Quantum logic as an implication algebra

Abstract

For the purpose of this paper a logic is defined to be a non-empty set of propositions which is partially ordered by a relation of logical implication, denoted by "s" , and which, as a poset, is orthocomplemented by a unary operation of negation.The negation of the proposition x is denoted by Nx and the least element in the logic is denoted by 0 , we write NO = 1 .

Quantum Implication Algebras

“…[3] The authors formulate an algebra based on the Dishkant implication previously considered by [10,1,6] and cited by [7,13]. There are also other quantum implication algebras given by [5,4,14,7,8,6,13] and others. In this paper we show how are all these algebras interrelated.…”

Residuated Semigroups and the Algebraic Foundations of Quantum Mechanics

“…The map ρ b is called the Sasaki projection. Finch (1970) considered this projection as a binary operation and proved the last claim. Having defined the binary operation & it is now easy to introduce the notion of compatibility and the center of an orthomodular lattice.…”

Logic of Quantum Mechanics for Information Technology Field

“…x ≤ y x∨(x´∧y) = y. Finch introduced logical conjunctions and implications that are defined on an orthomodular lattice [4], [5], and some logical structures with implication "→" were considered to describe the quantum logic. Chajda et al [6] proposed orthomodular implication algebras as another type of quantum logic.…”