and $B$. Let $\overline{Y}$ and $\overline{B}$ be the respective complementary of $Y$ and $B$ in $E$ of the same cardinal $n_{\overline{b}}= n-n_b$.
We will then say:
-Definition 1: $a \Rightarrow b$ is acceptable at confidence level
+
+\definition $a \Rightarrow b$ is acceptable at confidence level
$1-\alpha$ if and only if
$$Pr[Card(X\cap \overline{Y})\leq card(A\cap \overline{B})]\leq \alpha$$
\overline{B}))$ and the value it would have taken if there had been
independence between $a$ and $b$.
+\definition $$q(a,\overline{b}) = \frac{n_{a \wedge \overline{b}}- \frac{n_a.n_{\overline{b}}}{n}}{\sqrt{\frac{n_a.n_{\overline{b}}}{n}}}$$
+is called the implication index, the number used as an indicator of
+the non-implication of $a$ to $b$.
+In cases where the approximation is properly legitimized (for example
+$\frac{n_a.n_{\overline{b}}}{n}\geq 4$), the variable
+$Q(a,\overline{b})$ approximately follows the reduced centered normal
+distribution. The intensity of implication, measuring the quality of
+$a\Rightarrow b$, for $n_a\leq n_b$ and $nb \neq n$, is then defined
+from the index $q(a,\overline{b})$ by:
+
+\definition
+The implication intensity that measures the inductive quality of a
+over b is:
+$$\varphi(a,b)=1-Pr[Q(a,\overline{b})\leq q(a,\overline{b})] =
+\frac{1}{\sqrt{2 \pi}} \int^{\infty}_{ q(a,\overline{b})}
+e^{-\frac{t^2}{2}} dt,~ if~ n_b \neq n$$
+$$\varphi(a,b)=0,~ otherwise$$
+As a result, the definition of statistical implication becomes:
+\definition
+Implication $a\Rightarrow b$ is admissible at confidence level
+$1-\alpha $ if and only if:
+$$\varphi(a,b)\geq 1-\alpha$$
+
+
+It should be recalled that this modeling of quasi-implication measures
+the astonishment to note the smallness of counter-examples compared to
+the surprising number of instances of implication.
+It is a measure of the inductive and informative quality of
+implication. Therefore, if the rule is trivial, as in the case where
+$B$ is very large or coincides with $E$, this astonishment becomes
+small.
+We also demonstrate~\cite{Grasf} that this triviality results in a
+very low or even zero intensity of implication: If, $n_a$ being fixed
+and $A$ being included in $B$, $n_b$ tends towards $n$ ($B$ "grows"
+towards $E$), then $\varphi(a,b)$ tends towards $0$. We therefore
+define, by "continuity":$\varphi(a,b) = 0$ if $n_b = n$. Similarly, if
+$A\subset B$, $\varphi(a,b)$ may be less than $1$ in the case where
+the inductive confidence, measured by statistical surprise, is
+insufficient.
+
+{\bf \remark Total correlation, partial correlation}
+
+
+We take here the notion of correlation in a more general sense than
+that used in the domain that develops the linear correlation
+coefficient (linear link measure) or the correlation ratio (functional
+link measure).
+In our perspective, there is a total (or partial) correlation between
+two variables $a$ and $b$ when the respective events they determine
+occur (or almost occur) at the same time, as well as their opposites.
+However, we know from numerical counter-examples that correlation and
+implication do not come down to each other, that there can be
+correlation without implication and vice versa~\cite{Grasf} and below.
+If we compare the implication coefficient and the linear correlation
+coefficient algebraically, it is clear that the two concepts do not
+coincide and therefore do not provide the same
+information\footnote{"More serious is the logical error inferred from
+ a correlation found to the existence of a causality" writes Albert
+ Jacquard in~\cite{Jacquard}, p.159. }.
+
+The quasi-implication of non-symmetric index $q(a,\overline{b})$ does
+not coincide with the correlation coefficient $\rho(a, b)$ which is
+symmetric and which reflects the relationship between variables a and
+b. Indeed, we show~\cite{Grasf} that if $q(a,\overline{b}) \neq 0$
+then
+$$\frac{\rho(a,b)}{q(a,\overline{b})} = \sqrt{\frac{n}{n_b
+ n_{\overline{a}}}} q(a,\overline{b})$$
+With the correlation considered from the point of view of linear
+correlation, even if correlation and implication are rather in the
+same direction, the orientation of the relationship between two
+variables is not transparent because it is symmetrical, which is not
+the bias taken in the SIA.
+From a statistical relationship given by the correlation, two opposing
+empirical propositions can be deduced.
+
+The following dual numerical situation clearly illustrates this:
+
+
+\begin{table}[htp]
+\center
+\begin{tabular}{|l|c|c|c|}\hline
+\diagbox[width=4em]{$a_1$}{$b_1$}&
+ 1 & 0 & marge\\ \hline
+ 1 & 96 & 4& 100 \\ \hline
+ 0 & 50 & 50& 100 \\ \hline
+ marge & 146 & 54& 200 \\ \hline
+\end{tabular} ~ ~ ~ ~ ~ ~ ~ \begin{tabular}{|l|c|c|c|}\hline
+\diagbox[width=4em]{$a_2$}{$b_2$}&
+ 1 & 0 & marge\\ \hline
+ 1 & 94 & 6& 100 \\ \hline
+ 0 & 52 & 48& 100 \\ \hline
+ marge & 146 & 54& 200 \\ \hline
+\end{tabular}
+
+\caption{Numeric example of difference between implication and
+ correlation}
+\label{chap2tab1}
+\end{table}
+
+In Table~\ref{chap2tab1}, the following correlation and implications
+can be computed:\\
+Correlation $\rho(a_1,b_1)=0.468$, Implication
+$q(a_1,\overline{b_1})=-4.082$\\
+Correlation $\rho(a_2,b_2)=0.473$, Implication $q(a_2,\overline{b_2})=-4.041$
+
+
+Thus, we observe that, on the one hand, $a_1$ and $b_1$ are less
+correlated than $a_2$ and $b_2$ while, on the other hand, the
+implication intensity of $a_1$ over $b_1$ is higher than that of $a_2$
+over $b_2$ since $q1 <q2$.
+
+On this subject, Alain Ehrenberg in~\cite{Ehrenberg} writes: "The
+finding of a correlation does not remove the ambiguity between" when I do $X$, my brain is in state $Y$" and "if I do $X$, it is because my brain is in state $Y$", that is, between something that happens in my brain when I do an action.
+
+\remark Remember that we consider not only conjunctions of variables
+of the type "$a$ and $b$" but also disjunctions such as "($a$ and $b$)
+or $c$..." in order to model phenomena that are concepts as it is done
+in learning or in artificial intelligence.
+The associated calculations remain compatible with the logic of the
+proposals linked by connectors.
+
+\remark Unlike the Loevinger Index~\cite{Loevinger} and conditional
+probability $(Pr[B/A])=1$ and all its derivatives, the implication
+intensity varies, non-linearly, with the expansion of sets $E$, $A$
+and $B$ and weakens with triviality (see Definition 2.3).
+Moreover, it
+is resistant to noise, especially around $0$ for, which can only make
+the relationship we want to model and establish statistically
+credible.
+Finally, as we have seen, the inclusion of $A$ in $B$ does not ensure
+maximum intensity, the inductive quality may not be strong, whereas
+$Pr[B/A]$ is equal to $1$~\cite{Grasm,Guillet}.
+In paragraph 5, we study more closely the problem of the sensitivity
+and stability of the implication index as a function of small
+variations in the parameters involved in the study of its
+differential.
+
