# Quantum Logic

While when understood as kinds of logic that formalize the (meta-)language of quantum mechanics there are many proposal for ‘quantum logic’, e.g., the internal logic of Bohr topos, linear logic, etc., this entry is about the ‘quantum logic’, in its standard form, that was formulated following the 1936 Birkhoff-von Neumann proposal.

## Idea

Quantum logic is defined directly through its semantics, i.e., its class of models (and thus its theory).

### Classical mechanics and Boolean lattice

In the classical setting, elementary propositions correspond to measurable subsets of the phase space $M$ up to sets of measure 0. The equivalence classes of such sets define a Boolean lattice under tha partial order $A\leq B$ iff $A\subseteq B$ and the complementation $A’ = A^c = M\setminus A$ (or briefly, the Boolean lattice thus defined is the Lindenbaum–Tarski algebra of the theory).

The lattice operations are $$A\wedge B = A\cap B, A\vee B = A\cup B.$$

This lattice is distributive, and satisfies the law of the excluded middle

$$A\vee A’ = \top.$$

### Quantum mechanics from von Neumann algebras

A physical system is now a Hilbert space $\mathcal{H}$, and each elementary proposition is interpreted by a closed linear subspace $L\subseteq \mathcal{H}$. The set $\mathcal{L}(\mathcal{H})$ of such $L$ forms a lattice under the partial ordering given by inclusion.

The lattice operations are $$L\vee M = \overline{L+M}, L\wedge M = L\cap M.$$ where $\overline{L+M}$ is the closed linear span of $L$ and $M$.

The lattice is no longer distributive, but with the obvious orthocomplementation $L’= L^\bot$ (the orthogonal complement of $L$ in $\mathcal{H}$) it still satisfies the law of the excluded middle.

Let $B(\mathcal{H}$ be the algebra of all bounded operators on $\mathcal{H}$. Now generalize this by identifying the lattice $\mathcal{L}(\mathcal{H})$ of all closed subspaces of $\mathcal{H}$ with the lattice $P(B(\mathcal{H}))$ of all projections on $\mathcal{H}$, called projection lattice. If $M\subset B(\mathcal{H})$ is a von Neumann algebra, then its subset of projections $P(M)$ inherits the lattice structure of $PB(\mathcal{H})$. Each von Neumann algebra defines a quantum logic.

Remark. The quantum logic in the spirit of Birkhoff-von Neumann gives up distributivity but keeps the law of excluded middle makes it in some sense unnatural. The intuitionistic quantum logic tries to address this problem in the toposophy program.

## The original Birkhoff-von Neumann concept of quantum logic

The original Birkhoff-von Neumann idea of quantum logic is subtly different from what became later the standard view. The quantum logic one can extract from the Hilbert space formalism is non-modular, so in the standard view, quantum logic is an orthomorular lattice. while for Birkhoff and von Neumann quantum logic is required to be a modular lattice. This is due to the fact that BIrkhoff and Neumann were searching not only for a noncommutative logic, but also for a noncommutative generalization of classical probability theory.