An alternative derivation of common knowledge

In the last post I set out a puzzling passage from Lewis. That was the first part of his account of “common knowledge”. If we could get over the sticking point I highlighted, we’d find the rest of the argument would show us how individuals confronted with a special kind of state of affairs A—a “basis for common knowledge that Z”—would end up having reason to believe that Z, reason to believe that all others have reason to believe Z, reason to believe that all others have reason to believe that all others have reason to believe Z, and so on for ever.

My worry about Lewis in the last post was also a worry about the plausibility of a principle that Cubitt and Sugden appeal to in reconstructing his argument. What I want to do now is give a slight tweak to their premises and argument, in a way that avoids the problem I had.

Recall the idea was that we had some kind of “manifest event” A—in Lewis’s original example, a conversation where one of us promises the other they will return (Z).

The explicit premises Lewis cited are:

1. You and I have reason to believe that A holds.
2. A indicates to both of us that you and I have reason to believe that A holds.
3. A indicates to both of us that you will return.

I will use the following additional premise:

• A indicates to me that we have similar standards and background beliefs.

On Lewis’s understanding of indication, this says that if I had reason to believe that A obtained, I’d have reason to believe we are similar in the way described. It is compatible with my not having any reason to believe, antecedent to encountering A, that we are similar in this way. On the other hand, if I have antecedent and resilient reason to believe that we are similar in the relevant respects, the counterfactual will be true

That the reason to believe needs to be resilient is an important caveat. It’s only when the reasons to believe we’re similar are not undercut by coming to have reason to believe that A that my version of the premise will be true. So Lewis’s premise can be true in some cases mine is not.

But mine is also true in some cases his is not, and that seems to me a particular welcome feature, since these include cases that are paradigms of common knowledge. Assume there is a secret handshake known only to members of our secret society. The handshake indicates membership of the society, and allegiance to its defining goal: promotion of the growing of large marrows. But the secret handshake is secret, so this indication obtains only for members of the society. Once we share the handshake, and intuitively, establish common knowledge that each of us intends to to promote the growing of large marrows. But we lacked reason to believe that we were similar in the right way independent of the handshake itself.

Covering these extra paradigmatic cases is an attractive feature. And I’ve explained that we can also cite it in the other paradigmatic cases, the cases where our belief in similarity is independent of A, so this looks to me strictly preferable to Lewis’s premise.

(I should note one general worry however. Lewis’s official definition of indication wasn’t just that when one had reason to believe the antecedent, one would have reason to believe the consequent. It is that one would thereby have reason to believe the consequent. You might read into that a requirement that the reason one has to believe the antecedent has to be a reason you have for believing the consequent. That would mean that in cases where one coming to have reason to believe that A was irrelevant to your reason to believe that you were similar, we did not have an indication relation. I’m proposing to simply strike out the “thereby” in Lewis’s definition to avoid this complication–if that leads to trouble, at least we’ll be able to understand better why he stuck it in).

I claim that my premise allows us to argue for the following, for various relevant p:

• If A indicates to me that p then A indicates to me that (A indicates to you that p).

The case for this is as follows. We start by appealing to the inference pattern that I labelled I in the previous post, and that Lewis officially declared his starting point:

1. A indicates to x that p
2. x and y share similar standards and background beliefs.
3. Conclusion: A indicates to y that p.

I claim this supports the following derived pattern:

1. A indicates to x that A indicates to x that p
2. A indicates to x that x and y share similar standards and background beliefs
3. Conclusion: A indicates to x that A indicates to y that p.

This seems good to me, in light of the transparent goodness of I.

A bit of rearrangement gives the following version:

1. A indicates to x that x and y share similar standards and background beliefs
2. Conclusion: if A indicates to x that A indicates to x that p, then A indicates to x that A indicates to y that p.

The premise here is my first bullet point. Given Lewis’s counterfactual gloss on indication, the conclusion is equivalent to my second bullet point, as required. To elaborate on the equivalence: “If x had reason to believe that A, then if x had reason to believe A, then…” is equivalent to “If x had reason to believe that A, then…”, just because in standard logics of counterfactuals “if were p, then if were p, then…” is generally equivalent to “if were p, then…”. In the present context, that means that “A indicates to x that A indicates to x that…” is equivalent to “A indicates to x that”.

