentation” under which B thinks of A.

Stalnaker argues persuasively this is not general enough, pointing to cases of type 2 in our classification. There are all sorts of situations in which the mode of presentation under which a group member attributes belief to other group members is central. For example (drawing on Richard’s phone booth case) I might be talking to one and the same individual by phone that I also see out the window, without realizing they are the same person. I might attribute one set of beliefs to that person qua person-seen, and a different set of beliefs to them qua person-heard. That’s tricky in the standard formal models, since there will be just one accessibility relation associated with the person, where we need at least two. Stalnaker proposes to handle this by indexing the accessibility relations not to an individual but to an individual concept—a function from worlds to individuals—which will draw the relevant distinctions. This comes at a cost. Fix a world as actual, and in principle one and the same individual might fall under many individual concepts at that world, and those individual concepts will determine different belief sets. So this change needs to be handled with care, and more assumptions brought in. Indeed, Stalnaker adapts the formal model in various ways (e.g. he ultimately ends up working primarily with centred worlds). These details needn’t delay us, since my concern here isn’t with the formal model directly.  Rather, I want to point to the  desiderata that it answers to: that we make our theory of common belief sensitive to the ways in which we think about other individual group-members. It illustrates that the move to type 2 cases is a formally (and philosophically) significant step.

The same goes for common belief of type 3, where the subjects sharing in the common belief are characterized not individually but as members of a certain group. Here is an example of a type-3 case (loosely adapted from a situation Margaret Gilbert discusses in Political Obligation). We are standing in the public square, and the candidate to be emperor appears on the dais. A roar of acclaim goes up from the cloud—including you and I. It is public information among the crowd that the emperor has been elected by acclimation. But the crowd is vast—I don’t have any de re method of identifying each crowd member, nor do I have an individualized conception of each one. This situation is challenging to model in either the standard or Stalnakerian ways. But it seems (to me) a paradigm of common belief.

Though it is challenging to model in the multi-modal logic formal setting, other parts of the standard toolkit for analyzing common belief cover it smoothly. Analyses of common belief/knowledge like Lewis’s approach from Convention (and related proposals, such as Gilbert’s) can take it in their stride. Let me present it using the assumptions that I’ve been exploring in the last few posts. I’ll make a couple of tweaks: I’ll consider instances of the assumptions as they pertain to a specific member of the crowd (you, labeling u). I’ll make explicit the restriction to members of the crowd, C. The first four premises are then:

1. $A \supset B_u(A))$
2. $(A\Rightarrow_u [\forall y: Cy] B_y(A))$
3. $(A \Rightarrow_u q)$
4. $([A\Rightarrow_u [\forall y: Cy](x\sim y)]$

For “A”, we input a neutral description of the state of affairs of the emperor receiving acclaim on the dais in full view of everyone in the crowd. q is the proposition that the emperor has been elected by acclimation. The first premise says that it’s not the case that the following holds: the emperor has received acclaim on the dais in full view of the crowd (which includes you) but you have no reason to believe this to be the case. In situations where you are moderately attentive this will be true. The second assumption says that you would also have reason to believe that everyone in the crowd has reason to believe that the emperor has received acclaim on the dais in full view of the crowd, if you have reason to believe that the emperor has received such acclaim in the first place. That also seems correct. The third says if you had reason to believe this situation had occurred, you would have reason to believe that the emperor had been elected by acclimation. Given modest background knowledge of political customs of your society (and modest anti-sceptical assumptions) this will be true too. And the final assumption says that you’d have reason to believe that everyone in the crowd had relevantly similar epistemic standards and background knowledge (e.g. anti-sceptical, modestly attentive to what their ears and eyes tell them, aware of the relevant political customs), if/even if you have reason to believe that this state of affairs obtained.

