Parsons’s Mathematical Thought: Sec. 5

Parsons has been proposing the view that “speaking of objects just is using the linguistic devices of singular terms, predication, identity and quantification”. And the focus so far has been on first-order quantification. But what about generalizations about properties, the sort of generalization involved in familiar mathematical statements like the induction principle for arithmetic, or the separation axiom in set theory? Should we construe those as involving generalization over something like Frege’s “unsaturated” concepts, entities which aren’t objects? Or is the commitment here just to more objects? I’ll try to outline some of Parsons’s discussion (though I did not find it always easy to construe).

One way of perhaps resisting the Fregean line arises from noting that we can easily parlay quantification into predicate position into just more quantification into subject position (or so it seems). Suppose, using Parsons’s notation, we use ‘(Ox)Fx‘ to denote some object corresponding to the Fregean concept expressed by ‘F…’. And suppose we use ‘$’ for an appropriate copula (‘has’ if the object is a property/quality, ‘is a member of’ if the object is a set, etc.) Then we have Ft if and only if t $ (Ox)Fx. And so, given a context when we are minded to quantify into the position held by ‘F‘ we could instead first nominalize and then quantify into the position held by the singular term ‘(Ox)Fx‘ instead. It seems then that we can treat quantification over properties (as we might initially put it) as just more quantification over a kind of object. This after all seems common mathematical practice, as when we familiarly regiment second-order arithmetic as a theory of numbers and sets of numbers.

Still, at least two objections to the nominalizing strategy as an across-the-board way of eliminating ‘direct’ quantification into predicate position readily suggest themselves (as Parsons notes). First, the claim that Ft if and only if t $ (Ox)Fx is, itself, intended as a generalization, to express which we need to generalize into predicate position in a way that can’t be nominalized away. And second, that generalization in any case has to be restricted or else or we could instantiate with the predicate ‘¬x $ x‘, and paradox ensues.

However, that’s not yet game set and match to the Fregean. Can’t the force of the first objection be turned by adding the device of semantic ascent to our armoury? We can, for example, generalize about the possibility of nominalization by saying that for any predicate ‘F’ (and term ‘t’), ‘Ft‘ is true if and only if ‘t $ (Ox)Fx‘ is true.

Ah, it will be protested, the device of semantic ascent still doesn’t really allow us fully to capture what we want to say by means of quantifications over properties. Compare for example the familiar thought that the content of the full informal arithmetic induction axiom is not captured by semantically ascending and saying that all instances of the first-order schema are true. Reply: that familiar thought is true, if we confine the instances to a fixed language. But suppose we treat the schema in an open-ended way, available to be instantiated however we extend our language (as Parsons puts it, “In practice, in any language in which we talk about natural numbers, we are prepared to affirm induction for any predicate of that language”). Then, by treating the schema as open-ended we arguably recapture the intended sweep of the informal axiom still without taking on ontological commitments to Fregean concepts.

And as to second objection against the nominalizing strategy, the threat of paradox only arises if we take the reference of ‘(Ox)Fx‘ as an object that is, so to speak, already in the original domain of objects (i.e. of subjects of predication). But we could take the moral here to be that objects segregate into different types, the references of nominalized predicates being of a different type to the references of common-or-garden singular terms.

So where does this take us? Parsons summarizes: “the present discussion does show that considerations about predication do not lead inevitably to our taking second-order logic as our canonical framework and admitting, as values of our second-order variables, entities that are not objects.”

Three comments about all this. First, about semantic ascent and the open-ended nature of our commitment e.g. to the induction schema. Just why do we stand prepared to take on all-comers and instantiate the schema with any novel predicate we care to extend our language with? Kreisel suggested long since that we accept the instances of the induction schema because we already accept the full second-order induction axiom. I think there are issues about that claim (which I can’t pursue here and now). But the claim is a familiar one that many have found persuasive. And a fuller defence of the idea that we can avoid taking second-order quantifications at face value would require Parsons to say more about this.

Second, about avoiding paradox on the nominalizing strategy. The Fregean might well riposte that saying that the way to go is to segregate objects into different types just sounds like theft of Frege’s key insight rather than an alternative story. After all, speaking with the vulgar, the Fregean will say that what he is arguing for is precisely a distinction among “entities” between saturated and unsaturated types, between objects and concepts. So he has a principled type story to tell. And, he will add, once the distinction is made in the right way, the temptation to pursue the nominalizing strategy, putting all the work of unifying propositions into a copula, should evaporate. And what is the alternative principled story supposed to be?

Third, I’m left unclear exactly how Parsons thinks about the relationship between the two ways of avoiding second-order quantification that he discusses (i.e. the routes via nominalization and ascent). He does say that “The laws of logic have a certain dialectical character, in that the method of nominalization and the method of semantic ascent can both be used to state them, and neither can completely displace the other.” I’ve wrestled with this a bit, and I don’t have a clear grasp of the point. (And helpful comments on that here would be welcome!)

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