, Volume 188, Issue 1, pp 5–21 | Cite as

Newtonian determinism to branching space-times indeterminism in two moves



“Branching space-times” (BST) 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 (more or less) relativistic theory of spatio-temporal relations; this ability is essential for the representation of local (in contrast with “global”) indeterminism. This essay indicates how BST might be seen to grow out of Newton’s deterministic and non-relativistic theory by two independent moves: (1) Taking account of indeterminism, and (2) attending to spatio-temporal relationships in a spirit derived from Einstein’s theory of special relativity. Since (1) and (2) are independent, one can see that there is room for four theories: Newtonian determinism, branching time indeterminism, relativistic determinism, and (finally) branching space-times indeterminism.


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  1. Belnap, N. (1992). Branching space-time. Synthese, 92, 385–434. A postprint containing a number of additional explanations and a little restructuring is archived at
  2. Belnap N. (2002a) Double time references: Speech-act reports as modalities in an indeterminist setting. In: Wolter F., Wansing H., de Rijke M., Zakharyaschev M. (eds) Advances in modal logic, (Vol. 3, pp. 37–58). World Scientific, SingaporeGoogle Scholar
  3. Belnap N. (2002b) EPR-like “funny business” in the theory of branching space-times. In: Placek T., Butterfield J. (eds) Non-locality and Modality. Kluwer, Dordrecht, pp 293–315CrossRefGoogle Scholar
  4. Belnap, N. (2003). No-common-cause EPR-like funny business in branching space-times. Philosophical studies, 114, 199–221. A preprint may be obtained from
  5. Belnap N. (2005a) Agents and agency in branching space-times. In: Vanderveken D. (eds) Logic, thought and action. Springer, Dordrecht, pp 291–313CrossRefGoogle Scholar
  6. Belnap, N. (2005b). A theory of causation: Causae causantes (originating causes) as inus conditions in branching space-times. British journal for the philosophy of science, 56, 221–253. A preprint of this essay may be obtained from
  7. Belnap N. (2007a) In: Müller T., Newen A. (eds) Logik, Begriffe, Prinzipien des Handeln (Logic, concepts, principles of action). Mentis Verlag, Paderborn, pp 13–31Google Scholar
  8. Belnap, N. (2007b). Probabilities and propensities. Studies in the history and philosophy of modern physics, 38, 593–625. doi:10.1016/j.hpdb.2006.09.003. This essay contains a number of calculational errors. See Belnap 2010 for a corrected version.
  9. Belnap, N. (2010). Probabilities and propensities (revised). This postprint may be obtained as number 144x from
  10. Belnap N., Perloff M., Xu M. (2001) Facing the future: Agents and choices in our indeterminist World. Oxford University Press, OxfordGoogle Scholar
  11. Einstein, A., Lorentz, H. A., Weyl, H., & Minkowski, H. (1924). The principle of relativity. New York: Dover Publications Inc. A collection.Google Scholar
  12. Horty J. F. (2001) Agency and deontic logic. Oxford University Press, OxfordCrossRefGoogle Scholar
  13. James, W. (1884). The dilemma of determinism. In The will to believe. New York: Dover.Google Scholar
  14. Laplace, P. S. (1812–1820). Theorie analytique de probabilitiés: Introduction, VII, Oeuvres. Translation of passage taken from The Columbia world of quotations. New York: Columbia University Press (1996)Google Scholar
  15. Lewis D. (1986) On the plurality of worlds. Basil Blackwell, Oxford and New YorkGoogle Scholar
  16. Mackie J. L. (1974) The cement of the universe. Oxford University Press, OxfordGoogle Scholar
  17. Müller T. (2002) Branching space-time, modal logic and the counterfactual conditional. In: Placek T., Biutterfield J. (eds) Modality and non-locality. Kluwer, Dordrecht, pp 273–291CrossRefGoogle Scholar
  18. Müller T. (2005) Probability theory and causation: A branching space-times analysis. British journal for the philosophy of science 56: 487–520CrossRefGoogle Scholar
  19. Müller T. (2007) Branch dependence in the “consistent histories” approach to quantum mechanics. Foundations of physics 37(2): 253–276CrossRefGoogle Scholar
  20. Müller, T. (2009). Eliminating modality from the determinism debate? Models vs. equations of physical theories. In A. Hieke, & H. Leitgeb (Eds.), Reduction, abstraction, analysis. Proceedings of the 31th international Ludwig Wittgenstein symposium in Kirchberg (pp. 47–62). Frankfurt: Ontos Verlag.Google Scholar
  21. Müller T., Belnap N., Kishida K. (2008) Funny business in branching space-times: infinite modal correlations. Synthese 164(1): 141–159CrossRefGoogle Scholar
  22. Müller T., Placek T. (2001) Against a minimalist reading of Bell’s theorem: Lessons from Fine. Synthese 128: 343–379CrossRefGoogle Scholar
  23. Mundy B. (1986) Optical axiomatization of Minkowski space-time geometry. Philosophy of science 53: 1–30CrossRefGoogle Scholar
  24. Øhrstrøm P., Hasle P. (1995) Temporal logic: From ancient ideas to artificial intelligence. Kluwer Academic Publishers, DordrechtGoogle Scholar
  25. Placek T. (2000) Is nature deterministic?. Jagiellonian University Press, KrakówGoogle Scholar
  26. Placek T. (2004) Quantum state holism: A case for holistic causation. Studies in history and philosophy of modern physics 35: 671–692CrossRefGoogle Scholar
  27. Placek T. (2010) Possibilities without possible worlds/histories. Journal of Philosophical Logic 40: 1–29Google Scholar
  28. Placek, T., & Belnap, N. (2010). Indeterminism is a modal notion: branching spacetimes, and Earman’s pruning. Synthese, forthcoming.Google Scholar
  29. Placek T., Müller T. (2007) Counterfactuals and historical possibility. Synthese 154: 173–197CrossRefGoogle Scholar
  30. Placek T., Wroński L. (2009) On infinite EPR-like correlations. Synthese 167(1): 1–32CrossRefGoogle Scholar
  31. Prior A. N. (1967) Past, present, and future. Oxford University Press, OxfordCrossRefGoogle Scholar
  32. Stein H. (1991) On relativity theory and openness of the future. Philosophy of science 58: 147–167CrossRefGoogle Scholar
  33. Szabo L., Belnap N. (1996) Branching space-time analysis of the GHZ theorem. Foundations of physics 26(8): 989–1002CrossRefGoogle Scholar
  34. Thomason R. H. (1970) Indeterminist time and truth-value gaps. Theoria 36: 264–281CrossRefGoogle Scholar
  35. Thomson J. J. (1977) Acts and other events. Cornell University Press, IthacaGoogle Scholar
  36. Weiner M., Belnap N. (2006) How causal probabilities might fit into our objectively indeterministic world. Synthese 149: 1–36CrossRefGoogle Scholar
  37. Xu M. (1997) Causation in branching time (i): transitions, events and causes. Synthese 112: 137–192CrossRefGoogle Scholar

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© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.University of PittsburghPittsburghUSA

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