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\chapter{Paradigmatic overview of Implicative Statistical Analysis.
When? Why? How?}
\label{intro} % Always give a unique label
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\abstract{This prologue first of all presents the origin of the fundamental situation in which the need to organize students' response behaviours to a mathematics test appeared, illustrating the a priori complexity of exercises. 
Through a generative process, this led to the creation of an
implication index between response items, to evaluate quasi-rules such
as: "if a then generally b", then to representations of the partial
preorder obtained between responses.
The theory, called Statistical Implicative Analysis (SIA), was
developed under the impetus of the various applications encountered,
first by extending the nature of the behaviour variables to non-binary
variables of various types, then to the notion and representation of
rules of rules.
Finally, a dual topological relationship was established between the
subjects and the variables.
This development is ongoing.
It opens onto a vast space of paradigmatic research because it combines a set of problems to be studied and techniques specific to their study.}

\section{Introduction}
%\label{sec:1}
Statistical Implicative Analysis (SIA) was created by Régis Gras,
during his 66 years of service in the National Education system in
France (from kindergarten to university). During these years, he had
the opportunity to mix many ingredients into a jubilant and passionate
set.
With this 
composition of the pot, he proposes to tell you the genesis, the history
of the SIA, its purpose, its epistemological foundations, its
developments, without going into the technical details, in short by
appealing only to the common sense and intuition\footnote{Régis would like
  to cite
  M. Foucault in~\cite{Foucault} p. 103,
  that can be translated in English as "... there is no true knowledge except by intuition, that is, by a singular act of pure and attentive intelligence, and by deduction, which links the evidence together".} of the reader. In
fact, he would like to respect the advice of P. Valéry (Cahiers, II, La
Pleiade, p. 823) who reproaches the mathematician for not showing "the
origin of the choice of such an initial convention, which is more or
less justified by its use and subsequently. And further on (translated
in english): "... explain
what you want, what you will do and by what means". And yet this is Régis' will.



\section{Psycho-didactic problem}
%\label{sec:2}
During the 1970s, in France, as part of the Institutes of Research on
Mathematics Teaching, Régis once again attended French secondary
classes, particularly lower secondary classes (11 to 15 years old),
after his teenage years.
He conducted and evaluated a national pedagogical experience, while
participating in the in-service training of adults and the research of
teachers in these classes.
He has therefore witnessed the learning difficulties of students, both
young\footnote{Régis has even replaced teachers in each of the primary education levels.} and old, and sometimes the attempts to remedy the obstacles to
the assimilation of the concepts taught in mathematics.
Through missions carried out within the framework of French
Cooperation in French-speaking Africa, the Middle East, Latin America
and Canada; he noted that these obstacles were not always encountered
by all students, but that some of these obstacles were recurrent and
relatively shared.
Their nature was didactic but also often epistemological, opposing or
slowing down the acquisition of their knowledge\footnote{Régis would like
  to cite
  G. Bachelard in \cite{Bachelard} 
  that can be translated in English as "It is in terms of obstacles that the problem of scientific knowledge must be raised".}.
These regularities, these behavioural rules that exposed themselves,
referred, as Lévi-Strauss says in~\cite{Levi-strauss}, chapter 1, to both natural and cultural
processes, which it was a question of identifying, if possible
anticipating, analysing, understanding and on which it was ultimately
a question of acting.


Régis could observe these learning problems both directly through the
students' attitude or oral expression, but also through questionnaires
or written assignments.
Assuming that response attitudes or behaviours were generally
identifiable by these methods, he had data consisting of traces left
by the students.
When solving mathematical exercises or problems, a certain hierarchy
of difficulties segmented all the students interviewed and their
answers.
As the difficulty increased, the number of successes decreased, which
may seem like a tautology: indeed, we can expect that any student who
passes a test considered difficult, in a context that would be
comparable, would succeed a fortiori in what was easy.
What may {\bf surprise and challenge it} are the inconsistencies with this
expectation.
The idea that the difficulty a priori could be objectively defined by
Régis practice and the students' supposed knowledge no longer held
strictly as a predictor, even if it was most often respected.
Also, it was the potentially stable and relatively predictable
relationship between successes and failures, between response
behaviours, that interested him rather than the success or failure of
a given item.
This is in line with the opinion of H. Poincaré who says "that
mathematicians do not study objects but the relationships between
objects".



