Non-classical mereologies and their applications

The purpose of this dissertation is to develop new non-classical mereologies and explore their philosophical applications. Chapter 1 is primarily expository, laying out the basics of classical mereology as studied axiomatically, set-theoretically, and algebraically. In chapter 2, I give a non-antisymmetric mereology and show how it can serve as a unified response to puzzles about the extensionality of parthood. In chapter 3, I give a gappy and glutty mereotopology which is a natural alternative to extant theories of boundaries. Chapter 4 gives a formal exposition of the thesis of composition-as-identity, furnishing responses to its two main challenges. In chapter 5, I explore the previously unknown area of non-wellfounded mereology, giving two non-wellfounded systems; I then give various potential applications of the resulting systems.