scholarly journals Classical Logic in the Quantum Context

2020 ◽  
Vol 2 (4) ◽  
pp. 600-616
Author(s):  
Andrea Oldofredi
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Quantum Logic ◽  
Classical Logic ◽  
Theory Of Measurement ◽  
Propositional Calculus ◽  
Bohmian Mechanics ◽  
Physical Content ◽  
Present Essay ◽  
Classical Propositional Calculus

It is generally accepted that quantum mechanics entails a revision of the classical propositional calculus as a consequence of its physical content. However, the universal claim according to which a new quantum logic is indispensable in order to model the propositions of every quantum theory is challenged. In the present essay, we critically discuss this claim by showing that classical logic can be rehabilitated in a quantum context by taking into account Bohmian mechanics. It will be argued, indeed, that such a theoretical framework provides the necessary conceptual tools to reintroduce a classical logic of experimental propositions by virtue of its clear metaphysical picture and its theory of measurement. More precisely, it will be shown that the rehabilitation of a classical propositional calculus is a consequence of the primitive ontology of the theory, a fact that is not yet sufficiently recognized in the literature concerning Bohmian mechanics. This work aims to fill this gap.

scholarly journals Nonlocality in Bell’s Theorem, in Bohm’s Theory, and in Many Interacting Worlds Theorising

Entropy ◽  
2018 ◽  
Vol 20 (8) ◽  
pp. 567 ◽  
Author(s):  
Mojtaba Ghadimi ◽  
Michael Hall ◽  
Howard Wiseman
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Bohmian Mechanics ◽  
Bell’S Theorem ◽  
Bell's Theorem ◽  
Significant Progress ◽  
Technical Problems ◽  
New Approach ◽  
Weak Notion ◽  
Bohm’S Theory

“Locality” is a fraught word, even within the restricted context of Bell’s theorem. As one of us has argued elsewhere, that is partly because Bell himself used the word with different meanings at different stages in his career. The original, weaker, meaning for locality was in his 1964 theorem: that the choice of setting by one party could never affect the outcome of a measurement performed by a distant second party. The epitome of a quantum theory violating this weak notion of locality (and hence exhibiting a strong form of nonlocality) is Bohmian mechanics. Recently, a new approach to quantum mechanics, inspired by Bohmian mechanics, has been proposed: Many Interacting Worlds. While it is conceptually clear how the interaction between worlds can enable this strong nonlocality, technical problems in the theory have thus far prevented a proof by simulation. Here we report significant progress in tackling one of the most basic difficulties that needs to be overcome: correctly modelling wavefunctions with nodes.

Abacus logic: The lattice of quantum propositions as the poset of a theory

1994 ◽  
Vol 59 (2) ◽  
pp. 501-515
Author(s):  
Othman Qasim Malhas
Keyword(s):  
Hilbert Space ◽  
Quantum Logic ◽  
Hidden Variables ◽  
Propositional Calculus ◽  
Real Numbers ◽  
Quantum Observable ◽  
Quantum Observables ◽  
Classical Propositional Calculus ◽  
Classical Observable ◽  
Graphic Interpretation

AbstractWith a certain graphic interpretation in mind, we say that a function whose value at every point in its domain is a nonempty set of real numbers is an Abacus. It is shown that to every collection C of abaci there corresponds a logic, called an abacus logic, i.e.. a certain set of propositions partially ordered by generalized implication. It is also shown that to every collection C of abaci there corresponds a theory Jc in a classical propositional calculus such that the abacus logic determined by C is isomorphic to the poset of Jc. Two examples are given. In both examples abacus logic is a lattice in which there happens to be an operation of orthocomplementation. In the first example abacus logic turns out to be the Lindenbaum algebra of Jc. In the second example abacus logic is a lattice isomorphic to the ortholattice of subspaces of a Hilbert space. Thus quantum logic can be regarded as an abacus logic. Without suggesting “hidden variables” it is finally shown that the Lindenbaum algebra of the theory in the second example is a subalgebra of the abacus logic B of the kind studied in example 1. It turns out that the “classical observables” associated with B and the “quantum observables” associated with quantum logic are not unrelated. The value of a classical observable contains, in coded form, information about the “uncertainty” of a quantum observable. This information is retrieved by decoding the value of the corresponding classical observable.

scholarly journals Observability, Unobservability and the Copenhagen Interpretation in Dirac's Methodology of Physics

Quanta ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 68-87 ◽  
Author(s):  
Andrea Oldofredi ◽  
Michael Esfeld
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Close Contact ◽  
Copenhagen Interpretation ◽  
The Other ◽  
Interpretation Of Quantum Mechanics ◽  
Present Essay ◽  
Other Hand ◽  
Methodological Principles ◽  
The One

Paul Dirac has been undoubtedly one of the central figures of the last century physics, contributing in several and remarkable ways to the development of quantum mechanics; he was also at the centre of an active community of physicists, with whom he had extensive interactions and correspondence. In particular, Dirac was in close contact with Bohr, Heisenberg and Pauli. For this reason, among others, Dirac is generally considered a supporter of the Copenhagen interpretation of quantum mechanics. Similarly, he was considered a physicist sympathetic with the positivistic attitude which shaped the development of quantum theory in the 1920s. Against this background, the aim of the present essay is twofold: on the one hand, we will argue that, analyzing specific examples taken from Dirac's published works, he can neither be considered a positivist nor a physicist methodologically guided by the observability doctrine. On the other hand, we will try to disentangle Dirac's figure from the mentioned Copenhagen interpretation, since in his long career he employed remarkably different—and often contradicting—methodological principles and philosophical perspectives with respect to those followed by the supporters of that interpretation.Quanta 2019; 8: 68–87.

