Somehow Things Do Not Relate: On the Interpretation of Polyadic Second-Order Logic

Second-order logic, it has been alleged by W. V. Quine [17], is nothing but set theory in disguise, and hence not logic proper. The second-order quantifiers, so the allegation goes, have to range over some kind of “higher” entity or universal: properties or attributes at worst; sets at best, as sets at least are extensional. Moreover, a set-theoretic understanding may allow us to appreciate that second-order logic is consistent.1 But then second-order logic is ontologically committed to sets, although this commitment is cunningly hidden in the guise of quantification into predicate position. Operative here is Quine’s criterion of ontological commitment: to be is to be the value of a (bound) variable (see, e.g., Quine [16]). George Boolos, [1] and [2], has argued that this need not be so. While Boolos gives little reason to believe that he is particularly concerned about the boundary between logic proper and mathematics, he nevertheless finds it important to establish that the second-order quantifiers do not range over sets. Consider the case of second-order set theory. The statement