2. Lewis’ account

My question is whether the CK hierarchy could ever come about. There are various accounts of common knowledge available. These accounts differ in precisely how they define common knowledge, although typically it can be shown that the CK hierarchy follows from each definition of common knowledge. However, few of these definitions are couched in such a way that it will be possible to ask how it could be that a group of agents could come to possess common knowledge.Footnote ¹ The main exception is the definition proposed by Lewis. As Cubitt and Sugden put the point, Lewis’ account “doesn’t simply represent what individuals” would know, once they had achieved common knowledge, but instead seeks to “explain how those individuals come to know what they are represented as knowing” (Cubitt and Sugden Reference Cubitt and Sugden2003: 207).Footnote ² However, Lewis’ definition does not, in fact, generate the CK hierarchy itself, but a slightly different infinite hierarchy. In this Section, I’ll explain Lewis’ account. In Section 3, I explain why it needs to be modified. In the following Sections, I’ll turn to the question of what it would take to satisfy this set of modified conditions.

I emphasize at the outset that it might be possible for common knowledge properly so-called to come about even though the modified set of conditions I extract from Lewis is not satisfied. My claim is not that the definition I give of Strong Common Knowledge is the correct such definition, both necessary and sufficient for common knowledge, properly so-called. The question animating my inquiry is simply whether we can find any situations in which the CK hierarchy holds. I observe that Lewis’ account, when modified, yields a tractable set of conditions the satisfaction of which will entail the CK hierarchy. And I then claim that there are certain cases involving speech acts which (under certain assumptions about the nature of speech acts) satisfy the modified Lewisian conditions I propose, and which therefore would (if those assumptions were correct) count as cases of common knowledge, properly so-called.

I turn now to Lewis’ account. According to Lewis, C is (what he calls) “common knowledge” when:

Lewisian common knowledge

• LCK-1 Everyone in G has reason to believe that A holds.

• LCK-2 A indicates to everyone in G that everyone in G has reason to believe that A holds.

• LCK-3 A indicates to everyone in G that C.Footnote ³

“We can”, Lewis says, “call any such state of affairs A a basis” for common knowledge in G that C (Lewis Reference Lewis1969: 56).

In a moment, I’ll explain how Lewis’ definition of common knowledge works. I start with three prefatory observations. First, Lewis’ definition uses “reason to believe.” “Reason to believe” is, of course, not the same as “knowledge.” It is ultimately for this reason that Lewis’ account will need to be modified in order to characterize common knowledge, properly so-called.

Second, Lewis’ definition relies on the notion of indicating:

The definition of indicating is indexed to an individual: it may well be that what some particular A indicates to Gwendolen is different from what that particular A indicates to Cecily if Gwendolen and Cecily happen to have different “inductive standards and background information” (Lewis Reference Lewis1969: 53). Crucially, in any particular case, Lewis assumes that the members of G have “the same inductive standards and background information, at least nearly enough so that A will indicate the same things” to all the members of G (Lewis Reference Lewis1969: 53).

The definition of indicating also makes use of the word “thereby.” This is, as will become clear below, essential.Footnote ⁵

Third, the notion of a Lewisian basis is tractable. It is not, as we shall see, straightforward to understand. Nevertheless, it is possible to check, for any state of affairs, whether it counts as a Lewisian basis for common knowledge, as he defines that term.

In order to see how Lewis’ definition of common knowledge works, we can start with an example. Suppose that Gwendolen and Cecily are out hiking together. A large bear slowly ambles onto the trail ahead, follows the trail for a little while, then heads back into the woods. Wordlessly, Gwendolen and Cecily stop to watch the progress of the bear. Set:

• (A–1) Cecily and Gwendolen are hiking together. They both stop when a large bear appears and slowly follows the trail for a while before heading into the woods.

• (C–1) There is a bear nearby.

Now, let us see why A–1 is a basis for C–1 to be what Lewis calls “common knowledge.”

To see that LCK-1 holds, we must assume that each of the members of G has an ordinarily functioning perceptual apparatus, so let us make this assumption. We’ll also need to assume that Gwendolen and Cecily have relatively ordinary inductive standards and background information, so that bears are familiar enough animals that both Gwendolen and Cecily can readily identify them.

What about LCK-2? Since there are two individuals in our scenario, we need to check four things:

1. 2a. If Gwendolen has reason to believe that A–1 holds, then Gwendolen thereby has reason to believe that Cecily has reason to believe that A–1 holds.

2. 2b. If Gwendolen has reason to believe that A–1 holds, then Gwendolen thereby has reason to believe that Gwendolen has reason to believe that A–1 holds.

3. 2c. If Cecily has reason to believe that A–1 holds, then Cecily thereby has reason to believe that Gwendolen has reason to believe that A–1 holds.

4. 2d. If Cecily has reason to believe that A–1 holds, then Cecily thereby has reason to believe that Cecily has reason to believe that A–1 holds.

