On the Nature of Impure Sets : A Defense of Abstractness

This dissertation aims to clarify the nature of impure sets. The classic definition takes impure sets as sets having concrete members or elements in their transitive closure. Concreteness is often understood in terms of spatiotemporal location: something is concrete iff it is spatiotemporally located. My main purpose is to show that the definition of impure sets is misguided. Impure sets do not contain concrete objects but abstract ones. To corroborate this, I will first discuss and reject the idea that impure sets are abstract objects having concrete members or elements in their transitive closure. This idea is adopted by many authors, such as Cowling, Fine, and Parsons. I will focus on Parsons’ and Fine’s accounts which are the most detailed: Parsons takes impure sets as quasi-concrete objects, while Fine describes them as abstract objects having concrete elements as their parts. The arguments that I will raise against Parsons’ idea address his criterion of identity and his notion of quasi-concrete objects. Concerning Fine’s account, I will argue against the possibility of abstract objects composed of concrete ones. Once the idea of impure sets as abstract objects having concrete members or elements has been rejected, I will consider impure sets as fully concrete objects, i. e. concrete objects having concrete members or elements in their transitive closure. I will show that this thesis is misguided by focusing on Maddy’s framework which is divided into two different but related claims: the ontological claim according to which impure sets are spatiotemporally located where their members or elements in their transitive closure are located, and the epistemic claim according to which we perceive impure sets. I will show that both claims are problematic. Given that impure sets are neither abstract objects having concrete members or elements, nor fully concrete objects, I will conclude that impure are fully abstract objects (i. e., objects that are not spatiotemporally located having objects that are not spatiotemporally located as members or elements). I will state that the most compelling strategy to argue that members or elements of impure sets are abstract objects can be found by exploring the relations between impure sets and the notions of concept, conceptualization, and conceptualized objects. I will conclude by arguing that Urelements of impure sets can be considered conceptualized objects modeled using Zalta’s notion of individual concepts (or abstract copies) formulated in his Object Theory. Finally, I will discuss one application of this account of impure sets to the famous case of Fine’s counterexample with Socrates and {Socrates}. By exploring this application, I will further justify the idea of impure sets as fully abstract objects.