In the Feynman lectures on physics, Feynman gives an example where the Heisenberg uncertainty relation must hold true (http://www.feynmanlectures.caltech.edu/III_01.html). Specifically he writes: "Let us show for one particular case that the kind of relation given by Heisenberg must be true in order to keep from getting into trouble. We imagine..." (section 1.8 near the bottom of the page).
However, I am not sure I follow his argument, because in his argument, he invokes the uncertainty principle itself on the ability to measure the momentum of the wall. Specifically, he writes (in explaining why the proposed way of measuring the momentum of the electron won't work) "So when we measure the momentum after the electron goes by, we can figure out how much the plate’s momentum has changed. But remember, according to the uncertainty principle we cannot at the same time know the position...".
This seems to be a circular argument. The purpose of the example was to show why the uncertainty relation must hold in this situation. But in the argument, he invokes the uncertainty relation. And actually the argument does not in any way produce formulae or otherwise showing the uncertainty relation itself, it all rests on invoking the uncertainty relation itself. So it seems you could have substituted in almost any claim for an alternative uncertainty relation and then "proven" that this must hold in this situation by using this argument as a template, invoking the claimed alternative uncertainty principle when you come to the determination of the movement of the plate.
Surely I am missing something - but what is it?