Second-order propositional logic
A second-order propositional logic is a propositional logic extended with quantification over propositions. A special case are the logics that allow second-order Boolean propositions, where quantifiers may range either just over the Boolean truth values, or over the Boolean-valued truth functions.
Parigot, Michel (1997). Proofs of strong normalisation for second order classical natural deduction. Journal of Symbolic Logic 62(4):1461–1479.
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