The Paradigm of Complex Probability, Prognostic, and Dynamic Logic

What is it about?

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.

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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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