Can the Best-Alternative-Justification solve Hume's Problem?
|Table of Contents|
|2 Schurz' basic approach|
|3 Schurz' central results|
|4 Limitations of Schurz' approach: Confinement to the finite|
|5 Open Questions|
The technical framework within which Schurz derives his results consists of a series of events which can either be 1 or 0 (binary prediction game) or take any real value in the closed interval from 0.0 to 1.0 (real valued prediction game) and of a set of prediction strategies that have to get as many predictions of these events right as possible (in the binary prediction game) or that have to predict the coming events as closely as possible (in the real valued prediction game).
Two kinds of prediction strategies occur in Schurz' framework: Ordinary predictors that predict the next event by some arbitrary algorithm without looking at any of the other predictors and meta-inductivists that may - although they do not have to - base their own predictions on any of the ordinary predictor's predictions. The ordinary predictor's predictions are considered to be at least output-accessible, i.e.~the meta-inductivists get informed about the ordinary predictor's predictions before they place their own predictions. In order to chose between the ordinary predictors, the meta-inductivists may take into account the predictor's success rates as well as their previous predictions.
The ordinary predictors are symbolized by Schurz with capital “”s with an added index, e.g. , , etc. The meta-inductivists are symbolized as , where is a place holder for a string of characters that indicates the type of meta-inductivist and is, again, an index. There exists a kind of canonical ordinary predictor that Schurz calls the object-inductivist and which he denotes as . The object-inductivist's algorithm takes either - in the real valued prediction game - the mean value of all past events as its next prediction or - in the binary valued prediction game - the kind of event that had the higher frequency in the past.
If the sequence of events is a random sequence and if we exclude demonic predictors that know the coming world event ahead of time, there exists no strategy that can do better than the object inductivist. The meta-inductivist will consequently chose the object inductivist among the ordinary predictors, or, if no object inductivist is present, it will predict according to the object inductivist's algorithm by itself. At any rate the meta-inductivist will either be as good or better than any of the ordinary predictors.
However, in order for the meta-inductivist to be a “winning strategy” of the sort that is required to provide a solution to the problem of induction, it must also be optimal or, at least, approximately optimal in a “deceiving” world, where the event sequence or the other predictors or both “demonically” conspire against the meta-inductivist. The situation can be described as a game with the following rules:
 I am indepted to an anonymous referee for the precise formulation.