Talk:Second-order propositional logic

Difference between "Second-order propositional logic" and "Second-order logic"?

What's the difference? I thought "propositional logic" was logic without quantifiers, first-order logic adds quantifiers over elements in some domain of discourse to propositional logic, and second-order logic adds quantifiers over prepositions to first-order logic. I don't see where "second-order propositional logic" finds its niche. Jason Quinn (talk) 15:36, 28 May 2010 (UTC)

There are quantifiers but only over propositions. So there is still no universe of discourse and no variables for individuals. For example, there are sentences like

$\forall A\exists B(B\leftrightarrow \lnot A)$

and

$\forall P\forall Q((P\land Q)\lor \lnot P\lor \lnot Q)$

— Carl (CBM · talk) 17:29, 28 May 2010 (UTC)