Problem is, all these redefinitions of quantum mechanics via "quantum logic" or "quantum probability theory", while looking more fundamental and formal than the usual theory based on Schrödinger's equation and Born's rule, do not make it any more intuitive and do not allow anyone to say, "aha! now I get it!". This is because these new foundations themselves are just as unintuitive as quantum mechanics...

[Edit] On the other hand, I find the famous derivation of Schrödinger's equation by Feynman illuminating: http://fermatslibrary.com/s/feynmans-derivation-of-the-schro....