Wolfgang Schwarz

Lewis's argument for Humility

In "Ramseyan Humility", Lewis argues for a thesis he calls "Humility". He never quite says what that thesis is, but its core seems to be the claim that our evidence can never rule out worlds that differ from actuality merely by swapping around fundamental properties. Lewis's argument, on pp.205-207, is perhaps the most puzzling argument he ever gave.

Lewis begins with some terminology.

Let T be a theory stating the true laws of fundamental physics. Lewis seems to think of T not as a model of some kind, but as a set of sentences. The language of T contains T-terms denoting fundamental properties. "All the rest of our language" Lewis calls O-language. (This seems to imply that the T-terms are also part of "our language", which is not obvious, but plausible if the true laws of physics are approximately what we think they are.)

T can be axiomatized by a single, possibly infinite sentence. The Ramsey sentence of T is the sentence that results from that axiom by replacing all T-terms with existentially bound variables. "What we need to know about the Ramsey sentence", Lewis says (and proves in footnote 6), "is that it logically implies exactly the O-language sentences that are theorems of T."

Lewis also introduces a few more substantive assumptions at this stage.

First, he says that the T-terms are "implicitly defined by T". That is, if T says that mass does so-and-so, then 'mass' can be defined as whatever quantity does so-and-so.

Second, Lewis assumes that "no fundamental properties are named in O-language, except as occupants of roles", and that "O-language does suffice to express all possible observations". (The latter looks like a rather strong assumption, given that O-language was introduced as part of our language. One might have thought that not all differences between possible observations can be captured in English. But let's set that worry aside.)

Now there is a natural line of thought leading from these assumptions to Humility. Assume that there is a world w that is structurally just like the actual world but where mass has been replaced by some other quantity, schmass. (Lewis's arguments for that assumption is also surprisingly weak, but here I want to focus on the epistemic aspect of his argument.) How could we find out that w is not actual? Presumably by making some observations that would reveal to us that tables or chairs or electrons have mass and not schmass. However, we are not directly acquainted with mass: we only know mass as the occupant of a certain role. Suppose on the basis of some experiments we come to accept the claim that 'electrons have mass'. This only tells us that electrons have whatever property plays the mass role. But that much is also true in w. So no possible observation could reveal to us that w is not actual.

The argument has some gaps, but it doesn't look entirely unconvincing. Perhaps the largest gap is that there are countless other ways to rule out w besides discovering that tables or chairs or electrons have mass rather than schmass. For example, if water essentially contains hydrogen and hydrogen essentially contains a proton and protons essentially have mass, then it would also suffice to discover that there is water, since there is no water in w. However, it is not implausible that we also know water only as the occupant of a certain role.

In any case, the above argument is not the argument Lewis gives. Lewis's argument (on p.207) goes as follows. Quote.

O-language, we assumed, is rich enough to express all possible observations. Therefore any predictive success for T is equally a predictive success for the Ramsey sentence of T. Since the evidence for T consists in its record of predictive success, there is no way to gain evidence for T that is not equally evidence for the Ramsey sentence.

Though our theory T has a unique actual realization, I shall argue shortly that it has multiple possible realizations. Suppose it does indeed have multiple possible realizations, but only one of them is the actual realization. Then no possible observation can tell us which one is actual, because whichever one is actual, the Ramsey sentence will be true. There is indeed a true contingent proposition about which of the possible realizations is actual, but we can never gain evidence for this proposition, and so can never know it. If there are multiple possible realizations, Humility follows.

Every single step here is puzzling.

Let's work through the first paragraph. We have assumed that O-language is rich enough to express all possible observations. We also know that T and its Ramsey sentence logically entail the same O-sentences. "Therefore", Lewis claims, "any predictive success for T is equally a predictive success for the Ramsey sentence of T." This is correct if we take a "predictive success" for T to be a true observation sentence logically entailed by T. But on that interpretation it is completely implausible that "the evidence for T consists in its record of predictive success", as Lewis assumes.

First of all, the laws of microphysics by themselves make very few observable predictions. In order to test a physical theory, we generally need background assumptions about boundary conditions, measurement devices etc. That may not be a serious problem, for arguably these background assumptions are expressible in O-language. Then we can take the evidence for T to consist in O-sentences entailed by T in conjunction with some other O-sentences B stating the background assumptions. It is not hard to show that the conjunction of T and B logically entails the same O-sentences as the conjunction of the Ramsey sentence and B.

But that takes us to a more important problem: not all entailment is logical entailment. 'I have a hand' and 'I have something' logically entail the same sentences in a language without the word 'hand'. But only the former predicts that I have a limb. So the mere fact that T and the Ramsey sentence logically entail the same O-sentences (which Lewis proves in footnote 6) is not enough to show that they make the same observational predictions.

