Can the Best-Alternative-Justification solve Hume's Problem?
On the Limits of a Promising Approach

Eckhart Arnold

1 Introduction
2 Schurz' basic approach
3 Schurz' central results
4 Limitations of Schurz' approach: Confinement to the finite
5 Open Questions
6 Conclusion

5 Open Questions

Schurz' “New approach to Hume's problem” is both a research program on prediction strategies and a proposed answer to the problem of induction. As a research program it can be pursued more or less independently of its claim to offer a new approach to Hume's problem. Regarding this claim, the central question is of course: Can meta-inductivists solve the problem of induction? This question can be broken down into three separate questions.

  1. How is Schurz' model related to the problem of induction, if the latter is understood as the problem of justifying scientific inference?
  2. Does it matter that no single meta-inductivist but only a collective of meta-inductivists constitutes an optimal strategy in the (binary) prediction game?
  3. Is it acceptable that the best alternative justification works only when the number of predictors is finite?

In order to better understand these questions, each of them shall briefly be discussed:

1) Justification of scientific inference: Schurz answers the problem of induction within a technical framework that consists of a highly stylized world that produces a sequence of (binary or real-valued) events at discrete time intervals. Further interpretative work might be necessary to show that the problem of induction that has been solved within Schurz' technical framework matches the problem(s) of induction that occur in the real world. In particular, the kind of induction in Schurz' model is the induction from previous events to the next event - in contradistinction to the inductive or abductive inference from a number of single instances to a general rule. For the justification of scientific inference, the latter type of induction seems to be even more important. After all, we would like to know whether we can rely on a law of nature if it has been confirmed in a finite number of instances and never disconfirmed. Thus, the question would be whether and how Schurz' answer to Hume's problem can also be transferred to this case.

2) Admissibility of collectives of meta-inductivists: In the binary-valued game a collective of meta-inductivists is required to assure optimality in the prediction game. Now, if we are looking for a reliable method of induction to the next event, then we are faced with the somewhat puzzling fact that we do not get a single proposal but a multitude of proposals instead. As follows from the impossibility theorem for single meta-inductivists (impossibility theorem 1 above), it is impossible to melt down the different predictions of the collective of meta-inductivists to a single prediction without becoming vulnerable to deception by a demonic world. There is no grave problem involved if the prediction game is considered in a decision theoretical framework (Schurz 2008, p. 301ff.). For then the primary goal would not be to get the next prediction right, but rather to derive as much utility as possible from a number of predictions. However, this implies a shift of emphasis from the original problem of induction to a closely related decision theoretical problem.

3) Confinement to the finite: If only a finite number of prediction strategies is taken into account, then we exclude the overwhelming majority of possible prediction strategies from the game right from the beginning. For, given that the sequence of events is infinite, there exists an uncountably large number of posssible event sequences. And even if we take into consideration only those prediction strategies which can be described by an algorithm, there still remains a countable infinity of possible alternative prediction strategies. Unfortunately, neither a single meta-inductivist nor a collective of meta-inductivists can perform optimal in all possible worlds if the number of alternative predictors is infinite (see section 4.1).

Schurz deliberately restricts his investigation to prediction games with finetely many players, because he makes “the realistic assumption that xMI has finite computational means.” (Schurz 2008, p. 284). But in order to justify this restriction one would need to show that an infinite number of alternative players is impossible, rather than arguing that xMI cannot deal with an infinite number of alternative players. Otherwise the notice “that xMI has finite computational means” merely amounts to admitting that under this “realistic assumption” xMI simply cannot always perform optimal.

As there is no logical contradiction involved in the assumption of an infinite number of alternative players, the only grounds upon which it could be defended would be empirical grounds. But then it is hard to see how the general impossibility of an infinite number of alternative players can be demonstrated without silently or explicitly making use of inductive inference.

Very simply put, there seems to be no good reason why a theoretical framwork for answering Hume's problem that allows for “clairvoyants”, “deceivers”, “demonic” streams of world events, should not also admit infinite sets of alternative predictors. Surely, that an xMI which has only finite computational means does not work under this condition is not a sufficient reason.

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