As is well known, any preorder R on a set U induces an Alexandrov topology on U. In some interesting cases related to data mining an Alexandrov topology can be transformed into different types of logico-algebraic models. In some cases, (pre)topological operators provided by Pointless Topology may define a topological space on U even if R is not a preorder. If this is the case, then we call R a crypto-preorder. The paper studies the conditions under which a relation R (...) is a crypto-preorder and how to transform a crypto-preorder into a proper preorder. (shrink)

To model a (particular kind of) physical system, the perspective that encompasses preparations, tests and the interplay between them is crucial. In this paper, we employ the conceptual and technical framework presented by Buffernoir (2023) to model physical systems through this pivotal lens, utilizing Chu spaces. With some intuitive and operational axioms we manage to reproduce the following fundamental and abstract results, as well as (part of) the involved reasoning: (1) the states corresponding to a property form a (bi-orthogonally) closed (...) set; (2) the properties form an orthomodular lattice. Adding some idealistic axioms, we can derive: (1) the pure states form a quantum Kripke frame in the sense of Zhong (2017, 2021, 2023); (2) the properties form an irreducible propositional system in the sense of Piron (1976), isomorphic to the lattice of closed sets of pure states. Our axioms are different from those in Buffernoir (2023): on the one hand, they say less about the structure of mixed states; on the other hand, they are arguably more intuitive and operational. The work formalizes some important reasoning about quantum systems, reveals some implicit idealization behind the Hilbert space formalism of quantum theory and hints at other possible formalisms. Finally, it is argued that the framework can be applied to classical physics at an abstract level as well as naturally extended with probabilities. (shrink)