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.

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.

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 Labeled Sequent Calculus for Orthologic

2018 ◽  
Vol 47 (4) ◽  
Author(s):  
Tomoaki Kawano
Keyword(s):  
Quantum Logic ◽  
Classical Logic ◽  
Sequent Calculus ◽  
Sequent Calculi

Orthologic (OL) is non-classical logic and has been studied as a part of quantumlogic. OL is based on an ortholattice and is also called minimal quantum logic. Sequent calculus is used as a tool for proof in logic and has been examinedfor several decades. Although there are many studies on sequent calculus forOL, these sequent calculi have some problems. In particular, they do not includeimplication connective and they are mostly incompatible with the cut-eliminationtheorem. In this paper, we introduce new labeled sequent calculus called LGOI, and show that this sequent calculus solve the above problems. It is alreadyknown that OL is decidable. We prove that decidability is preserved when theimplication connective is added to OL.

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 Quantum Uncertainties and Holism Seem to Render Irrelevant Qudit-Semantics

Entropy ◽  
2021 ◽  
Vol 23 (6) ◽  
pp. 735
Author(s):  
Roberto Leporini
Keyword(s):  
Quantum Logic ◽  
Classical Logic ◽  
Density Operator ◽  
Weak Form ◽  
Quantum Circuit ◽  
Point Of View ◽  
Computational Logic ◽  
Quantum Formalism ◽  
New Form

We consider a semantics based on the peculiar holistic features of the quantum formalism. Any formula of the language gives rise to a quantum circuit that transforms the density operator associated to the formula into the density operator associated to the atomic subformulas in a reversible way. The procedure goes from the whole to the parts against the compositionality-principle and gives rise to a semantic characterization for a new form of quantum logic that has been called “ukasiewicz quantum computational logic”. It is interesting to compare the logic based on qubit-semantics with that on qudit-semantics. Having in mind the relationships between classical logic and ukasiewicz-many valued logics, one could expect that the former is stronger than the fragment of the latter. However, this is not the case. From an intuitive point of view, this can be explained by recalling that the former is a very weak form of logic. Many important logical arguments, which are valid either in Birkhoff and von Neumann’s quantum logic or in classical logic, are generally violated.

scholarly journals Minimal Systems of Temporal Logic

Axioms ◽  
2020 ◽  
Vol 9 (2) ◽  
pp. 67 ◽  
Author(s):  
Dariusz Surowik
Keyword(s):  
Temporal Logic ◽  
Intuitionistic Logic ◽  
Classical Logic ◽  
Basic Properties ◽  
Logic Systems ◽  
T System ◽  
Logic System

The article discusses minimal temporal logic systems built on the basis of classical logic as well as intuitionistic logic. The constructions of these systems are discussed as well as their basic properties. The K t system was discussed as the minimal temporal logic system built based on classical logic, while the IK t system and its modification were discussed as the minimal temporal logic system built based on intuitionistic logic.

JOSEPHSON DEVICES FOR QUANTUM COMPUTING

2003 ◽  
Vol 17 (04n06) ◽  
pp. 675-679
Author(s):  
F. CHIARELLO ◽  
M. G. CASTELLANO ◽  
R. LEONI ◽  
G. TORRIOLI ◽  
C. COSMELLI ◽  
...  
Keyword(s):  
Quantum Computing ◽  
Quantum Logic ◽  
Classical Logic ◽  
Experimental Results ◽  
Quantum Gates ◽  
Josephson Devices ◽  
The Moment

Computing tools, all based on classical logic at the moment, present intrinsic limitations that can be overcome by using quantum logic. In this direction, superconducting Josephson devices have been proved to be very suitable candidates for the realization of quantum computing tools. We present some basic elements of quantum computing, possible strategies for the implementation of quantum gates by using Josephson devices, and recent experimental results in this field.

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