# 3

Logical InductionLogical Uncertainty
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Sam Eisenstat asked the following interesting question: Given two logical inductors over the same deductive process, is every (rational) convex combination of them also a logical inductor? Surprisingly, the answer is no! Here is my counterexample.

We construct two logical inductors over PA, , and .

Let be any logical inductor over PA.

We consider an infinite sequence of sentences .

Let be computable in time.

We construct as in the paper, but instead of using all traders computable in time polynomial in , we use all traders computable in time polynomial in time. Since this also includes all polynomial time traders, is a logical inductor.

However, since the truth value of is computable in time, if the difference between and the indicator of did not converge to 0, a trader running in time polynomial in f(n) can easily exploit . Thus,

and

Now, consider the market

Observe that

so

Now, consider the trader who exploits by repeatedly either buying a share of when the price is near 3/4, or selling a share when the price is near 1/4, waiting for that sentence to be resolved, and then repeating. Eventually, in each cycle, this trader will make roughly 1/4 of a share, because eventually the price will always be close enough to either 1/4 or 3/4, and all shares that this trader buys will be true, and all shares that this trader sells will be false.

Thus is not a logical inductor.