Now, because we are assuming that Gwendolen and Cecily are likewise situated with respect to the relation of indication, whatever reasoning can convince us that 2a is true will, mutatis mutandis, convince us that 2d is true, and vice versa (and likewise for 2b and 2c). Furthermore, the description I have given of A–1 shows that Gwendolen and Cecily each stand in the same epistemic relation to A–1 as does the other. So, given that they have the same inductive standards and background information, whatever reasoning can convince us that 2a is true will also convince us that 2c is true, and vice versa (and likewise for 2b and 2d).Footnote ⁶ So, in this particular case, we can convince ourselves that all of 2a–2d are true simply by convincing ourselves that one of them is true.Footnote ⁷

To see that 2a holds, we need to show that if Gwendolen has reason to believe that A–1 holds, then she thereby has reason to believe that Cecily has reason to believe that A–1 holds. It is plausible to think that this is so. For if Gwendolen has reason to believe that Cecily was hiking with her and they both stopped upon the appearance of the bear, then Gwendolen will thereby have reason to believe that Cecily will also (a) have reason to believe that they were both hiking together but stopped, and (b) have reason to believe that a bear just up ahead followed the trail for a while.

LCK-3 is, on the assumption that Cecily and Gwendolen both have relatively ordinary inductive standards and background knowledge, also straightforward. What we need to see is (a) if Cecily has reason to believe that A–1 holds, she would thereby have reason to believe that there is a bear nearby and (b) if Gwendolen has reason to believe that A–1 holds, she would thereby have reason to believe that there is a bear nearby. Now, since we are assuming that the indication relation is the same for each of them, then we only need to check (a). And (a) is surely a reasonable inference for most people. I, therefore, propose to stipulate that Cecily’s background knowledge and inductive standards are such as to ensure the satisfaction of LCK-3.

Lewis goes on to show that the satisfaction of the clauses of Lewisian Common Knowledge will generate what we can call the RTB hierarchy: both Gwendolen and Cecily have reason to believe C–1, both Gwendolen and Cecily have reason to believe that both Gwendolen and Cecily have reason to believe C–1, and so on. I omit the details, since these can be found elsewhere (Cubitt and Sugden Reference Cubitt and Sugden2003: 204–206). But it is worth identifying the central ideas of Lewis’ demonstration. Lewis establishes a key lemma:

Recall that LCK-2 states that “A indicates to everyone in G that everyone in G has reason to believe that A holds.” This is an instance of clause (i) of the lemma. And LCK-3 states that “A indicates to everyone in G that C.” This is an instance of clause (ii). So, we can use the lemma to combine LCK-2 and LCK-3 to yield:

• IND-1 A indicates to everyone in G that everyone in G has reason to believe that C.

But now, IND-1 is itself an instance of clause (ii) of the lemma. So, we can use the lemma to combine LCK-2 with IND-1 to yield:

• IND-2 A indicates to everyone in G that everyone in G has reason to believe that everyone in G has reason to believe that C.

We can continue in this way indefinitely, at each step applying LCK-2 and Lewis’ lemma to the previous result.

At the heart of this indefinite series of inferences is the fact that when A is a basis, it has the feature that A indicates to everyone in G that everyone in G has reason to believe that A itself holds. In other words, it is true of every individual X in G that if X had reason to believe that A holds, then X would thereby have reason to believe that everyone in G has reason to believe that A holds. To put the point yet another way: A is such that one of the things one has reason to believe, when one has reason to believe A, is that all the members of G also have reason to believe A.

3. A modification to Lewis’ account

Lewis’ account of common knowledge works with the notion of “reason to believe.” “Reason to believe” is not the same as “knowledge,” and it is not straightforward to see how to convert a reason to believe $p$ into knowledge that $p$ . In fact, as I understand Lewis’ account, C can be (what he calls) common knowledge without any of the members of G knowing C, or even believing C. How might this come about?

One might think the answer is straightforward: somebody can have reason to believe something, but, being finite and busy, simply not take the time to come to believe it. But this is not my worry about Lewis’ use of “reason to believe.” I think that it is possible to believe infinitely many things without having taken time to think about each of them. I’ll return to this point briefly at the end of this Section. Before that, I describe why the notion of “reason to believe” doesn’t give an adequate analysis of the notion of common knowledge.

To see the problem, consider the following case: Gatsby has offered to have Nick’s grass mown because Gatsby is planning to meet Daisy at Nick’s house. Gatsby arrives for the meeting, and Nick narrates the following exchange between himself and Gatsby, who speaks first:

As I understand the case, Nick knows perfectly well that Gatsby can distinguish between a well-mown lawn and an overgrown lawn. But although Gatsby is looking out the window at the lawn, and although Gatsby even says that the lawn “looks very good,” “That the lawn looks good” is not common knowledge between Nick and Gatsby, for Nick has reason to doubt that Gatsby has, in fact, seen the lawn and determined that it looks good.

First, consider: does Gatsby have reason to believe that the lawn looks good? We know that he cares about the appearance of the lawn, we assume that he is a competent judge of such things, and he is looking directly at the lawn. Crucially, the situation seems precisely parallel to the situation described in A–1 (Cecily, we’ve assumed, has the ability to identify bears, and she stops and looks directly at one on the path ahead). Just as we said that Cecily has reason to believe A–1, my own inclination would be to say that Gatsby does indeed have reason to believe that the lawn looks good.Footnote ⁸

But now consider Nick’s situation. Does Nick have reason to believe that Gatsby has reason to believe that the lawn looks good? I’ve just suggested that because Cecily and Gatsby are in parallel situations, we have reason to believe that Gatsby has reason to believe that the lawn looks good. But all of this information about Gatsby is also available to Nick. So, my own inclination is to think that Nick does indeed have reason to believe that Gatsby has reason to believe that the lawn looks good. However, it would be a mistake for Nick to come to believe that Gatsby believes that the lawn looks good, for Nick also has reason to believe – in this case, stronger reason to believe – that Gatsby hasn’t paid any attention to the lawn.

