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Justified belief and possible worlds

Dante statue

05/12/2024

Here is a promising idea: whether a belief is justified depends not only on whether it is true in the actual world, but also on what would have happened in other possible worlds. Indeed, this idea is so promising that there are a number of well-known philosophical theories based on it. In this blog post, Dr Nick Makins suggests that these theories are vulnerable to a new version of an old problem: Bertrand’s Paradox.

According to a popular theory known as “reliabilism”, a belief is justified if and only if it was produced by a reliable process. A belief-forming process is reliable if and only if it produces a high enough ratio of true beliefs to false beliefs across a range of possible worlds. One motivation for this turn to possible worlds in reliabilism and related theories, is that it allows us to include important properties like truth in an account of justified belief, without being beholden to flukey truth or falsity in the actual world. Even a process that produces a false belief in the actual world may be considered reliable if it would have produced enough true beliefs in other possible worlds.

Despite its promise, I think that this approach cannot get off the ground, because there is no consistent way to calculate the required truth ratios. This can be shown by considering a famous puzzle known as Bertrand’s Paradox.

Bertrand’s paradox presents a challenge to an idea in probability theory known as the Principle of Indifference. This principle states that all possibilities should be assigned equal probability, in the absence of reason to assign unequal probability. The probability of a subset of possibilities therefore equates to the ratio of that subset to the total set of possibilities. For example, if the spin of a roulette wheel has 37 equally possible outcomes, then we should assign each possible outcome a probability of 1/37. And the probability that the winning number lies between 10 and 20 is 10/37.

There are numerous versions of Bertrand’s paradox, but all involve showing that the same set of possibilities can be described in multiple ways that yield different probabilities when the Principle of Indifference is applied. For example, imagine a factory that produces cubic boxes. The boxes can have any volume ≤8m3, so the area of any box’s face is ≤4m2, and the length of any box’s edge is ≤2m. For a box made by the factory, about which we know nothing more, consider three questions:

  1. What is the probability that its volume is ≤1m3? That range of possibilities makes up 1/8 of the total range, so applying the Principle of Indifference, the answer is 1/8.
  2. What is the probability that its faces have area ≤1m2? That range of possibilities makes up ¼ of the total range, so applying the Principle of Indifference, the answer is ¼.
  3. What is the probability that its edges have length ≤1m? That range of possibilities makes up ½ of the total range, so applying the Principle of Indifference, the answer is ½.

The problem is that these questions are equivalent, since any box with volume ≤1m3 just is a box whose faces have area ≤1m2 and whose edges have length ≤1m. But applying the Principle of Indifference to answer them yields inconsistent results. This problem has led many to reject the Principle of Indifference altogether and its application to a wider range of views may warrant rejection of them too.

The problem for theories like reliabilism is that they require us to calculate proportions of possibilities in much the same way as the Principle of Indifference. Therefore, we can construct Bertrand-style cases in which different descriptions of the same possibilities yield inconsistent results.

For example, consider a subject who is told only that a box has been produced by the factory described above. The way that they reason in this sort of case leads them to believe that the box has edges with length <1m. In some possible worlds this belief is true and in some it is false. How reliable is this belief-forming process? What is the ratio of true to false beliefs produced by it across all of the ways things could have been? This ratio depends on the proportion of possibilities in which the box indeed has edges with length ≤1m. But, as we have seen, this range of possibilities can be redescribed as the range of possibilities in which the box has volume ≤1m3, or the range of possibilities in which the box has faces with area ≤1m2, and these different ways of describing the same range of possibilities suggest different proportions of the total.

This means that we are unable to answer the questions of whether the process was reliable and, therefore, whether the belief was justified. This shows that (1) Bertrand’s paradox has far wider implications than have previously been recognised and (2) proponents of the theories in question will have to reckon with this challenge if they are to succeed.

Dr Nick Makins is a British Academy Postdoctoral Research Fellow in the School of Philosophy, Religion and History of Science at the University of Leeds. Most of his research is related to the reasoning that underlies decisions made under uncertainty, and draws on ideas from metaethics, decision theory, and philosophy of science.