  The structure of a state property system was introduced to formalize in a
complete way the operational content of the Geneva-Brussels approach to the
foundations of quantum mechanics, and the category of state property systems
was proven to be equivalence to the category of closure spaces. The first
axioms of standard quantum axiomatics (state determination and atomisticity)
have been shown to be equivalent to the $T_0$ and $T_1$ axioms of closure
spaces, and classical properties to correspond to clopen sets, leading to a
decomposition theorem into classical and purely nonclassical components for a
general state property system. The concept of orthogonality, very important for
quantum axiomatics, had however not yet been introduced within the formal
scheme of the state property system. In this paper we introduce orthogonality
in a operational way, and define ortho state property systems. Birkhoff's well
known biorthogonal construction gives rise to an orthoclosure and we study the
relation between this orthoclosure and the operational orthogonality that we
introduced.