I just claimed that Nick has reason to believe that Gatsby has reason to believe that the lawn looks good. But if this is right, then ultimately the Gatsby case is parallel to A–1, and Lewis’ account of common knowledge would predict that “The grass looks good” is common knowledge between Nick and Gatsby.Footnote ⁹ But this can’t be right, because Nick knows perfectly well that Gatsby doesn’t believe that the lawn looks good. The problem is that on the way I’ve proposed to interpret “reason to believe,” one can have reason to believe $p$ and yet also recognize that one has better reason not to believe $p$ .

One could certainly try to dodge this problem by arguing that my preferred way of understanding “reason to believe” is incorrect. One could insist that we must understand “reason to believe,” so that it will be true to say that Gatsby does not have reason to believe that the grass looks good, and Nick does not have reason to believe that Gatsby has reason to believe that the grass looks good. Call this the stronger interpretation of “reason to believe.”

The problem now is that if we adopt the stronger interpretation of “reason to believe”, my argument (from Section 2) that A–1 can serve as a basis for Lewisian common knowledge that C–1 is rendered implausible. Just as Gatsby can look at the lawn without seeing it, surely we can imagine Cecily, in A–1, being sufficiently distracted such that she can look at the bear without seeing it. A–1, as described, tells us nothing to rule out the possibility that she is thus distracted. If we insist on ruling out the possibility of such a distraction in order for it to be true to say that Cecily has reason to believe that there is a bear on the path, then A–1 is not, as it stands, a Lewisian basis for common knowledge.

Neither of these choices seems satisfactory. A–1 seems plausible as a Lewisian basis for common knowledge if we interpret “reason to believe” in my preferred sense. But then, we cannot rule out the Gatsby case, which is clearly not a case of common knowledge. But if we interpret “reason to believe” more strongly, A–1 seems significantly less plausible, as it stands, as a basis for Lewisian common knowledge.

I suggest, therefore, that “reason to believe” does not yield an account of common knowledge. I propose instead that we shift attention directly to the notion of knowledge. Can we somehow strengthen Lewis’ account so as to directly yield the CK hierarchy?

The modification I propose is relatively simple: to replace “reason to believe” with “knows” in Lewis’s account of common knowledge.

Strong common knowledge

• SCK-1 Everyone in G knows that A holds.

• SCK-2 A K-indicates to everyone in G that everyone in G knows that A holds.

• SCK-3 A K-indicates to everyone in G that C.

• K-Indicating A K-indicates to someone X that ${\rm{\Delta }}$ if and only if, if X knew that A held, X would thereby know that ${\rm{\Delta }}$ .

We could then derive the CK hierarchy in essentially the same way that Lewis derives the RTB hierarchy, changing “reason to believe” to “knows” as appropriate.Footnote ¹⁰

One might object that it is simply psychologically impossible for the CK hierarchy ever to come about, for the CK hierarchy attributes to each subject in G infinitely many pieces of knowledge. I concede that if you think that, in order for $S$ to know $p$ , $S$ must have taken time to contemplate $p$ , or to have taken time to infer $p$ from her stock of existing knowledge, then you must reject the CK hierarchy. However, as Lederman points out, there is a familiar use of the word “know” (and indeed “believe”) according to which I know and believe infinitely many things:

Of course, there are many questions about how an account of belief and knowledge adequate to account for the fact that I know infinitely many propositions about unicorns should go. But I believe that nothing I say in what follows relies on any philosophical claims stronger than those that would be required for such an account.

4. Strong common knowledge

My question now is: Can we find some state of affairs which will constitute a basis for Strong Common Knowledge?Footnote ¹¹ Stock examples of common knowledge tend to involve cases in which something is publicly perceptible to a group of individuals, all of whom have the standard perceptual apparatus. For example:

• (A–2) Gwendolen and Cecily are hiking together. A bear appears on the trail ahead.

• (C–2) There is a bear up ahead.

Although this case might seem to be paradigmatic, it does not satisfy our characterization of Strong Common Knowledge. We’ve already seen (in Section 3) the essential reason why this is so. To establish SCK-2, we’ll need to show, among other things, that:

1. (1) If Gwendolen knows that A–2 holds, then she thereby knows that Cecily knows that A–2 holds.

But we cannot establish (1). First, observe that Gwendolen might not know that Cecily knows what a bear is, or how to identify them (this problem becomes more pressing if we replace “bear” with, for example, “kingsnake”). But even if we can fix this (perhaps by appropriately modifying how we handle our background assumptions), there is a second, more pressing, problem. It is perfectly possible that, on this particular occasion, Cecily has simply failed properly to exercise her perceptual capacity to recognize bears: even though it is in plain sight, on this occasion, she just hasn’t recognized it. So, although she has reason to believe there is a bear on the path, she hasn’t seen it, and so doesn’t know it is there.

I am not here claiming that publicly perceptible facts can never yield an example of Strong Common Knowledge, as I have defined it. Perhaps there is some way that this can come about. My claim is simply that it is not at all straightforward to see how this could be. Nor am I claiming that publicly perceptible facts can never yield the CK hierarchy. For perhaps there is some other way to specify a tractable set of conditions the satisfaction of which will yield the CK hierarchy. My claim is simply that publicly perceptible facts do not furnish straightforward examples of Strong Common Knowledge.

