It is not defined on all X—i.e. P is agnostic about some things
It has P(X) < P(X and Y) for at least one pair X and Y—i.e. P sometimes falls for the conjunction fallacy
It has P(X) = 1 for all mathematically provable statements X—i.e. P is an oracle.
Taken literally, your P falls for 1. For instance it doesn’t have an opinion about whether it will be sunny tomorrow or the Riemann hypothesis is true. If you wish avoid this problem by encoding the universe as a string of symbols to feed to your induction machine, why wouldn’t you let me encode “X and Y” at the same time?
The trichotomy is:
Taken literally, your P falls for 1. For instance it doesn’t have an opinion about whether it will be sunny tomorrow or the Riemann hypothesis is true. If you wish avoid this problem by encoding the universe as a string of symbols to feed to your induction machine, why wouldn’t you let me encode “X and Y” at the same time?