Abstract
The PI holds that, given no reason for preferring any of a set of alternatives over any other, all are equally probable. There are cases in which the PI can seemingly be used to justify incompatible probability assignments, due to different ways of conceptualizing the same set of possibilities. These conflicts can generally be resolved by appealing to the principle that the PI should be applied to the most explanatorily fundamental hypotheses. This has applications to the Problem of Induction as well as the mystery of entropy.
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- 1.
This might be entailed by the original formulation, because for any pair of equal-sized intervals, we would have no reason to think the true value more likely to fall into one interval than the other. (For different-sized intervals, we have a reason to think one is more likely to contain the true value – namely, that it’s larger.) So equal-sized intervals must in general contain equal probability, so the probability density must be uniform.
- 2.
This example is from Keynes 1921, p. 43. Keynes’ intention, however, is not to refute the Principle of Indifference but to refine it (pp. 55–64).
- 3.
From Keynes 1921, p. 44.
- 4.
From Fumerton 1995, p. 215.
- 5.
From van Fraassen 1989, p. 303.
- 6.
Bertrand 1889, pp. 4–5.
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Taking evidence broadly, to include everything that might serve as a source of epistemic justification for or against a proposition.
- 11.
Exception: if A has probability zero simply because A is one of a continuous infinity of possibilities, this does not mean that one’s evidence refutes A.
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- 13.
See, for example, Rawls 1999, pp. 145–50, explaining why the parties in his original position thought experiment allegedly cannot rationally rely on the Principle of Indifference (which Rawls refers to as “the principle of insufficient reason”) when making decisions.
- 14.
This is not quite correct: since we know who the members of our sample are, we needn’t assign probabilities to different hypotheses about who they are. The correct statement is too complicated for the main text: let n be some particular possible number of Platonists that the population might have. Consider each possible way of distributing n instances of “being a Platonist” across the 10,000 members of the population. Assign equal probability to each of those. It will turn out that most of those ways would result in our sample of 400 being roughly representative. This will turn out to be true for each n between 0 and 10,000. Thus, the prior probability of our sample being roughly representative is high, no matter what probability distribution we give to n. This is essentially the basis for David Stove’s (1986, pp. 55–75) probabilistic defense of induction, though Stove does not bring out the dependence on the Principle of Indifference explicitly.
- 15.
This is explained and defended further in Huemer 2009.
- 16.
Cf. Keynes’ (1921, p. 59) treatment of the case.
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- 18.
Here is another one: select a random position in the circle, then select a random angle, then construct a chord that passes through the chosen point, with the chosen angle to the vertical. See Marinoff 1994 for more methods.
- 19.
More precisely, inductive reasoning is reasoning that non-demonstratively infers, from the fact that certain things of kind A have feature B, that other things of kind A also have B. The fact that certain A’s have B need not be observed; it might be, say, intuited or discovered by inference. For instance, from the fact (i) that each of the first billion even numbers can be written as the sum of two prime numbers, one might inductively infer (ii) that all even numbers can be written as the sum of two prime numbers. Nevertheless, I shall continue to speak of observed and unobserved objects, since the most common and widely discussed examples of induction involve inference from the unobserved to the observed.
- 20.
For more on the character of the honey badger, see Randall 2011.
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This is shown in my 2009, pp. 24–6.
- 24.
If the wave function collapse in quantum mechanics is real, the law governing it is also temporally asymmetric. Also, some elementary particle interactions violate temporal symmetry.
- 25.
For further explanation, see Hurley 1986.
- 26.
Why is Fig. 8.3 drawn as it is? The farther we are from thermal equilibrium, the higher the proportion of possible transitions that lead toward equilibrium rather than further away. Thus, the lower entropy is, the higher the rate at which entropy increases. So as we move forward from the low-entropy time, entropy increases at a decreasing rate, i.e., the curve is concave down. The curve in the past direction is just the mirror image of the future-oriented curve.
- 27.
Though perhaps I should reserve judgment until Peter Unger has weighed in.
- 28.
Thanks to Iskra Fileva for this analogy.
- 29.
I thank Randall McCutcheon (p.c.) for pointing this out.
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Huemer, M. (2018). The Principle of Indifference. In: Paradox Lost. Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-319-90490-0_8
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