+meaning of a whole not reduced to the sum of its
+parts~\cite{Seve}\footnote{This is what the philosopher L. Sève
+ emphasizes :"... in the non-additive, non-linear passage of the
+ parts to the whole, there are properties that are in no way
+ precontained in the parts and which cannot therefore be explained by
+ them" }, i.e. operating as a complex non-linear system.
+For example, it is well known, through usage, that the meaning of a
+sentence does not completely depend on the meaning of each of the
+words in it (see the previous chapter, point 4).
+
+Let us return to what we believe is fertile in the approach we are
+developing.
+It would seem that, in the literature, the notion of implication index
+is also not extended to the search for subjects and categories of
+subjects responsible for associations.
+Nor that this responsibility is quantified and thus leads to a
+reciprocal structuring of all subjects, conditioned by their
+relationships to variables.
+We propose these extensions here after recalling the founding
+paradigm.
+
+
+\section{Implication intensity in the binary case}
+
+\subsection{Fundamental and founding situation}
+
+A set of objects or subjects E is crossed with variables
+(characters, criteria, successes,...) which are interrogated as
+follows: "to what extent can we consider that instantiating variable\footnote{Throughout the book, the word "variable" refers to both an isolated variable in premise (example: "to be blonde") or a conjunction of isolated variables (example: "to be blonde and to be under 30 years old and to live in Paris")} $a$
+implies instantiating variable $b$?
+In other words, do the subjects tend to be $b$ if we know that they are
+$a$?".
+In natural, human or life sciences situations, where theorems (if $a$
+then $b$) in the deductive sense of the term cannot be established
+because of the exceptions that taint them, it is important for the
+researcher and the practitioner to "mine into his data" in order to
+identify sufficiently reliable rules (kinds of "partial theorems",
+inductions) to be able to conjecture\footnote{"The exception confirms the rule", as the popular saying goes, in the sense that there would be no exceptions if there were no rule} a possible causal relationship,
+a genesis, to describe, structure a population and make the assumption
+of a certain stability for descriptive and, if possible, predictive
+purposes.
+But this excavation requires the development of methods to guide it
+and to free it from trial and error and empiricism.
+
+
+\subsection{Mathematization}
+
+To do this, following the example of the I.C. Lerman similarity
+measurement method \cite{Lerman,Lermanb}, following the classic
+approach in non-parametric tests (e. g. Fischer, Wilcoxon, etc.), we
+define~\cite{Grasb,Grasf} the confirmatory quality measure of the
+implicative relationship $a \Rightarrow b$ from the implausibility of
+the occurrence in the data of the number of cases that invalidate it,
+i.e. for which $a$ is verified without $b$ being verified. This
+amounts to comparing the difference between the quota and the
+theoretical if only chance occurred\footnote{"...[in agreement with
+ Jung] if the frequency of coincidences does not significantly
+ exceed the probability that they can be calculated by attributing
+ them solely by chance to the exclusion of hidden causal
+ relationships, we certainly have no reason to suppose the existence
+ of such relationships.", H. Atlan~\cite{Atlana}}.
+But when analyzing data, it is this gap that we take into account and
+not the statement of a rejection or null hypothesis eligibility.
+This measure is relative to the number of data verifying $a$ and not
+$b$ respectively, the circumstance in which the involvement is
+precisely put in default.
+It quantifies the expert's "astonishment" at the unlikely small number
+of counter-examples in view of the supposed independence between the
+variables and the numbers involved.
+
+Let us be clear. A finite set $V$ of $v$ variables is given: $a$, $b$,
+$c$,...
+In the classical paradigmatic situation and initially retained, it is
+about the performance (success-failure) to items of a questionnaire.
+To a finite set $E$ of $n$ subjects $x$, functions of the type : $x
+\rightarrow a(x)$ where $a(x) = 1$ (or $a(x) = true$) if $x$ satisfies
+or has the character $a$ and $0$ (or $a(x) = false$) otherwise are
+associated by abuse of writing.
+In artificial intelligence, we will say that $x$ is an example or an
+instance for $a$ if $a(x) = 1$ and a counter-example if not.
+
+
+The $a \Rightarrow b$ rule is logically true if for any $x$ in the
+sample, $b(x)$ is null only if $a(x)$ is also null; in other words if
+set $A$ of the $x$ for which $a(x)=1$ is contained in set $B$ of the
+$x$ for which $b(x)=1$.
+However, this strict inclusion is only exceptionally observed in the
+pragmatically encountered experiments.
+In the case of a knowledge questionnaire, we could indeed observe a
+few rare students passing an item $a$ and not passing item $b$,
+without contesting the tendency to pass item $b$ when we have passed
+item $a$.
+With regard to the cardinals of $E$ (of size $n$), but also of $A$ (or
+$n_a$) and $B$ (or $n_b$), it is therefore the "weight" of the
+counter-examples (or) that must be taken into account in order to
+statistically accept whether or not to keep the quasi-implication or
+quasi-rule $a \Rightarrow b$. Thus, it is from the dialectic of
+example-counter-examples that the rule appears as the overcoming of
+contradiction.
+
+\subsection{Formalization}
+
+To formalize this quasi-rule, we consider any two parts $X$ and $Y$ of
+$E$, chosen randomly and independently (absence of a priori link
+between these two parts) and of the same respective cardinals as $A$
+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
+$1-\alpha$ if and only if
+$$Pr[Card(X\cap \overline{Y})\leq card(A\cap \overline{B})]\leq \alpha$$
+
+\begin{figure}[htbp]
+ \centering
+\includegraphics[scale=0.34]{chap2fig1.png}
+ \caption{The dark grey parts correspond to the counter-examples of the
+ implication $a \Rightarrow b$}
+\label{chap2fig1}
+\end{figure}
+
+It is established \cite{Lermanb} that, for a certain drawing process,
+the random variable $Card(X\cap \overline{Y})$ follows the Poisson law
+of parameter $\frac{n_a n_{\overline{b}}}{n}$.
+We achieve this same result by proceeding differently in the following
+way:
+
+Note $X$ (resp. $Y$) the random subset of binary transactions where
+$a$ (resp. $b$) would appear, independently, with the frequency
+$\frac{n_a}{n}$ (resp. $\frac{n_b}{n}$).
+To specify how the transactions specified in variables $a$ and $b$,
+respectively $A$ and $B$, are extracted, for example, the following
+semantically permissible assumptions are made regarding the
+observation of the event: $[a=1~ and~ b=0]$. $(A\cap
+\overline{B})$\footnote{We then note $\overline{v}$ the variable
+ negation of $v$ (or $not~ v$) and $\overline{P}$ the complementary
+ part of the part P of E.} is the subset of transactions,
+counter-examples of implication $a \Rightarrow b$:
+
+Assumptions:
+\begin{itemize}
+\item h1: the waiting times of an event $[a~ and~ not~ b]$ are independent
+ random variables;
+\item h2: the law of the number of events occurring in the time
+ interval $[t,~ t+T[$ depends only on T;
+\item h3: two such events cannot occur simultaneously
+\end{itemize}
