Abstract
Bayesian epistemology has struggled with the problem of regularity: how to deal with events that in classical probability have zero probability. While the cases most discussed in the literature, such as infinite sequences of coin tosses or continuous spinners, do not actually come up in scientific practice, there are cases that do come up in science. I shall argue that these cases can be resolved without leaving the realm of classical probability, by choosing a probability measure that preserves “enough” regularity. This approach also provides a resolution to the McGrew, McGrew and Vestrum normalization problem for the fine-tuning argument.
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Notes
I am grateful to Robin Collins for this point.
I am grateful to an anonymous reader for bringing up indifference.
For simplicity, suppose our range is the non-negative reals \([0,\infty )\). Fix a finite length a of an interval and suppose that \(P([x,x+a))\) is the same for every \(x\ge 0\). Then for any finite n we have \(1\ge P([0,na)) = P([0,a))+P([a,2a))+\dots +P([(n-1)a,na)) = n P([0,a))\) by finite additivity, and hence \(0\le P([0,a)) \le 1/n\) for all finite n, which implies that \(P([0,a))=0\).
I am grateful to an anonymous reader for this point.
I am grateful to an anonymous reader for pointing in this direction.
Suppose P(H) is an infinitesimal \(\alpha >0\) and \(P(E)\ge r\) for some strictly positive real number r. Then \(P(H\mid E)=P(H\cap E)/P(E)\le \alpha \cdot (1/r)\), which is infinitesimal since an infinitesimal multiplied by any positive real yields an infinitesimal.
Note, however, that often when one talks of non-measurable sets, one talks of Lebesgue non-measurable sets. I have chosen to work with measures on the \(\sigma \)-algebra of Borel sets rather than the larger \(\sigma \)-algebra Lebesgue sets in order to allow for the possibility of more measures in our set \(V_\mathcal B\). The trade-off here is that these measures will measure fewer sets. One can also go for the other trade-off and replace the Borel sets with the Lebesgue measurable sets throughout the paper. It is a question for further investigation which approach is superior.
In set theory, real numbers, probability measures and other such objects are all sets.
If \(f:A\rightarrow B\) with A countable, then A bijects with a subset of the natural numbers, so there is a well-ordering \(\prec \) on A derived from arithmetic ordering on the naturals. For y in the range of f, let g(y) be the \(\prec \)-least x such that \(f(x)=y\). Then g is a one-to-one function from B to a countable set, so B is countable.
Normally, we would define a wff as true provided it has no free variables and is satisfied. Above, however, we are only interested in the satisfaction of formulas with a free variable. But the satisfaction of formulas with a free variable can also define truth: a wff \(\phi \) is true provided that it has no free variables and something satisfies \( \phi \& (x=x)\).
For instance, if quantum mechanics is the correct theory of the world, then we have gained reference to some cardinal \(\kappa \) as the cardinality of the Hilbert space of the global wavefunction. That cardinality is often thought to be \(\aleph _0\), which is definable in set theory without parameters, but it might turn out to be some higher cardinal, in which case our expressive resources might increase when we use \(\kappa \).
Additionally, if we opt for the Lévy hierarchy solution to the problems with expressibility in Sect. 7, we need a choice of a level at which to cut off the hierarchy. However, the approximation argument in that section suggests that in practice we can hope to get approximate answer by simply cutting off the hierarchy.
One might think that projectibility or naturalness do not apply to mathematical predicates. However, since inductive arguments do have some epistemic force in mathematics—the verification of Goldbach’s conjecture for all numbers less than \(4\times 10^{18}\) is some evidence of its truth in general—one would expect the Goodmanian distinction to have some applicability in mathematics as well.
I am grateful to three anonymous readers whose comments have significantly improved this paper, to Yoaav Isaacs for some helpful suggestions, and to William Wood for encouraging me to continue to think about normalizability.
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Pruss, A.R. A classical way forward for the regularity and normalization problems. Synthese 199, 11769–11792 (2021). https://doi.org/10.1007/s11229-021-03311-4
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DOI: https://doi.org/10.1007/s11229-021-03311-4