Popper Functions, Uniform Distributions and Infinite Sequences of Heads

Popper functions allow one to take conditional probabilities as primitive instead of deriving them from unconditional probabilities via the ratio formula P(A|B) = P(A ⋂ B)/P(B). A major advantage of this approach is it allows one to condition on events of zero probability. I will show that under pla...

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Bibliographic Details
Published inJournal of philosophical logic Vol. 44; no. 3; pp. 259 - 271
Main Author Pruss, Alexander R.
Format Journal Article
LanguageEnglish
Published Dordrecht Springer 01.06.2015
Springer Netherlands
Springer Nature B.V
Subjects
Algebra
Education
Logic
Mathematical models
Philosophy
Symmetry
Group action
Banach-tarski paradox
Infinity
Independence
Probability
Conditional probability
Rotation
Popper function
Symmetry
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Summary:Popper functions allow one to take conditional probabilities as primitive instead of deriving them from unconditional probabilities via the ratio formula P(A|B) = P(A ⋂ B)/P(B). A major advantage of this approach is it allows one to condition on events of zero probability. I will show that under plausible symmetry conditions, Popper functions often fail to do what they were supposed to do. For instance, suppose we want to define the Popper function for an isometrically invariant case in two dimensions and hence require the Popper function to be rotationally invariant and defined on pairs of sets from some algebra that contains at least all countable subsets. Then it turns out that the Popper function trivializes for all finite sets: P(A|B) = 1 for all A (including A = Ø) if B is finite. Likewise, Popper functions invariant under all sequence reflections can't be defined in a way that models a bidirectionally infinite sequence of independent coin tosses.
ISSN:0022-3611
1573-0433
DOI:10.1007/s10992-014-9317-7