Lewis himself suggests a more promising place to look for clear examples of Strong Common Knowledge. He asks us to consider a situation in which two people have been talking. Ernest tells Cecily that he intends to return to the same place tomorrow to continue the conversation, and as a result, Cecily comes to expect that Ernest will return.Footnote ¹² So, let us set:

• (A–3) Ernest and Cecily have been talking. Ernest tells Cecily that he intends to return to the same place at the same time tomorrow to continue the conversation.

• (C–3) Ernest intends to return to the same place at the same time tomorrow to continue the conversation.

I’ll argue that, on certain assumptions, A–3 can satisfy clauses SCK-1 and SCK-2 of the definition of Strong Common Knowledge. The A–3/C–3 pair does not satisfy SCK-3 as they stand, but I’ll fix this at the end of the section by modifying the example.Footnote ¹³

The assumptions we need come from Austin. Recall that Austin argues that the performance of an illocutionary act (such as telling) involves the securing of what he calls “uptake”:

I claim that if Austin, interpreted in a sufficiently strong way, is correct, then A–3 would satisfy clauses SCK-1 and SCK-2 of the definition of Strong Common Knowledge.

To show that A–3 satisfies SCK-1, we need to show that, on the assumption that A–3 is true, both Ernest and Cecily know that A–3 holds. So, what we need to show is:

1. i If A–3 is true, then Cecily knows it is true.

2. ii If A–3 is true, then Ernest knows it is true.

I assume that if Ernest and Cecily are engaged in a conversation, then they both know that this is so. The difficulty comes when we ask whether each knows that Ernest has told Cecily that he intends to return tomorrow to continue the conversation.

With respect to (i), observe first that if Austin is wrong about the nature of illocutionary acts, then (i) will not obviously hold. For if Austin is wrong, then Ernest can “tell” Cecily that $r$ without uptake being secured. That is, Ernest can “tell” Cecily that $r$ without Cecily knowing that Ernest has “told” her that $r$ . But this means that A–3 could be true without Cecily knowing it to be true.

However, if Austin is right, then Ernest cannot have told Cecily $r$ without Cecily “understanding the meaning and the force of the illocution.” It is reasonable to think (and this is how I propose to interpret Austin’s remark) that this means that Cecily will know that Ernest has told her (force) that he intends to return to the same place at the same time tomorrow to continue the conversation (meaning). Notice that in specifying what Cecily knows, I specified that she’ll know that Ernest has told her such-and-such. It won’t be enough for Cecily simply to know that some speech act of telling has occurred, and what the content of that speech act is. She’ll need to know that it was she who was told, and by whom she was told. So, if Austin is right, (i) holds.

With respect to (ii), notice that if Austin is wrong, then (ii) is relatively easy to establish. For if Austin is wrong, then Ernest can “tell” Cecily that $r$ even if uptake is not secured. He’ll count as “telling” Cecily that $r$ so long as his utterance has an appropriate meaning and he has the intention to bring about in Cecily an understanding of the force and content of his utterance. And it is reasonable to think that if he has done this, then he’ll know that he has done so, as (ii) requires.

To establish that (ii) can be true if Austin is right to insist that performing an illocution requires the securing of uptake, we’ll need it to be the case that when one secures uptake, one knows that one has done so. So, I’ll make use of the following principle:

1. (2) If X intentionally told Y that $r$ , then X knows that Y knows that Y has been told $r$ by X.

I think it is reasonable to think that (2) follows from Austin’s account. For Austin, the successful performance of an illocutionary act involves “the securing of uptake”. So, assuming that Ernest has intentionallyFootnote ¹⁴ told Cecily that $r$ , this will have required him intentionally to have secured in Cecily an understanding of the force and the content of his illocution. And this – intentionally securing in one’s audience an understanding of the force and the content of one’s illocutionary act – is not something one could do without knowing one has done so.

There is a natural way of attempting to resist the claim that when one intentionally performs an Austinian speech act, one does so knowingly. Since illocutions require uptake, one’s attempts to perform an illocution will be unsuccessful if one fails to secure uptake. So, wouldn’t it be more correct for Austin to say that all Ernest was knowingly able to do was to try to tell Cecily that $r$ ? But this objection can’t be right as an interpretation of Austin, for it elides the distinction between illocutions and perlocutions. To see why, observe that the objection characterizes Ernest’s action as an attempt to bring about a particular consequence. Whether or not his action succeeds depends on whether or not those consequences at which Ernest was aiming (viz., uptake) happen to come about. But this is a description of a perlocutionary act. The contrast between perlocutionary and illocutionary acts depends crucially on the fact that uptake is not a “downstream” consequence of the illocutionary act, but is instead a constitutive component of the act itself, and so must be part of what Ernest intentionally did.

The point here is contentious, both philosophically and as an interpretation of Austin. But since my goal is to try to describe a situation in which Strong Common Knowledge is possible, I’ll simply record here the assumptions on which, I’ve claimed, we can find a case of Strong Common Knowledge.

