For the sake of this answer, "dimension" should be interpreted as "number of variables."
In quantum logic, four is the smallest $n$ such that a classically unsatisfiable propositional formula in $n$ variables can be satisfied by substituting quantum propositions in a meaningful way. One example of such a proposition is $$((a\oplus b)\oplus(c\oplus d))\oplus((a\oplus c)\oplus(b\oplus d)),$$ where $\oplus$ is exclusive-or. Note that the grouping of expressions here is crucial: in quantum logic, two propositions can only be meaningfully combined by a logical connective if the corresponding projection operators (or equivalently, the "spin" operators) commute. (For example, the proposition "I have position X and momentum Y" is not meaningful.) One "satisfying assignment" for the formula above is given by (using the spin operator convention) $a=\sigma_x\otimes 1, b=1\otimes\sigma_x, c=\sigma_z\otimes 1, d=1\otimes\sigma_z$.
(Basically, boolean algebras are to classical logic as partial boolean algebras are to quantum logic, and every 3-generator partial boolean algebra can be embedded in a boolean algebra, so there are no such formulas with three or fewer variables.)
