In the second chapter of Convention, Lewis seeks to refine the account of conventions given in the first chapter. The most significant of these adjustments is the introduction of the notion of common knowledge. In this post I will focus on the most technically intricate section on common knowledge (section 1) and mention how Lewis uses this in his final analysis of a convention. Joshua has written an interesting post on interesting comments that Lewis makes about language use in the middle of the chapter. Perhaps later in the week I’ll post on other adjustments and his reasons for those.

Lewis noted in the first chapter that a pervasive feature of many conventions is that they include higher order expectations. A question arises: where do these come from? How is it that they can get up to an arbitrarily high order? Here is a possible answer. First, a definition: a situation A indicates to x that p iff x has reason to believe that A holds, x thereby has reason to believe that p.

Lewis thinks that there are three premises plus some background assumptions in order to generate the systematic higher order expectations that p for some group G in a situation A. The notion he introduces here to get the generation started is called common knowledge.

It is common knowledge in G that p iff and only if there is a situation A such that the following hold:

Background: Every member M of a group G share background information and inductive standards.

1. Every M in G has reason to believe A holds.

2. For every M in G, A indicates to M that every M’ in G has reason to believe that A holds.

3. For every M in G, A indicates to M that p.

Let’s see a simple two person example. Suppose m and n come to a four-way stop at the same time and have the background information that the traffic law that the person on the right has to go first. Say that m is the person on the right and the above premisses hold where G={m,n}, A is the traffic situation, and p is that m has to go first. Unwinding definitions, we get:

a) m and n have reason to believe that m has to go first. (1, 3)

b) m and n have reason to believe that both m and n have reason to believe that A holds. (1, 2)

c) m and n have reason to believe that both m and n have reason to believe that m has to go first. (b, 3)

d) m and n have reason to believe that both m and n have reason to believe that m and n have reason to believe that A holds. (b, 2)

…

The trick, of course, is in 2. This is what allows arbitrary nesting of the “x has reason to believe that p” operator. At each step we are also implicitly using the common inductive standards condition, since we are assuming, for instance, that m knows that A will indicate the same thing to n as it does to herself. If we get rid of the “indicates”, we can see this even more clearly:

1′ Every M in G has reason to believe A holds.

2′ For every M in G, M has reason to believe that A holds iff M has reason to believe that every M’ in G has reason to believe that A holds.

3′ For every M in G, M has reason to believe A holds iff M has reason to believe that p.

With some simple FOL, these are entail:

2” For every M in G, M has reason to believe that every M’ in G has reason to believe that A holds.

3” For every M in G, M has reason to believe that p.

The common inductive standards might be formalized as the following schema:

B For every M in G and situation A, if A indicates to M that p, then for all M’ in G, A indicates to M’ that p.

As above, we could eschew talking in terms of situations here, but I’ll leave the more condensed statement of the condition. This might seem implausibly strong, since surely we all disagree to a certain extent about what a certain situation gives us reason to believe. In the situations we’ll be interested in, however, I don’t think there are any particularly implausible consequences.

So now, what do we have? We have situations that give agents reasons for belief. But on their own, these reasons don’t do anything. What we were trying to generate were some actual beliefs that would be able to ground conventions. To get these, Lewis claims “Anyone who has reason to believe something will come to believe it, provided that he has a certain degree of rationality.” (55) The idea, then, is to go from these arbitrarily complex reason for belief, all that we need to do is plug in some rationality for each of the members of the group. Otherwise, while I would form the belief that you have reason to believe that p, I need not believe that you believe that p, since you might not be (sufficiently) rational.

This allows Lewis to explain why we only have relatively low orders of expectations, for instance, no 917th order beliefs. So, although we have reason to form this 917th order belief, we do not actually form it because we do not have reason to ascribe to sufficient rationality to ourselves and others. “The generating process stops when the ancillary premises give out.” (56)

I want to spend the rest of the post trying to figure out what Lewis can mean by rationality in these last two paragraphs. I think of rationality as, first and foremost, a dispositional property of individuals. When I say someone is rational, I do not mean to say that they are currently forming beliefs or assessing evidence on the basis of reasons. Rather, I mean to say that they have the capacity to do so. Such a conception of rationality, however, is not at issue here because having such a disposition just does not guarantee in any particular case that I actually act according to the disposition.

“Aha!” you might be thinking, “Justin has not been paying attention to the quote from page 55. It did not require just that the person is rational, but sufficiently rational.” Sadly, this is a red herring. Adding “more rationality”, if we are working with the sense above, just means either adding more dispositions or strengthening the current dispositions to form certain kinds of beliefs. What we get at the end of the process are not actualizations of dispositions but still dispositions. Even the most perfectly rational agent may not form a particular belief because she was just thinking about something else. Nor is this a case of limited cognitive resources, for these are again something potential. I could have enough resources to do a computation but still (rationally) not perform it, just because I have better things to be doing. Adding more computational resources by itself does not make me perform certain computations!

Therefore, instead of this purely dispositional notion of rationality, I suggest we understand what is needed to generate the actual beliefs as a complex, robust notion of rationality, which includes dispositions, states, and in some cases, even particular cognitive acts. Lewis himself hints at this when he talks of how we “manifest a modicum of rationality”. (57) This means, I think, that rational people not only that we have the correct dispositions, but also that they activate them in certain circumstances. Perhaps one way of saying this is that what Lewis needs to get to beliefs from reasons for belief is reasoning or rational thought.

This gives us what I take to be a pretty plausible view: having reasons to belief + thinking rationally on the basis of those reasons = actual rational beliefs. And bringing this all together with the account of common knowledge, higher order expectations are generated by higher order reasons to believe as characterized in 1-3 and higher order ascriptions of reasoning. When we can no longer ascribe a bit of reasoning to someone, our higher order expectations then give out, explaining why we only form relatively low order beliefs even though we have arbitrarily high order reasons to believe. The answer now is not that we lack rationality of some special kind, but rather that we just don’t think about it.

Now we are able to see the biggest fix Lewis gives to his account of conventions in chapter 1. I will bold the change:

A regularity R in the behavior of members of a population P when they are agents in a recurrent situation S is a convention iff in any instance of S it is common knowledge among members of P that:

1) Everyone conforms to R

2) Everyone expects everyone else to conform to R

3) Everyone prefers to conform to R on condition that the others do, since S is a coordination problem and uniform conformity to R is a coordination equilibrium in S.

Before, 1-3 merely had to be true, but now it must be common knowledge. If we assume factivity of knowledge and the KK principle for common knowledge, then we get the interesting idea that a population has a convention if and only if it is common knowledge among members of that population that they have a convention.

Why is this an improvement to the account? The answer, I think, has to do with the third clause. If we all mistakenly believe that everyone else conforms to R just because it is the first thing that occurs to them, all three conditions would be true, but not common knowledge. Lewis does not want this to be counted as a convention, I think because the group members do not seem to be really working together to solve the coordination problem. His new definition excludes it because in that case, it is not common knowledge that everyone prefers to conform to R on the condition that the others do.