A Survey of Fuzzy Intuitionistic Logics in Quantum Mechanics

Abstract

The structure of Brouwer-Zadeh (or BZ)-poset has been first investigated in Cattaneo and Nisticò [1] drawing inspiration from the examples of generalized characteristic functionals (fuzzy sets) on a reference space and of generalized orthogonal projections (fuzzy operators) on a Hilbert space [2-4]. A characteristic feature of these partially ordered structures is a splitting of the standard orthocomplementation mapping into two forms of non-usual ortocomplementations: The first of these orthocompementations’ , called the Zadeh orthocomplementation, is the algebraic generalization of usual orthocomplementation of fuzzy set theory (fuzzy-like orthocomplementation) and the second one ~, called the Brouwer orthocomplementation, generalizes the orthocomplementation obtained from any Brouwerian lattice (intuitionistic-like orthocomplementation). In conclusion, a BZ poset can be summarized as a structure (Σ, ≤, ´, ∼,0,1) of poset bounded by the minimum element 0 and the maximum element 1 and equipped with the two non-usual orthocomplementations’ : Σ→Σ (the fuzzy-like such that a = a“; a ≤ b implies b’ ≤a’; a ≤a’ and b’ ≤b imply a ≤b) and ∼: Σ→Σ (the intuitionistic-like such that a ≤ a^∼∼ ; a ≤ b implies b^∼ ≤ a^∼; a ∧ a∼] =0) interconnected by the rule ∀ ∈ Σ, a^∼’ = a^∼∼