Henkin on Nominalism and Higher-Order Logic

In this paper a proposal by Henkin of a nominalistic interpretation for second and higher-order logic is developed in detail and analysed. It was proposed as a response to Quine’s claim that second and higher-order logic not only are (α) committed to the existence of sets, but also are (β) committed to the existence of more sets than can ever be referred to in the language. Henkin’s interpretation is rarely cited in the debate on semantics and ontological commitments for these logics, though it has many interesting ideas that are worth exploring. The detailed development will show that it employs an early strategy of using substitutional quantification in order to reduce ontological commitments. It will be argued that the perspective adopted for the predicate variables renders it a natural extension of Quine’s nominalistic interpretation for first-order logic. However, we will argue that, with respect to Quine’s nominalistic program and his notion of ontological commitment, (α) still holds and thus Henkin’s interpretation is not nominalistic. Nevertheless, it will be seen that (β) is addressed successfully and this provides further insights on the so-called “Skolem Paradox”. Moreover, the interpretation is ontologically parsimonious and, in this respect, it arguably fares better than a recent proposal by Bob Hale.