the other hand, this notion is not extended to other types of
variables, to extraction and representation according to a rule graph
or a hierarchy of meta-rules; structures aiming at access to the
-meaning of a whole not reduced to the sum of its parts \footnote{ICI }, 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).
+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}
--- /dev/null
+<?xml version="1.0" encoding="UTF-8" standalone="no"?>
+<!-- Created with Inkscape (http://www.inkscape.org/) -->
+
+<svg
+ xmlns:osb="http://www.openswatchbook.org/uri/2009/osb"
+ xmlns:dc="http://purl.org/dc/elements/1.1/"
+ xmlns:cc="http://creativecommons.org/ns#"
+ xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#"
+ xmlns:svg="http://www.w3.org/2000/svg"
+ xmlns="http://www.w3.org/2000/svg"
+ xmlns:xlink="http://www.w3.org/1999/xlink"
+ xmlns:sodipodi="http://sodipodi.sourceforge.net/DTD/sodipodi-0.dtd"
+ xmlns:inkscape="http://www.inkscape.org/namespaces/inkscape"
+ width="210mm"
+ height="297mm"
+ viewBox="0 0 210 297"
+ version="1.1"
+ id="svg8"
+ inkscape:version="0.92.1 r15371"
+ sodipodi:docname="chap2fig1.svg"
+ inkscape:export-filename="/home/couturie/latex/book_chic/figures/chap2fig1.svg.png"
+ inkscape:export-xdpi="96"
+ inkscape:export-ydpi="96">
+ <defs
+ id="defs2">
+ <linearGradient
+ id="linearGradient964"
+ osb:paint="solid">
+ <stop
+ style="stop-color:#ffffff;stop-opacity:1;"
+ offset="0"
+ id="stop962" />
+ </linearGradient>
+ <linearGradient
+ id="linearGradient913"
+ osb:paint="solid">
+ <stop
+ style="stop-color:#ffffff;stop-opacity:1;"
+ offset="0"
+ id="stop911" />
+ </linearGradient>
+ <linearGradient
+ inkscape:collect="always"
+ xlink:href="#linearGradient913"
+ id="linearGradient915"
+ x1="113.8378"
+ y1="114.96667"
+ x2="148.62884"
+ y2="114.96667"
+ gradientUnits="userSpaceOnUse"
+ gradientTransform="translate(-4.2333333,-0.26458333)" />
+ </defs>
+ <sodipodi:namedview
+ id="base"
+ pagecolor="#ffffff"
+ bordercolor="#666666"
+ borderopacity="1.0"
+ inkscape:pageopacity="0.0"
+ inkscape:pageshadow="2"
+ inkscape:zoom="1"
+ inkscape:cx="81.214286"
+ inkscape:cy="560"
+ inkscape:document-units="mm"
+ inkscape:current-layer="layer1"
+ showgrid="false"
+ inkscape:snap-center="true"
+ inkscape:snap-grids="true"
+ inkscape:window-width="2809"
+ inkscape:window-height="1581"
+ inkscape:window-x="1370"
+ inkscape:window-y="625"
+ inkscape:window-maximized="0" />
+ <metadata
+ id="metadata5">
+ <rdf:RDF>
+ <cc:Work
+ rdf:about="">
+ <dc:format>image/svg+xml</dc:format>
+ <dc:type
+ rdf:resource="http://purl.org/dc/dcmitype/StillImage" />
+ <dc:title></dc:title>
+ </cc:Work>
+ </rdf:RDF>
+ </metadata>
+ <g
+ inkscape:label="Layer 1"
+ inkscape:groupmode="layer"
+ id="layer1">
+ <ellipse
+ style="opacity:1;fill:none;stroke:#000000;stroke-width:0.5;stroke-linejoin:round;stroke-miterlimit:4;stroke-dasharray:none;stroke-opacity:1"
+ id="path12"
+ cx="107.81772"
+ cy="118.80312"
+ rx="63.186649"
+ ry="37.786648" />
+ <ellipse
+ style="opacity:1;vector-effect:none;fill:#000000;fill-opacity:0.26353796;fill-rule:evenodd;stroke:#000000;stroke-width:0.5;stroke-linecap:butt;stroke-linejoin:round;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1;paint-order:stroke fill markers"
+ id="path845-6-7-5"
+ cx="146.05"
+ cy="113.37917"
+ rx="8.2234831"
+ ry="17.4839" />
+ <ellipse