All of these seem very reasonable: and notice, they are perfectly consistent with utter anonymity of the crowd. There are a couple of caveats here, about the assumption that all members of the crowd are knowledgable or attentive in the way that the premises presuppose. I come back to that later

Together with five other principles I set out previously (which I won’t go through here: the modifications are obvious and don’t raise new issues) these deliver the following results (adapted to the notation above):

• $A \Rightarrow_u q$
• $A\Rightarrow_u [\forall y: Cy] B_y(q)$
• $A\Rightarrow_u [\forall z : Cz] B_z([\forall y: Cy] B_y(q))$
• $\ldots$

And each of these with a couple more of the premises entails:

• $B_u q$
• $B_u [\forall y : Cy] B_y(q)$
• $B_u [\forall z : Cz] B_z([\forall y: Cy] B_y(q))$
• $\ldots$

It’s only at this last stage that we then need to generalize on the “u” position, reading the premises as holding not just for you, but schematically for all members of the crowd. We then get:

• $[\forall x : Cx] B_x q$
• $[\forall x : Cx] B_x [\forall y :Cy] B_y(q)$
• $[\forall x : Cx] B_x [\forall z : Cz] B_z([\forall y\in C] B_y(q))$
• $\ldots$

If this last infinite list of iterated crowd-reasons-to-believe is taken to characterize common crowd-belief, then we’ve just derived this from the Lewisian assumptions. And nowhere in here is any assumption about identifying crowd members one by one. It is perfectly appropriate for situations of anonymity.

(A side point: one might explore ways of using rather odd and artificial individual concepts to apply Stalnaker’s modelling to this case. Suppose, for example, there is some arbitrary total ordering of people, R. Then there are the following individual concepts: the R-least member of the crowd, the next-to-R-least member of the crowd, etc. And if one knows that all crowd members are F, then in particular one knows that the R-least crowd member is F. So perhaps one can extend the Stalnakerian treatment to the case of anonymity through these means. However: a crucial question will be how to handle cases where we are ignorant of the size of the crowd, so ignorant about whether “the n-th crowd member in the crowd” fails to refer. I don’t have thoughts to offer on this puzzle right now, and it’s worth remembering that nobody’s under any obligation to extend this style of formal modelling to the case of anonymous common belief.)

Type-3 cases allow for anonymity among the subjects of common belief. But remember  that it needs to be assumed that all members of the crowd are knowledgable and attentive. In small group settings, where we can monitor the activities of each other group member, each can be sensitive to whether others have the relevant properties.  But this seems in principle impossible in situations of anonymity. On general grounds, we might expect most of the crowd members to have various characteristics, but as the numbers mount up, the idea that the characteristics are universally possessed would be absurd. We would be epistemically irresponsible not to believe, in a large crowd, that some will be distracted (picking up the coins they just dropped and unsure what the sudden commotion was about) and some will lack the relevant knowledge (the tourist in the wrong place at the wrong time). The Lewisian conditions for common belief will fail; likewise, the first item on the infinite list characterizing common belief itself will fail—the belief that q will not be unanimous.

So we can add to earlier list a fourth kind of situation. In a type-4 situation, the crowd is not just anonymous, but also contains the distracted and ignorant. More generally: it contains unbelievers.

A first thought about accommodating type 4 situations is to replace the quantifiers, replacing the universal quantifiers “all” with “most” (or: a certain specific fraction). We would then require that the state of affairs indicates to most crowd members that the emperor was elected by acclimation; that it indicates to most that most have reason to believe that the emperor was elected by acclimation, and so on. (This is analogous to the kind of hedges that Lewis imposes on the initially unrestricted clauses characterizing convention in his book). But the analogue of the Lewis derivation won’t go through. Here’s one crucial breaking point. One of the background principles that is needed in getting from Lewis’s premises to the infinite lists was the following: If all have reason to believe that A, and for all, A indicates that q, then all have reason to believe that q. Under the intended understanding of “indication”, this is underwritten by modus ponens, applied to an arbitrary member of the group in question–and then universal generalization. But if we replace the “all” by “most”, we have something invalid: If most have reason to believe that A, and for most, A indicates that q, then most have reason to believe that q. The point is that if you pool together those who don’t have reason to believe that A, and those for whom A doesn’t indicate that q, you can find enough unbelievers that it’s not true that most have reason to believe that q.

A better strategy is the analogue of one that Gilbert suggests in similar contexts (in her book Political Obligation). We run the original unrestricted analysis not for the crowd but for some subgroup of the crowd: the attentive and knowledgeable. Let’s call this the core crowd. You are a member of the core crowd, and the Lewisian premises seem correct when restricted to the core crowd (for example: the public acclaim indicates to you that all attentive and knowledgable members of the crowd have reason to believe that he public acclaim occurred). So the derivation can run on as before, and established the infinite list of iterated reason-to-believe among members of the core crowd.

