Classical and Non Classical Limiting Cases of Quantum Logic

María Luisa Dalla Chiara

Sonja Smets

Marcelo Losada, Sebastián Fortín, Federico Holik

Arkady Bolotin

Contrary to classical semantics, the disjunction of two experimental propositions relating to pure states of a quantum system ("quantum propositions" for short) can be true even in the case where neither disjunct is true. This suggests that in such case either both disjuncts are false and so the distributive laws are not applicable to quantum propositions (this inference is accepted in quantum logic) or the disjuncts are not bivalent, i.e., neither true nor false, therefore the principle of bivalence is not applicable to quantum propositions. But, to accept the latter inference, one must explain how quantum propositions become bivalent in the classical limit. This paper shows the emergence of bivalence through the interaction between a quantum system and its environment and compares the environmentally induced bivalence with the classical limit of quantum logic.

Viktoras Rimsha, Laimontas Rimsha

In this paper we propose a unified conceptual framework in which quantum probability, quantum logic, and classical probability theory are understood as different structural levels of a single theoretical hierarchy. Starting from a statistical interpretation of quantum interference, previously developed by the authors, we argue that quantum probability is not an alternative to classical Kolmogorov probability, but a more general structure. Classical probability theory emerges as a limiting case corresponding to the disappearance of interference and the effective decomposition of the whole into independent elements. Within this framework, the Birkhoff–von Neumann quantum logic appears as a natural logical expression of quantum probability rather than as an abstract algebraic construction detached from physical meaning. The priority of the whole over its parts plays a central role in both quantum probability and quantum logic, while classical logic and probability arise when this priority becomes trivial. We emphasize that the proposed interpretation does not reject existing quantum formalism, but reinterprets its logical and probabilistic foundations. This approach may have implications for the conceptual understanding of quantum algorithms and quantum information processing.

Heinrich Saller