Austin*:

1. a. Performing an illocutionary act requires securing in one’s audience an understanding of the force and content of the illocution;

2. b. If X has intentionally performed some illocutionary act, then X knows that her audience knows that X has performed that illocutionary act with them as the audience.Footnote ¹⁵

I conclude that if Austin* is correct, (ii) will hold. So, I’ve established that if Austin* is correct, A–3 satisfies SCK-1.

I turn now to the claim that if Austin* holds, then A–3 will satisfy SCK-2. To show this, we’ll need to establish the following:

1. i If Ernest knows that A–3 is true, then he thereby knows that Cecily knows that A–3 is true.

2. ii If Cecily knows that A–3 is true, then she thereby knows that Ernest knows that A–3 is true.Footnote ¹⁶

To see that (i) is true if Austin* holds: in my discussion of SCK-1, I argued that in order for Ernest to tell Cecily that $r$ , Ernest must see to it that she understands the force and the content of his illocutionary act, that is, he must see to it that she knows she has been told $r$ . That means that one of the things he knows, on the assumption that he knows that he has told her that $r$ , is that she knows that she has been told $r$ . And this is what (i) requires.

To see that (ii) is true if Austin* holds: let us suppose that Cecily knows that A–3 is true. So, Cecily knows that Ernest has successfully performed a particular speech act, which, we are assuming, can only have been performed intentionally (see footnote 14). But if intentionally performed speech acts are performed knowingly, then for the speech act to have occurred, Ernest must know that he has successfully performed it. But this means that Cecily cannot know that the speech act has occurred if she doesn’t know that Ernest knows he has successfully performed it, for to the extent that she harbors any doubt about whether Ernest knows he has succeeded, she must accordingly doubt whether the speech act has occurred. In other words, if Cecily does know that the speech act has occurred, then one of the things that she will thereby know is that Ernest knows he has performed it. And this is precisely what (ii) requires.

I’ve defended a conditional claim: if Austin* holds, then A–3 will satisfy SCK-1 and SCK-2. I emphasize that my claim is conditional. I have not taken it upon myself to give a full defense of Austin*.

SCK-3 is more troublesome. Even if Austin* is correct, Ernest can succeed in telling Cecily that he intends to return tomorrow without Cecily coming to believe him. I’ll improve the example in a moment.

Austin’s claim (let alone Austin*) is controversial.Footnote ¹⁷ If Austin’s opponents are correct, then one can successfully perform an illocutionary act without securing uptake. And if this is so, then I see little hope that A–3 could be a basis for Strong Common Knowledge. For if Ernest can tell Cecily that $r$ without her recognizing the force and content of the illocution, then Ernest can know that he has told Cecily that $r$ and yet not be in a position to know that Cecily knows she has been told that $r$ . In that case, we would not be able to establish SCK-2. Thus, I claim that if Austin is wrong, then we will not be able to establish SCK-2.

This argument might seem too fast. One might object: suppose that Austin is wrong, and Ernest can successfully tell Cecily that $r$ without uptake being secured. Let us call such an action a telling ${\rm{'}}$ . And now suppose that in our description of A, we simply add that uptake has successfully occurred. Thus, we set:

• (A–4) Ernest and Cecily have been talking. Ernest tells ${\rm{'}}$ Cecily that he intends to return to the same place at the same time tomorrow to continue the conversation.

• (C–4) Cecily has heard and understood what Ernest told ${\rm{'}}$ her.

Surely, (the objection proceeds) if A–3 can serve as a basis for Strong Common Knowledge if Austin is right, then A–4 can serve as a basis for Strong Common Knowledge if Austin is wrong. In other words, if we can have Strong Common Knowledge if Austin is right, then we can have Strong Common Knowledge even if Austin is wrong. Remarkably, this is not so. To show that A–4 satisfies SCK-2, we’d need to establish:

1. i′ If Ernest knows A–4, then Ernest thereby knows that Cecily knows A–4.

2. ii′ If Cecily knows A–4, then Cecily thereby knows that Ernest knows A–4.

I take it that i ${\rm{'}}$ is true. ii ${\rm{'}}$ , however, fails. The problem is that even if Cecily knows that she has heard and understood what Ernest has told ${\rm{'}}$ her, she does not thereby know that Ernest knows that she has heard and understood. So, we do not have SCK-2 in the case of A–4.Footnote ¹⁸ I conclude that Austin* must be correct for A–3 to serve as a basis for Strong Common Knowledge.

I’ve indicated that we can modify Lewis’ example so as to satisfy clause SCK-3. Suppose we set:

• (A–6) Ernest and Cecily have been talking. Ernest promises Cecily that he will return to the same place tomorrow with his pet lizard.

• (C–6) Ernest has promised to bring a reptile to the same place tomorrow.

Promises are illocutionary acts. If we adopt Austin*, then the argument of this section will show that A–6 satisfies clauses SCK-1 and SCK-2 of the definition of Strong Common Knowledge. What about clause SCK-3? This requires that it is true of both Ernest and Cecily that if they know A–6, they thereby know C–6. Recall that we are allowed to assume that Ernest and Cecily both have relatively ordinary background knowledge. Thus, we assume that each of them knows that lizards are reptiles. Thus, it will be true of each of them that in knowing A-6, they thereby know C–6.Footnote ¹⁹ So, assuming Austin*, A–6/C–6 is an example of Strong Common Knowledge.

