Two types of indefinites: Hilbert & Russell

This paper compares Hilbert’s terms and Russell’s approach to indefinite descriptions, Russell’s indefinites for short. Despite the fact that both accounts are usually taken to express indefinite descriptions, there is a number of dissimilarities. Specifically, it can be shown that Russell indefinites—expressed in terms of a logical operator—are not directly representable in terms of their corresponding terms. Nevertheless, there are two possible translations of Russell indefinites into epsilon logic. The first one is given in a language with classical terms. The second translation is based on a refined account of epsilon terms, namely indexed terms. In what follows we briefly outline these approaches both syntactically and semantically and discuss their respective connections; in particular, we establish two equivalence results between the (indexed) epsilon calculus and the proposed -term approach to Russell’s indefinites.