## 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.

It seems to me that the two views have the following advantages/disadvantages:

Lewis holds that the world comes pre-packaged with a domain of individuals and a domain of relations. That certain instances are related by R is a primitive fact. Our predicate 'R' refers to R, because the latter is a 'reference magnet' (unlike some arbitrary set of sequences of individuals). The problem with this is that our practice of using the predicate 'R' may not match up with the real extension of 'R'. The result is a type of incorrigible error.

The other view is that 'R' refers to R because the total theory that contains this predicate is made true by a sequence of 'arbitrary' (whatever that means) sets of sequences of individuals. This is global descriptivism. The problem with this view, it appears, is that our practice of using the predicate 'R' lines up too easily with the extension of R. I'm not sure if this is Chalmers' view, but he's closer to it than Lewis.