Unified Interpretation of Quantum and Classical Logics

Chan

Quantum mechanics is empirically unrivalled, yet the origin of its distinctive non-classical features remains a foundational question. This paper proves, within the unchanged standard quantum formalism, that these features—collectively termed “quantumness”—arise if and only if certain configurations of the system are indistinguishable in principle. Quantumness is defined as any deviation from classical probability distributions, i.e., probabilities that require coherent amplitude addition rather than incoherent probability addition. Three theorems are established using only the standard Hilbert-space formalism and unitary evolution: 1. Indistinguishability of configurations is necessary and sufficient for quantumness.2. Indistinguishability of identical particles is neither necessary nor sufficient.3. An equivalent restatement in precise operational terms: quantumness occurs exactly when a variable lacks physical definition (distinguishability in principle by macroscopic outcomes). These results are presented and interpreted within the Logical Interpretation of quantum mechanics, a minimalist framework that adds no new postulates and applies classical logic strictly to physically defined facts. The interpretation provides the conceptual tools—particularly the precise notion of “indistinguishability in principle”—that make the theorems sharp and unified. This work unifies the origin of quantumness and serves as the conceptual prequel to the full Logical Interpretation developed in the author’s companion paper.

Claudio Garola

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

Andrei Galiautdinov

Alexander Stahl, Ingo Schmitt

This article provides a detailed explanation of the BBQ-Tree, a unified logic-based model that integrates both classical Decision Trees and Quantum-Logic Decision Trees into a generalized framework for classification and regression. As it combines these paradigms, the BBQ-Tree effectively addresses problems with both linear and curved decision boundaries while prioritizing interpretability. We provide a detailed description of the underlying concepts, a possible training algorithm, experimental evaluations and the incorporation of regression functionality, broadening its applicability beyond classification tasks. Strategies for efficient training and model optimization are also presented. Experimental results demonstrate that the BBQ-Tree produces compact, interpretable models capable of revealing data trends, while achieving accuracy comparable to Decision Trees. Furthermore, its new regression capabilities highlight its versatility and performance across a wider range of tasks.