The idea of associating a non-classical logic with the Hilbert-space model of quantum mechanics goes back to the early days of the development of this physical theory. The success has been modest. Propositions about a quantum physical system correspond to the closed subspaces of a complex Hilbert space and hence their inner structure may be described by an orthomodular lattice. There are calculi that are reasonably called logics of orthomodular lattices. However, a convincing logic doing justice to the canonical model has not been found. Rather, undecidability results have become known.
The situation is different if one replaces the framework of logic by the framework of category theory. More specifically, what we propose is to consider a dagger category whose objects are orthosets and whose morphisms are adjointable maps between them. The notion of an orthoset is entirely built on the notion of orthogonality, which in turn stands for mutual exclusion. As in logic, we deal with the interrelations between objects that are to be interpreted as Hilbert spaces. Orthoclosed subspaces, which model yes-no properties in quantum logic, correspond to dagger monomorphisms. Sasaki projections, which interpret in certain quantum logics the implication connective, correspond in the categorical approach to adjoints of inclusion maps. The conjunction of mutually exclusive propositions in quantum logics might finally be seen as corresponding to the formation of categorical biproducts.
In the talk, we shall present five non-technical assumptions about a dagger category of orthosets that characterise uniquely the standard model of quantum mechanics.