Abstract
Feynman pointed out a logic and mathematical paradox in particle physics. The paradox is that we get for the same entity only local dependence and global dependence at the time. This contradiction is coming from the dual nature of the particle viewed as a wave. In the first capacity it has only local dependence; in the second (wave) capacity it has a global dependence. The classical logic has difficulties in resolving this paradox. Changing the classical logic to logic makes the paradox apparent. Particle has the local property or zero dependence with other particles, media has total dependence so it is a global unique entity. Now, in set theory, any element is independent from the other so disjoint set has no elements in common. With this condition we have known that the true/ false logic can be applied and set theory is the principal foundation. Now with conditional probability and dependence by copula the long distance dependence has an effect on any individual entity that now is not isolate but can have different types of dependence or synchronism (constrain) whose effect is to change the probability of any particle. So particle with different degree of dependence can be represented by a new type of set as fuzzy set in which the boundary is not completely defined or where we cannot separate a set in its parts as in the evidence theory. In conclusion the Feynman paradox and Bell violation can be explained at a new level of complexity by many valued logics and new types of set theory.
Highlights
Bell InequalityThese sets have the following Bell inequality
The Bell inequality is based on the classical set theory that is connected with the classical logic
Feynman pointed out a logic and mathematical paradox in particle physics [1]
Summary
These sets have the following Bell inequality. The set theory assumes empty overlap (as a form of independence) of elementary elements which is the basis for the Bell inequality. 2. Dependence and Independence in the Double Slit Experiment as Physical Image of Copula and Fuzzy. The joint probability p ( x1, x2 , , xn ) is represented via multiple conditional probabilities to express the dependence between variables. The copula approach introduces a single function c(u1, u2) denoted as density of copula as a way to model the dependence or inseparability of the variables with the following property in the case of two variables. The copula allows representing the joint probability p(x1, x2) as a combination (product) of single dependent part c(u1, u2) and independent parts: probabilities p(x1) and p(x2).
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