Branching space-times, general relativity, the Hausdorff property, and modal consistency

The logical theory of branching space-times (BST; Belnap, Synthese 1992), which is intended to provide a framework for studying objective indeterminism, remains at a certain distance from the discussion of space-time theories in the philosophy of physics. In a welcome attempt to clarify the connection, Earman has recently found fault with the branching approach and suggested “pruning some branches from branching space-time” (2008). The present note identifies the different—order theoretic vs. topological—points of view of both discussion as a reason for certain misunderstandings, and tries to remove them. Most importantly, we give a novel, topological criterion of modal consistency that usefully generalizes the order-theoretic criterion of directedness, and we introduce a differential-geometrical version of BST based on the theory of non-Hausdorff (generalized) manifolds. Branching space-times (BST; Belnap, 1992) is a logical theory that allows for the representation of objective indeterminism in a space-time setting. It deviates from the mainstream representation of indeterminism in the Lewis tradition, in which wholly separate possible worlds are taken to signal indeterminism if they are partially isomorphic. In BST, the world is allowed to contain different complete possible courses of events, called histories, whose past overlap and future branching grounds indeterminism. Arguably this accords better with the notion of objective (rather than epistemic) indeterminism, but various objections have been raised against such a branching conception of indeterminism. ∗Copyright by the author. Comments welcome; send to [email protected]