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.

The most widely known formalism is the intuitionistic logic with impredicative quantification, System F. Parigot (1997) showed how this calculus can be extended to admit classical logic.

See also

• True quantified Boolean formula
• Second-order arithmetic
• Second-order logic
• Type theory

References

• Parigot, Michel (Dec 1997). "Proofs of strong normalisation for second order classical natural deduction". Journal of Symbolic Logic (published 12 March 2014). 62 (4): 1461–1479. doi:10.2307/2275652. ISSN 0022-4812. JSTOR 2275652.