+ style="opacity:0.75;vector-effect:none;fill:url(#linearGradient915);fill-opacity:1;fill-rule:evenodd;stroke:#000000;stroke-width:0.5;stroke-linecap:butt;stroke-linejoin:round;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1;paint-order:normal"
+ id="path845-6"
+ cx="126.99999"
+ cy="114.70209"
+ rx="17.145519"
+ ry="16.880936" />
+ <ellipse
+ style="opacity:1;vector-effect:none;fill:#000000;fill-opacity:0.26353832;fill-rule:evenodd;stroke:#000000;stroke-width:0.5;stroke-linecap:butt;stroke-linejoin:round;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1;paint-order:stroke fill markers"
+ id="path845-6-7-5-3"
+ cx="83.60833"
+ cy="128.19583"
+ rx="8.2234831"
+ ry="17.4839" />
+ <ellipse
+ style="opacity:0.75;vector-effect:none;fill:#ffffff;fill-opacity:1;fill-rule:evenodd;stroke:#000000;stroke-width:0.5;stroke-linecap:butt;stroke-linejoin:round;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1;paint-order:normal"
+ id="path845-6-2"
+ cx="85.460419"
+ cy="120.52291"
+ rx="17.145519"
+ ry="16.880936" />
+ <text
+ xml:space="preserve"
+ style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:4.93888855px;line-height:125%;font-family:'Liberation Serif';-inkscape-font-specification:'Liberation Serif';letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:0.26458332px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1"
+ x="70.114586"
+ y="99.356247"
+ id="text974"><tspan
+ sodipodi:role="line"
+ id="tspan972"
+ x="70.114586"
+ y="99.356247"
+ style="font-size:8.46666622px;stroke-width:0.26458332px"
+ dy="0 0 0 2.3812499 -2.3812499">B(nb)</tspan></text>
+ <text
+ xml:space="preserve"
+ style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:4.93888855px;line-height:125%;font-family:'Liberation Serif';-inkscape-font-specification:'Liberation Serif';letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:0.26458332px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1"
+ x="89.398186"
+ y="147.89256"
+ id="text974-9"><tspan
+ sodipodi:role="line"
+ id="tspan972-0"
+ x="89.398186"
+ y="147.89256"
+ style="font-size:8.46666622px;stroke-width:0.26458332px"
+ dy="0 0 0 2.3812499 -2.3800001">A(na)</tspan></text>
+ <text
+ xml:space="preserve"
+ style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:4.93888855px;line-height:125%;font-family:'Liberation Serif';-inkscape-font-specification:'Liberation Serif';letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:0.26458332px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1"
+ x="107.91898"
+ y="94.446983"
+ id="text974-8"><tspan
+ sodipodi:role="line"
+ id="tspan972-1"
+ x="107.91898"
+ y="94.446983"
+ style="font-size:8.46666622px;stroke-width:0.26458332px"
+ dy="0 0 0 2.3812499 -2.3812499">Y(nb)</tspan></text>
+ <text
+ xml:space="preserve"
+ style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:4.93888855px;line-height:125%;font-family:'Liberation Serif';-inkscape-font-specification:'Liberation Serif';letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:0.26458332px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1"
+ x="135.25992"
+ y="138.89697"
+ id="text974-9-3"
+ inkscape:transform-center-x="-1.5875"
+ inkscape:transform-center-y="-3.4395833"><tspan
+ sodipodi:role="line"
+ id="tspan972-0-1"
+ x="135.25992"
+ y="138.89697"
+ style="font-size:8.46666622px;stroke-width:0.26458332px"
+ dy="0 0 0 2.3812499 -2.3800001">X(na)</tspan></text>
+ </g>
+</svg>
