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Introductions such as the one from SEP state that Montague semantics is a higher-order logic. I understand that Montague makes heavy use of higher-order functions, but does he ever use higher-order quantifiers?

In all the examples I have seen, the interpretation of a grammatical sentence is always a closed first-order logic formula. Am I mistaken thinking that Montague is in fact higher-order computation of first-order modal logic?

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  • Part of that may be due to the fact that first order quantified logic is known to be consistent, while second order quantified logic is not.
    – jlawler
    Mar 18, 2022 at 13:41

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