Abstract
Incomplete knowledge about the state of a system gives rise to statements about probability. There is more than one kind of probability; most are not to be reckoned in numbers. The notion of equal possibility can be applied to ranges of possible events (or possible things). A range theory of probability is developed; a complex of ranges is known as a range configuration. Statements about the equality of ranges may be compared. The comparability of ranges is requisite to numeric probability. Ranges must be indifferent to be compared, meaning they have parts assigned to be equal in value. Those ranges should not be reducible to other ranges, meaning they are original. Numeric probability only makes sense where probability statements span original ranges which are indifferent and comparable in magnitude, and where the statements cover all possibilities. An example of numeric probability is given for two series of processes known not to influence one another.
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Notes
- 1.
Compare the more thorough critique of this procedure in Chap. 6, Section 5 ff., which is only touched on here.
- 2.
Poisson uses this example, which underlines the arbitrariness of the ordinary procedure especially clearly, without a second thought to its explanation. (Recherches sur la probabilité des jugements, p. 96) [Poisson, S.-D. (1837). Recherches sur la probabilité des jugements en matière criminelle et en matière civile. Paris: Bachelier, Imprimeur-Libraire.]
- 3.
In the context of the so-called Bayesian principle.
- 4.
It is easily seen that if different varieties of behavior cannot be combined with one another completely and freely, then the product s1• s2 • s3 … is to be replaced by the multiple integral âˆâ€¦â€‰. ds1 • ds2 • ds3… . , which spans the totality of all admissible combinations. Things are somewhat similar, should we consider the rectangular coordinates XYZ for the location of an individual point, instead of parts of space. The probability that that point would be found in a certain space, would be measured by the size of the space, and would have the value âˆdx dy dz. It would be measured by the product x • y • z instead only if the various values of the three coordinates were entirely and freely combinable with one another, that is, if the whole range were described as a die or hexahedron. It is unnecessary at this point to expound further on this generalization.
- 5.
One may find it easier to understand this measurement of magnitude – just as one may find the later expositions easier – by tying it to spatial representations. One may consider the range configuration as a manifold similar to a space which may have very many dimensions, and whose independent coordinates could represent various values which stand for the arrangement defining the system. Under the analogy, the determination of distributed magnitudes takes the place of measurement of spatial elements.
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Niall, K.K. (2023). The Convention of Equally Warranted Premises. 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_2
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DOI: https://doi.org/10.1007/978-3-031-36506-5_2
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