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

Eckhart Arnold

## 2 Schurz' basic approach

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 “$P$”s with an added index, e.g. $P_1$, $P_2$, etc. The meta-inductivists are symbolized as $xMI_i$, where $x$ is a place holder for a string of characters that indicates the type of meta-inductivist and $i$ is, again, an index. There exists a kind of canonical ordinary predictor that Schurz calls the object-inductivist and which he denotes as $OI$. 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:

1. In each round, first, the ordinary predictors predict what the next event in the event sequence will be. For making their predictions the ordinary predictors have access to the following information:
1. Complete knowledge about the past of the game, i.e.~the past event sequence and the past predictions by all other ordinary and meta-inductivist predictors.

2. Deceiving predictors know if and by which meta-inductivists their output will be accessed (see point 2). However, they have to deliver their predictions before the meta-inductivists do. As a consequence, deceivers that always have a reliable forenowledge of the predictions of the meta-inductivists cannot exist. For, the case may arise where a deceiver would have to base its evaluation of which prediction to deliver on a meta-inductivist's prediction which is in turn based on the very results of this evaluation. Now, assume a deceiver $A$ predicts 0 whenever a meta-inductivist $MI$ is going to predict 1 and 1 if $MI$ is going to predict 0 and at the same time $MI$ predicts 0 when $A$ has predicted 0 and 1 when $A$ has predicted 1. Then, the prediction of the deceiver would be undefined.

3. Deceiving predictors may have the capability of clairvoyance that is, in a non deceiving world they know beforehand what the next event will be. For reasons similar as in 2, permanent clairvoyants cannot exist in a deceiving world.

2. Then, the meta-inductivists make their predictions. In doing so, they may access the “output”, i.e.~the predictions, of any non meta-inductivist. Also, the meta-inductivists may - just like the ordinary predictors - take into account the complete information about all past events and predictions.

3. Finally, the world event occurs. In a deceiving world, the world event may depend in an arbitrary way on what predictions the predictors have made.

4. A meta-inductivist “wins”, i.e. is long run optimal, if in the long run the success loss of MI as compared to the best player at the given time converges to zero or a negative number.[2] (Or, simply put, if in the long run it performs at least almost as good as the best player.) Otherwise, the meta-inductivist “looses” the game.

[2] I am indepted to an anonymous referee for the precise formulation.