Hence the idea, in order to help teachers in the evaluation of a level
of acquisition of a given mathematical concept and in a project to
construct calibrated tests, to model presupposed levels of
acquisition, in a {\bf taxonomy of cognitive objectives}.
This one, like the one of Bloom (Bloom B. et al, "Taxonomie des objectifs
pédagogique", Education Nouvelle, Volume 1, Domaine Cognitif, 1969),
the best known and major tool in objective-based pedagogy, aimed to
organize a priori, according to an order of increasing complexity\footnote{Régis temporarily confuses the two concepts: "difficulty" and "complexity" which, although comparable and related, fall into different fields, one subjective, the other objective. }
after and through a task analysis, the control or appropriation of a
concept (and not the moments of its learning).
For example, an objective expressed in terms of the use of an
algorithm would be considered less complex than an objective requiring
the construction of a counter-example.
A causal relationship would underlie this hypothesis: the cognitive
tools of a higher objective would be sufficient to those mobilized by
the student for a lower-level objective, such as a "consequence" or
"effect" would be the result of a "cause".
In other words: "solving a complex exercise" would imply "solving a
less complex exercise" and its proven success would make it a {\bf good
predictor}.


Let us immediately note the difference between the information
instilled by correlation, a tool of symmetrical statistical proximity,
and the one whose function Régis refers to above, which is not
symmetrical.
Let us read again, for example, together a paragraph from E. Todd's
book ("Où en sommes nous?", p. 148, Seuil, 2017), on his desk today
(translated in english): ``through a cartographic approach, we observe
the coincidence in Europe of 3 elements of social structure between
1900 and 1930: the founding family, the Lutheran religion and a high
educational level. However, we cannot stop at this point and decree,
for example, that the family has fostered the emergence of
Protestantism, which in turn has demanded that people learn to read.
We need to imagine and describe more complex historical interactions.
This is indeed what we are aiming for, in another situation, through taxonomy''. 

With regard to a questionnaire composed of items specifying each of
the cognitive objectives of taxonomy, in theory, one could have
expected successes organized linearly according to the complexity a
priori.
Which, as Régis said, has not been observed.
The presumed total order has been replaced by a {\bf partial preorder}, as
defined by Piaget's differential stages of child development.
We approach this question with more attention in chapters 10 and 11 of
this book.
The observation of this partial preorder indicates that students could
in some cases and for some of them, succeed at an item considered
difficult while failing at an item b considered easier.
This is without calling into question the assertion that "generally
success at a is accompanied by success at b" and without its
reciprocal being necessarily true.
Régis' interest will then focus on this type of non-symmetrical
relationship, trying to weight, by an inductive operation, the quality
of its approximate character and to organize if possible all the pairs
of items variables at stake of this partial preorder\footnote{A preorder on a set E is a reflexive (all x of E is in relation to itself) and transitive (if x is in relation to y and y in relation to z then x is in relation to z) binary relationship.  It is partial if some elements are not linked to any other. This is the case, for example, of a preferential relationship.}.


What {\bf statistical tools} were then available to Régis to qualify
and quantify this non-symmetrical relationship between two items,
i.e. two variables?
A parametric test that is not symmetrical? but to refute what
hypothesis? that the students who responded to $a$ also reponded to
$b$? what to do with the refutation? put it away for further study?
draw up a list of rejected or accepted cases? no, because of the
absence of an overall structure expected from such a list.
Use the link measurement between two variables based on their {\bf
  correlation}?
But this measure quantifies the quality of
co-occurrences, the concomitance and is therefore symmetrical and
therefore ineffective and inappropriate in this case.
Use a {\bf
  multidimensional data analysis method} to organize relationships
into a whole?
Régis had then collaborated with J.P. Benzecri~\cite{Benzecri} on {\bf Correspondence
  Analysis (C.A.)} and I.C. Lerman~\cite{Lermanb} on {\bf hierarchical
  classification} and, very often, taught and used their analysis
methods.
But their theoretical foundations are essentially {\bf symmetrical}.
Régis' reservation regarding their methods was the same as for
correlation.
Moreover, emphasizing the fundamental difference in points of view, a
set metaphor illuminated the difference and facilitated the intuitive
understanding of the problem of the quasi-implication of the variable
$a$ on the variable $b$.
Here it is: among the population of subjects concerned by the study of
involvement "if $a$ then a generally $b$", subset A of the subjects
who satisfy $a$ is almost contained in subset B of the subjects who
satisfy $b$.
We know that if involvement is strict A is contained in B.
There remained the {\bf Bayesian theory} which offers the means to
calculate what is called the "causal probabilities".
It is very effective but, with reservations because it seems to Régis
to have less sensitivity to sample sizes (embarrassing in statistics)
and somewhat overwhelms rare cases.
Circumstances that SIA will avoid and that Yves Kodratoff~\cite{Kodratoff} has
expressed through the research, which he considers necessary, "nuggets
of knowledge".

