The paper by Garrett Birkhoff and John von Neumann explores the logical structure of quantum mechanics, which differs from classical logic. It argues that quantum mechanics requires a propositional calculus that is formally similar to the calculus of linear subspaces, with operations like set products, linear sums, and orthogonal complements. This calculus resembles the usual propositional logic with operations like and, or, and not. The authors first present heuristic arguments suggesting this structure is appropriate for quantum mechanics, then reconstruct the calculus axiomatically. They compare quantum mechanics with classical mechanics throughout to clarify the discussion. The paper concludes with tentative conclusions based on the presented material. The concept of a physical system is central, with an observation being the recording of measurements. The most general prediction about a system is that the measured values will lie in a subset of the observation-space. The paper then discusses phase-spaces, a concept shared by quantum theory and classical mechanics. In classical mechanics, a phase-space is a region of 2n-dimensional space, representing positions and momenta. In electrodynamics, it is a function-space of infinitely many dimensions. In quantum theory, it is a function-space, typically Hilbert space, where points correspond to wave-functions. The law of propagation, whether in classical mechanics or quantum theory, is seen as inducing a steady fluid motion in the phase-space.The paper by Garrett Birkhoff and John von Neumann explores the logical structure of quantum mechanics, which differs from classical logic. It argues that quantum mechanics requires a propositional calculus that is formally similar to the calculus of linear subspaces, with operations like set products, linear sums, and orthogonal complements. This calculus resembles the usual propositional logic with operations like and, or, and not. The authors first present heuristic arguments suggesting this structure is appropriate for quantum mechanics, then reconstruct the calculus axiomatically. They compare quantum mechanics with classical mechanics throughout to clarify the discussion. The paper concludes with tentative conclusions based on the presented material. The concept of a physical system is central, with an observation being the recording of measurements. The most general prediction about a system is that the measured values will lie in a subset of the observation-space. The paper then discusses phase-spaces, a concept shared by quantum theory and classical mechanics. In classical mechanics, a phase-space is a region of 2n-dimensional space, representing positions and momenta. In electrodynamics, it is a function-space of infinitely many dimensions. In quantum theory, it is a function-space, typically Hilbert space, where points correspond to wave-functions. The law of propagation, whether in classical mechanics or quantum theory, is seen as inducing a steady fluid motion in the phase-space.