In §3.3 of Conceptions of Set, Luca discusses what he calls the ‘no semantics’ objection to the iterative conception. He sums up the supposed objection like this:
Consider the case of iterative set theory, which for present purposes will be our base theory Z+. Since the set-theoretic quantifier is standardly taken as ranging over all sets, it seems that one of the interpretations quantified over in the definition of logical validity for L [the standard first-order language of set theory] – the intended interpretation – will have the set of all sets as its domain. But there can be no set of all sets in Z+, on pain of contradiction. Hence, the objection goes, if we take all sets to be those in the hierarchy, we cannot give the usual model-theoretic definition of logical validity.
Or rather, that is how the objection starts. Of course, the further thought that is supposed to give the consideration bite is that, if we can’t apply the usual model-theoretic definition of logical validity, then we are bereft of a story to tell about why we can rely on the inferences we make in our set theory when we quantify over all sets.
As Luca immediately remarks, this challenge is not especially aimed at the iterative conception: any conception of the universe of sets that rules out there being a set of all sets will be open to the same prima facie objection. It looks too good to be true!
Graham Priest is mentioned as a recent proponent of this objection. But as Luca point out, Kreisel over fifty years ago both mentions the issue raised in the quote and has a response to what I called the further thought which is supposed to make the issue a problem. For Kreisel’s ‘squeezing argument’ is designed precisely to show that we have a perfectly good warrant for using standard first-order logic as truth-preserving over all structures, not just the ones that can be formally regimented in the usual model-theoretic way. I’ve defended Kreisel’s argument, properly interpreted, e.g. here: so I’m more than happy to go along with Luca’s endorsement.
Luca does, however, have other things to say about the ‘no semantics’ objection before turning to Kreisel’s way out. As he notes, we can hold onto the idea that we can sensibly quantify over all sets, and hold on to the core of the classical Tarskian definition of validity, by denying that domains have to be taken to be sets. Of course, there is little point in arm-wavingly talking about classes instead of sets as if that by itself resolves anything. What we need to do is to reject what Richard Cartwright calls the All-in-One Principle which tells us that to quantify over some things presupposes that there’s a set (or proper class, or other single object-in-its-own-right) to which they all belong. We can speak, if you like, of virtual classes, classes-as-many, i.e. we can use a singular idiom for talking about objects, plural. But better, we should just go straight to giving the semantics for FOL in a plural metalanguage, saying that quantifiers range over objects (one or many), interpretations assign e.g. monadic predicates some of these objects (zero, one or many), and so on. We know this can be done, and the plural metalanguage itself formalized — Oliver and Smiley show how do this in all the detail you could want in their Plural Logic. I’m really not sure why Luca doesn’t mention the possibility of taking the plural route [added meaning something like the Oliver/Smiley version] here, as it would surely give him [added a simple and direct] rebuttal of the further thought that drives the ‘no semantics’ objection.
What Luca does discuss is another way of giving up the idea that the domain of quantification is an object which he finds in work by Rayo, Williamson and Uzquiano. They propose a higher-order semantics where our metatheory is to be regimented in a second-order way. [Added The second-order semantics, as Luca points out, can be given a plural interpretation, but I don’t find the second-order version here] especially natural, and it is not at all clear to me why going via higher-order logic should be thought a better bet than staying first-order but allowing plurals.
2 thoughts on “Luca Incurvati’s Conceptions of Set, 6”
[From Luca Incurvati] Note that I say that “models are taken to be given by the *objects* which a monadic second-order variable I is true of. On this higher-order semantics, the ‘domain’ is […] not construed as a set, but as whatever is in the range of our quantifiers: when we speak of the domain, we are not speaking of an object of some kind; rather, we are speaking of the *objects* which are the values of I” [emphases not in the original]. So although various interpretations of the higher-order variables are possible, it is the plural interpretation that I have in mind in the section on higher-order semantics. (Indeed, that’s also the official interpretation Rayo and Uzquiano work with in their 1999 paper, for instance). So I *am* taking the plural route here (although I’m also keeping things general enough, so that the reader can see what’s really doing the work, namely rejecting the assumption that the domain is an object). Logically, what’s behind all of this is that, as Boolos showed, monadic second-order logic (interpreted as involving quantification into predicate position) and first-order plural logic are equi-interpretable (and the differences between monadic and polyadic second-order logic don’t really matter here, since we have enough set theory in the background to be able to replace quantification over relations using ordered pairs).
Thanks for this! I wasn’t as clear as I should have been, and I’ve added a few words to the original comments. Oliver and Smiley, as I recall, explicitly say that one reason for adopting their plural first-order framework is that it deals with the quantifying-over-all-sets issue. My thought was just that this seems a more direct and natural way of tackling the issue than going second-order and then (controversially?) treating second order quantifiers via plurals.