The Paradigm of Complex Probability and the Novel Dynamic Logic – The Simulations

What is it about?

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”.

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Why is it important?

Computing probabilities is the main work of classical probability theory. Adding new dimensions to the stochastic experiments will lead to a deterministic expression of probability theory. This is the original idea at the foundations of this work. Actually, the theory of probability is a nondeterministic system in its essence; that means that the events outcomes are due to chance and randomness. The addition of novel imaginary dimensions to the chaotic experiment occurring in the real set R will yield a deterministic experiment and hence a stochastic event will have a certain result in the complex probability set C. If the random event becomes completely predictable then we will be fully knowledgeable to predict the outcome of stochastic experiments that arise in the real world in all stochastic processes. Consequently, the work that has been accomplished here was to extend the real probabilities set R to the deterministic complex probabilities set C = R + M by including the contributions of the set M which is the imaginary set of probabilities. Therefore, since this extension was found to be successful, then a novel paradigm of stochastic sciences and prognostic was laid down in which all stochastic phenomena in R was expressed deterministically in C. I called this original model ‘The Complex Probability Paradigm’ that was initiated and illustrated in my previous 25 research publications. Hence, this original probability paradigm will be applied in this work to the novel dynamic logic which is a development of the ordinary static logic and that was accomplished after adding the dimension of time to the classical system of axioms of logic.

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