To fully solve this problem requires answering the question of how the placebo effect physically works, which requires answering the question of what a belief physically is, to have that physical effect.
However, no-one yet knows the answers to those questions, which renders all of these logical arguments about as useful as Zeno's proof that arrows cannot move. The problem of how to knowingly induce a placebo response is a physical one, not a logical one. Nature has no paradoxes.
The first part is wrong, the second is obvious and I never said anything to contradict it. We don't need to know exactly how beliefs are implemented just approximately how they behave.
Of coarse this is a physical problem and of coarse we don't know every detail enough to give an exact answer, the math can still be useful for solving the problem.
If it's worth saying, but not worth its own post (even in Discussion), then it goes here.
If continuing the discussion becomes impractical, that means you win at open threads; a celebratory top-level post on the topic is traditional.