And having said that I would write about Linnebo’s paper next, I find myself rather regretting that promise, and this will have to be a non-comment!
Linnebo begins by announcing that the “strongest argument against the coherence of unrestricted generalization” is Williamson’s variant Russell paradox about interpretations; and he then takes the most promising line of reply on the market to involve adopting a kind of hierarchical type theory. Then Linnebo locates what he thinks to be a problem with the usual kind of “type-theoretic defences”. So he changes tack, and offers a different, more revisionary response to Williamson’s argument, which depends on rethinking the very idea of an interpretation (so that now predicates get not extensions but rather properties as their semantic values). He then needs a whole framework for talking about properties as well as sets (which has, he says, some similarity with Fine’s deviant project in his 2005 paper “Class and membership”), and this gives us a new sort of hierarchy of semantic theories. As with the old-style type-theoretic defences, we can now use this new hierarchical apparatus to blunt Williamson’s argument.
Now, how exciting/illuminating you find all this — given our interest here is in absolute generality — will depend, in part, on whether you think that the Williamson variant on Russell’s paradox gets to the very heart of a certain kind of argument against the possibility of absolute general quantification, or alternatively think that it somewhat muddies the waters by dragging in tangential issues (e.g. in its talk of interpretations as objects, etc.). My hunch so far has been the latter, though of course I’m open to persuasion; and the very complexities of Linnebo’s excursus don’t do much to dislodge that hunch so far.
I might return to Linnebo later, since Rayo’s paper seems to take us back into similar territory. But at the moment, I really don’t think I have anything useful to say. So let’s for now rapidly move on to consider Parsons’s paper.