Moreover, not all evidence is conclusive evidence. Why should we accept that the evidence for T consists solely in true observation sentences strictly entailed by T (perhaps in combination with background assumptions)? Couldn't T be also supported by observations that are only made much more probable by T than by non-T?

Relatedly, why must an observational prediction be entailed by T in order to count as evidence -- even conclusive evidence -- for T? After all, the fact that A entails B generally doesn't imply that B entails A. Couldn't there be an O-sentence A that entails T, although neither T nor its Ramsey sentence entail A?

In sum, it is utterly unclear how the argument in the first paragraph is supposed to support its conclusion, that "there is no way to gain evidence for T that is not equally evidence for the Ramsey sentence".

There is a simple argument for that conclusion which Lewis didn't give, but which would seem to sidestep all the problems just mentioned. The argument exploits the fact that on Lewis's account of theoretical terms, T and its Ramsey sentence not only logically entail the same O-sentences, they are analytically equivalent. The postulate of T logically entails the Ramsey sentence by existential generalization. The converse entailment, from the Ramsey sentence to the postulate, is ensured by the analyticity of the Carnap sentence of T, the material conditional with the Ramsey sentence as antecedent and the postulate as consequent. On the Ramsey-Carnap-Lewis account of theoretical terms, the Carnap sentence is indeed analytic. That's what Lewis means when he says that the theoretical terms are "implicitly defined" by T. And now, if T and the Ramsey sentence are analytically equivalent, then of course no empirical evidence can support one over the other.

We'll see why Lewis didn't exploit the analytic equivalence in a moment, when we turn to the second paragraph.

As above, I'll assume that there are indeed different possible realizations of T: worlds that differ by replacing one fundamental property with another. Then, Lewis claims in the second paragraph, "no possible observation can tell us which one is actual, because whichever one is actual, the Ramsey sentence will be true".

What comes after 'because' here is evidently right: whichever realization of T is actual, the Ramsey sentence of T is true. But how is that meant to entail that no possible observation can tell us which realization is actual? Presumably the conclusion of the previous paragraph is relevant here: there is no way to gain evidence for T that is not equally evidence for the Ramsey sentence. So Lewis seems to assume that if there were evidence specifically for the actual realization of T, then that would be evidence for T that is not equally evidence for the Ramsey sentence.

Note that this assumption would make no sense if T and the Ramsey sentence were equivalent formulations of one and the same hypothesis. For the purposes of his argument here, Lewis seems to assume that T is significantly stronger than its Ramsey sentence, so that if we could come to know that T is true, we would know which realization of T is actual.

Let's grant that for the moment and not worry about how it fits Lewis's views about theoretical terms. It is still unclear how the argument in the second paragraph is meant to go. Suppose some O-sentence A entails T. Since T entails the Ramsey sentence, A also entails the Ramsey sentence. In that sense, it seems that A is "equally evidence" for T and for the Ramsey sentence. Nonetheless, A tells us which realization is actual.

Finally, "if there are multiple possible realizations, Humility follows". -- Why is that? Why does Humility follow from the fact that we cannot find out which realization of the final physical theory T is actual? To be sure, if we could find out which realization is actual, Humility would be false. But the contrary is far from obvious.

I can't think of any way to construe Lewis's argument -- in line with what he is saying -- on which it doesn't contain several serious mistakes.

Some aspects of the argument make slightly more sense if we assume that T doesn't just state the true laws of nature, but the complete truth about everything (making nonsense of Lewis's claim in footnote 5 that we will hopefully discover T "someday soon"). The proposition expressed by T, call it T*, is then the singleton of the actual world. By contrast, the proposition R* expressed by the Ramsey sentence of T is the set of merely quidditistic alternatives to the actual world. Speaking in terms of propositions, it is plausible that any evidence for T* must be "logically entailed" by T* in the sense that it must be a superset of T*. For if a proposition isn't a superset of T*, then it is incompatible with T*, and surely nothing can be evidence for a hypothesis if it is incompatible with that hypothesis. Moreover, if Humility were false, we could come to know a proposition that is true at the actual world but not at some quidditistic alternative. That is, there should be evidence for T* that excludes other possibilities in R* and in that sense supports T* more strongly than it supports R*. If T* and R* entail the very same observation propositions, then that is impossible.

But why should we believe that T* and R* entail the same observation propositions? (Isn't that just a restatement of Humility?) Somehow the assumptions about theoretical terms and O-language would have to be relevant here, but it is not clear how. For one thing, it may be somewhat plausible that 'mass' is implicitly defined by some physical theory, but it is implausible that it is implicitly defined by the complete truth about everything. Moreover, Lewis's proof that T and its Ramsey sentence logically entail the same O-sentence is still beside the point.