My claim in this Section has been twofold. First, I’ve argued that if we assume Austin*, then A–3 satisfies SCK-1 and SCK-2, and A–6/C–6 satisfy SCK-1, SCK-2, and SCK-3. Second, I’ve argued that if we reject Austin*, neither A–3 nor A–6 can be a basis for Strong Common Knowledge. Austin’s conception of speech acts seems tailor-made to satisfy the “thereby” clause of SCK-2.

5. Illocutions addressed to larger audiences

I’ve just argued that if we accept a particularly strong version of Austin’s claim that the performance of an illocutionary act involves the securing of uptake, then we have a class of situations which can count as bases for Strong Common Knowledge: situations in which two people are talking, and one, addressing the other, intentionally performs some illocutionary act. However, the situation is more complicated if we seek a basis for Strong Common Knowledge in a group consisting of more than two.

• (A–7) Ernest, Gwendolen, and Cecily have been talking. Ernest promises them that he will return to the same place tomorrow with his pet lizard.

• (C–7) Ernest has promised to bring a reptile to the same place tomorrow.

To see why an audience of more than one person complicates the situation, recall that in order to establish SCK-2, we’ll need to establish that:

1. (3) It is true of every person X in G (where G is a set whose members are, in this case, Ernest, Gwendolen, and Cecily) that: if X knows that A–7 holds, then X will thereby know that everybody in G knows that A–7 holds.

For example, we’ll need to be sure, among other things, that it is true that:

1. (4) If Gwendolen knows A–7, then Gwendolen thereby knows that Cecily knows A–7.

I propose to take for granted that the first sentence of A–7 (“Ernest, Gwendolen, and Cecily have been talking”) satisfies SCK-2. The difficulty lies in establishing the component of SCK-2, which concerns Ernest’s speech act.

To see why there is a difficulty here, we must ask what is involved in what Austin describes as “bringing about the understanding of the meaning and of the force of the locution” in this case. What, exactly, does Gwendolen know when she knows that Ernest has “promised them to $\varphi $ ”? We might try the following analysis:

In effect, CA1 assumes that Austin is correct in thinking that for a promise to be made, uptake must be secured, and then treats “Ernest has promised Gwendolen and Cecily to $\varphi $ ” as a simple conjunction of “Ernest has promised Gwendolen to $\varphi $ ” and “Ernest has promised Cecily to $\varphi $ .”

But although CA1 is already a stronger account of the speech act of promising than Austin’s opponents would be prepared to countenance, CA1 is both unlikely as an interpretation of what is involved in the speech act of promising specified in A–7, and it is also insufficient to render A–7 a basis for common knowledge. The problem is that CA1 does not ensure that Gwendolen knows that Cecily knows that Ernest has made the promise in question, nor vice versa. And this means that (4) will not be satisfied. For in knowing A–7 so understood, Gwendolen will know that Cecily knows that a promise has been made to her (Cecily), but Gwendolen will not know whether Cecily knows that a promise has been made to Gwendolen. So, Gwendolen won’t know that Cecily knows A–7.

Not only does CA1 prevent us from establishing (4), but CA1 also strikes me as an inadequate analysis of what is involved in the speech act described in A–7. Intuitively, when Ernest makes a promise to Gwendolen and Cecily, Gwendolen and Cecily will each know that the other is aware of the promise (the promise is, at least paradigmatically, public). Ordinarily, this is part of what it is to promise them that he will $\varphi $ .

I propose that we can make A–7 satisfy SCK-2 if we insist that, in promising Gwendolen and Cecily that he will $\varphi $ , Ernest brings it about that both of them know that he has promised them to $\varphi $ .

If this condition is satisfied, then if Gwendolen knows A–7, she will thereby know that Cecily knows A–7: for if Gwendolen knows that Ernest has promised them that he will $\varphi $ , then she thereby knows that Cecily knows that he has promised them that he will $\varphi $ , which is precisely what is required for SCK-2.

I’ve italicized “them” to emphasize the role that this word is playing. The idea, in effect, is that in order to secure uptake for certain speech acts directed at an audience of more than one person, it is necessary not simply that the audience grasp the force and the content of the utterance, but that they recognize themselves as the audience for the utterance. The act of recognition is essentially plural. We cannot represent what the audience grasps with any finite conjunction of propositions without making use of plural pronouns.

One might think my insistence on such irreducibly plural knowledge is too hasty, and that we can ensure that SCK-2 is satisfied by generating a more complicated conjunction of facts that do not essentially involve plural pronouns.

One might, for example, propose the following:

So, now let us suppose that Gwendolen knows A–7, and so she knows all four clauses of CA-2. Will this be enough to ensure the satisfaction of (4)? In other words, does Gwendolen thereby know that Cecily knows all four of these? The answer to this is “No,” for suppose that Gwendolen (who, we are supposing, knows A–7), in trying to determine whether Cecily knows A–7, asks herself: “Does Cecily know clause b of CA-2?” But as far as Gwendolen knows, that which Cecily knows is recorded in clauses c and d of CA-2, and neither of these puts Gwendolen in a position to know that Cecily knows that Gwendolen knows that Ernest has made a promise to Cecily.