[edit: wait… that last move doesn’t quite work does it? “A indicates that (A indicates B)” translates to: “If x had reason to believe A, then x would have reason to believe (if A had reason to believe A, then A would have reason to believe B)”. It’s not just the counterfactual move, because there’s an extra operator running interference. Still, it’s what I need for the proof….

But still, the counterfactual gloss may allow the transition I need. For consider the closest worlds where x has reason to believe that B. And let’s stick in a transparency assumption: that in any situation x has reason to believe p, x has reason to believe x has reason to believe p. Given transparency, at these closest worlds, x has reason to believe that she has reason to believe A, ie reason to believe that the closest world where she has reason to believe A is the world in which she stands. But in the world in which she stands Transparency entails she has reason to believe she has reason to believe p. So she has reason to believe the relevant counterfactual is true, in those worlds. And that means we have derived the double iteration of indication from the single iteration. Essentially, suitable instances of transparency for “reason to believe” gets us analogous instances of transparency for “indication”.  ]

The final thing I want to put on the table is the good inference pattern VI from the previous post. That is:

1. A indicates that [y has reason to believe that A holds] to x.
2. A indicates that [A indicates Z to y] to x.
3. Conclusion: A indicates that [y has reason to believe that Z] to x.

This looked good, recall, because the embedded contents are just an instance of modus ponens when you unpack them, and it’s pretty plausible in worlds where x has reason to believe the premises of modus ponens, then x has reason to believe the conclusion—which is what the above ends up saying. (As you’ll see, I’ll actually use a form of this in which the embedded clauses are generalized, but I think that doesn’t make a difference).

This is enough to run a variant of the Lewis argument. Let me give it to you in a formalized version. I use $\Rightarrow_x$ for the “indicates-to-x” relation, and $B_x$ for “x has reason to believe”.  I’ll state it not just for the two-person case, but more generally, with quantifiers x and y ranging over members of some group, and a,b,c ranging over propositions. Then we have:

1. $\forall x (A\Rightarrow_x \forall yB_y(A))$ (the analogue of Lewis’s second premise, above).
2. $\forall x (A \Rightarrow_x Z)$ (the analogue of Lewis’s third premise, above)
3. $\forall x ([A \Rightarrow_x Z]\supset [A \Rightarrow_x(\forall y[A\Rightarrow_y Z])]$ (an instance of the formalization of the bullet point I argued for above).
4. $\forall x [A \Rightarrow_x(\forall y[A\Rightarrow_y Z])]$ (by logic, from 2,3).
5. $\forall x [A \Rightarrow_x(\forall yB_y (Z))]$ (by inference pattern VI, from 1,4).

Line 5 tells us that not only does A indicate to each of us that Z (as Lewis’s premise 2 assures us) but that A indicates to each of us that each has reason to believe Z. The argument now loops, by further instances of the bullet assumption and inference pattern VI, showing that A indicates to each of us that each has reason to believe that each has reason to believe that Z, and so on for arbitrary iterations of reason-to-believe.

As in Lewis’s original presentation, the analogue of premise 1 allows us to detach the consequent of each of these indication relations, so that in situations where we all have reason to believe that A holds, we have arbitrary iterations of reason to believe Z.

(To quickly report the process by which I was led to the above. I was playing around with versions of Cubitt and Sugden’s formalization of Lewis, which as mentioned used the inference pattern that I objected to in the last post. Inference pattern VI is what looked to me the good inference pattern in the vicinity—the thing that they label A6, and the bullet pointed principle is essentially the adjustment you have to make to another premise they attribute to Lewis—one they label C4—in order to make their proof go through with VI rather than the problematic A6. From that point, it’s simply a matter of figuring out whether the needed change is a motivated or defensible one).

So I commend the above as a decent way of fixing up an obscure corner of Lewis’s argument. To loop around to the beginning, the passage I was finding obscure in Lewis, had him endorsing the following argument (II):

1. A indicates that [y has reason to believe that A holds] to x.
2. A indicates that Z to x.
3. x has reason to believe that x and y share standards/background information.
4. Conclusion: A indicates that [y has reason to believe that Z] to x.

The key change is to replace II.3 with the cousin of it introduced above: that A indicates to x that x and y share standards/background information. Once we’ve done this, I think the inference form is indeed good. Part of the case for this is indeed the argument that Lewis cites, labelled I above. But as we’ve seen, there’s seems to be quite a lot more going on under the hood.