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|
|4.1 Why Schurz' approach cannot be extended to the infinite case|
|4.2 A sidenote: Limitations of “one favorite” meta-inductivists|
|5 Open Questions|
Schurz explicitly restricts his investigation to prediction games with finitely many prediction strategies (Schurz 2008, p. 284). In this case the number of meta-inductivists may even be much smaller than the number of alternative predictors. It will now be shown why a similar argument cannot be made in case the number of alternative predictors is infinite and why, therefore, Schurz' optimality argument is confined to the finite.
Impossibility Theorem 2: If there is an infinite number of alternative predictors, then even a collective of meta-inductivists cannot perform approximately optimal in all possible worlds in the binary prediction game.
Proof: Consider the following scenario: Let there be an arbitrary number of meta-inductivists. As there are only two possible events, namely 1 and 0, at least half of the meta-inductivists predicts the same event. Obviously, in each round there exists an event that is not picked by a majority of meta-inductivists. Now, assume a demonic world where the world event is always an event that is not predicted by a majority of meta-inductivists. Then the average success rate of the meta-inductivists never exceeds 50%.
Now we only need to show that there can exist at least one (demonic) predictor that achieves a higher success rate. For this purpose, split the (infinite) set of alternative predictors into two infinite sets in the first round. The predictors in the first set predict 1, the predictors in the other set predict 0. In the following rounds take the (infinite) set of predictors that have always predicted true so far, split it into two infinite sets and again let all predictors from the first set predict 1 and all predictors from the second set predict 0. In any round of the game there is thus an infinite number of predictors left that has a success rate of 100%. Since the meta-inductivist's average success is significantly lower (smaller or equal 50%), their strategy is not optimal.
And, as may be expected, there is a similar impossibility theorem for the real valued prediction game:
Impossibility Theorem 3: If there is an infinite number of alternative predictors then no meta-inductivist can be approximately optimal in all possible worlds in the real valued prediction game.
Proof: Assume a demonic world, where the world event is always 0 or 1, whichever of these two numbers is further away from the predicion that makes. As to the infinite number of alternative predictors: In the first round, let half of them predict 1 and the other half 0. In the following rounds let half of the alternative predictors that have always predicted correctly so far predict 1 and the other half 0. Then at any point in time , there exist some predictors with complete success, while the average success of does not exceed 50%.
Hence, the conclusion: Neither in the binary nor in the real valued prediction game exists an optimal meta-inductive strategy if the number of alternative predictors is infinite.