[Revised entry by Jouko Väänänen on August 31, 2024.
Changes to: Main text, Bibliography]
Second-order logic has a subtle role in the philosophy of mathematics. It is stronger than first order logic in that it incorporates “for all properties” into the syntax, while first order logic can only say “for all elements”. At the same time it is arguably weaker than set theory in that its quantifiers range over one limited domain at a time, while set theory has the universalist approach in that its quantifiers range over all possible domains. This stronger-than-first-order-logic/weaker-than-set-theory duality is the…

Read the full article which is published on Stanford Encyclopedia of Philosophy (external link)