(Aside: Suppose we stuck with the original restriction to members of the crowd, but replaced the quantifiers for “all” not with some “most” or fractional quantifier, but with a generic quantifier. The premises become something like: given A,  crowd members believe A; A indicates to crowd members that crowd members believe A; A indicates to crowd members that q; crowd members have reason to believe that crowd members are epistemically similar to themselves, if/even if they have reason to believe A. These will be true if generically, crowd members are attentive and knowledgable in the relevant respects. Now, if the generic quantifier is aptly represented as a restricted quantifier—say, restricted to “typical” group members—then we can derive an infinite list of iterated reason-to-believe principles by the same mechanism as with any other restricted quantifier that makes the premises true. And the generic presentation makes the principles seem cognitively familiar in ways in which explicit restrictions do not. I like this version of the strategy, but whether it works turns on issues about the representation of generics that I can’t explore here.)

Once we allow arbitrary restrictions into the characterization of common belief, it makes it potentially pretty cheap (I think this is a point Gilbert makes—she certainly emphasizes the group-description-sensitivity of “common knowledge” on her understanding of it). For an example of cheap common belief, consider the group: those in England who have reason to believe sprouts are tasty (the English sprout-fanciers). All English sprout fanciers have reason to believe that sprouts are tasty. That is analytically true! All English sprout fanciers have reason to believe that all English sprout fanciers have reason to believe that sprouts are tasty, since they have reason to believe things that are true by definition. And all English sprout fanciers have reason to believe this last iterated belief claim, since they have reason to believe things that follow from definitions and platitudes of epistemology. So on, all the way up the hierarchy.

So there seems to be here a cheap common belief among the English sprout fanciers that sprouts are tasty. It’s cheap, but useless, given that I, as an English sprout fancier, am not in a position to coordinate with another English sprout fancier—we can meet one in any ordinary context and not have a clue that they are one of the subjects involved in this common belief is shared. (Contrast if the information that sprouts are tasty were public among a group of friends going out to dinner). It seems very odd to call the information that sprouts are tasty public among the English sprout fanciers, since all that’s required on my part to acquire all the relevant beliefs in this case is one idiosyncractic belief and a priori reflection. Publicity of identification of subjects among whom public information is possessed seems part of what’s required for information to be public in the first place. Type 1 and type 2 common beliefs build this in. Type 3 common beliefs, if applied to groups membership of which is easy to determine on independent grounds, don’t raise many concerns about this. But once we start using artificial, unnatural, restrictions under pressure from type 4 situations, the lack of any publicity constraint on identification becomes manifest, dramatized by the cases of cheap common belief.

Minimally, we need to pay attention to whether the restrictions that we put into the quantifiers that characterize type 3 or 4 common belief undermine the utility of attributing common belief among the group so-conceivedBut it’s hard to think of general rules here. For example, in the case characterized above of the emperor-by-acclamation, the restriction to the core crowd–the attentitive and knowledgeable crowd members—seems to me harmless, illuminating and useful. On the other hand, the same restrictions in the case in the next paragraph gives us common belief that while not as cheap as the sprout case earlier, is prima facie just as useless.

Suppose that we’re in a crowd milling in the public square, and someone stands up and shouts a complex piece of academic jargon that implies (to those of us with the relevant background) that the government has fallen. This event indicates to me that the government has fallen, because I happened to be paying attention and speak academese. I know that the vast majority of the crowd either weren’t paying attention to this speech, and haven’t wasted their lives obtaining the esoteric background knowledge to know what it means. Still, I could artificially restrict attention to the “core” crowd, again defined as those that are attentive and knowledgable in the right ways. But now this “core” crowd are utterly anonymous to me, lost among the rest of the crowd in the way that English sprout fanciers are lost among the English more generally. The core crowd might be just me, or it could consist of me and one or two others. I don’t have a clue. Again: it is hardly public between all the core crowd (say, three people) that they share this belief, if for all each of them know, they might be the only one with the relevant belief. And again: this case illustrates that the same restriction that provides useful common belief in one situation gives useless common belief in another.

