Contextual logic for quantum systems

Abstract

In this work we build a quantum logic that allows us to refer to physical magnitudes\ud pertaining to different contexts from a fixed one without the contradictions\ud with quantum mechanics expressed in no-go theorems. This logic arises from considering\ud a sheaf over a topological space associated with the Boolean sublattices of\ud the ortholattice of closed subspaces of the Hilbert space of the physical system.\ud Different from standard quantum logics, the contextual logic maintains a distributive\ud lattice structure and a good definition of implication as a residue of the\ud conjunctio

Fuzzy Propositional Logic Associated with Quantum Computational Gates

“…In Domenech and Freytes (2005), we presented a logical formalization associated with contextuality using Heyting structures of the intuitionistic calculus. They are a particular case of residuated lattices (Kowalski and Ono, 2000), which are the algebraic counterpart of some systems of fuzzy logic.…”

Modal‐type orthomodular logic

“…As we have proved in [8], the KS theorem in the algebraic terms of the previous definition rules out this possibility:…”

Semilattices global valuations in the topos approach to quantum mechanics

“…In the framework of orthomodular logic, a classical proposition is usually represented by a Boolean (also called central) element of an orthomodular lattice (Domenech and Freytes 2005). In particular, propositions about classical systems are represented by a Boolean algebra.…”

“…Kochen-Specker (KS) Theorem rules out this possibility (Kochen and Specker 1967). In algebraic terms, this theorem is expressed as follows Domenech and Freytes 2005): This impossibility to assign values to the properties simultaneously satisfying compatibility conditions is a weighty obstacle for the interpretation of the formalism. The problem of contextuality has been studied from different approaches.…”