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

Eckhart Arnold

## 1 Introduction

In a recent article on “The Meta-inductivist's Winning Strategy in the Prediction Game: A New Approach to Hume's Problem” (Schurz 2008) Gerhard Schurz proposes the Best-Alternative-Justification as a new approach to the problem of induction. As acknowledged by Schurz, the original idea goes back to Hans Reichenbach (Reichenbach 1935, Reichenbach 1938). But Schurz furnishes this idea with a new technical approach relying on a certain class of prediction strategies which he calls “meta-inductivists”. Given that all attempts to solve the problem of induction have hitherto failed (Howson 2000), what should induce us to reconsider the sceptical conclusion that there is no solution to the problem of induction? Prima facie, Gerhard Schurz has a convincing answer to this question: Most of the proposed solutions to the problem of induction tried to prove the reliability of the inductive procedure. But Schurz, following Reichenbach, merely tries to show the optimality of a specific inductive strategy, namely his “meta-inductivist” strategy. Demonstrating the optimality of an inductive prediction strategy is a less ambitious task than demonstrating its reliability, because for an inductive strategy to be reliable one would have to prove that it works in any possible world. But an inductive strategy that is merely optimal is allowed to fail in some possible worlds, as long as in the worlds where it does not work all other possible prediction strategies are bound to fail, too. Schurz does not raise the claim that he has solved the problem of induction literally, but the closing paragraph of his article suggests that he believes that he has at least provided a very good candidate for a solution to the problem of induction (Schurz 2008, p. 304).[1]

Schurz discusses the problem of induction within the technical framework of prediction games, where a number of players have to predict the next event in a series of binary valued (0 or 1) or real valued (any real number from 0.0 to 1.0) world events. By proving two theorems regarding the optimality of the prediction strategies of the “weighted meta inductivist” and the “collective weighted meta inductivist”, Schurz is at least able to give a partial solution for the problem of induction that accounts for the case of finitely many prediction strategies. But, as shall be demonstrated in the following, Schurz' technical approach meets insurmountable limits once one tries to pass from a finite number to an infinite number of prediction strategies. This raises the philosophical question whether an optimality result demonstrated for a finite number of prediction strategies might suffice to answer the problem of induction. If not, then providing a full solution to Hume's problem remains an open challenge.

In the following, I am going to briefly restate Schurz' central results and then demonstrate that the results for prediction games with finitely many strategies cannot be extended to prediction games with infinitely many strategies. Finally, there will be a brief discussion of open questions regarding Schurz' answer to Hume's problem.

[1] Regarding this claim, see also an earlier presentation of Schurz' ideas on the 5th GAP conference (Schurz 2003, p. 256).