I was looking at how did von Neumann and Birkhoff formulate their Quantum Logic formalism back in 1936. To solve some questions, I contacted via email a philosopher who studied this topic.
I thought that Von Neumann and Birkhoff's original formalism was strongly related to projective geometry (and that basically, Quantum Logic was fundamentally based on projective geometry, as they said in their original paper:
Hence, we conclude that the propositional calculus of quantum mechanics has the same structure as an abstract projective geometry
I also thought that Von Neumann disliked the Hilbert space formulation of quantum mechanics before publishing his Quantum Logic with Birkhoff, since in a letter to Birkhoff in 1935, Von Neumann said:
I would like to make a confession which may seem immoral: I do not believe absolutely in Hilbert space any more.
— John von Neumann, letter to Garrett Birkhoff, 1935
Finally, in 1954, almost two decades after his original paper with Birkhoff, he still studied and considered the non-boolean/non-commutative/non-distributive characteristics of quantum logic, since according to Jeffrey Bub:
They (probabilities) are “uniquely given from the start” as a feature of the non-Boolean structure, to quote von Neumann, related to the angles in Hilbert space, not measures over states as they are in a classical or Boolean theory.
 John von Neumann, “Unsolved problems in mathematics,” an address to the International Mathematical Congress, Amsterdam, September 2, 1954.
The thing is that the philosopher I contacted with contradicted all of this. He said:
In the strict sense, quantum logic is a non-distributive consequence relation, not a geometry or an algebra. In this sense of quantum logic, von Neumann abandoned his interest in it (and in the Hilbert state formulation of quantum mechanics more generally), after 1936, and shifted it to type II factor algebras (which we now call von Neumann algebras).
I have a few questions about this:
Is this philosopher right? Or on the contrary, Von Neumann and Birkhoff's quantum logic is fundamentally based in an abstract projective geometry?
Didn't Von Neumann already dislike or abandon the Hilbert space formalism before 1936 (according to the letter that von Neumann sent to Birkhoff in 1935)?