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Book Publisher International (a part of SCIENCEDOMAIN International)

The Paradigm of Complex Probability, Prognostic, and Dynamic Logic

Abdo Abou Jaoudé

Published Mar 15, 2024• 0 citations
Publisher

Abstract

The system of axioms for probability theory laid in 1933 by Andrey Nikolaevich Kolmogorov can be extended to encompass the imaginary set of numbers and this by adding to his original five axioms an additional three axioms. Therefore, we create the complex probability set C, which is the sum of the real set R with its corresponding real probability, and the imaginary set M with its corresponding imaginary probability. Hence, all stochastic experiments are performed now in the complex set C instead of the real set R. The objective is then to evaluate the complex probabilities by considering supplementary new imaginary dimensions to the event occurring in the ‘real’ laboratory. Consequently, the corresponding probability in the whole set C is always equal to one and the outcome of the random experiments that follow any probability distribution in R is now predicted totally in C. Subsequently, it follows that, chance and luck in R is replaced by total determinism in C. Consequently, by subtracting the chaotic factor from the degree of our knowledge of the stochastic system, we evaluate the probability of any random phenomenon in C. My innovative Complex Probability Paradigm (CPP) will be applied to the established theory of logic in order to express it completely deterministically in the probability universe C = R + M. Therefore, after adding the time dimension, we will relate and join this original paradigm to a newly defined logic that I called ‘Dynamic Logic’ and it will be also implemented to pipeline prognostic with the aim of illustrating CPP and this novel kind of logic.

DOI
10.9734/bpi/mono/978-81-971164-9-0/ch0

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Book Publisher International (a part of SCIENCEDOMAIN International)•Mar 15, 2024•1 citation

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The system of axioms for probability theory laid in 1933 by Andrey Nikolaevich Kolmogorov can be extended to encompass the imaginary set of numbers and this by adding to his original five axioms an additional three axioms. Therefore, we create the complex probability set C, which is the sum of the real setR with its corresponding real probability, and the imaginary setM with its corresponding imaginary probability. Hence, all stochastic experiments are performed now in the complex setC instead of the real setR. The objective is then to evaluate the complex probabilities by considering supplementary new imaginary dimensions to the event occurring in the “real” laboratory. Consequently, the corresponding probability in the whole set C is always equal to one and the outcome of the random experiments that follow any probability distribution in R is now predicted totally inC. Subsequently, it follows that chance and luck in R is replaced by total determinism in C. Consequently, by subtracting the chaotic factor from the degree of our knowledge of the stochastic system, we evaluate the probability of any random phenomenon in C. My innovative complex probability paradigm (CPP) will be applied to the established theory of quantum mechanics in order to express it completely deterministically in the universe C=R+M.

