The Mathematical Apparatus for Quantum-Theories: Based on the Theory of Boolean Lattices

Otton Martin Nikodým

Gang Liu

This paper lays out the philosophical foundation of mesoscopic information theory. Taking S4 modal logic and the M₃ diamond modular lattice as its mathematical skeleton, this work puts forward four modal‑probability bridging axioms which form a minimally‑sufficient axiom set: the modal necessity axiom, the modal‑probability consistency axiom, the modal‑probability regularity axiom, and the distributive‑law breaking axiom. This constraint system realizes a controlled transition from the Boolean base of S4 to the non‑Boolean M₃ lattice, embedding modal operators and probability measures within one unified formal framework. Global completeness is not pursued; only internal self‑consistency and traceable reasoning are required. This axiomatic apparatus yields a unified philosophical account of mesoscopic‑level features including modal tension, quasi‑quantum‑like behaviors and informational phase transitions. Empirical adequacy and real‑world case studies are reserved for future versions.

Hermann Haken, Hans Christoph Wolf

Nabila Nassiief Mikhaeel, Basim Samir Labib

We investigate subadditive measures on orthomodular lattices. We show as the main result that the Boolean algebra, the special metric orthomodular lattice and the orthomodular lattice which is unital with respect to subadditive states are equivalent. This result may find an application in the foundation of quantum theories and mathematical logic.

Hermann Haken, Hans Christoph Wolf

Gerald W. Charlwood

Beginning with the work of Birkhoff and von Neumann, the lattice of closed subspaces of Hilbert space has been studied by mathematicians, physicists, and philosophers. Since the 1960's, work of Gudder and of Kochen has focussed on partial structures extracted from, or related to, the lattice mentioned above. This thesis follows in the same tradition. Starting from a new point of view, a particular sort of partial algebra is constructed. The relations of these structures, quasi boolean algebras, to Gudder's work on various sorts of posets and lattices are studied. Their relation to the partial boolean algebras studied by Kochen is also described. In this connection, his definition of partial boolean algebras is refined. A study of quasi boolean algebras themselves is started. Various properties of these partial algebras are isolated and studied. The theory of relations between these partial algebras is then investigated. This theory is modelled after both standard developments in quantum mechanics proper, and some recent work by Kochen. In particular, a theory of "projections" and "interactions" in the context of these algebras is begun.