Interlude: Some Alternative Interpretations

2017 ◽  
Author(s):  
Richard Healey
Keyword(s):  
Decision Making ◽  
Quantum Mechanics ◽  
Quantum Theory ◽  
Bohmian Mechanics ◽  
Natural Phenomena ◽  
Fundamental Theory ◽  
Relativistic Invariance ◽  
The World ◽  
And Control ◽  
Non Locality

An understanding of quantum theory is manifested by the ability successfully and unproblematically to use it to further the scientific goals of prediction, explanation, and control of natural phenomena. An Interpretation seeks further to formulate or reformulate it as a fundamental theory that provides a self-contained description of the world. I critically review three prominent but radically different Interpretations of quantum theory (Bohmian mechanics, non-linear theories, Everettian quantum mechanics) and give my reasons for rejecting each as a way of understanding quantum theory. These include problems associated with non-locality, failure of relativistic invariance, empirical inaccessibility, and decision-making. We can achieve a satisfactory understanding of quantum theory and how it successfully advances the goals of science without providing an Interpretation of the theory.

MATHEMATICAL ASPECTS OF QUANTUM PHASE

1995 ◽  
Vol 09 (20) ◽  
pp. 2597-2687 ◽  
Author(s):  
D.A. DUBIN ◽  
M.A. HENNINGS ◽  
T.B. SMITH
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Function Theory ◽  
Theory Of Measurement ◽  
Quantum Phase ◽  
Phase Operator ◽  
Current State ◽  
Classical Function ◽  
Quantum Theory Of Measurement

We consider the current state of the quest for a quantum phase operator, which started in the earliest days of quantum mechanics.4–7 Particular emphasis has been placed on analysis of the structure of the several distinct theories, both physical and mathematical, which has led us from classical function theory to the quantum theory of measurement.

THE EINSTEIN–PODOLSKY–ROSEN PROPOSITION ON QUANTUM THEORY AND ITS IMPLICATIONS FOR INTUITIVE CLASSICAL LOGIC

2010 ◽  
Vol 19 (06) ◽  
pp. 799-807 ◽  
Author(s):  
ALI ESKANDARIAN
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Information Content ◽  
Classical Logic ◽  
Quantum States ◽  
Einstein Podolsky Rosen ◽  
Completeness Of Quantum Mechanics ◽  
Shed Light ◽  
Intense Research

Einstein, Podolsky and Rosen raised foundational questions about the completeness of quantum mechanics, if certain intuitive logical statements regarding the nature of reality were assumed to be true. These questions are ultimately of significance to the information content of the theory, which is currently the focus of intense research. In this presentation, selected investigations that have made progress in addressing the EPR concerns and that shed light on the nature of quantum states are surveyed. The implications for intuitive classical logic are speculated in the concluding remarks.

Quantum logic and the classical propositional calculus

1987 ◽  
Vol 52 (3) ◽  
pp. 834-841 ◽  
Author(s):  
Othman Qasim Malhas
Keyword(s):  
Quantum Logic ◽  
Hyperbolic Geometry ◽  
Euclidean Plane ◽  
Propositional Calculus ◽  
Classical Propositional Calculus

AbstractIn much the same way that it is possible to construct a model of hyperbolic geometry in the Euclidean plane, it is possible to model quantum logic within the classical propositional calculus.

Popper and the Quantum Theory

1995 ◽  
Vol 39 ◽  
pp. 163-176
Author(s):  
Michael Redhead
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Quantum Logic ◽  
Scientific Discovery ◽  
Von Neumann ◽  
Interpretation Of Probability ◽  
Propensity Interpretation

Popper wrote extensively on the quantum theory. In Logic der Forschung (LSD) he devoted a whole chapter to the topic, while the whole of Volume 3 of the Postscript to the Logic of Scientific Discovery is devoted to the quantum theory. This volume entitled Quantum Theory and the Schism in Physics (QTSP) incorporated a famous earlier essay, ‘Quantum Mechanics without “the Observer”’ (QM). In addition Popper's development of the propensity interpretation of probability was much influenced by his views on the role of probability in quantum theory, and he also wrote an insightful critique of the 1936 paper of Birkhoff and von Neumann on nondistributive quantum logic (BNIQM).

scholarly journals On quantum implication

2019 ◽  
Vol 1 (1-2) ◽  
pp. 53-63 ◽  
Author(s):  
Yousef Younes ◽  
Ingo Schmitt
Keyword(s):  
Quantum Mechanics ◽  
Quantum Computing ◽  
Quantum Theory ◽  
Quantum Logic ◽  
Algebraic Structure ◽  
Modus Ponens ◽  
System State ◽  
Modus Tollens ◽  
Implication Operator ◽  
Area Of Interest

AbstractLogic is an algebraic structure that defines a set of abstract rules which govern an area of interest. The abstraction property of the rules makes them reusable tools to model different problems and to reason with them. The proliferation of quantum theory brought attention to quantum logic which is a lattice of projectors and it is of importance to quantum computing. Unfortunately, basic tools like implication are not sufficiently studied in that logic, which prevents us from exploiting the power of quantum mechanics in reasoning. This note investigates the implication issue in quantum logic and defines a quantum implication operator for compatible events as well as for incompatible events. The suggested operator depends both on the angle between the vector sub-spaces of the involved events and the angles between the system state and the vector sub-spaces. It differentiates between three cases depending on the angle between the events’ sub-spaces. The article further shows through an example that some classical reasoning rules such as Modus Ponens and Modus Tollens hold given the suggested implication.

Sign in / Sign up

Export Citation Format

Share Document