Abstract
In order to comprehend the world around us and construct explaining theories for this purpose, we need a conception of physical probability, since we come across many (apparently) probabilistic phenomena in our world. But how should we understand objective probability claims? Since pure frequency approaches of probability are not appropriate, we have to use a single case propensity interpretation. Unfortunately, many philosophers believe that this understanding of probability is burdened with significant difficulties. My main aim is to show that we can treat propensity as a theoretical concept that exhibits many similarities to other theoretical concepts, and its difficulties are not insuperable if we make explicit some general presuppositions of scientific practice and apply them to propensities. At least this is true if we formulate the right bridge principle for propensity and rely on further methodological rules in dealing with propensity assertions to make them empirically testable.
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Notes
Here it seems appropriate to make one remark about the question of determinism and its role in the explication of physical probability. To get rid of unnecessary problems I assume merely that we find nontrivial probabilities in instances of genuinely indeterministic processes. Quantum theory and the experiments on violations of Bell’s inequality give us good reason to think that such processes exist and can have effects even on the macro level. But what if nearly all macroprocesses are of a deterministic nature? I prefer to remain agnostic with respect to determinism on the macrolevel. Many such processes certainly appear indeterministic. But whether they would turn out to be deterministic—supposing we knew all initial conditions with infinite precision—seems to me an open question. Furthermore, we cannot follow up in this paper the question concerning the conception of a continuous space–time that underlies many deterministic theories and whether it ought to be interpreted realistically or not (cf. Bartelborth 1994). After all, even the treatment of deterministic processes by objective probabilities may seem reasonable in some examples. We can use a conception of probability as a proportion of an initial state space having further properties for this purpose (cf. Rosenthal 2004, Chap. 3).
In fact, even interpreting geometry is not unproblematic since nothing in reality has all the properties of a mathematical straight line.
Here and in what follows I treat coin tossing as a genuine chance mechanism. If the reader thinks it is actually deterministic, she should replace these cases with a genuine chance mechanism of her choice.
In order to be able to refer meaningfully to concrete events a in the future, we have to specify these events by the instantiation of a certain property A at time t and place x. But here I will make no further special assumptions about the inner structure of complex events or situations.
If we work with basic propositions a, b,… stating the occurrence of certain events and logical combinations thereof, we can be content with the axioms: P(t) = 1 for all tautologies t, P(non-a) = 1 − P(a), and P(a∨b) = P(a) + P(b) for all propositions a and b that exclude each other. For the sake of simplicity I will only speak directly about the events in question. For a definition of P(a|b), see the remark on conditional propensities in Sect. 6.
In general, a confidence interval I = [U(Z),O(Z)] with new chance variables U and O is constructed in such a way that P p (p ∈ [U(Z),O(Z)]) ≥ 1 − α for all p ∈ [0,1], with P p as the probability distribution with respect to the parameter p. If we use BPP we already have a general estimation for p in which p(1 − p) is roughly evaluated as equal to or smaller than 1/4. This estimation is of course too crude to be used in the practice of statistics.
This intuitive interpretation is even more plausible if we work with regular probabilities for which P(E) = 0 implies that E does not occur, which is at least a suggesting option for finite domains. Then we can see that P(E) ≥ 0.99 is a claim near this limiting case.
The effect of a growing number of factors can be seen in a simple model that simulates the random selection of the experimental and control groups. For example, think of 20 Persons and n factors f i which are randomly distributed among the persons. If we assume that these factors don’t interact and are independently distributed and all have positive effects on E, and if we represent them numerically in each case by adding +1, then we get a first impression of the possible differences between the two groups by adding up the factors for every person. These numbers are binomially distributed among the persons. Now compare the overall sum of factors in two randomly selected groups (with the help of a spread sheet program, for example) for n = 2, n = 4, n = 6 and so on. The simulation soon shows that these sums will tend to differ more and more as n increases.
The characterization “transcendental” should not imply that the assumption is a priori. I merely mean a pragmatic presupposition that we ordinarily subscribe to in science, which does not exclude the possibility of obtaining empirical evidence that opposes it.
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I thank two anonymous referees for their helpful comments on an earlier version of the article.
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Bartelborth, T. Propensities and Transcendental Assumptions. Erkenn 74, 363–381 (2011). https://doi.org/10.1007/s10670-010-9258-7
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DOI: https://doi.org/10.1007/s10670-010-9258-7