Lewis: "Humean Supervenience Debugged" (1994)

In this paper, David Lewis wrings his hands at a phenomenon he calls "undermining." He considers a probabilistic model as "undermined" if it assigns a positive probability to a data set that would cause a rational agent to adopt a different model.

"Contradiction!"

To say this in a vocabulary closer to Lewis', suppose that C(F | E) is the posterior probability a rational believer would assign to the event F in light of the evidence E. Suppose further that we are looking at the specific case in which E reveals the actual parameters of the world (e.g., "This coin has bias 0.3"), and F is a possible future which would produce a different subjective belief in our hypothetical observer (e.g., "The empirical frequency will be 0.4")

The question is then: Given E, does the possible future F have zero probability, or positive probability? Without giving any argument, Lewis asserts that

there is some present chance that events would would go in such a way as to complete a chancemaking pattern that would make the present chances different from what they actually are. (p. 482)

But he contrasts that with the following "argument":

But F is inconsistent with E, so C(F/E) = 0. Contradiction. (p. 483)

The former of these quotes seem to indicate that he is thinking about a finite sample from the model (consistent with the example he gives on p. 488). The latter argument, on the other hand, seems to assume that he is talking about a limiting frequency from an ergodic process or something like that — unless he seriously believes that empirical frequencies cannot differ from parameter values.

The Super-Objectivist

But this way of putting the argument is of course alien to Lewis. He has no concept of a statistical model, and he thinks that the credence of a rational agent is a unique and well-defined concept that doesn't require any assumptions:

Despite appearances and the odd metaphor, this is not epistemology! You're welcome to spot an analogy, but I insist that I am not talking about how evidence determines what's reasonable to believe about laws and chances. Rather, I'm talking about how nature—the Humean arrangement of qualities—determines what's true about laws and chances. Whether there are any believers living in the lawful and chancy world has nothing to do with it. (pp. 481–82)

This is even stronger and more absurd than classical objectivism. Instead of just discarding certain models as inconsistent with the evidence, Lewis assumes that the evidence suggests a single optimal model out of its own accord. For no apparent reason, he also wants "nature" to do this in a retrospective manner even though there is no reason to, given that he has expelled all subjective observers from the universe.

The Big Flip

Lewis' own solution to the "paradox" is to say that credences should be conditioned on "theories" as well as data — but "theory" doesn't quite mean what it sounds like. This is evident from the example he gives towards the end of the paper.

In this example, he assumes that a coin has exhibited a frequency of 2/3 heads in the past, and he assumes that this means that our hypothetical rational agent estimates its bias to be 2/3.

The "theory" T that he wants us to consider is then that the next 10,002 coin flips exhibit a frequency of exactly 2/3 heads, i.e., 6,668 heads and 3,334 tails. This event has the binomial probability

Pr(T) = B(6,668; 10,002, 2/3).

He then asks us to consider a possible future A in which the next four coin flips come up heads. Still using the parameter estimate of 2/3, this has the binomial probability

Pr(A) = B(4; 4, 2/3).

What is the conditional probability Pr(A | T)? Since the "theory" T did not change the parameter estimate 2/3, one might think that it equals the unconditional probability Pr(A). But for no apparent reason, Lewis decides to take the four coin flips in A from the coin flips in T, producing an amputated event T' with 3 fewer heads and 1 fewer tails. Even more oddly, he computes Pr(A, T') as if the two events were independent even though the observation of either would clearly change the parameter estimate used to compute the conditional probability of the other.

So according to his logic, the "joint probability" of A and T' is then

Pr(A, T') = B(4; 4, 2/3) B(6,665; 9,998, 2/3).

By dividing this by Pr(T), he supposedly finds the "conditional probability" of A given T.

This computation is, of course, completely absurd. If the parameter had been 1/3 instead of 2/3, it would have produced a "probability" larger than 1. So I'm afraid the example isn't doing much good.

Lewis: Convention (1969)

Lewis' definition of "convention" (p. 78) crucially relies on the concept of common knowledge as well as rationality: For a equilibrium R to be a convention, it must be common knowledge to everyone that the majority conforms to R.

So mere behavioral adaption without a mutual ascription of rationality doesn't count as "convention" (cf. p. 59). Every player has to think that every other player is rational and has the same knowledge as him- or herself. You can't think that you're the only one who's not a robot and still call it a "convention" according to Lewis.

He also excludes solutions to trivial coordination games (p. 78, item 5). There has to be at least two distinct equilibria available so that the fixation in one or the other becomes truly contingent. He doesn't seem to entertain the possibility that there could be something like degrees of contingency.

Lewis on bias and analogy in Convention (1969)

David K. Lewis' short book Convention: A Philosophical Study (1969) is an attempt to explain meaning in terms of equilibria in coordination games. The concepts that do the most of the heavy lifting are precedent, analogy, and salience.

The Ambiguity of Precedent
Lewis is aware that no two situations are ever alike, and that agent thus have to engage in some kind of extrapolation:

Suppose not that we are given the original problem again, but rather that we are given a new coordination problem analogous somehow to the original one. Guided by whatever analogy we notice, we tend to follow precedent by trying for a coordination equilibrium in the new problem which uniquely corresponds to the one we reached before. (p. 37)

A consequence of this is ambiguity. He immediately continues:

There might be alternative analogies. If  so, there is room for ambiguity about what would be following precedent and doing what we did before. Suppose that yesterday I called you on the telephone and I called back when we were cut off. We have a precedent in which I called back and a precedent—the same one—in which the original caller called back. But this time you are the original caller. No matter what I do this time, I do something analogous to what we did before. Our ambiguous precedent does not help us. (p. 37)

Or, that is exactly the big question: Whether and when precedent decides or even strictly determines what a construction means in some new situation.

Finding A Relevant Precedent
We then have a two-dimensional similarity space (me/you differences and caller/receiver differences). If these dimensions have the same weight, then both analogies yield the same average fit:

The big question is then how convention is possible at all, if everything is similar to everything else in some respects. Some sense of immediate similarity must be picked up along the way or have been there all along.

Lewis doesn't seem to have any psychological theory of salience or of similarity between new and old situations. However, one could construct a similarity measure based on possible courses of action, so that situations are similar when their elements can be handled in a similar way.

For instance, electricity can be much like water because many of our intuitions about how it behaves are reliable. Thus source elements such as "source," "direction," "pressure," etc. can be paired with certain target elements without needing much behavioral adjustment.

Similarly, source elements as "front" and "back" can be paired with the screen-side and the wall-side of a television in either of two ways. These pairings will on average require less behavioral adjustment than, say, pairing "head" and "feet" with screen-side and wall-side. A few candidate analogies are thus consistent with our prior knowledge, although no single analogy takes absolute priority.