The logic of paradox, , is a first-order, three-valued logic that has been advocated by Graham Priest as an appropriate way to represent the possibility of acceptable contradictory statements. Second-order is that logic augmented with quantification over predicates. As with classical second-order logic, there are different ways to give the semantic interpretation of sentences of the logic. The different ways give rise to different logical advantages and disadvantages, and we canvass several of these, concluding that it will be extremely difficult to appeal to second-order for the purposes that its proponents advocate, until some deep, intricate, and hitherto unarticulated metaphysical advances are made.
"Second-Order Logic of Paradox." Notre Dame J. Formal Logic 59 (4) 547 - 558, 2018. https://doi.org/10.1215/00294527-2018-0011