The Many and the One, Ch. 5 & Ch. 6

I confess that I have never been able to work up much enthusiasm for mereology. And Florio and Linnebo’s Chapter 5, in which they compare ‘Plurals and Mereology’, doesn’t come near to persuading me that there is anything of very serious interest here for logicians. I’m therefore quite cheerfully going to allow myself to ignore it here. So let’s move on to Chapter 6, ‘Plurals and Second-Order Logic’. The broad topic  is a familiar one ever since Boolos’s classic papers of — ye gods! — almost forty years ago: though oddly enough F&L do not directly discuss Boolos’s arguments here.

In §6.1, F&L give a sketchy account of second-order logic, and then highlight its monadic fragment. Note, they officially treat the second-order quantifiers as ranging over Fregean concepts. And they perhaps really should have said more about this — for can the intended reader be relied on to have a secure grasp on Frege’s notion? Indeed, what is a Fregean concept?

The following point seems relevant to F&L’s project. According to Michael Dummett’s classic discussion (in his Frege, Philosophy of Language, Ch. 7), Fregean concepts are extensional items: while (for type reasons) we shouldn’t say that co-extensive concepts are identical, the relation which is analogous to identity is indeed being coextensive. So the concept expressions ‘… is a creature with a heart’ and ‘… is a creature with a kidney’ have the same Fregean concept as Bedeutung. I take it that Dummett’s account is still a standard one (the standard one?). For example, Michael Potter in his very lucid Introduction to the Cambridge Companion to Frege — while noting Frege’s reluctance to talk of identity in this context — writes (without further comment)

Concepts, for Frege, are extensional, so that, for instance, the predicates ‘x is a round square’ and ‘x is a golden mountain’ refer to the same concept (namely the empty one).

But now compare F&L. They write

Two coextensive concepts might be discerned by modal properties. Assume, for example, that being a creature with a heart and being a creature with a kidney are coextensive. Even so, these two [sic] concepts can be discerned by a modal property such as possibly being instantiated by something that lacks a heart.

Which seems to suggest that, contra Dummett and Potter’s Frege, co-extensive predicates can have distinct concepts as Bedeutungen. That’s why I really do want more elaboration from F&L of their story about the Fregean concepts which, according to them, are to feature in an account of the semantics of second-order quantification.

§6.2 describes how theories of plural logic and monadic second order logic can be interpreted in each other. And, analogously to §4.3, a question then arises: can we eliminate pluralities in favour of concepts, or vice versa?

So §6.3 discusses the possibility of using second-order language to eliminate first-order plural terms, as once suggested by Dummett. As F&L note, this suggestion has already come in for a lot of criticism in the literature; but they argue that there is some wriggle room for defenders of (something like) Dummett’s line to avoid the arguments of e.g. Oliver and Smiley and others. I’m not really convinced. For example, F&L suggest that a manoeuvre invoking events proposed by Higginbotham and Schein will help the cause — simply ignoring the extended critique of that manoeuvre already in Oliver and Smiley’s Plural Logic.  In the end, though, F&L think that there is a more compelling argument against the elimination of pluralities in favour of concepts on the basis of their respective modal behaviour (but note, F&L are here seemingly relying  on their departure from the standard Dummettian construal of Fregean concepts — or if not, we need to hear more).

§6.4 then looks at the possibility of an elimination going the other way, reducing second-order logic to a logic of plurality. But so far we have only been offered a way of interpreting  monadic second order logic using plurals; the obvious first question is — how can we interpret full second-order logic with polyadic predicates, quantifying over polyadic concepts? Perhaps we can do the trick if we help ourselves to a pairing function for the first-order domain (so, for example, dyadic relations get traded in for monadic properties of pairs). F&L raise this familiar idea: but suggest — again very briefly — that there is another modal objection: “while a plurality of ordered pairs can model the extension of a dyadic relation, it cannot in general represent all of its intensional features.” Tell us more! We also get a promissory note forward to discussion of a different objection to eliminating second-order logic.

There’s a short summary §6.5. But, to my mind, this is again a somewhat disappointing chapter. As it happens, my inclinations are with F&L’s conclusion that both plural logic and second order logic can earn their keep (without one being reduced to the other). But I do rather doubt that anyone who already  takes a different line will find themselves compelled to change their minds by the arguments so far outlined here.

To be continued, but after a break.