The way I suggest tackling this is to start with the straightforward analysis of common belief that allows for cheap common belief, but then start building in suitable context-specific anti-anonymity requirements as part of an analysis of an account of the conditions under which common belief is useful. In the original crowd situation for example, it’s not just that the manifest event of loud acclaim indicated to all core crowd members that all core crowd members have reason to believe that the emperor was elected by acclaim. It’s also that it indicated to all core crowd members that most of the crowd are core crowd. That means that in the circumstances, it is public among the core crowd that they are the majority among the (easily identifiable) crowd. Even though there’s an element of anonymity, all else equal each of us can be pretty confident  that a given arbitrary crowd member is a member of the core crowd, and so is a subject of the common belief. In the second scenario given in the paragraph above, where the core crowd is a vanishingly small proportion of the crowd, it will be commonly believed among the core that they are a small minority, and so, all else equal, they have no ability to rationally ascribe these beliefs to arbitrary individuals they encounter in the crowd.

We can say: a face to face useful common belief is one that where there are face-to-face method of categorizing the people we encounter (independently of their attitudes to the propositions in question) within a certain context as a G*, where we know that most G*s are members of the group among which common belief prevails.

(To tie this back to the observation about generics I made earlier: if generic quantifiers allow the original derivation to go through, then there may be independent interest in generic common belief among G*s, where this only requires the generic truth that G* members belief p, believe that G* members belief p, etc. The truth of the generic then (arguably!) licenses default reasoning attributing these attitudes to an arbitrary G*. So generic common belief among a group G*, where G*-membership is face-to-face recognizable, may well be a common source of face-to-face useful common belief).

Perhaps only face-to-face useful common beliefs are decent candidates to count as information that is “public” among a group. But face-to-face usefulness isn’t the only kind of usefulness. The last example I discuss brings out a situation in which the characterization we have of a group is purely descriptive and detached from any ability to recognize individuals within the group as such, but is still paradigmatically a case in which common beliefs should be attributed.

Suppose that I wield one of seven rings of power, but don’t know who the other bearers are (the rings are invisible so there’s no possibility of visual detection–and anyway, they are scattered through the general population). If I twist the ring in a particular way, then in the case that all other ring bearers do likewise, then the dark lord will be destroyed, if he has just been reborn. If he has not just been reborn, or if not all of us twist the ring in the right way, everyone will suffer needlessly. Luckily, there will be signs in the sky and in the pit of our stomachs that indicate to a ring bearer when the dark lord has been reborn. All of us want to destroy the dark lord, but avoid suffering. All of us know these rules. When the distinctive feelings and signs arise, it will be commonly believed among the ring bearers that the dark lord has been reborn. And this then sets us up for the necessary collective action: we twist each ring together, and destroy him. This is common belief/knowledge among an anonymous group where there’s no possibility of face-to-face identification. But it’s useful common belief/knowledge, exactly because it sets us up for some possible coordinated action among the group so-characterized.

I don’t know whether I want to say that the common knowledge among the ring-bearers is public among them (if we did, then clearly face to face usefulness can’t be a criterion for publicity…). But the case illustrates that we should be interested in common beliefs in situations of extreme anonymity—after all, there’s no sense in which I have de re knowledge even potentially of the other ring-bearers. Nor have I even any way getting an informative characterization of larger subpopulations to which they belong, or even of raising my credence in the answer to such questions. But despite all this, it seems to be a paradigmatic case of common belief subserving coordinated action—one that any account of common belief should provide for. Many times, cooperative activity between a group of people requires they identify each other face-to-face, but not always, and the case of the ring bearers reminds us of this.

Stepping back, the upshot of this discussion I take to be the following:

• We shouldn’t get too caught up in the apparent anti-anonymity restrictions in standard formal models of common belief, but we should recognize that they directly handle on a limited range of cases.
• Standard iterated characterizations generalize to anonymous groups directly, as do Lewisian ways of deriving these iterations from manifest events.
• We can handle worries about inattentive and unknowledgable group members by the method of restriction (which might include as as special case: generic common belief).
• Some common belief will be very cheap on this approach. And cheap common belief is a very poor candidate to be “public information” in any ordinary sense.
• We can remedy this by analyzing the usefulness of common belief (under a certain description) directly. Cheap common belief is just a “don’t care”.
• Face-to-face usefulness is one common way in which common belief among a restricted group can be useful. This requires that it be public among the restricted group that they are a large part (e.g. a supermajority, or all typical members) of some broader easily recognizable group.
• Face-to-face usefulness is not the only form of usefulness, as illustrated by the extreme anonymity of cases like the ringbearers.