 The mapping \(p \rightarrow[p\)] is order preserving. For both the classical and quantum-mechanical models considered above, it is in fact an order-isomorphism. Whenever this is the case, \(\Pi\) will inherit from \(\mathbf{L}\) the structure of a complete lattice, which will then automatically be orthomodular by Lemma 4.3. In other words, in such cases we have only one logic, which is a complete orthomodular lattice. While it is surely too much to expect that [ ] will be an order-isomorphism every conceivable physical system—indeed, we can easily construct toy examples to the contrary—the condition is at least reasonably transparent in its meaning.