Faux-boolean algebras and classical models

W. Michael Dickson

Veeravalli S. Varadarajan

Veeravalli S. Varadarajan

Weng-Long Chang, Athanasios V. Vasilakos

Pieter E. Vermaas

The different modal interpretations all advance a core property ascription {〈P j , C α j 〉} j . This chapter is about how this core property ascription fixes the full property ascription. I start with some logic and algebra. I then present two existing proposals for determining the full property ascriptions as well as four conditions one can impose on them. Lastly I give my own proposal and end by discussing how the full property ascription leads to a value assignment to magnitudes. Some logic and algebra To prepare the ground for discussing the full property ascription, I firstly define the logical connectives (negation, conjunction and disjunction) for properties. Then because, as discussed in Section 4.1, all the modal interpretations which I consider give up on the idea that thefull property ascription assigns definite values to all properties, I secondly define two types of subsets of properties: Boolean algebras and faux-Boolean algebras. I have already assumed that every property Q α pertaining to a system α is represented by one and only one projection Q α defined on the Hilbert space ℋα associated with α. Each projection Q α in its turn corresponds one-to-one to the subspace of ℋα, denoted by 2α, onto which it projects. One thus has a bijective mapping from a property Q α to a projection Q α to a subspace 2α. For the set of subspaces of a Hilbert space one can in a natural way define an orthocomplement, a meet and a join. I now choose 45 to define the logical connectives for properties pertaining to a system by means of this orthocomplement, meet and join for subspaces and the bijective mapping between properties and subspaces (by means of an isomorphism, more shortly).

Atul Kumar Singh, Akhilesh Kumar Singh

This paper examines how logical structures represent sharp and unsharp propositions in classical and quantum systems. Classical sharp logic is modeled by Boolean algebras, where propositions have bivalent truth values, complements are globally defined, and distributivity holds. Quantum sharp logic is modeled by orthomodular lattices of closed Hilbert-space subspaces or projection operators; it retains definite yes–no outcomes for ideal measurements but rejects the Boolean assumptions of global truth valuation and distributivity. Classical unsharp logic uses fuzzy membership or probabilistic truth values to describe vagueness and incomplete information, whereas quantum unsharp logic is expressed through effects and effect algebras, which capture noisy, inefficient, and generalized measurements. The paper’s central question is which algebraic structure is appropriate for each combination of classical or quantum behavior and sharp or unsharp measurement. The analysis shows that the four cases form a coherent hierarchy: Boolean algebras describe deterministic classical events, orthomodular lattices describe ideal quantum propositions, fuzzy or probabilistic models describe classical partial truth, and effect algebras describe realistic quantum effects. This classification clarifies the mathematical and physical role of unsharpness and explains why effect algebras are needed beyond projection-based quantum logic.