you might be able to do something like "predict the next prime number" or "predict the next zero of the Riemann zeta function".
you could try something like this for statements in a formal axiomatic system, but know that you're running up against things like the halting problem / entscheidungsproblem / godel incompleteness. so it may be possible to train a neural net to decide the veracity of a statement and do so more quickly than a human might, but you would inevitably be running up against things that are truly undecidable in nature. which is not like go or chess where although they are difficult, they are decidable.
Yes, but it isn't useful. A conversation also has three outcomes (neutral, you like the person more, or less) and yet it doesn't make conversations similar to chess.
It takes O(n) to check if a chess game follows the rules, and to check who wins. Same for checking a proof written in CoQ. What do you mean by "check?"
you could try something like this for statements in a formal axiomatic system, but know that you're running up against things like the halting problem / entscheidungsproblem / godel incompleteness. so it may be possible to train a neural net to decide the veracity of a statement and do so more quickly than a human might, but you would inevitably be running up against things that are truly undecidable in nature. which is not like go or chess where although they are difficult, they are decidable.