Common-Knowledge-Based Pragmatics

Abstract

Suppose a speaker S and an audience A are in a communication coordination problem. That is, for some proposition p, they each prefer that S mean that p and that A believe p in response. How do they coordinate their thought and action to solve the problem? The Gricean answer is that they reason their way to the solution. Pragmatics makes a similar assumption. “Pragmatics involves perception augmented by some species of ‘ampliative’ inference … a sort of reasoning” (Kepa Korta and John Perry, “Pragmatics”). There are two objections. The first is that it is not plausible to attribute such reasoning to speakers and audiences. The second objection grants, for the sake of argument, that speakers and audiences reason in the required ways. The objection is that this is not sufficient to solve coordination problems since a speaker and an audience may not know how each has reasoned. The problem is well-known in game theory, which typically solves it by assuming the parties’ preferences are common knowledge. Common knowledge is the recursive belief state in which people are know something, know they know it, know they know they know it, ad infinitum. I propose a similar solution. When people interact as speaker and audience, common knowledge arises from their perceptions of each other as fulfilling the roles of speaker and audience. The account of how common knowledge arises entails Grice’s general characterization of conversational implicatures. More precisely, it entails speakers and audiences involved in communication coordination problems typically commonly know that a relevant instance of the conditions holds. The common knowledge approach requires attributing reasoning to speakers and audiences, but it keeps those attributions to a plausible minimum. Thus the common knowledge approach is a viable alternative to current approaches that attribute extensive reasoning to speakers and audiences.

A common knowledge-based approach to pragmatics would answer at least the following question: What informative generalizations are possible about the role of contexts in generating common knowledge relevant to conversational implicatures? The question is important. “Much has been learned about these domains of psychology from a focus on the problem of altruistic cooperation and the mechanisms of reciprocity. We hope that comparable insights are waiting to be discovered by psychologists as they investigate the problem of mutualistic cooperation, and the mechanisms of common knowledge are—as we might say—put out there” (Thomas et al., “The Psychology of Coordination and Common Knowledge”).

Professor of Law, Chicago-Kent College of Law. This article is a companion piece to “Meaning, Reasoning, and Common Knowledge,” (Warner 2019b). I am gratefully indebted to Stephen Schiffer for his comments and encouragement and to Steven Wagner for many insightful conversations over the years.

Appendix

Assume the banner is flying over the stadium. Assume (1) everyone paying sufficient attention sees and thereby comes to know the banner is flying over the stadium, and (2) everyone knows (1). To see how (1) and (2) give rise to an infinite sequence of knowledge levels, suppose Alice and Bob are sitting together in the stadium, and they are looking up at the sign. Then:

First level::

Alice knows₁ that Bob sees the sign.

Bob knows₁ that Alice sees the sign.

Second level::

Alice knows₂ Bob knows₁ that Alice sees the sign.

Bob knows₂ Alice knows₁ that Bob sees the sign.

Third level::

Alice knows₃ Bob knows₂ Alice knows₁ Bob sees the sign.

Bob knows₃ Alice knows₂ Bob knows₁ Alice sees the sign.

How do Alice and Bob take this infinite series of steps?

Start with the first level. Alice knows that Bob sees the sign by reasoning this way: “I see Bob looking up toward the sign. He has normal perceptual abilities, so I can conclude that he sees the sign.” Bob reasons the same way about Alice. This gives us the first level: Alice knows that Bob see the sign, and Bob knows that they see the sign. They get to the second level by reasoning about each other’s reasoning at the first level. Alice reasons: “At the first level, I started from the fact I saw Bob’s looking at the sign, and I reached the conclusion that Bob sees the sign. Bob sees looking at the sign, and he has normal perceptual and reasoning capacities, so he will have reasoned just as I did to his conclusion that I see the sign.” For Alice to realize that fact about Bob is for her to reach the second-level conclusion: “I know₂ Bob knows₁ I see the sign.” Bob will reason in the same way to his second-level conclusion that he knows₂ Alice knows₁ he sees the sign. Thus: Alice knows₂ Bob knows₁ Alice sees the sign, and Bob knows₂ Alice knows₁ Bob sees the sign. Alice and Bob get to the rest of the levels the way they get from the first level to the second: by reasoning about their reasoning at the level below. For any level n, Alice reasons about Bob’s n−1 level reasoning to reach the conclusion that Alice knows_n Bob knows_n−1 … that Alice sees the sign, and Bob reasons in the same way to reach the conclusion that that Bob knows_n Alice knows_n−1 …. that Bob sees the sign.

There is an obvious problem of course. No one reasons in this way. It is extremely unlikely that Alice and Bob reason in a way that even approximates the reasoning we have attributed to her. The answer is that Alice and Bob do not need to derive the infinite sequence. They commonly know that they see the sign as long as each knows (1) that the other sees the sign and thereby comes to know the banner is flying over the stadium, and (2) each knows that the other knows (1). Then, to start with Alice, for any number n, she has the capacity to generate the sequence Alice knows_n Bob knows_n−1 … that Alice sees the sign, and Bob has the capacity to generate the sequence Bob knows_n Alice knows_n−1 … that Bob sees the sign. It was convenient to characterize that capacity by representing Alice and Bob as reaching successive knowledge levels through explicit reasoning. But it is the capacity that matters. It means that Alice and Bob are capable of ruling out any possibility of doubt or deception with regard to their seeing the sign at any level of knowledge.