As an epistemological starting point, Régis therefore had to establish
a measure, an {\bf index}, between 0 and 1 for example, capable of
accounting for "deviance", the gap between {\bf prediction} and {\bf contingency},
i.e. between what was expected from the order a
priori\footnote{Translation in English of~\cite{Levy-Leblondb} p. 35 "Contrary to common opinion, the great affair of science is less the production of absolute and universal truths or the recognition of redhibitory errors, than the delimitation of the conditions for the validity of statements..."}  (A is
included in B in the overall metaphor) and what was actually observed,
namely a {\bf rule of quasi-implication} "{\bf if a then generally b}".
The strategy he used at that time, in 1978, was to take into
consideration the dissatisfaction with the involvement, which, as we
know, is satisfied as soon as $a$ is true, $b$ is false (for example: "it
is raining and I am not taking my umbrella").
It is therefore the {\bf counter-examples} to the implicit rule "if $a$ then
$b$" (for example, "if it rains then I take my umbrella"), to which my
attention will be drawn.
As Régis wished it an inductive virtue, he had to take into account
the size of the populations concerned: total number of pupils, number
of those who satisfy a and do not satisfy b.
Moreover, without information on the population or on the existence of
a relationship between the variables studied, he assumed that there
was {\bf no a priori link} between them, as I.-C. did. Lerman (1981), as is
common in non-parametric tests and as expressed by H. Atlan and
B. d'Espagnat\footnote{Translated in English from \cite{Atlana} p.160
  "...[in agreement with Jung] if the frequency of coincidences does
  not significantly exceed the probability that they can be calculated
  by attributing them solely to chance, excluding hidden causal
  relationships, we certainly have no reason to suppose the existence
  of such relationships". In \cite{Espagnat}, p. 186 ", Bernard would
  speak of (translated in English) " influence in these terms: "Events A significantly influence events B if and only if the frequency with which events B take place is (significantly) different depending on whether or not events A are imposed to exist"}.
The objective is, if possible, to preserve and qualify it by
highlighting the low probability of derogation from the rule on
statistical bases.
A good illustration of the adage: "{\bf exception that proves the rule}". 

\section{A statistical measure of quasi-implication. Associated representations}

Let's move on. The strategy was therefore as follows: if the pass-fail
variables, {\bf boolean} in this case, were random and independent,
the number of random counter-examples to the quasi-implication rule
would follow a certain probability law, definable on the basis of
sample sizes.
If the number of counter-examples expected with probability $p$ is
greater than the number of counter-examples observed, the rule with a
{\bf quality measure} $p$ reflecting the difference between what is expected
and what is actually observed is accepted.
This probability has been called {\bf implication intensity}, as such,
it is identified with a probability scale, a property that other
numerical indices such as the Guttman scale, Loevinger or Shapiro
indices, for example, do not possess.
It represents a kind of {\bf statistical astonishment}, and therefore
of an anthropological nature\footnote{This is also what René Thom says
  in~\cite{Thoma} p. 130  (translated in English):"...the problem is not to describe reality, the problem is much more to identify in it
what makes sense to us, what is surprising in all the facts. If the
facts do not surprise us, they do not bring any new element to the
understanding of the universe: we might as well ignore them" and
further on: "...which is not possible if we do not already have a
theory" }, in the face of the small number of counter-examples
  compared to those expected in the theory.
  
