On the Logicality of Second-Order Logic in Terms of Plural Arbitrary Reference

The aim of this chapter is to argue that: (a) our semantics of acts of choices (SAC), as developed in Chap. 2 , defends second-order logic from claims of ontological commitment; (b) understanding our semantics does not require any prior mathematical concepts; and (c) although SAC is not universally applicable, it still offers significant applicability, especially in mathematics. We conclude the chapter arguing that second-order logic, as interpreted through our semantics, can indeed be considered a genuine logic. One might object that, since only concrete objects can be referred to by arbitrary acts of choice, SAC lacks the logical requirement of universal applicability. However, such limitation is superseded by the structuralist conception of mathematics, according to which arbitrary entities, independently of their very nature, can play the role of mathematical objects.