Denoting Concepts and Ontology in Russell's Principles of Mathematics

Bertrand Russell’s Principles of Mathematics (1903) gives rise to several interpretational challenges, especially concerning the theory of denoting concepts. Only relatively recently, for instance, has it been properly realised that Russell accepted denoting concepts that do not denote anything. Such empty denoting concepts are sometimes thought to enable Russell, whether he was aware of it or not, to avoid commitment to some of the problematic non-existent entities he seems to accept, such as the Homeric gods and chimeras. In this paper, I argue first that the theory of denoting concepts in Principles of Mathematics has been generally misunderstood. According to the interpretation I defend, if a denoting concept shifts what a proposition is about, then the aggregate of the denoted terms will also be a constituent of the proposition. I then show that Russell therefore could not have avoided commitment to the Homeric gods and chimeras by appealing to empty denoting concepts. Finally, I develop what I think is the best understanding of the ontology of Principles of Mathematics by interpreting some difficult passages