Received 10 December 2015; accepted 15 January 2016; published 19 January 2016
1. Bell Inequality
Let be a set of elements with joint probabilities
For example, consider and a power set
In the classical probability calculus this power set can be rewritten as
Also in the classical probability theory 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 Figure 1.
These sets have the following Bell inequality.
Now we introduce dependence between events. Consider an event with property A and another event with a negated property AC. These events can be called dependent (correlated). This dependence takes place for particles.
Figure 1. Set theory intersections or elements.
Consider an event with property that is with both properties A and CC 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 as two events: event eA with property A and with the property C in the opposite state (negated). The num-
ber of pairs of events is the same as the number of events with the superposition of A and CC,. In this condition 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.
2. 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 slit experiment  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 Figure 2(A) and Figure 2(B)  , where points in Figure 2(B) show particles (elementary probability event) that pass slits.
Figure 2(C) shows theoretical result of the double slit expreiment when only the set theory is used to combine events: one event e1 for one slid and another event e2 for the second slid. In this set-theoretical appro- ach it is assumed that events e1 and e2 are elementary events that do not overlap (have empty intersection, “in- compatible”, completely independent). In this case, the probability that either one of these two events will occur is
Figure 2. (A) Double slit experiment design; (B) Result of double slit experiment: electron buildup over time; (C) Distribution of independent particles (events).
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(x1, x2) as a product of independent probabilities, p(x1)p(x2) 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 x1 and x2 in the probability theory. The joint probability 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). 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 u1 and u2 and the corresponding joint distribution function.
3. Conditional Probability, Dependence in Probability Calculus and Copula
A joint probability distribution
e.g., for two variables. A function is a density of copula
A cumulative function C with inverse functions xi(ui) as arguments:
where and respectively inverse functions
An alternative representation of a cumulative function C
General n-D case
In literature commonly is denoted as copula and is denoted as a density of copula.
4. Examples of Copula and Dependence
When u(x) is a marginal probability F(x), u(x) = F(x) and u is uniformly distributed then the inverse function x(u) is not uniformly distributed, but has values concentrated in the central part as the Gaussian distribution. The inverse process is represented graphically in Figures 3-5.
Figure 3. Relation between marginal probability F(x) and the random variable x.
Figure 4. Symmetric joint probability and copula.
Figure 5. Asymmetric joint probability and symmetric copula.
Consider another example where a joint probability density function p is defined in the two dimensional interval (0, 2) ´ (0, 1) as follows,
Then the marginal function in this interval is
Next we change the reference
and use the marginal probabilities
This allows us computing the inverse function to identify variables x and y as functions of the marginal functions u1 and u2:
Then these values are used to compute the copula C in function (1)
5. Physical Paradox and Physical Meaning of Copula and Fuzzy Theory
Feynman’s argument  involves the idea that classically we think in terms of two distinct and incompatible concepts, particles and waves. These concepts are incompatible because particles are localized and waves are not. To see this, let us start with a point particle or elementary event. In classical mechanics, particles are objects localized in space, and therefore, can only interact with systems that local for them. If a particle then collides with another particle, say constituent of a wall placed in the way of the original particle, an interaction will occur. However, as soon as the particle loses contact with the wall, the interaction ceases to exist. In other words, particles interact locally or have local not global dependence. The second basic concept is the concept of waves. Historically, the physics describing a point particle was extended to include the description of continuous media, and, more importantly to our current discussion, the vibrations of such media in the form of waves. Therefore, waves were considered vibrations of a medium made out of several point particles, and the local interactions between two neighboring particles would allow for a perturbation in one point of the medium to be propagated to another point of the medium. More importantly, such effect depends not only on the position of the particle, but also possibly of all other particles or elementary events that make up the medium, and also on all interactions or boundary conditions that such particles need to satisfy. In other words, waves interact non-locally.
Thus, a media and the wave give an example of total (global) dependence in contrast with the particle. The paradox is that an element (a particle) has a property (global dependence) of the whole media. This is impossible in the classical logic. The global dependence (non-local interaction of the whole system) is a property of the structure of the media. An element cannot have such a property of the whole system because an element has no structure. To explain why the paradox is only apparent we start from Kolmogorov’s probability measure that is defined at the level of propositional classical logic and set theory.
Let be a finite set, F be an algebra over and p be a real-valued function, p:. Then is a probability space, and p a probability measure, if and only if:
The elements of Ω are called elementary probability events or simply elementary events. The elementary events are disjoint sets. Given two sets of elementary probability events A and B the intersection of the two events is given by the expression
When the two sets of events are independent we have
with a trivial density of copula, c(A,B) = 1.
Now when the events ar dijoint one with the other we have
The real joint probability for double slit experiment by quantum mechancis is
for which copula is
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.
Extension of the set theory by evidence theory (Figure 6) in quantum mechanics can be found in the paper of Germano Resconi and others International Journal of Modern Physics C. Vol. 10 No 1 (1999) 29-62.
Figure 6. Set theory intersections or elements with dependence.
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.