Abstract
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 \(A=\varnothing \)) 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.
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Notes
The arguments in [13] suggest that sometimes denial is the right move.
That said, the methods of [16, Section 4] can be used to show that if there is a Popper function on some set of cardinality equal to the continuum for which all countable subsets are normal (in a sense that will be defined shortly), then that yields a weak version of the Axiom of Choice that is nonetheless strong enough to generate the Hausdorff and Banach-Tarski paradoxes. The proof in that paper is given for the case of a Popper function where all subsets are normal, but works just as well if only the countable ones are assumed to be normal.
In [1, p. 5] there is a claimed proof that weak invariance is equivalent to strong invariance in this case, but the putative proof is incomplete. Note also that while the Axiom of Choice is assumed at the outset of [17], it is not needed for the proof that not-(b) implies not-(a) in Theorem 1 of that paper, and that implication applied in the case where \(G=\Omega =\mathbb {Z}\) (acting on itself by addition) will generate a weakly but not strongly invariant Popper function on an algebra \(\mathcal {F}\) of subsets of \(\mathbb {Z}\) with all members of \(\mathcal {F}\) normal, contrary to [1].
The difference may seem small, namely whether one also requires invariance under reflections—and, in fact, under any single fixed reflection, since all reflections in \(\mathbb {R}^{2}\) can be generated by a single fixed reflection combined with appropriate rigid motions. But nonetheless the difference may be significant. The proof of Theorem 1 uses a trick similar to that used in [15] to show that there is no isometrically-invariant preordering on the subsets of the circle that extends strict inclusion. But [15] also showed that there is such a preordering when isometries are replaced with rigid motions. So we know that sometimes there is a significant difference between isometries and rigid motions in respect of the generation of invariant functions or relations.
Though see [8] for an interesting critique of this equivalence condition, and in general of symmetry-based reasoning. Nonetheless, abandoning symmetry in this way appears a very high cost.
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Pruss, A.R. Popper Functions, Uniform Distributions and Infinite Sequences of Heads. J Philos Logic 44, 259–271 (2015). https://doi.org/10.1007/s10992-014-9317-7
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DOI: https://doi.org/10.1007/s10992-014-9317-7