scholarly journals From Classical Logic to Fuzzy Logic and Quantum Logic: A General View

2021 ◽  
Vol 16 (1) ◽  
Author(s):  
Sorin Nadaban
Keyword(s):  
Fuzzy Logic ◽  
Quantum Logic ◽  
Hilbert Spaces ◽  
Classical Logic ◽  
Mathematical Concept ◽  
General View

The aim of this article is to offer a concise and unitary vision upon the algebraic connections between classical logic and its generalizations, such as fuzzy logic and quantum logic. The mathematical concept which governs any kind of logic is that of lattice. Therefore, the lattices are the basic tools in this presentation. The Hilbert spaces theory is important in the study of quantum logic and it has also been used in the present paper.

scholarly journals A New Approach for Dynamic Stochastic Fractal Search with Fuzzy Logic for Parameter Adaptation

2021 ◽  
Vol 5 (2) ◽  
pp. 33
Author(s):  
Marylu L. Lagunes ◽  
Oscar Castillo ◽  
Fevrier Valdez ◽  
Jose Soria ◽  
Patricia Melin
Keyword(s):  
Fuzzy Logic ◽  
Fuzzy Inference ◽  
Dynamic Parameter ◽  
Mathematical Concept ◽  
Parameter Adaptation ◽  
Inference Models ◽  
Fractal Particles ◽  
Novel Method ◽  
Dynamic Stochastic ◽  
Stochastic Fractal Search

Stochastic fractal search (SFS) is a novel method inspired by the process of stochastic growth in nature and the use of the fractal mathematical concept. Considering the chaotic stochastic diffusion property, an improved dynamic stochastic fractal search (DSFS) optimization algorithm is presented. The DSFS algorithm was tested with benchmark functions, such as the multimodal, hybrid, and composite functions, to evaluate the performance of the algorithm with dynamic parameter adaptation with type-1 and type-2 fuzzy inference models. The main contribution of the article is the utilization of fuzzy logic in the adaptation of the diffusion parameter in a dynamic fashion. This parameter is in charge of creating new fractal particles, and the diversity and iteration are the input information used in the fuzzy system to control the values of diffusion.

A THEOREM OF SOLÉR, THE THEORY OF SYMMETRY AND QUANTUM MECHANICS

2012 ◽  
Vol 09 (02) ◽  
pp. 1260005 ◽  
Author(s):  
GIANNI CASSINELLI ◽  
PEKKA LAHTI
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Quantum Logic ◽  
Hilbert Spaces ◽  
Classical Problem ◽  
Partial Solution ◽  
Space Realization ◽  
Symmetry Transformations ◽  
Axiomatic Quantum Mechanics

A classical problem in axiomatic quantum mechanics is deducing a Hilbert space realization for a quantum logic that admits a vector space coordinatization of the Piron–McLaren type. Our aim is to show how a theorem of M. Solér [Characterization of Hilbert spaces by orthomodular spaces, Comm. Algebra23 (1995) 219–243.] can be used to get a (partial) solution of this problem. We first derive a generalization of the Wigner theorem on symmetry transformations that holds already in the Piron–McLaren frame. Then we investigate which conditions on the quantum logic allow the use of Solér's theorem in order to obtain a Hilbert space solution for the coordinatization problem.

scholarly journals Classical Logic and Quantum Logic with Multiple and Common Lattice Models

2016 ◽  
Vol 2016 ◽  
pp. 1-12 ◽  
Author(s):  
Mladen Pavičić
Keyword(s):  
Quantum Logic ◽  
Classical Logic ◽  
Orthomodular Lattice ◽  
Lattice Models ◽  
Quantum Space ◽  
Technical Terms

We consider a proper propositional quantum logic and show that it has multiple disjoint lattice models, only one of which is an orthomodular lattice (algebra) underlying Hilbert (quantum) space. We give an equivalent proof for the classical logic which turns out to have disjoint distributive and nondistributive ortholattices. In particular, we prove that both classical logic and quantum logic are sound and complete with respect to each of these lattices. We also show that there is one common nonorthomodular lattice that is a model of both quantum and classical logic. In technical terms, that enables us to run the same classical logic on both a digital (standard, two-subset, 0-1-bit) computer and a nondigital (say, a six-subset) computer (with appropriate chips and circuits). With quantum logic, the same six-element common lattice can serve us as a benchmark for an efficient evaluation of equations of bigger lattice models or theorems of the logic.

