Beyond Set Theory in Bell Inequality

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.

Example, consider

{ }
, , 1 2 3 X x x x = and a power set 1 2 1 3 2 3 1 2 3 In the classical probability calculus this power set can be rewritten as Also in the classical probability theory (

) ( ) ( ) ( ) p A B p A p B p A B
∪ = + − ∩ and the probability of a set is the sum of probabilities of its elements: because the intersection of elementary events is empty.
In a graphic way it is shown in Fig.
1 ( ) ( ) Now we introduce a dependence between events.Consider an event with property A and another event with a negated property A C .These events can be called dependent (correlated).This dependence takes place for particles.Consider an event with property C A C ∩ that is with both properties A and C C at the same time.We cannot measure the two properties by using one instrument at the same time, but we can use the correlation to measure the second property if two properties are correlated.We can also view an event with property C A C ∩ as two events: event e A with property A and with the property C in the opposite state (negated).The number of pairs of events ( , ) is the same as the number of events with the superposition of A and C C , ∩ .In this d'Espagnat explains the connection between the set theory and Bell's inequality.It is known that the Bell's inequality that gives us the reality condition is violated .Conclusion.The Bell inequality is based on the classical set theory that is connected with the classical logic.The set theory assumes empty overlap (as a form of independence) of elementary elements which is the basis for the Bell inequality.Thus the logic of dependence can differ from the logic of independence.Thus we must use a theory beyond the classical set theory.

Dependence and independence in the double slit experiment as physical image of copula and fuzzy
The goal of this section is to analyze the double slits experiment [Feynman, 1988] as a demonstration of the need to build a separate theory to deal with dependent/related evens under uncertainty.The design and results of the double slits experiment is outlined in Fig. 5a Fig. 5c shows theoretical result of the double slit expreiment when only the set theory is used to combine events: one event e 1 for one slid and another event e 2 for the second slid.In this settheoretical approach it is assumed that events e 1 and e 2 are elementary events that do not overlap (have empty intersection, "incompatible", completely independent).In this case, the probability that either one of these two events will occur is In classical logic it is always true that variable is self-dependent (that is the repeat of the process produces the same result).In the probability calculus it is not the case.The random factors can change the output when the situation is repeated.Quite often the probabilistic approach is applied to study frequency of independent phenomena.In the case of dependent variables we cannot derive p(x 1 ,x 2 ) as a product of independent probabilities, p(x 1 )p(x 2 ) and must use multidimensional probability distribution with dependent valuables.The common technique for modeling it is a Bayesian network.In the Bayesian approach the evidence about the true state of the world is expressed in terms of degrees of belief in the form of Bayesian conditional probabilities.The conditional probability is the main element to express the dependence or inseparability of the two states x 1 and x 2 in the probability theory.The joint probability p(x 1 ,x 2 ,…,x n ) is represented via multiple conditional probabilities to express the dependence between variables.The copula approach introduces a single function 1 2 ( , ) c u u 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(x 1 ,x 2 ) as a combination (product) of single dependent part c(u 1 ,u 2 ) and independent parts: probabilities p(x 1 ) and p(x 2 ).The investigation of copulas and their applications is a rather recent subject of mathematics.From one point of view, copulas are functions that join or 'couple' one-dimensional distribution functions u 1 and u 2 and the corresponding joint distribution function.

Conditional probability ,dependence in probability calculus and copula
A joint probability distribution where u 1 =∫p(x 1 ) dx 1 and u 1 =∫p(x 2 ) dx 2 .
A cumulative function C with inverse functions x i (u i ) as arguments: and respectively inverse functions 1 1

2-D case
This copula is tabulated as follows: Now for the dependence element as copula we have that set theory is not sufficient because two disjoint sets can have a probability ( evidence ) different from the traditional formula In a graphic way we see the traditional set theory with dependences by arrows

Conclusion
Feynman pointed out on a logic and mathematical paradox in particle physics [1].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 to resolve 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 is a global unique entity.Now, in set theory, any element is independent from the other so disjoint set has not element in common.With this condition we have 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 effect on any individual entity that now is not isolate but can have different type of dependence or synchronism ( constrain ) which 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 are 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 logic and new type of set theory.

Figure 4 .
Figure 4. Set theory intersections or elements with dependence