First-Order Logic, Second-Order Logic, and Completeness

Both firstand second-order logic (FOL and SOL, respectively) as we use them today were arguably created by Frege in his Begriffsschrift – if we ignore the notational differences. SOL also suggests itself as a natural, and because of its much greater strength, desirable extension of FOL. But at least since W. V. Quine’s famous claim that SOL is “set theory in sheep’s clothing” it is widely held that SOL is not proper logic – whatever this is taken to be by different authors – but some kind of mathematics. Even contemporary advocates of SOL like Stewart Shapiro point out its mathematical character, albeit without regarding this as problematic. Recent criticisms focus both on the ontological commitment of SOL, which is believed to be to the set-theoretic hierarchy, and on the allegedly problematic epistemic status of the second-order consequence relation.