Data Dependencies in Codd's Relational Model with Similarities

2011 ◽  
pp. 634-657
Author(s):  
Radim Belohlavek ◽  
Vilem Vychodil
Keyword(s):  
Fuzzy Logic ◽  
Classical Logic ◽  
Reference Model ◽  
Relational Model ◽  
Future Research ◽  
Functional Dependencies ◽  
Narrow Sense ◽  
Data Dependencies ◽  
Similarity Relations ◽  
Conceptual Clarity

This chapter deals with data dependencies in Codd’s relational model of data. In particular, we deal with fuzzy logic extensions of the relational model that consist of adding similarity relations to domains and consider functional dependencies in these extensions. We present a particular extension and functional dependencies in this extension that follow the principles of fuzzy logic in a narrow sense. We present selected features and results regarding this extension. Then, we use this extension as a reference model and compare it to several other extensions proposed in the literature. We argue that following the principles of fuzzy logic in a narrow sense, the same way we can follow the principles of classical logic in the case of the ordinary Codd relational model, helps achieve transparency, versatility, conceptual clarity, and theoretical and computational tractability of the extension. We outline several topics for future research.

Mathematics Based on Fuzzy Logic

2017 ◽  
Author(s):  
Radim Bělohlávek ◽  
Joseph W. Dauben ◽  
George J. Klir
Keyword(s):  
Fuzzy Logic ◽  
Mathematical Analysis ◽  
Classical Logic ◽  
Mathematical Reasoning ◽  
Narrow Sense ◽  
And Mathematics ◽  
Truth Degrees

Mathematical reasoning is governed by the laws of classical logic, based on the principle of bivalence. With the acceptance of intermediate truth degrees, the situation changed substantially. This chapter begins with a characterization of mathematics based on fuzzy logic, an identification of principal issues of its development, and an outline of this development. It then examines the role of fuzzy logic in the narrow sense for developing mathematics based on fuzzy logic and the main approaches developed toward its foundations. Next, some selected areas of mathematics based on fuzzy logic are presented, such as the theory of sets and relations, algebra, topology, quantities and mathematical analysis, probability, and geometry. The chapter concludes by examining various semantic questions regarding fuzzy logic and mathematics based on it.

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 Fuzzy Logic versus Classical Logic: An Example in Multiplicative Ideal Theory

2016 ◽  
Vol 2016 ◽  
pp. 1-4
Author(s):  
Olivier A. Heubo-Kwegna
Keyword(s):  
Fuzzy Logic ◽  
Classical Logic ◽  
Ideal Theory

We discuss a fuzzy result by displaying an example that shows how a classical argument fails to work when one passes from classical logic to fuzzy logic. Precisely, we present an example to show that, in the fuzzy context, the fact that the supremum is naturally used in lieu of the union can alter an argument that may work in the classical context.

scholarly journals Logical Paradoxes in Non-Classical Logic Systems

2021 ◽  
Author(s):  
Serge Dolgikh
Keyword(s):  
Quantum Logic ◽  
Classical Logic ◽  
Logic Systems ◽  
Logical Paradoxes

It is shown that well-known logical paradoxes such as Barber paradox can be interpreted differently in non-classical logic systems such as multi-valued, continuous and quantum logic with possibility of solutions of the paradox. The results of this research can have applications in investigations of completeness of logic systems.

Concluding Remarks: Classical Logic and Quantum Logic

Quantum Logic ◽  
1978 ◽  
pp. 140-143
Author(s):  
Peter Mittelstaedt
Keyword(s):  
Quantum Logic ◽  
Classical Logic
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