Disjunctive Quantum Logic in Dynamic Perspective

Sonja Smets

Alexandru Baltag, Sonja Smets

Bob Coecke, David Moore, Sonja Smets

This paper surveys some recent developments towards a dynamic quantum logic and outlines its explicite construction -- some analogies and contrasts with other logics of dynamics are indicated. Abstract: The development of ``(static) operational quantum logic" points out that classical boolean structures are too rigid to describe the actual and potential properties of quantum systems. On the other hand, an intuitionistic perspective on operational quantum logic, guides us in the direction of incorporating dynamics logically by reconsidering the primitive propositions required to describe the behavior of a quantum system, in particular in view of the emergent disjunctivity due to the non-determinism of quantum measurements. A further elaboration on "intuitionistic quantum logic" emerges into a "dynamic operational quantum logic", which allows us to express dynamic reasoning in the sense that we can capture how actual properties propagate, including their temporal causal structure, and provides a unified logical description of systems which evolve or which are submitted to measurements. This setting reveals that even static operational quantum logic bears a hidden dynamic ingredient in terms of what is called "the orthomodularity" of the lattice-structure. Focusing on the quantale semantics for dynamic operational quantum logic, we delineate some points of difference with the existing quantale semantics for (non)-commutative linear logic.

María Luisa Dalla Chiara, Roberto Giuntini

James H. McGrath

To the memory of John D. Trimmer This paper assesses the impact of disjunctive facts on the quantum logic read off procedure. The purpose of the procedure is to transfer a significant quantum structure to a set of propositions; its first step is an attempt to discover that structure. Here I propose that disjunctive facts as traditionally conceived have blocked the procedure at its first step and have therefore subverted the best-known attempts to read off quantum logic. Recently however Allen Stairs has proposed a view of disjunctive facts which re-establishes the possibility of reading off quantum logic. Both the traditional conception and Stairs’ revision of disjunctive facts are interesting in their own right, independent of quantum propositional logic. Too many things are called ‘quantum logic’. The term is disentangled and notation is fixed in this preliminary step which relies on basic lattice theory.

Sonja Smets