Psychological origin of quantum logic: An orthomodular lattice derived from natural-born intelligence without Hilbert space

While the role of quantum theory becomes increasingly important through molecular computing and quantum cognition, it is still unclear how quantum mechanics can be involved in macroscopic phenomena. We previously argued the origin of quantum logic in terms of lattice theory; however, we do not refer to Hilbert space very much. Here, we show the psychological origin of quantum logic (i.e., orthomodular lattice) by not using Hilbert space. After showing how Hilbert space plays a role in constructing an orthomodular lattice, we show the idea of natural-born intelligence, in which a binary opposition pair can constitute not only positive but negative antinomy. In taking an object outside and an image inside a brain as that binary opposition pair, the structure entailing positive and negative antinomy is expressed as a specific binary relation consisting of multiple diagonal relations, which is called traumatic relation. Finally, we show that the traumatic relation leads to an orthomodular lattice without Hilbert space.