If relational quantification is reinterpreted as first-order quantification over set-theoretic proxies, the generalization of CL need not be genuinely second-order in logical kind.

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Generalization of CL(in logic): An expansion or broadening of 'classical logic' (the standard system of reasoning with true/false statements) to cover more complex situations.
Relational quantification(in logic and mathematics): A way of making logical statements about relationships between things, where you're saying something like 'for some things related to X' rather than just 'for some things.'
Set-theoretic proxies(in mathematics and logic): Using collections of items (called sets) to stand in for or represent something else, like using a mathematical container to represent a real-world concept.
first-order quantification(Cited as a motivation for using reification operators instead of higher-type Montagovian semantics): Quantification ranging only over individuals, as opposed to higher-order quantification which ranges over predicates or functions of higher types

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