  

Here is an illustration of its subjective nature (of common sense) and
its relationship to the workforce at stake in the experience:
\begin{itemize}
\item on a set of 10 individuals, attributes $a$ and $b$ are checked
  respectively 6 and 8 times, without counter-example to the rule "if
  $a$ then $b$". This is logically acceptable; the frequency of b
  knowing a is 1,
\item on a set of 1,000 individuals, attributes $c$ and $d$ are
  verified respectively 600 and 800 times, with only one
  counter-example to the rule "if $c$ then $d$". The logical rule is
  no longer acceptable since the frequency of $d$ knowing $c$ is no
  longer equal to 1.
\end{itemize}

Which one would surprise you?
To which would you give the best predictive quality?
In the first case, the rule is strict but confidence in it is
fragile.
In the second case, it is the opposite, the astonishment is greater
despite the lower value of the conditional frequency.
This paradox relating to the acceptability of the rule is soluble in
subjectivity.
SIA will restore an objective component of it.
It can be noted that the {\bf statistical concept of confidence} of the
association rule $a\Rightarrow b$ (ratio of the frequency of $b$
knowing $a$) does not coincide with the {\bf psychological
  confidence}\footnote{Shapin in~\cite{Shapin} quote J. Locke "Before
  approving or disapproving [trust], the mind determined to proceed rationally must consider all the motives of probability and see how they orient themselves, more or less in favour of or against a given proposal; and, by balancing the whole, reject or accept it with more or less firm agreement, i. e. on the predominance of one or other motive of probability on either side"} just
mentioned, which weakens, if not somewhat discredits, the data mining
methods based on the statistical acceptance of confidence.
We will come back to this in the discussion of some of these methods
(Chapter 2).

Certainly, the intensity of implication is all the closer to 1 as the
perceived quality of involvement is high.
The relationship it establishes between variables is frequently
expressed in terms of causality.
But this causal explicability of a variable by one or more others
(J.-M. Levy Leblond speaks of a "{\bf causal field}" in "Aux contraires")
that the intensity of implication evaluates, is in no way
{\bf deterministic}, it just brings {\bf predictability}.
Moreover, it is not {\bf transitive} as we later formalised by the study of
{\bf exception rules} where the rules $a \Rightarrow b$ and $b \Rightarrow
c$ are not accompanied by the rule $a\Rightarrow c$ or if
$a\Rightarrow c$ and $b\Rightarrow c$, the conjunction $a$ and $b$
does not imply $c$. Nor is it a matter of {\bf probabilistic determinism} as
it is often misinterpreted: if 0.95 is an intensity of implication of
$a$ over $b$, this does not mean that $b$ is achieved with the
probability 0.95 if $a$ is achieved.


Thus, starting from the search for strict rules, respecting Platonic
logic, Régis traded and looked for quasi-rules, no longer respecting
formal logic because of its counter-examples.
This approach illustrates -- we will return to it -- the way of
thinking that we call dialectical\footnote{Régis would like to
  quote~\cite{Seve}: "...dialectic only comes into play to examine and solve logical difficulties at a higher level"} because it accepts contradictions
("the improbability of the false") and integrates them pragmatically,
while enriching knowledge.
We will return to this point in Chapter 13 by examining the specific
nature of the SIA as formalizing a logic of everyday life into a
so-called paracoherent logic.
This one, stripped of the Aristotelian non-contradiction, nevertheless
makes it possible to provide a formal tool of thought making {\bf
the
contingent  necessary}.
This approach is similar to that of Greek Sophists (5th to 4th centuries A.D.) as we will see in chapter 13 of this book. 


Suppose, therefore, that the intensity of implication for each pair of
variables in the experimental situation is calculated.
Of course, only the one relating to the couple leading to the greatest
intensity of implication is retained for each pair.
What to do with the table of all these values?
How can we identify the main lines of force that would structure them,
as a factorial experiment does?
Having a set of evaluated binary relationships, Régis chose to
represent it by a weighted and non-cyclical oriented graph, an image
more easily apprehensible by the expert user of the field, for example
didactics in its evaluation component, psychology, sociology,
medicine, etc.
In general, it is not reduced to a linear path since several "effects"
can be associated with the same "cause" and an "effect" can be the
result of several "causes".
The problem of ontology representation is then compatible with this
framework (work in collaboration with Jérôme David~\cite{Davida}).
The following graph, called the implicative graph (Fig.~\ref{chap1fig1}), illustrates this situation. 

\begin{figure}[htbp]
  \centering
\includegraphics[scale=1]{chap1fig1}
 \caption{An example of implicative graph}
\label{chap1fig1}       % Give a unique label
\end{figure}


