Second-order Logic: Properties, Semantics, and Existential Commitments

Abstract

Quine’s charge against second-order logic is that it carries massive existential commitments. This chapter argues that if we interpret second-order variables as ranging over properties construed in accordance with an abundant or deflationary conception, Quine’s charge can be resisted. This need not preclude the use of model-theoretic semantics for second-order languages; but it precludes the standard semantics, along with the more general Henkin semantics, of which it is a special case. To that extent, the approach of this chapter has revisionary implications; it is, however, compatible with the different special case in which second-order variables are taken to range over definable subsets of the first-order domain, and with respect to such a semantics, important metalogical results obtainable under the standard semantics may still be obtained. Finally, the chapter discusses the relations between second-order logic, interpreted as recommended, and a strong version of schematic ancestral logic promoted in recent work by Richard Kimberly Heck.