Substitutional Quantification and Arithmetical Truth

ABSTRACT

In the mid‐20th century, there was a debate about whether the quantifiers in first‐order logic should be understood objectually or substitutionally. This debate typically focused on ontology and assumed that we could use only one of these two types of quantification. After Kripke and others influentially argued that the two kinds of quantification were logically equivalent, defenses of substitutional quantification were rare. Following Gottlieb and Field, I argue that Kripke was wrong and that substitutional consequence is language‐bound and non‐compact. While of no help with ontology, I also argue that so‐understood, substitutional quantification can be applied to help explain our grasp of a determinate notion of arithmetical truth. To argue for this, I give an account of the metasemantics of the substitutional quantifiers and prove that adding a single, very natural, mixed axiom to weak theories of arithmetic results in a categorical theory. This approach promises to explain arithmetical truth on a less controversial basis than appeals to either second‐order logic or the omega rule, thus making progress on the project of explaining our grasp of arithmetic in a naturalistically acceptable fashion.