I’ve claimed that we cannot give an account of what is involved in the speech act described in A–7, which will satisfy SCK-2 by specifying a finite conjunction of facts that do not make essential use of plural pronouns. Observe, however, that we can do so with an infinite conjunction:

I’ve claimed that the proper extension of Austin’s notion of uptake to an audience of more than one person requires us to maintain that Ernest will have failed to promise Gwendolen and Cecily that $\varphi $ unless he brings it about that Gwendolen and Cecily know that he has promised them to $\varphi $ , where this them cannot be finitely analyzed. So understood, A–7 can serve as a basis for common knowledge, properly so-called, that C–7.

This claim extends to other illocutionary acts directed at more than one person. Suppose that Ernest is attempting to warn Cecily and Gwendolen of some imminent threat (we suppose the three of them are within earshot of each other). It is surely possible that he could succeed in warning one but fail to warn the other. Furthermore, I think it is also possible that he could succeed in warning each of them, but not bring it about that each knows that the other has been warned (perhaps he employs a gesture or phrase known to each, but which each doubts the other knows). In this case, the correct thing to say is that although he has warned both of them, he has failed to warn them. And this difference matters. Suppose that if Ernest succeeds in warning Cecily and Gwendolen of the impending danger in such a way that the warning becomes common knowledge in the manner I’ve been describing, then they will have time to turn and flee. Whereas (to continue the hypothetical) if Gwendolen recognizes the warning but is in some doubt about whether Cecily has recognized the warning, then Gwendolen will pause to try to rescue Cecily from danger, a pause which, we suppose, makes the difference between life and death. In this case, although Ernest managed to warn them several times, he did not succeed in warning them, with fatal consequences.Footnote ²⁰

Not every case in which one performs an illocution with an audience of more than one person will fit this pattern, for it is crucial that the target of the illocutionary act be the audience considered collectively. Suppose, for example, that Ernest, Cecily, and Gwendolen have been talking. Ernest (with Cecily still present) orders Gwendolen to $\varphi $ . It is perfectly plausible to imagine a situation in which Ernest successfully orders Gwendolen to $\varphi $ without Cecily knowing that he has done so (Ernest might, for example, disguise the order by using a special code known only to himself and Gwendolen, or Cecily might take it as a request, not noticing the tone of voice which Ernest uses). But in this case, Ernest made no attempt to order them to do anything, so there is no counterexample to my claim.

However, in a case in which Ernest is talking to Cecily and Gwendolen and orders them to $\varphi $ , I need it to be the case that he will have failed to order them to $\varphi $ if each takes themselves to have been ordered, but is in doubt about whether the other has been so ordered. And there is indeed a difference here. Suppose Ernest attempts to order them to load a hundred crates of such-and-such into the truck. Each mistakenly thinks this order has been issued to them severally, so that night each loads a hundred crates. And now, two hundred crates leave the warehouse, instead of the one hundred Ernest had intended. I think it is correct in this case to say that although he somehow managed to issue an order to each of them, he failed to issue the intended order to them.

6. Does common knowledge ever come about in this way?

In Section 3, I modified Lewis’ original definition of common knowledge so as to define what I called Strong Common Knowledge. The satisfaction of that set of conditions would entail the satisfaction of the CK hierarchy. In Section 4, I observed that cases which are often taken to be paradigmatic for common knowledge, in which some everyday object is in the public view of a group of ordinarily equipped perceivers, do not appear to meet these conditions. I suggested a different example ( ${\rm{\S}}$ 4– ${\rm{\S}}$ 5): I’ve claimed that, on a strong interpretation of Austin’s claim (Austin*) that the performance of an illocution involves securing uptake, then common knowledge can be achieved in the course of a conversational exchange. I’ve also claimed that if Austin* is wrong, then Strong Common Knowledge cannot come about as the result of a conversational exchange.

I have not, so far, argued that the conditions which define Strong Common Knowledge can ever, in fact, be satisfied. For I have not given an argument defending Austin*. If you are one of the many who reject Austin’s claim, then my argument in fact provides reason to be sceptical that common knowledge, properly so-called, could ever come about.

I do not have a direct argument to make in defense of Austin*. However, in this Section, I draw attention to the striking fact that one of the so-called paradoxes of common knowledge bears directly on the issue of illocutionary acts and common knowledge.

According to the “Coordinated Attack Problem,” we are to imagine two army divisions encamped on either side of an enemy.Footnote ²¹ If both divisions attack at the same time, the enemy will be vanquished. If only one division attacks, that division will suffer a catastrophic defeat. So, the general of each division will attack only if he knows that the other will do so. Unfortunately, the generals can communicate only via a messenger, who must sneak through enemy lines. Thus, each time a message is sent, there is a nonzero chance that the messenger is intercepted. We are to suppose that the generals believe the messenger.

To get a feeling for the problem, suppose that General Moncrieff sends a message to General Cardew outlining his plans to attack.Footnote ²² Suppose this message makes it through enemy lines and Cardew believes it. Now, we’ve assumed that each general will attack only if he knows that the other will. And since Moncrieff does not know whether Cardew has received his message, he does not know whether Cardew will attack. So, Moncrieff will not attack. So, suppose that Cardew writes back to confirm that she has received the message and intends to attack. Suppose that this message gets through and Moncrieff believes it. But now, Cardew will not attack. For she knows that Moncrieff will not attack unless Moncrieff knows that she has received his message and intends to attack, and (since there is a chance her message was lost) she does not know that Moncrieff knows that she plans to attack. Recognizing this, Moncrieff sends a message back to Cardew confirming that her message has been received. But now it won’t be rational for Moncrieff to attack unless he has confirmation that Cardew has received his message, for he knows that Cardew won’t attack unless she has received his message, and since there is some chance his message has been lost, he won’t attack unless he receives confirmation from her.

