A critical appraisal of second-order logic
Ignacio Jané · History and Philosophy of Logic · 1993
Because of its capacity to characterize mathematical concepts and structures—a capacity which first-order languages clearly lack—second-order languages recommend themselves as a convenient framework for much of mathematics, including set theory. This paper is about the credentials of second-order logic:the reasons for it to be considered logic, its relations with set theory, and especially the efficacy with which it performs its role of the underlying logic of set theory