Orthomodular lattice obtained from addressing a fixed point

One of us previously proposed a model of internal measurement that is not only an interaction between internal elements but also involves the notion of an indefinite observer measuring the interaction. In this model, all interactions encompass the notion of detection. The latter can be expressed as a process generating a contradiction. Hereby a fixed point becomes invalid after inducing the contradiction. Instead of embedding this process as an instance along the time axis, we here represent the logical jump via a contradiction (fixed point). Specifically a new operation is defined which replaces the operation by which a fixed point is induced, and we show that both a set as a fixed point and this particular operation on a set can give rise to an orthomodular lattice that is a model of quantum logic. The new procedure is a mixture of inducing a fixed point and addressing a fixed point. It can be compared to mixing up indicating an element with indicating a set consisting of elements, that is, to Russel's paradox. Finally, we point out that orthomodularity can ubiquitously be found in the evolutionary complex systems because it is generally relevant to resolving a paradox or a logical jump.