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Annalen der Physik

Quantum Logics and Relative Lindenbaum Property

Roberto Giuntini

Published Jan 1, 1989• 2 citations
Publisher

Abstract

Abstract The relative Lindenbaum property is considered in orthomodular quantum logic and in partial class ical logic. The properties of these models are connected with the possibility of hidden‐variable reconstruction of particular (1/2‐spin particle e.g.) quantum physical systems.
DOI
10.1002/andp.19895010408

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This paper will take into account the Lindenbaum property in Ortho modular Quantum Logic (OQL) and Partial Classical Logic (PCL). The Lindenbaum property has an interest both from a logical and a physical point of view since it has to do with the problem of the completeness of quantum theory and with the possibi? lity of extending any semantically non-contradictory set of formulas to a semanti? cally non-contradictory complete set of formulas. The main purpose of this paper is4 to show that both OQL and PCL cannot satisfy the Lindenbaum property.
A many–valued logic from an old neglected Lukasiewicz paper is adopted to describe results of physical (esp. quantum) experiments. Truth–values of propositions are interpreted as probabilities of their experimental confirmation. It is shown that the Lindenbaum algebra of the obtained logic is an orthomodular partially ordered set, i.e. a “quantum logic” in the Birkhoff–von Neumann sense. Short outline of the historical development of applications of many–valued logics in foudations of quantum mechanics is given.
A class of quantum logics (orthomodular atomic lattices) is defined and studied. The obtained results suggest that the covering property acts as a selection criterion for theories in the sense that, given a set of orthomodular atomic lattices, those which may be theories for a fixed set of systems may be selected by verifyng if they have the covering property or not. A relation between physical meanings of the covering property, lattice - algebraic operations and the commutativity relation is also suggested
This article employs a descriptive-analytical method with an algebraic and axiomatic approach to address quantum logic. Through several thought experiments, it is demonstrated that the distributive law for propositions does not hold in quantum mechanics. Therefore, the logic governing this domain of science is not classical logic. In fact, the algebra corresponding to the logic of quantum propositions is an orthomodular lattice, which is why the distributive law is not necessarily valid for such propositions. In quantum logic, various definitions are offered for the implication operator. Since none of these operators satisfy the deduction theorem, a suitable implication operator with this property must be defined. This objective is achieved by defining a unified quantum logic. By constructing a Lindenbaum-Tarski algebra for this unified quantum logic, it can be shown that it is indeed a quantum logic. Dishkant presented an embedding of quantum logic into the Br-modal logic system. This provides an appropriate semantics for quantum logic. A suitable model for this logic is the closed subspaces of a Hilbert space.