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The first lines of the Wikipedia entry Quantum logic give an impression. To cut it short, the essence of the example is this: You have a particle smeared out in some box of length $d$. If you split $d$ into two parts, $d_\text{Left}$ and $d_\text{Right}$, then "the particle is in the union of $d_\text{Left}$ and $d_\text{Right}$" is true by definition, while ...

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The most elementary formulation of Quantum Mechanics (the one usually formulated in Hilbert spaces) can be constructed starting form a lattice of all the elementary propositions which can be tested on a given quantum system obtaining, as the outcome, YES or NOT. This can be similarly done for classical mechanics and, in that case the elementary ...

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Consider for example, a plane vector and two orthogonal unit vectors $\hat x$ and $\hat y$. Any vector in the plane can be expressed as $$\vec v = (\vec v \cdot \hat x) \;\hat x + (\vec v \cdot \hat y) \; \hat y = v_x\; \hat x + v_y\; \hat y$$ So, you're correct, $\vec b \cdot \hat a$ is the component of $\vec b$ in the $\hat a$ direction. And further, ...

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