IntechOpen eBooks•Nov 28, 2022•4 citations
The system of axioms for probability theory laid in 1933 by Andrey Nikolaevich Kolmogorov can be extended to encompass the imaginary set of numbers and this by adding to his original five axioms an additional three axioms. Therefore, we create the complex probability set C, which is the sum of the real set R with its corresponding real probability, and the imaginary set M with its corresponding imaginary probability. Hence, all stochastic experiments are performed now in the complex set C instead of the real set R. The objective is then to evaluate the complex probabilities by considering supplementary new imaginary dimensions to the event occurring in the ‘real’ laboratory. Consequently, the corresponding probability in the whole set C is always equal to one and the outcome of the random experiments that follow any probability distribution in R is now predicted totally in C. Subsequently, it follows that, chance and luck in R is replaced by total determinism in C. Consequently, by subtracting the chaotic factor from the degree of our knowledge of the stochastic system, we evaluate the probability of any random phenomenon in C. My innovative Complex Probability Paradigm (CPP) will be applied to the established theory of logic in order to express it completely deterministically in the probability universe C = R + M. Therefore, after adding the time dimension, we will relate and join this original paradigm to a newly defined logic that I called ‘Dynamic Logic’ and it will be also implemented to pipeline prognostic with the aim of illustrating CPP and this novel kind of logic.
The set of imaginary numbers is taken into account by extending the probability system of five axioms of Andrey Nikolaevich Kolmogorov which was put forward in 1933. This is achieved by adding three new and supplementary axioms. Hence, any random experiment can thus be performed in the extended complex probability set C = R + M which is the sum of the real set R of real probabilities and the imaginary set M of imaginary probabilities. The objective here is to determine the complex probabilities by encompassing and considering additional new imaginary dimensions to the event that occurs in the “real” laboratory. The outcome of the stochastic phenomenon in C can be foretold perfectly whatever the probability distribution of the input random variable in R is since the corresponding probability in the whole set C is permanently and constantly equal to one. Thus, the consequence that follows indicates that randomness and chance in R is substituted now by absolute determinism in C. This is the result of the fact that the probability in C is computed after the subtraction of the chaotic factor from the degree of our knowledge of the nondeterministic experiment. This novel complex probability paradigm will be applied to a newly defined logic that I called “Dynamic Logic”.
The system of axioms for probability theory laid in 1933 by Andrey Nikolaevich Kolmogorov can be extended to encompass the imaginary set of numbers and this by adding to his original five axioms an additional three axioms. Therefore, we create the complex probability set C, which is the sum of the real set R with its corresponding real probability, and the imaginary set M with its corresponding imaginary probability. Hence, all stochastic experiments are performed now in the complex set C instead of the real set R. The objective is then to evaluate the complex probabilities by considering supplementary new imaginary dimensions to the event occurring in the ‘real’ laboratory. Consequently, the corresponding probability in the whole set C is always equal to one and the outcome of the random experiments that follow any probability distribution in R is now predicted totally in C. Subsequently, it follows that, chance and luck in R is replaced by total determinism in C. Consequently, by subtracting the chaotic factor from the degree of our knowledge of the stochastic system, we evaluate the probability of any random phenomenon in C. My innovative Complex Probability Paradigm (CPP) will be applied to the established theory of quantum mechanics in order to express it completely deterministically in the universe C = R + M as well as to the quantum uncertainty principle in order to verify it and to extend it to the universes M and C.
In the year 1933, the Russian mathematician Andrey Nikolaevich Kolmogorov put forward the system of axioms of modern probability theory. By adding to Kolmogorov’s original five axioms and an additional three axioms, this established system can be extended to encompass the imaginary set of numbers. Accordingly, the complex probability set C will be created and which is the sum of its corresponding real probability belonging to the real set R and of its corresponding imaginary probability belonging to the imaginary set M. Thus, all random phenomena do not occur now in the real set R but in the general complex set C that encompasses both R and M. Hence, we take into consideration supplementary new imaginary dimensions to the event occurring in the ‘real’ laboratory to evaluate the complex probabilities. This is consequently the objective of this novel paradigm. Subsequently, the outcome of the stochastic experiments that follow any probability distribution in R is now predicted perfectly and totally in C and the corresponding probability in the whole set C is always equal to one. Afterward, it follows that luck and chance in R are substituted by absolute determinism in C. Therefore, we evaluate the probability of any probabilistic phenomenon in C by subtracting the chaotic factor from the degree of our knowledge of the random system. My groundbreaking Complex Probability Paradigm (or CPP) will be applied to the well-known theory of Markov Chains Transition Matrices in order to express it perfectly and absolutely in a deterministic way in the universe C = R + M as well as to extend it to the probabilities’ universes M and C.
The system of axioms for probability theory laid in 1933 by Andrey Nikolaevich Kolmogorov can be extended to encompass the imaginary set of numbers and this by adding to his original five axioms an additional three axioms. Therefore, we create the complex probability set C, which is the sum of the real set R with its corresponding real probability, and the imaginary set M with its corresponding imaginary probability. Hence, all stochastic experiments are performed now in the complex set C instead of the real set R. The objective is then to evaluate the complex probabilities by considering supplementary new imaginary dimensions to the event occurring in the ‘real’ laboratory. Consequently, the corresponding probability in the whole set C is always equal to one and the outcome of the random experiments that follow any probability distribution in R is now predicted totally in C. Subsequently, it follows that, chance and luck in R is replaced by total determinism in C. Consequently, by subtracting the chaotic factor from the degree of our knowledge of the stochastic system, we evaluate the probability of any random phenomenon in C. My innovative Complex Probability Paradigm (CPP) will be applied to the established theory of quantum mechanics in order to express it completely deterministically in the universe C = R + M.
The five fundamental axioms of classical probability theory were put forward in 1933 by Andrey Nikolaevich Kolmogorov. Encompassing new imaginary dimensions with the experiment real dimensions will make the work in the complex probability set C totally predictable and with a probability permanently equal to one. This is the original idea in my complex probability paradigm. Therefore, this will make the event in C = R + M absolutely deterministic by adding to the real set of probabilities R the contributions of the imaginary set of probabilities M. It is of great importance that stochastic systems become totally predictable since we will be perfectly knowledgeable to foretell the outcome of all random events that occur in nature. Consequently, by calculating the parameters of the new prognostic model, we will be able to determine the chaotic factor, the magnitude of the chaotic factor, the degree of our knowledge, the real and imaginary and complex probabilities in the probability sets R and M and C and which are all subject to chaos and random effects. Accordingly, my purpose here is to link my complex probability paradigm to logic. Hence, after adding the time dimension, we will apply this novel paradigm to a newly defined logic that I called ‘Dynamic Logic’.
The system of axioms for probability theory laid in 1933 by Andrey Nikolaevich Kolmogorov can be extended to encompass the imaginary set of numbers and this by adding to his original five axioms an additional three axioms. Therefore, we create the complex probability set C, which is the sum of the real setR with its corresponding real probability, and the imaginary setM with its corresponding imaginary probability. Hence, all stochastic experiments are performed now in the complex setC instead of the real setR. The objective is then to evaluate the complex probabilities by considering supplementary new imaginary dimensions to the event occurring in the “real” laboratory. Consequently, the corresponding probability in the whole set C is always equal to one and the outcome of the random experiments that follow any probability distribution in R is now predicted totally inC. Subsequently, it follows that chance and luck in R is replaced by total determinism in C. Consequently, by subtracting the chaotic factor from the degree of our knowledge of the stochastic system, we evaluate the probability of any random phenomenon in C. My innovative complex probability paradigm (CPP) will be applied to the established theory of quantum mechanics in order to express it completely deterministically in the universe C=R+M.