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B The physical motivation for this particular lattice structure comes mainly from realistic and operational considerations. At first sight, the mathematical demands of orthomodularity and covering law look rather technical. They are usually justified by taking a more active (and ideal) point of view with respect to the physical meaning of the elements in the property lattice (for an overview, see 1 3 ). 2 A Single Spin 1/2 System To illustrate the physical meaning of these mathematical considerations, we shall treat some relatively simple particular cases in extenso.

We say that a property a € £ is a classical property if for allp 6 S we have p € n(a) or p G n(a') (19) The set of all classical properties we denote by C. Again considering our two examples, it is easy to see that for the quantum case, hence for £ = V(H), we have no nontrivial classical properties. Indeed, for any closed subspace A € H, different from 0 and H, we have rays of "H that are neither contained in A nor contained in A'. These are exactly the rays that correspond to states that are superposition states of states in A and states in A'.

There did exist at that time versions of operational quantum axiomatics that incorporated right from the start experiments with any number of possible outcomes as basic 39 operational concepts, as for example the approach elaborated by Randall and Foulis 28 > 29 ' 30 . Cattaneo and Nistico proved in 2 6 that, by considering only yes/no-experiments as an operational basis for the construction of the property lattice of the compound entity consisting of separated entities, some of the possible experiments that can be performed on this compound entity are overlooked.