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Reference.org
Second-order propositional logic
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<p>A second-order propositional logic is a <a href="/facts/Propositional_logic/MNt5v31V">propositional logic</a> extended with <a href="/facts/Quantification_(logic)/bgmvwxoo">quantification</a> over propositions. A special case are the logics that allow second-order Boolean propositions, where quantifiers may range either just over the Boolean <a href="/facts/Truth_value/SY4e2w8l">truth values</a>, or over the Boolean-valued <a href="/facts/Truth_function/aH5kBhQ7">truth functions</a>. </p><p>The most widely known formalism is the <a href="/facts/Intuitionistic_logic/zI7ulCnM">intuitionistic logic</a> with impredicative quantification, <a href="/facts/System_F/odpwVpAG">System F</a>. Parigot (1997) showed how this calculus can be extended to admit <a href="/facts/Classical_logic/ls62MAms">classical logic</a>. </p>