The theorem that isomorphic structures agree on the truth values of all sentences is one of those theorems that cuts across normal logical boundaries. It is true for first-order logics, second-order logics, and higher-order logics. It is also true in infinitary logics. Certain monotony properties for quantifiers are like this as well, as pointed out by van Benthem. The medieval logicians knew about the latter; they probably were not aware of the former. Is there an “order-free” framework in which to state and prove these theorems? Is category theory capable of that? I’m curious since this makes two theorems that modern logical theorizing blinds us to the generality of.

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