## Dissertations, Theses, and Capstone Projects

#### Title

Metaphysical Dependence and Set Theory

2013

Dissertation

Ph.D.

Philosophy

Graham Priest

#### Committee Members

Richard Mendelsohn

Arnold Koslow

Melvin Fitting

Kit Fine

Philosophy

#### Abstract

In this dissertation, I articulate and defend a counterfactual analysis of metaphysical dependence. It is natural to think that one thing x depends on another thing y if had y not existed, then x wouldn't have existed either. But counterfactual analyses of metaphysical dependence are often rejected in the current literature. They are rejected because straightforward counterfactual analyses fail to accurately capture dependence relations between objects that exist necessarily, like mathematical objects. For example, it is taken as given that sets metaphysically depend on their members, while members do not metaphysically depend on the sets they belong to. The set {0} metaphysically depends on 0, while 0 does not metaphysically depend on {0}. The dependence is asymmetric. But if counterfactuals are given a possible worlds analysis, as is standard, then the counterfactual approach to dependence will yield a symmetric dependence relation between these two sets. Because the counterfactual analysis fails to accurately capture dependence relations between sets and their members, most reject this approach to metaphysical dependence.

To generate the desired asymmetry, I argue that we should introduce impossible worlds into the framework for evaluating counterfactuals. I review independent reasons for admitting impossible worlds alongside possible worlds. Once we have impossible worlds at our disposal, we can consider worlds where, e.g., the empty set does not exist. I argue that in the worlds that are ceteris paribus like the actual world, where 0 does not exist, {0} does not exist either. And so, according to the counterfactual analysis of dependence, {0} metaphysically depends on 0, as desired. Conversely, however, there is no reason to think that every world that is ceteris paribus like the actual world, where {0} does not exist, is such that 0 does not exist either. And so 0 does not metaphysically depend on {0}. After applying this extended counterfactual analysis to several set-theoretic cases, I show that it can be applied to account for dependence relations between other mathematical objects as well. I conclude by defending the counterfactual analysis, extended with impossible worlds, against several objections.