I'm reading a book that uses first- and second-order logic. The author defines first-order logic normally, but then defines second-order logic as "quantification on relations." Almost everywhere else I've looked on the internet, folks use "quantification over sets". What's the difference? Are these interpretations equivalent, and how?
I prefer the relation semantics because it doesn't bring in set theory (quantification over sets of sets). I have not found any definition of third-order logic that does not involve sets. Is there a way to define third-order logic in terms of relations/predicates?