Chapter 6 of MacFarlane’s dissertation is a study of the permutation invariance criterion of logicality, which says that an operation is logical iff it is invariant under all permutations of the domain. As an historical note, MacFarlane mentions that the idea of permutation invariance comes from Mauntner and Tarski and originates in Klein’s Erlangen program. The Erlangen program was an attempt to classify geometries by the classes of transformations under which their basic notions were invariant. For example (examples from MacFarlane’s thesis actually), the notions of Euclidean geometry are invariant under similarity transformations and those of topology under bicontinuous transformations. Since the logical is presumably broader than the geometrical, its notions should be preserved under a larger class of transformation. While Tarski was a logician and philosopher, Mautner was more a mathematician and Klein was certainly a mathematician. Resting our notion of logicality on the notion of permutation, as in the permutation invariance criterion, seems to put the mathematical before the logical, i.e. requires permutations to get a grip on (or to explain, to be a bit less loaded) the logical notions. Since this idea is a grandchild of a mathematical program, it is not surprising that it would give rise to various approaches to logic that rely so much on mathematical notions. (This is by no means a bad thing. I’m all for it.)

There is one thing I want to note about the permutation invariance criterion. If the criterion is necessary and sufficient for logicality, then the logical is tied to the representational approach, that is, there must be a domain of things represented that is invariant under permutations (at least for the first-order case; although it looks like a similar point could be made about, say, truth functions in the propositional case.). This seems to put it in tension with inferentialism. This may not be the case if the inferentialist is not giving a criterion for logicality. I’m not sure that anyone would be happy with this. An inferentialist might claim that the inferential rules give the meaning of the logical constant whereas the permutation invariance criterion tell us which operations are logical. This doesn’t seem promising either for two reasons. First, this raises the further problem of explaining when a certain constant designates or otherwise represents a logical operation. Second, (and putting the first to one side) if we already have a the logical operations, what further use is there for a distinct meaning given by the inference rules? It would appear none. This leaves the inferentialist looking a little left out in the cold. Of course, it would likely be illuminating if some link between an inferentialist criterion and the permutation invariance criterion could be found.