Unsurprisingly, this pattern of reasoning iterates. It can be shown formallyFootnote ²³ that: the generals will attack only if they commonly know that each plans to attack, and that: no matter how many messages are sent, the generals cannot achieve such common knowledge. It follows that no matter how many messages they exchange, the generals will never be in a position to attack the enemy. The “paradox” arises from the clash between this result and the intuitive thought that surely, after messengers have gone back and forth between the generals a few times, each general can be confident about the intentions of the other, and it can be perfectly sensible for the generals to attack.

To see how the coordinated attack problem bears on the question of the nature of speech acts, consider what difference it makes that the generals are on opposite sides of the enemy and able to communicate only via messenger. Suppose instead that the generals are in the same room when they discuss their plans. Once their plans are hatched, each will depart for their respective divisions to prepare to attack (or not). We continue to assume that neither will attack unless they know that the other intends to attack, from which it will again follow that the generals will attack only if it is common knowledge that they intend to do so.

In the original version of the problem, in which the generals are situated on opposite sides of the enemy, common knowledge cannot be achieved. I’ve argued that if we accept Austin*, then it is possible for two or more people engaged in a face-to-face conversation to achieve common knowledge. If that argument is correct, then it follows that if Austin* is a correct analysis of illocutionary acts and such illocutionary acts are possible, then when the generals are in the same room, there will be no paradox.Footnote ²⁴ That is, if illocutionary acts, as described by Austin*, are possible, then the face-to-face generals can rationally act in a coordinated fashion on the basis of their common knowledge of a plan of attack.Footnote ²⁵

However, if Austin is wrong, then the paradox remains even if the generals are in the same room. For if Austin is wrong to claim that the performance of an illocutionary act requires the securing of uptake, then it will be possible for Ernest to “tell” Cecily that $r,$ without it being the case that she knows that she has been “told” $r$ by Ernest. It is precisely this possibility that is modeled, in the coordinated attack problem, by the possibility that the messenger does not make it through enemy lines. Just as Moncrieff can send a message to Cardew without being sure that Cardew has received it, so too (if Austin is wrong) can Ernest “tell” Cecily that $r$ without being assured that Cecily has heard and understood what she has been “told,” even when Ernest and Cecily are face-to-face. In other words, if Austin is wrong, then there is always a small chance that any “message,” which an agent has “sent” in the course of a face-to-face conversational exchange, will fail to be “received.” So, if Austin is wrong, then even if Ernest and Cecily are in the same room, in order for it to become common knowledge that Ernest has “told” Cecily that $r$ , he’ll need some indication from her that she knows that he has “told” her that $r$ . But confirming that one has understood what one has been “told” – even when one is in the same room – is itself a speech act. And since (on the assumption that Austin is wrong) it is possible to perform such a speech act without uptake being secured, then in order for Cecily to be sure that Ernest has understood her confirmatory speech act, it’ll have to be the case that Ernest confirms to Cecily that he has understood her confirmatory speech act. And so on indefinitely.Footnote ²⁶

The current claim is, therefore, that if Austin is wrong, then the plight of any two interlocutors, even two who are in the same room, is exactly the same as the plight of the two generals on opposite sides of the opposing army. At best, Cecily can know what it is that Ernest is attempting to “tell” her, Ernest can know that Cecily knows this, Cecily can know that Ernest knows that Cecily knows, and so on, but we can continue the series only for as long as their patience at exchanging confirmatory remarks holds out.Footnote ²⁷ If Austin is wrong, one can successfully perform a speech act but still be in some doubt about whether one’s audience has recognized the force and content of that speech act. In that case, as far as achieving common knowledge is concerned, the plight of any two conversationalists is the same as the plight of the two generals on opposite sides of the enemy.Footnote ²⁸

My view is, therefore, that the coordinated attack problem in fact gives us some reason in favor of thinking that Austin’s account of illocution is correct. If all communicative exchanges must be modeled on the case of the two generals encamped on opposite sides of the army, then common knowledge can never be achieved through communication. But the coordinated attack problem then forces us to confront the question: How can it ever be rational to engage in coordinated action if common knowledge can never be achieved?Footnote ²⁹ An alternative becomes available once we recognize that the plight of the two generals is not the correct way to model all communicative exchanges. I’ve argued that genuine common knowledge can be achieved in the course of a conversation if we accept Austin*. This doesn’t itself provide a solution to the problem confronting the generals encamped on opposite sides of the enemy. The point is that, on the one hand, if we accept Austin*, interlocutors can generate common knowledge in the course of an in-person conversation and thereby rationally engage in coordinated action. On the other hand, if Austin is wrong about uptake, then genuine common knowledge cannot be achieved, and any communicative exchange, even one between people in the same room, must resemble that between the generals in the coordinated attack problem, continually sending one-way messages to each other and never able to secure common knowledge. The most straightforward path through these claims is to accept that sometimes coordinated action is rational only if genuine common knowledge can come about, but to hold that in-person communication, deploying speech acts as described by Austin*, ensures that common knowledge can, at least in some circumstances, arise.Footnote ³⁰