The edge $c \rightarrow a$ represents the rule\footnote{
     From now on, "rules" will mean both strict rules and
     quasi-rules. We will only specify when there may be ambiguity or
     to emphasize the specificity of quality.} "if the variable $c$ is
chosen then generally the variable $a$ is also" i.e. $c \Rightarrow
a$.
Similarly, the chain $e \rightarrow c \rightarrow a$ is an implicative
path.
The expert then analyses and interprets the different paths in
conceptual terms by giving meaning in his field of expertise to paths
in the graph, to networks of paths as Marc Bailleul (ESPE\footnote{in
  French Ecole Supérieure du Professorat et de l’Education, in English
it can be translated in Higher School for Professor and Education.} of Caen)
says, to an ascending-descending cone of a variable to the role of
attractor that it plays (Fig.~\ref{chap1fig2} and~\ref{chap1fig3}), but also to the connections or their
absence.
The graph relating to the questionnaire constructed according to cognitive taxonomy has almost validated it by generally, and therefore not always, organizing in the expected preorder the 5 classes and the 20 or so subclasses of this taxonomy (see Chapter 2). 


Here are two other examples of implicative graphs where the colors of
the edges are associated with different levels of intensity of
implication.
Here, red edges mean a higher quality of implication than blue edges and themselves than green edges. 

\begin{figure}[htbp]
  \centering
\includegraphics[scale=0.4]{chap1fig2}
 \caption{A second example of an implicative graph}
\label{chap1fig2}       % Give a unique label
\end{figure}



\begin{figure}[htbp]
  \centering
\includegraphics[scale=0.6]{chap1fig3}
 \caption{An example of an implicative graph with the mode cone}
\label{chap1fig3}       % Give a unique label
\end{figure}



Let us go further. In child development psychology according to
Piaget, the notion of reflective abstraction describes the passage
from a level of conceptualization to a higher level, each level
consisting of rules about objects, then operations on these objects,
then operations on these operations, etc... (e. g. diagrams,
procedures, design, method,...).
We find, moreover, in Darwinian theory of evolution, but also in any
mathematical theory, these same expansions when we move from a theorem
to a corollary such as, for example, in the study of functions to
that, in functional analysis, of function functions.
Hence the idea of building a second plan of implicit relations, that
of rules of rules according to a hierarchy called cohesive because of
the so-called cohesion index which makes it possible to generate
classes oriented by such rules.
Like some data analysis methods, the interest of moving, non-linearly,
from the relational level to the hierarchy level appeared to Regis and
his team therefore extended their work for rules to rules of rules or
meta-rules or generalized rules.

\begin{figure}[htbp]
  \centering
\includegraphics[scale=0.5]{chap1fig4}
 \caption{Oriented hierarchy graph}
\label{chap1fig4}       % Give a unique label
\end{figure}



The rule, illustrated in Fig.~\ref{chap1fig4}, can be expressed by the sentence: the
"theorem" generally results in the "theorem".
For example, from behavioural theorems, we can identify a trait or a
conception (e.g. Dominique Lahanier-Reuter's~\cite{Lahanierb} conceptions of chance).
A statistical index, cohesion, also makes it possible to identify the
hierarchical levels corresponding to the statistical significance of
the classes formed at these levels (in red in Fig.~\ref{chap1fig5}).
We construct and detail the mathematical model that underlies the
cohesive hierarchy in Chapter 3.


\begin{figure}[htbp]
  \centering
\includegraphics[scale=0.3]{chap1fig5}
 \caption{Another oriented hierarchy graph}
\label{chap1fig5}       % Give a unique label
\end{figure}


\section{A detour into the philosophy of science}

Through these two types of representations of simple rules and then
generalized rules, which reveal two types of structures in all
variables, we respond to the structuralist philosophy which considers
that the "whole" is richer than the sum of its "parts".
Régis quotes, on this subject, two excerpts from L. Sève's book
(p. 58, translated in English)"...the whole consists of nothing other
than its parts, and yet it presents, as a whole, properties that do
not belong to any of its parts.
In other words, 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 therefore cannot be explained by them" and to continue
"Everything happens as if a spontaneous generation of properties of
the whole occurs... This is the paradox of the emergence".


