Wolfgang Schwarz

Second-order logic and Newman's problem

How much can you say about the world in purely logical terms? In first-order logic with identity, one can construct formulas like '(Ex)(Ey)~(x=y)'. But arguably, this doesn't yet mean anything. As we learned in intro logic, formulas of first-order logic have no fixed interpretation; they mean something only once we provide a domain of quantification and an assignment of values to predicate and function symbols. As it happens, '(Ex)(Ey)~(x=y)' doesn't contain any non-logical predicate and function symbols, so to make it mean anything we just need to specify a domain of quantification. For example, if the domain is the class of Western black rhinos, then the formula says that there are at least two Western black rhinos.

Perhaps every way a world might be comes with a built-in universal domain: the domain of all individuals. We could then stipulate that formulas of first-order logic are always to be evaluated relative to the universal domain. In this dialect of first-order logic, which enjoys some popularity among philosophers, '(Ex)(Ey)~(x=y)' is true iff the universal domain contains at least two individuals. In general, every formula of pure first-order logic (without non-logical terms) is then equivalent to a statement about the cardinality of the universal domain.

This interpretation actually comes in several flavours, depending on how we cash out the domain of "individuals" associated with a way a world might be. Shall we include numbers and sets? Arbitrary mereological fusions? Merely possible objects? Shall we count Lumpl and Goliath as one or as two? On an attractively simple view, all such questions correspond to different ways a world might be (epistemically speaking): a world might be such that there are numbers or such that there are no numbers, and if there are no numbers, it would be wrong to include them in the domain of individuals. I don't like this view, as I think it conflates substantive questions about the world with pragmatic or verbal questions about how to talk about the world. But I can still make sense of the idea that (substantively) different ways a world might be are associated with different domains of individuals. It's just that there is less variation than on the simple view: either all domains include numbers or none of them do; either all include gerrymandered fusions or none; and so on. Even on the simple view, one can of course restrict the domain associated with a way a world might be to, say, fundamental individuals (whatever that means) -- not as a substantial claim to the effect that nothing else exists but simply as a convention for the interpretation of purely logical formulas.

Now consider second-order logic. What can we say about the world with a pure second-order statement like '(ER)(Ex)(Ey)(Rxy)'? So far, we had to assume that any way a world might be determines a universal domain of individuals relative to which pure formulas are interpreted. What do we need for the second-order statement? It depends on the semantics of second-order logic.

On the "standard" semantics, the first-order domain of individuals is still all we need, since the second-order quantifiers range over sets of sequences of individuals. As a consequence, all pure second-order formulas are still equivalent to statements about the cardinality of the first-order domain; the only news is that we can now distinguish between different infinite cardinalities.

But in the present context, this interpretation of second-order formulas looks very artificial. A more natural and general idea is to assume that each way a world might be somehow determines both a domain of individuals and a domain of relations. '(ER)(Ex)(Ey)(Rxy)' is true iff the domain of relations contains a binary element that relates elements from the domain of individuals. If we really wanted, we could then choose to include relations for arbitrary sets of sequences of individuals, and we could choose to identify relations whenever they have the same extension, but we don't have to make these unnatural choices. In particular, we can limit the second-order domain to fundamental properties and relations (whatever that means).

This interpretation is especially natural on a combinatorial account of possibility on which all ways a world might correspond to different distribution patterns of fundamental properties and relations. Here it would be absurd to individuate properties and relations extensionally as sets of sequences of individuals.

Suppose we endorse the more natural (Henkin-type) interpretation of second-order formulas. Then it is no longer true that the only thing one can say by pure second-order formulas is how many individuals there are. What can be said in this austere language is now an interesting open question. I think Lewis is committed to the striking claim that all truths we can entertain are equivalent to purely logical truths (although in a slightly extended logical language that includes plurals and mereology, which effectively adds the expressive power of "standard" monadic third-order order logic to the Henkin-type second-order logic discussed here). In his Constructing the World, Dave Chalmers dismisses this idea on the grounds that the only truths we could entertain would then be truths about the cardinality of individuals -- as Newman famously objected to Carnap. But Newman's problem rests on a controversial and arguably implausible assignment of truth-conditions to second-order formulas.