New scientific theories have been and will always be required for the progress of science. Two examples of new theories that were advanced in the 20th century are quantum mechanics and fuzzy set theory. Both deal with the concept of uncertainty and both use mathematical tools that are essentially different from those used in classical probability theory. In both, the introduction of new concepts (theoretical terms) was necessary because phenomena and processes observed in the real world could not be explained with existing mathematical concepts. This paper examines the possibilities of (re)interpreting uncertainty in quantum mechanics with the aid of fuzzy sets and systems
New scientific theories have been and will always be required for the progress of science. Two examples of new theories that were advanced in the 20th century are quantum mechanics and fuzzy set theory. Both deal with the concept of uncertainty and both use mathematical tools that are essentially different from those used in classical probability theory. In both, the introduction of new concepts (theoretical terms) was necessary because phenomena and processes observed in the real world could not be explained with existing mathematical concepts. This paper examines the possibilities of (re)interpreting uncertainty in quantum mechanics with the aid of fuzzy sets and systems
In this paper we consider physical systems and the concept of their states in the context of the theory of fuzzy sets and systems. In section 1 we give a brief sketch on the fundamental difference between the theories of classical physics and quantum mechanics. In section 2 and 3 we introduce very shortly systems and their states in classical and quantum mechanics, respectively. Section 4 presents the concept of fuzzy systems. We propose to fuzzify the classical systems in section 5 and quantum systems in section 6. In section 7 we start to consider a fuzzy interpretation of the uncertainty principle.
In this paper we discuss the principles of two fundamental theories of physics: mechanics and quantum mechanics. First, we consider two philosophical positions of the German physicist Heinrich Hertz. He established one of the both in the introduction of his well known Principles of Mechanics. This view - Hertz's "picture" concept - which has a long tradition in philosophy, particularly with regard to physics and later to other empirical sciences, serves as the starting point of an interpretation of the relationship between real systems and theoretical structures of modern science using the concept of fuzziness. This "fuzzy view" on philosophy of science gives particularly an interpretation of Hertz's earlier lecture on The Constitution of matter. Second we consider some principles of quantum mechanics, particularly Werner Heisenberg's uncertainty relation of subatomic object's position and momentum, and Max Born's probabilistic interpretation of the quantum mechanical wave function. These principles are parts of the so-called Copenhagen interpretation of quantum mechanics. Third, we argue that the "fuzzy approach" is fruitful for a "fuzzy" interpretation of quantum mechanics because fuzziness fills the gap between systems and phenomena in reality and the scientific theory of quantum mechanics - as every other scientific theory.
Recognizing that syntactic and semantic structures of classical logic are not sufficient to understand the meaning of quantum phenomena, we propose in this paper a new interpretation of quantum mechanics based on evidence theory. The connection between these two theories is obtained through a new language, quantum set theory, built on a suggestion by J. Bell. Further, we give a modal logic interpretation of quantum mechanics and quantum set theory by using Kripke's semantics of modal logic based on the concept of possible worlds. This is grounded on previous work of a number of researchers (Resconi, Klir, Harmanec) who showed how to represent evidence theory and other uncertainty theories in terms of modal logic. Moreover, we also propose a reformulation of the many-worlds interpretation of quantum mechanics in terms of Kripke's semantics. We thus show how three different theories — quantum mechanics, evidence theory, and modal logic — are interrelated. This opens, on one hand, the way to new applications of quantum mechanics within domains different from the traditional ones, and, on the other hand, the possibility of building new generalizations of quantum mechanics itself.
Human interaction with the world is dominated by uncertainty. Probability theory is a valuable tool to face such uncertainty. According to the Bayesian definition, probabilities are personal beliefs. Experimental evidence supports the notion that human behavior is highly consistent with Bayesian probabilistic inference in both the sensory and motor and cognitive domain. All the higher-level psychophysical functions of our brain are believed to take the activities of interconnected and distributed networks of neurons in the neocortex as their physiological substrate. Neurons in the neocortex are organized in cortical columns that behave as fuzzy sets. Fuzzy sets theory has embraced uncertainty modeling when membership functions have been reinterpreted as possibility distributions. The terms of Bayes' formula are conceivable as fuzzy sets and Bayes' inference becomes a fuzzy inference. According to the QBism, quantum probabilities are also Bayesian. They are logical constructs rather than physical realities. It derives that the Born rule is nothing but a kind of Quantum Law of Total Probability. Wavefunctions and measurement operators are viewed epistemically. Both of them are similar to fuzzy sets. The new link that is established between fuzzy logic, neuroscience, and quantum mechanics through Bayesian probability could spark new ideas for the development of artificial intelligence and unconventional computing.
The goal of this study is to extend quantum theory to the field of fuzzy phenomena. Using the idea and method of fuzzy mathematics, the quantum probability theory of fuzzy event is proposed, with the form of fuzzy event route integral, membership degree amplitude, fuzzy field function, Green function, physical quantity and fuzzy diagram. This theory changes quantum mechanics, making it a special instance of the former. Under some conditions, this theory breaches unitarity, gauge invariance, probability conservation, and information conservation, rendering these concepts approximate. This new theory, which does not require renormalization and can naturally produce meaningful results that are consistent with the experiments, is the proper theory for describing microscopic high-speed phenomena, whereas quantum mechanics is only a proper theory for describing microscopic low-speed phenomena. Because this theory is not divergent under the conditions that there is no renormalization and an infinite number of offsetting terms, it can constitute the theoretical framework required for gravity quantization.
It is known the quantum mechanics is different from the classical mechanics.The reason is the principle of Heisenberg's Uncertainty in quantum mechanics.The uncertainty as principle of mechanics result in wave function in it, square of which is probability.Therefore principal stochasticity of microworld which noted Bohr in famous dispute with Einstein about a God playing dice results in the quantum logic.In frame this logic the Schrodinger cat is alive and dead the same time.The probability theory with its elementary events in its topological probability space cannot extend its definitions.Fuzzy logic and fuzzy sets of Zadeh with focal elements in topological possibility space can makes it.