Yes, the probability that the Socrates syllogism would be valid, given that Socrates is mortal, is the same as the probability that it would be valid given that he immortal.
On the other hand, the probability that “I observe a valid syllogism for the conclusion that Socrates is mortal, starting from statements that I believe to be true,” given that Socrates is mortal, is not the same as its probability given that he is immortal—at least if your beliefs have more than zero correlation with reality. So observing a valid syllogism for some conclusion from statements that you believe to be true is indeed Bayesian evidence that the conclusion is true.
This is an important point, because Eliezer seems to have misled himself in the past by noticing the first point but not the second point, for example in his use of these parables in arguing against Robin Hanson regarding disagreement.
Yes, the probability that the Socrates syllogism would be valid, given that Socrates is mortal, is the same as the probability that it would be valid given that he immortal.
On the other hand, the probability that “I observe a valid syllogism for the conclusion that Socrates is mortal, starting from statements that I believe to be true,” given that Socrates is mortal, is not the same as its probability given that he is immortal—at least if your beliefs have more than zero correlation with reality. So observing a valid syllogism for some conclusion from statements that you believe to be true is indeed Bayesian evidence that the conclusion is true.
This is an important point, because Eliezer seems to have misled himself in the past by noticing the first point but not the second point, for example in his use of these parables in arguing against Robin Hanson regarding disagreement.