Suppose the bridge is safe iff there’s a proof that the bridge is safe. Then you would forbid the reasoning “Suppose I cross. I must have proven it’s safe. Then it’s safe, and I get 10. Let’s cross.”, which seems sane enough in the face of Löb.
Not good enough. If I know that the bridge isn’t safe, and that I’m not going to cross it, then I know that the bridge is safe iff I’m going to cross it, but not crossing it is still the correct decision.
Suppose the bridge is safe iff there’s a proof that the bridge is safe. Then you would forbid the reasoning “Suppose I cross. I must have proven it’s safe. Then it’s safe, and I get 10. Let’s cross.”, which seems sane enough in the face of Löb.
Then you can prove the bridge is safe without any reference to your own actions.
Suppose the bridge is safe iff A() would decide to cross?
Not good enough. If I know that the bridge isn’t safe, and that I’m not going to cross it, then I know that the bridge is safe iff I’m going to cross it, but not crossing it is still the correct decision.