Nuel Belnap

(? - 2024)

“Branching space-times” is intended as a representation of objective, event-based indeterminism. As such, BST exhibits both a spatio-temporal aspect and an indeterministic “modal” aspect of alternative possible historical courses of events. An essential feature of BST is that it can also represent spatial or space-like relationships as part of its relativistic theory of spatio-temporal relations; this ability is essential for the representation of local indeterminism. This essay indicates how BST might be seen to grow out of Newton ’s deterministic and non-relativistic theory by two independent moves: Taking account of indeterminism, and attending to spatio-temporal relationships in a spirit derived from Einstein’s theory of special relativity. Since and are independent, one can see that there is room for four theories: Newtonian determinism, branching time indeterminism, relativistic determinism, and branching space-times indeterminism.

Branching space-time is a simple blend of relativity and indeterminism. Postulates and definitions rigorously describe the causal order relation between possible point events. The key postulate is a version of everything has a causal origin; key defined terms include history and choice point. Some elementary but helpful facts are proved. Application is made to the status of causal contemporaries of indeterministic events, to how splitting of histories happens, to indeterminism without choice, and to Einstein-Podolsky-Rosen distant correlations.

Stit theory (a logic of seeing-to-it-that) is applied to cases involving many agents. First treated are complex nestings of stits involving distinct agents. The discussion is driven by the logical impossibility of "a sees to it that b sees to it that Q" in the technical sense, even though that seems to make sense in everyday language, Of special utility are the concepts of "forced choice", of the creation of deontic states, and of probabilities, Second, joint agency, both plain and strict (every participant is essential) is given a rigorous treatment. A central theorem is that strict joint agency is itself agentive.

In spite of a powerful tradition, more than two thousand years old, that in a valid argument the premises must be relevant to the conclusion, twentieth-century logicians neglected the concept of relevance until the publication of Volume I of this monumental work. Since that time relevance logic has achieved an important place in the field of philosophy: Volume II of Entailment brings to a conclusion a powerful and authoritative presentation of the subject by most of the top people working in the area. Originally the aim of Volume II was simply to cover certain topics not treated in the first volume--quantification, for example--or to extend the coverage of certain topics, such as semantics. However, because of the technical progress that has occurred since the publication of the first volume, Volume II now includes other material. The book contains the work of Alasdair Urquhart, who has shown that the principal sentential systems of relevance logic are undecidable, and of Kit Fine, who has demonstrated that, although the first-order systems are incomplete with respect to the conjectured constant domain semantics, they are still complete with respect to a semantics based on "arbitrary objects." Also presented is important work by the other contributing authors, who are Daniel Cohen, Steven Giambrone, Dorothy L. Grover, Anil Gupta, Glen Helman, Errol P. Martin, Michael A. McRobbie, and Stuart Shapiro. Robert G. Wolf's bibliography of 3000 items is a valuable addition to the volume.

The conditional,if an agent did something, then the agent could have done otherwise, is analyzed usingstit theory, which is a logic of seeing to it that based on agents making choices in the context of branching time. The truth of the conditional is found to be a subtle matter that depends on how it is interpreted (e.g., on what otherwise refers to, and on the difference between could and might) and also on whether or not there are busy choosers that can make infinitely many choices in a finite span of time.

Following von Wright, ``transitions'' are needed for understanding agency. I indicate how von Wright's account of transitions should be adapted to take account of objective indeterminism, using the idea of branching space-time. The essential point is the need to locate transitions not merely in space-time, but concretely amid the indeterministic, causally structured possibilities of our (only) world. (This is a ``postprint'' of Belnap 1999, as cited in the paper. The page numbers do not, of course, match those of the original.).