There are two main theories of persistence on the market: three- and four-dimensionalism. It is generally considered that these theories are competitors, and thus that at most, one can be true. In this thesis I argue that these theories are not competitors at all. Rather, they are metaphysically equivalent: they are two different ways of describing the same underlying metaphysics. I formulate an account of metaphysical equivalence in terms of theories being correctly inter-translatable, and more importantly I formulate a set of diagnostic criteria for determining when theories are equivalent in this manner. These diagnostic criteria include empirical and explanatory equivalence, hermeneutic principles as well as a principle that rejects the ad hoc positing of certain kinds of truth makers.
I then make use of these criteria to argue that for any version of three-dimensionalism, there is an analogous version of four-dimensionalism that is metaphysically equivalent and vice versa. Pairs of theories are analogous in this sense just if they embrace the same set of ancillary metaphysical commitments. Thus we can think of theories in this sense, as being bundles of metaphysical views. Part of any bundle might include a particular account of persistence, another part might include a particular account of the nature of time, or of composition. Thus any two bundles are analogous just if they differ with respect to only one aspect of the bundle, in this case the aspect of persistence.
I argue that the appearance of substantial dissimilarity between three- and four-dimensionalism is the result of comparing three-dimensionalist accounts that have one set of ancillary metaphysical commitments, with four-dimensionalist accounts that have a different set of ancillary metaphysical commitments. Once we separate out each of these ancillary views, we see there are many different versions of three- and four-dimensionalism, each committed to a different set of such ancillary views-that is, there are different bundles of views that we can think of as being three- or four-dimensionalist in nature. I argue that when we compare each analogous pair of theories we find that each meet the diagnostic criteria of metaphysical equivalence. From there I argue that inference to the best explanation tells us that we should conclude that theories meeting these diagnostic criteria are metaphysically equivalent. Thus if each pair of analogous theories is equivalent, then we can conclude that three- and fourdimensionalism are equivalent simpliciter.