Different logical inductors will give different probabilities for each ϕn. The logical induction criterion does not require any answer in particular.

Any particular deterministic algorithm for finding a logical inductor (such as the one in the paper) will yield a logical inductor that gives particular probabilities for these statements, which are close to fixed points in the limit. The algorithm in the paper is parameterized over some measure on Turing machines, and will give different answers depending on this measure. You could analyze which measures would lead to which fixed points, but this doesn’t seem very interesting.

Different logical inductors will give different probabilities for each ϕn. The logical induction criterion does not require any answer in particular.

Any particular deterministic algorithm for finding a logical inductor (such as the one in the paper) will yield a logical inductor that gives particular probabilities for these statements, which are close to fixed points in the limit. The algorithm in the paper is parameterized over some measure on Turing machines, and will give different answers depending on this measure. You could analyze which measures would lead to which fixed points, but this doesn’t seem very interesting.