How Causal Probabilities Might Fit into Our Objectively Indeterministic World

We suggest a rigorous theory of how objective single-case transition probabilities fit into our world. The theory combines indeterminism and relativity in the ‘‘branching space–times’’ pattern, and relies on the existing theory of causae causantes (originating causes). Its fundamental suggestion is that (at least in simple cases) the probabilities of all transitions can be computed from the basic probabilities attributed individually to their originating causes. The theory explains when and how one can reasonably infer from the probabilities of one ‘‘chance set-up’’ to the probabilities of another such set-up that is located far away. 1. TRANSITION PROBABILITIES HERE AND THERE Imagine two ‘‘chance set-ups’’ that are separated by perhaps millions of miles. When and how could the transition probabilities of two such chance set-ups be related? ð1Þ We suggest a rigorous theory of objective single-case event–event transition probabilities that gives a modestly partial answer to question (1). The theory only makes sense if one takes into account some aspects of the indeterministic and spatio-temporal structure of our world. We shall suggest an answer to (1) under the proviso that there is an absence of Bell-like strange stochastic correlations coming from quantum mechanics. Our chief purpose, however, is not so much to answer (1) as to lay down a general framework for no-nonsense discussions of how causal probabilities might fit into our indeterministic and spatiotemporal world. The basic proposal is that causal probabilities for any transition are inherited exclusively from probabilities ingredient in the causae causantes or originating causes of that transition. We begin with a story involving a simple flip of a coin, so simple that although its telling requires indeterminism, spatio-temporal complications may be downplayed. Later we bring in a second chance set-up that is located far away from the first, at Synthese (2006) 149: 1–36 Springer 2006 DOI 10.1007/s11229-004-6240-4 which point we shall need explicitly to consider spatio-temporal relations as well. 1.1. The Clock Story The Marshall Fields Clock sits at the corner of State and Randolph in Chicago. Imagine that we are situated there at 3:00 p.m. on a certain Saturday. A trick coin was flipped under the Clock an hour ago, at 2:00 p.m. The altered balance of the coin favored – but did not guarantee – that the coin would land heads-up on the sidewalk. In detail, the chances of the coin showing heads on just that flip were 0.6 instead of the figure of 0.5 suggested by the symmetries. As it turned out, however, the coin landed tails, even though the chances of such were only 0.4. It helps the story if you picture the Heads-face of the coin as Hot pink, and the Tails-face as Turquoise. Perhaps our world is as deterministic as Kant or Hume would have it, so that such talk of ‘‘chances’’ is mere mythology: The coin came up tails, and there’s an end on it. Let us, however, explore the option that our world is in part truly and objectively indeterministic, and in particular let us suppose that the distribution of chances 0.4 vs. 0.6 among the Chicago coin-flip outcomes was entirely objective. That is, at any time in the causal past of the 2:00 p.m. flip, there was no settled fact of which outcome would ensue. At those earlier times, there was only the 0.4 vs. 0.6 probability distribution on ‘‘after 2:00 p.m. the coin will lie heads up and hot pink.’’ In contrast, after 2:00 p.m. under the Marshall Fields Clock it was a definite matter that the coin lay tails, and that therefore anyone standing under the Clock saw turquoise. There was, that is, a transition under that Clock on that Saturday from 0.4 vs. 0.6 as to hot pink vs. turquoise to determinate or settled turquoise.