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Good stuff. I particularly appreciated the section on Benacerraf.
Do you have anywhere where you elaborate on the metaphysics of structures? Here you say that they get defined by propositional functions, but I'm curious as to whether you've fleshed out the metaphysics further.
(Structures are just standard mathematical objects with a signature, domain, and distinguished relations) The article is instead about abstract structure, i.e., what all isomorphic copies "have in common". And these are explained as propositional functions (which are categorical in the appropriate sense; they only have one model-up-to-isomorphism). They're very closely related to what Carnap called "state descriptions". So, the metaphysical question concerning the metaphysics of propositional functions, whose argument places are second-order. Such a propositional function is a function which takes relations (in intension) as arguments and yields a "saturated" proposition as the value.Jeff