Abstract
In the range theory, knowledge of objective import is the basis for statements of probability. The range theory provides a satisfactory explanation of the law of large numbers, and it clarifies the notion of likelihood. Statements about probability are associated with essential premises; pride of place is given to the assumption of independence among cases. Two types of factors determine probability: nomological and ontological. Nomological statements concern generally valid laws; ontological statements have singular meaning. It is our ignorance of ontological relations which makes outcomes seem uncertain. An event is called random, if more precise ontological determinants (withheld from our knowledge) are constraints on which an event depends. The notion of a normal distribution is introduced, by the example of series of draws from an urn. Different procedures for the draws are introduced to illustrate hypernormal and hyponormal dispersion for distributions. Hyponormal dispersion occurs when successive cases are not independent.
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Notes
- 1.
A fully precise report of numeric probability might be possible in ideal cases, as in a game of chance in which various results alternate in an infinitely small period, such as for a bowling-game where alternating black and white stripes are infinitely thin.
- 2.
From this it can be seen that the label “the Law of large numbers” is really not very suitable. That is because there is basically nothing here except a proposition of mathematics that exhibits certain peculiarities which follow from the size ratios of ranges and consequently from the expectations based on them. It is true that the Law of large numbers has often been taken for a proposition of far greater import. That is, one thinks it legitimate to claim that in the phenomena in any area there must be found a repetition of cases of the same kind, so that for a great number of such repetitions, an arbitrary characteristic may always be observed which occurs with approximately the same relative frequency. Of course that is something altogether different than a mere mathematical relation; that is a claim about a very specific regularity of nature. Yet as we shall see later on, this proposition is not at all valid in general, and the general proofs which have been sought for it are based on fallacies.
- 3.
This is the same condition that Poisson characterizes when he calls that circumstance a cause “in the larger sense” of the event in question.
- 4.
Given the appreciable vagueness in terminology which attends literature on the probability calculus, it is clear that the word ‘chance’ is used often enough purely to mean probability, and not used in the particular sense that has been given here. Nevertheless it is always ‘chance’ that is spoken of where the general validity of a probability is to be emphasized. Then since a short expression is altogether necessary for the concept which has been elaborated above, I believe the term ‘chance’ should be applied to it.
- 5.
Windelband, W. (1870). Die Lehren vom Zufall. Berlin: F. Henschel, 80 pp. (Diss. Univ. Göttingen).
- 6.
Cf. Chap. 6, Section 6 on this topic.
- 7.
Here I adopt the term used by Wilhelm Lexis (1877, Zur Theorie der Massenerscheinungen in der menschlichen Gesellschaft), who has introduced some exceptionally important applications of the concept of dispersion, as will be discussed later. [Lexis, W. (1877). Zur Theorie der Massenerscheinung in der menschlichen Gesellschaft. Freiburg i. B.: Fr. Wagner.]
- 8.
The description of definite dispersion, be that of an expected dispersion or be it of an observed dispersion, always presumes a determination in advance of the principles by which individual cases are aggregated into series. It may not be too much to stress this explicitly once again. In the first of our examples, the succession of draws gives us this principle; in the latter example, the rules of the game give us definite bounds for the series. And if one wanted to form series by the random aggregation of single cases, then naturally one would always obtain a normal dispersion.
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Niall, K.K. (2023). The Special Theory of Probability. In: Johannes von Kries: Principles of the Probability Calculus. Studies in History and Philosophy of Science, vol 59. Springer, Cham. https://doi.org/10.1007/978-3-031-36506-5_4
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