Since Benacerraf’s ‘What Numbers Could Not Be, ’ there has been a growing interest in mathematical structuralism. An influential form of mathematical structuralism, modal structuralism, uses logical possibility and second order logic to provide paraphrases of mathematical statements which don’t quantify over mathematical objects. These modal structuralist paraphrases are a useful tool for nominalists and realists alike. But their use of second order logic and quantification into the logical possibility operator raises concerns. In this paper, I show that the work of both these elements can be done by a single natural generalization of the logical possibility operator.

There has been a groundswell of interest in the account of modality that Kant sets forth in his pre-Critical Only Possible Argument. Andrew Chignell's reconstruction of Kant's theistic argument in terms of what he calls ‘real harmony’ has a prima facie advantage in that it appears to be able to block the plurality objection (namely, that even if every modal fact presupposes some ground, this does not entail that all modal facts share the same ground). I argue that it is both textually and philosophically problematic to interpret Kant's argument in terms of real harmony. Then, I set forth an alternative response to the plurality objection which does not require the adoption of the problematic notion of real harmony. Instead, I argue that the objection can be overcome by observing that the argument seeks to ground modal facts as a totality and that, according to Kant, such relations can be accounted for only by their schematization in a single intellect.