MacFarlane’s project in his dissertation requires that he make sense of quantifiers in terms of his presemantics. The initial suggestion is to assign quantifiers the type ((O => V) => V), where O is the basic object type and V is the truth value type. Two problems arise for this. The quantifiers could receive interpretations that are sensitive to the domain of objects. Regardless the quantifiers receive different interpretations as the domains vary. This leads to the second problem. What do the variable domains represent? Why do we use them? MacFarlane follows Etchemendy here. Etchemendy says that there are two competing ways of understanding the variable domains, neither of which is satisfactorily captured by variable domains. The first is understanding the variable domains as representing the things that exist at each possible world, with models representing worlds. Three objections to this are given, only two of which I will mention. One is that it seems to make the strong metaphysical claim that for any set of objects at all, the world could have contained just those objects. There might be ways to respond to this; MacFarlane cites a couple of attempts, one of which appeals to “subworlds.” The other objection, which seems promising, is that this is hard to square this with the use of frames in modal logic. If the various domains are parts of worlds in different frames, then we must make sense of ways the very structure of possibility could have been, in MacFarlane’s phrase. This seems like a problem. I think some people have objected along these lines to David Lewis’s modal realism. Making sense of the moving parts of modal logic is hard.

The other way of understanding variable domains is as picking out different meanings for terms and quantifiers. This gets around some of the problems with the possible worlds understanding. It runs into a problem with cross-term restrictions, restrictions put on one class of terms by another class. It becomes unclear on this understanding why the same domain is used for both the universal and existential quantifier. It seems like one could stipulate the usual interdefinability. Etchemendy’s point leads to the question of why the same domain is used for singular terms as well as the quantifiers. It isn’t clear what a principled response to this would be. MacFarlane and Etchemendy both seem to find it decisive.

In response to these worries, MacFarlane suggests that the proper way of understanding the variable domains is not either of these two. Rather, he thinks that it is as “a specification of a presemantic type: the type O, from which semantic values for singular terms is to be drawn in an interpretation.” (p. 199) Using variable domains is just using different basic presemantic types. Before proceeding to the point that I really wanted to focus on, I want to comment on this. This is a better explanation of the variable domains if we have a good grip on what a presemantic type is. I’m not sure that I’ve enough of a grip on it that it would explain variable domains. The types are sets of things (or functions, constant functions or not) and functions on those (or sets of those). Does this really explain or give us a better understanding of the use of variable quantifier domains? It seems like we would want to appeal to the same intuitions used in variable domains, i.e. the possible meanings and possible worlds intuitions above, for the basic types. I’m not sure how the types fair with respect to the objections to the possible worlds view. It seems to get around the objections to the possible meanings view since all the types are defined with respect to the basic types and so build in the cross-term restrictions.

MacFarlane continues by saying that the basic types are indexical. That is, the basic types in the presemantic ontology are functions from contexts (or points of evaluation; MacFarlane does not use this term.) to domains or types. This is clearly based on work in the philosophy of language, such as Kaplan’s work. The basic type O is relative to an index, namely the sortal or set of sortals that specify object in a given interpretation. The idea, from Brandom, being that “object” and “thing” are presortals, which depend on context for complete specification since they do not carry with them the criteria of individuation that other sortals carry with them. I think this is something that he picks up from Quine, Evans, Strawson, Gupta, and others. The understanding of variable domains then depends on this view about sortals. Granted, MacFarlane points out that treating all the basic types as indexical in this way results in a simpler presemantic theory. Other logics can vary the type V of sentential values, say, assigning different sets of propositions in different settings.

Later on MacFarlane says that semantics should answer to postsemantics, which is rooted squarely in the notions of assertion and inference, again notions from the philosophy of language. MacFarlane suggests that a coherent, useful philosophy of logic should be rooted in other philosophical views, in particular from the philosophy of language. It might be useful to extend this to the philosophy of mind since logical notions on MacFarlane’s view are supposed to be normative for thought as such. I’m not sure how one would not hae to engage in some philosophy of mind with a claim like that.

This idea strikes me as quite sensible but I want to register a concern. If the justification for some views in the philosophy of logic come from some particular views in philosophy of language, then there is a worry about circularity arising when logical views are used to adjudicate disputes in the philosophy of language. Quine comes to mind as someone whose logical views figured prominently in his views on language. It is not necessary that a circularity arise; the justifying views in the philosophy of language may not be the ones that the particular philosophy of logic is being used to defend. Of course, this sort of dependence might not strike one as bad at all if one adopts a more coherentist outlook on things. It had surprised me since I had thought that logic and by extension the philosophy of logic were more foundational. As such, this sort of dependence on other areas of philosophy did not arise. If the foundational aspirations for the philosophy of logic are abandoned, then this is even less of a problem. This requires allowing more possible answers to the question: what can justify a view about the nature of logic? Possible answers to this question, both historically and in MacFarlane’s work seem rather restricted though so there can’t be that much widening. One could respond that while logic might play some foundational role, the philosophy of logic need not, and so justification in that domain can come from a much wider range of sources, even though it has not historically. To abuse a metaphor, while logic is near the center of one’s web of belief, the philosophy of logic stands farther out. I’m doubtful this can be right since one, especially post-Tarskian philosophers, would expect the philosophy of logic to answer, or address, the demarcation question: what is a logical constant? The answer to this changes what the logic at the core of that web looks like, and by continued abuse of metaphor, how large chunks of the rest of the web look. This suggests a bit more of a foundational role for the philosophy of logic.