  Quantum logic aims to capture essential quantum mechanical structure in
order-theoretic terms. The Achilles' heel of quantum logic is the absence of a
canonical description of composite systems, given descriptions of their
components. We introduce a framework in which order-theoretic structure comes
with a primitive composition operation. The order is extracted from a
generalisation of C*-algebra that applies to arbitrary dagger symmetric
monoidal categories, which also provide the composition operation. In fact, our
construction is entirely compositional, without any additional assumptions on
limits or enrichment. Interpreted in the category of finite-dimensional Hilbert
spaces, it yields the projection lattices of arbitrary finite-dimensional
C*-algebras. Interestingly, there are models that falsify standardly assumed
correspondences, most notably the correspondence between noncommutativity of
the algebra and nondistributivity of the order.
