The Hole Argument and Determinism(s)

This paper first offers an interpretation and clarification of Halvorson and Manchak (2025)’s recent argument that there is no mathematical basis for the Hole Argument; the upshot of this is that it depends on whether the notion of “rigidity” discussed in that paper can be understood as a form of determinism. It then reviews various notions of determinism that have been put forward in the Hole Argument literature, to argue that rigidity cannot be so understood—but that rigidity does stand in a close relationship to certain species of determinism.