Skolemizing Laws

Suppose you’re a Humean best-system theorist about laws of nature. For S to be a law of nature is for S to feature (as an axiom? As a theorem?) in the best theory that predicts local matters of fact.

Suppose we have some candidate theory T—mentioning perhaps material objects, their physical properties, and locations in space-time. T is interpreted—-perhaps via interpretation I with domain D. Now apply the Loewenheim-Skolem theorem to this, to produce a countable domain D* with pretty much the same interpretation I. (That is: any names have the same referents, and the extension assigned to predicates is as on the original except intersected with the new domain. If you thought of the interpretation as first assigning properties, whose extension relative to a domain was just the set of things within that domain that instantiate the property, they we can leave the interpretation untouched). D*, I will make true exactly the same sentences in the language of T as D,I does. And in particular, it makes true the sentences that figure in T. So let T continue to stand for the interpreted theory, whose quantifiers range over D, and T* stand for the distinct interpreted theory, whose quantifiers range over D*. T and T* are syntactically identical (hence have the same syntactic consequences). And since their interpretations differ only wrt the quantifiers, if we concentrate on their predictions for “local matters of fact” (which I’ll assume for now can be expressed using quantifier-free vocabulary) we won’t find any differences in how these consequences are interpreted. So it looks like they “fit” the data to exactly the same extent.

If T is our best theory, then the laws will be exceptionless regularities. If T* is the best theory, then the laws will be restricted. After all, the domain of T*, and hence the scope of the quantifiers featuring in its generalizations, is countable, and so only ranges over e.g. countably many regions of space-time.  Datum: the scientific laws are those that T projects, not the cunning skolemite restrictions that T* talks about. How does the Humean secure this?

In Lewis’s story, we get this via simplicity considerations. Simplicity is measured for Lewis by economy of expression in a canonical language. And (on the way I read Lewis) the canonical language will contain an unrestricted quantifier (or at least, something unrestricted enough to range over all physically relevant stuff). But there’s a weirdness here. The principled fix that Lewis gives us on the canonical language concerns the predicates it contains (those corresponding to natural properties, rather than artificial stuff like “being such that T obtains”). But it’s really not clear what the principled reason is for selecting the rest of the canonical language—-why it should contain these connectives, those quantifiers, or whatever. Ted Sider notes this sort of lacuna, and proposes extending the Lewisian natural/non-natural distinction so it covers all categories of entity. That would do the trick here, if with Sider you think that the unrestricted quantifier is perfectly natural. But this is a considerable metaphysical commitment, both in extending the naturalness ideology and in endorsing the particular thesis about its application to unrestricted quantification (that’s partly Sider’s point). Instead of going Sider’s way, perhaps the line will be just that the canonical language contains unrestricted quantification, and there’s *no* deeper explanation for why. That’s just the way it has to be for the Humean proposal to return the extensionally correct results. Ok: but a bit disappointing if that’s the end of the story.

Another thing to focus on here is whether T* is as informative as T. After all, telling us only about how countably many things behave *sounds* less informative than telling us about how *all* things behave. But there’s an issue with how we unpack this idea. First, as noted, syntactically they have the same consequences (and in the quantifier-free case there’s no relevant difference of interpretation).

But there’s a more semantic way of understanding “informativeness”. It could be measured by, e.g. the volume of worlds at which the theory is true (smaller volume=more informative). In general, modal approaches to informativeness (whether framed in terms of volume, or the shape of the region defined in modal space, or anything else) will turn on whether the theory is true in this world vs. that world. To get a grip on this, we need a theory equipped not merely with an extensional interpretation, but with an intensional one. We can run the skolemite trick to build up something like this too (walk round from world to world, and at each world w, apply the skolemite construction to the extensions induced at w by the intended unrestricted interpretation. You now have countable domains for each w, such that holding the rest of the interpretation untouched, you get exactly the same L-sentences coming out true at w on the restricted as on the unrestricted interpretation. Now you can piece together a property P, whose extension at w is the skolemite domain you constructed at w. And by keeping the original intended intensional interpretation, except restricting the quantifier to P, we guarantee that that the resulting intensional interpretation will assign the same sentence-intensions to L-sentences as the intended one did. Net result: if we take T* to be the interpreted theory with quantifiers restricted to P, we have the same informativeness as T by any modal criterion. And of course, at every world, T* ranges over only countably many things.

So simplicity might knock T* out, but at least on Lewis’s proposal, only because we built unrestricted quantification rather than skolemite quantification into the canonical language from which simplicity is assessed from the get-go. Informativeness I don’t think would discriminate T from T*, if we extend the skolemite construction as indicated (this takes a bit of thinking through—at first it seems *obvious* that restricted claims are less informative than unrestricted ones—it’s well worth thinking through exactly how the skolemite construction works to undercut that natural thought). Is there some more satisfying way for the Humean to establish that laws of nature are truly general and unrestricted?

Here’s one idea : in order to derive *any* predictions about local matters of fact, it looks like we need in the language of the theory singular terms for local circumstances (the *local* prediction isn’t that any scenario with such-and-such qualitative characteristics will have thus-and-so features; but that *this* scenario has thus-and-so features, *because* it has such-and-such characteristics). And there’s a question about what singular terms do feature in the  language of the theory. The pure language of science (or Lewis’s canonical language) may not contain demonstratives—but to account for applications then it’s natural to supplement it with just such resources. If we just supplement the canonical language with just that stuff we actually use in applications, then presumably there’ll only be countably many such singular terms, and the argument can run as previously. If we supplement it with terms for each scenario whatsoever, then the argument is blocked—-for there’ll be more than countably-many local scenarios, and each such named scenario will have to feature in the domain.

(Thanks to Jason Turner for discussion of some of these ideas; though he shouldn’t be blamed for deficiencies in the above. Literature question: are there places out there where this issue is discussed?)