In nearly all (if not all) projective geometry texts I have bumped into the following theorem:


"Principle of duality: If in a theorem in $\mathfrak{P}$ one switches the word point for the word line and the corresponding incidence relations once again one obtains a theorem of $\mathfrak{P}$."


So far so good. Then I found this awesome list by G. Eric Moorhouse:
<http://www.uwyo.edu/moorhouse/pub/planes/> ([Wayback Machine](http://web.archive.org/web/20171019085548/http://www.uwyo.edu/moorhouse/pub/planes/))


I noted the distinction between a Hall Plane and its dual. So looking a bit into the matter I kept running into the claim "Hall Planes are non-Desarguian and non-self-dual" and the modified version of the principle of duality, which claims the dualized theorem is true *on the dual plane* (this makes much more sense in my mind).


My question is twofold:


1. How does one prove that Hall Planes are not self dual? I haven't managed to find a proof of this fact!
2. What would be the true duality principle? If duality holds in the dual plane it should not hold in Hall Planes (as they're not self dual), yet all texts I've read claim that duality holds in any projective plane.


Thanks in advance!


P.S.
Any good references on the concept of duality in projective geometry from a geometrical point of view would be much appreciated!



On duality on finite projective planes
I'd expect that, in the duality principle that you quoted from "most (if not all) projective geometry texts", the symbol $\mathfrak P$ refers to the theory of projective planes, not to an arbitrary particular projective plane. One reason for this expectations is that theories, not planes, are the sort of entity that can "have" theorems (planes can satisfy statements, including theorems). Another reason is the observation you noted in your question; it's possible for a projective plane to satisfy a statement but not the dual statement. A final reason in favor of my expected interpretation of $\mathfrak P$ is that it makes the principle of duality true.


