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, $\left\{P_n\right\}$, and $\left\{P_n^{\prime }\right\}$.

Let $\left\{P_n\right\}$ be any logical inductor over PA.

We consider an infinite sequence of sentences ${\varphi }_n:\iff "P_n\left({\varphi }_n\right)<1/2"$.

Let $P_n\left({\varphi }_n\right)$ be computable in $f\left(n\right)$ time.

We construct $\left\{P_n^{\prime }\right\}$ as in the paper, but instead of using all traders computable in time polynomial in $n$, we use all traders computable in time polynomial in $f\left(n\right)$ time. Since this also includes all polynomial time traders, $\left\{P_n^{\prime }\right\}$ is a logical inductor.

However, since the truth value of ${\varphi }_n$ is computable in $f\left(n\right)$ time, if the difference between $P_n^{\prime }\left({\varphi }_n\right)$ and the indicator of ${\varphi }_n$ did not converge to 0, a trader running in time polynomial in f(n) can easily exploit $\left\{P_n^{\prime }\right\}$. Thus,

$$P_n\left({\varphi }_n\right){\eqsim }_{n}1/2,$$

and

$$P_n^{\prime }\left({\varphi }_n\right){\eqsim }_{n}Th{m}_{PA}\left({\varphi }_n\right).$$

Now, consider the market

$$\{P_n^{\prime \prime }=\frac{P_n+P_n^{\prime }}{2}\}.$$

Observe that

$$P_n^{\prime \prime }\left({\varphi }_n\right){\eqsim }_n\frac{1+2\cdot Th{m}_{PA}\left({\varphi }_n\right)}{4},$$

so

$$|P_n^{\prime \prime }\left({\varphi }_n\right)-1/2|{\eqsim }_{n}1/4.$$

Now, consider the trader who exploits $\left\{P_n^{\prime \prime }\right\}$ by repeatedly either buying a share of ${\varphi }_n$ 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 $\left\{P_n^{\prime \prime }\right\}$ is not a logical inductor.