Implicative logical nature of quantum correlations

Kenji Tokuo

Aleksandar Bakalov

This paper presents a novel framework for examining causality through a fundamentally non-temporal perspective, grounded in mathematical logic and category theory. We demonstrate that standard logical operations, including implication, conjunction, disjunction, and quantification, are inherently non-temporal, suggesting that causality itself might be better understood as a logical rather than temporal relationship. Our framework provides: (1) A rigorous mathematical foundation for non-temporal causality, based on set theory and mathematical logic; (2) A reinterpretation of quantum mechanical phenomena, including wave function collapse and entanglement, as manifestations of non-temporal logical relations; (3) A new perspective on causal set theory that challenges the assumed primacy of temporal ordering in discrete spacetime approaches. The framework naturally addresses several persistent puzzles in physics, including the measurement problem in quantum mechanics, the emergence of time in quantum gravity, and the apparent tension between locality and quantum correlations. We develop formal proofs showing how temporal causation might emerge as a special case of more general non-temporal logical relations, with implications for quantum foundations, relativity theory, and the unification of quantum mechanics with gravity. This approach suggests new directions for understanding the foundations of physics, the nature of time, and the relationship between logic and physical reality.

Nikos A. Tambakis

Sonja Smets

Research within the operational approach to the logical foundations of physics has recently pointed out a new perspective in which quantum logic can be viewed as an intuitionistic logic with an additional operator to capture its essential, i.e., non-distributive, properties. In this paper we will offer an introduction to this approach. We will focus further on why quantum logic has an inherent dynamic nature which is captured in the meaning of “orthomodularity” and on how it motivates physically the introduction of dynamic implication operators, each for which a deduction theorem holds with respect to a dynamic conjunction. As such we can offer a positive answer to the many who pondered about whether quantum logic should really be called a logic. Doubts to answer the question positively were in first instance due to the former lack of an implication connective which satisfies the deduction theorem within quantum logic.

Allen Stairs, Jeffrey Bub

John Harding, Zhenghan Wang