What’s So Bad About Second-Order Logic?

Second-order logic is generally thought problematic by the philosophical populace. Philosophers of mathematics and logic may have sophisticated reasons for rejecting second-order logic, but ask the average philosopher-on-the-street what’s wrong with second-order logic and they will probably mumble something about Quine, ontological commitment, and set theory in sheep’s clothing. In this paper, I try to get more precise about exactly what might be behind these mumblings. I offer four potential arguments against second-order logic and consider several lines of response to each. Two arguments target the coherence of second-order quantification generally, and stem from concerns about ontological commitment. The other two target the expressive power of ‘full’ (as opposed to ‘Henkin’) second-order logic, and give content to the concern that second-order logic is in fact “set theory in sheep’s clothing”. My aim is to understand the dialectic, not take sides; still, second-order logic comes through looking more promising than we might have initially thought.