Régis insists on this specific property of the SIA where the
cohesive hierarchy satisfies in an original way through the extension
of relationships between variables.
It is through a qualitative leap, generally produced by a threshold
effect in quantity (e.g. group psychology, water vaporization, etc.),
that everything, at the cost of a synthesis, makes sense.
This is extracted from the truly dialectical reading of the {\bf non-linear
all/part ratio}\footnote{Régis would like to add that "as shown by the
  non-linear differential equations of the fundamental index with
  respect to the cardinal parameters of the observations, unlike what
  is observable with other competing indices". He also would like to quote K. Mark~\cite{Marx} 
"Society is not made up of individuals, but expresses the sum of the
  relationships, the relationships in which these individuals are
  situated in relation to each other". And finally we would like to
  add that "A street is not the sum of the houses in it. Likewise a city is not the sum of its streets, etc."} (non-proportionality and non-additivity of the cause on
the effect).
With the SIA, there is then a paradox between opposites, link and
absence of link, because the whole, constituted (one should say
organized) of parts in a dynamic system, has properties that the parts
do not possess and which are generally of higher level.
In the same way, and metaphorically, in linguistics, the meaning of a
sentence is not determined by the analysis of each of its words but by
their interaction\footnote{Régis Gras would like to
  quote~\cite{Gaudin} "There is neither additivity nor proportionality
  between the meaning of the units (words) and that of the
  sentence. We see a topology of meaning emerging" and "... the
  isolated word of the Chinese language has in truth neither meaning
  nor existence apart, each one receives its meaning only from the
  word itself (from intonation, etc....), taken in isolation it has
  ten, even forty meanings,...; if we subtract this word from the
  totality, it gets lost in a hollow infinity. " from~\cite{Schelling}.}.


The logic behind this all/parts relationship is dialectical (and not
dichotomous) because it interactively reconciles contradictions: rule
and non-rule, instability of a dynamic system and structural
stability.
It is based on the importance of the rarity of the counter-example and
on the negation of entropy, disorder in the couple rule and non-rule.
In this respect, dialectical logic is opposed to strict logic (of the
mathematician) without, of course, being a paradox.
The fertility and originality of the SIA are due to this character,
particularly in the hierarchical analysis, which is clearly non-linear
and where the whole is based on its meaning, not by adding the
properties of its parts (subclasses) but by synthesizing the
inferential interactions.

It is through the notion of {\bf significant level} that we can highlight
the phenomenon of {\bf emerging property}.
In this sense, the SIA appears as a kind of avatar of non-linear
knowledge learning, learning made of ruptures and dialectical
reconstruction (e.g. the epistemology of G. Bachelard or Lev Vygotsky
and Chapter 10 on conceptualization). Régis will come back to these
properties of the SIA in the chapter devoted to paracoherent logic
(chapter 13).
In contrast to SIA, the all-part ratio would be {\bf linear} in the case
of nesting classes as in clustering based on similarity up to its
terminal node.




From the first applications, in the didactics of mathematics and on
human management problems, it appeared that the first index, the
implication intensity, had a problem in its discrimination when the
number of subjects increased.
Hence the need, for this reason and to get as close as possible to the
causal interpretation of the rules, to integrate into this intensity
the information provided by the {\bf contraposed of implication}.
Thus, not only will the measurement of implication take into account
the relationship "if $a$ then generally $b$" but also, dialectically,
its counterpart "{\bf if not $b$ then generally no $a$}"\footnote{ In
  everyday life and in most scientific approaches, the causal
  relationship between "cause $\rightarrow$ effect" is reinforced by observation "if no effect then no cause". This is how a so-called negative experiment is carried out, in which a presumed cause is removed in order to deduce from the lack of effect what is missing for it to occur. }.
This new index is based on the notion of entropy, therefore
information of experiences evaluating the two rules.
This choice allows for more in-depth action to access "knowledge
nuggets", rules that would be rejected or ignored by a method based on
support and trust (the Agrawal algorithm \emph{a priori}~\cite{Agrawal} seems to be
linear thinking).
In Chapter 2, we will come back to another way of integrating
intensity of involvement and trust while taking into account the
contraposition.
A brief aside: these different challenges to a system in formalized
equilibrium, these readjustments, without questioning the fundamental
philosophy, are always frustrating, even irritating for the researcher
who must revisit his work.
But isn't that what is expected of a scientist if not an attitude that
leaves room for doubt, while demanding modesty, honesty and
obstinacy.

Although we seek relationships between variables through non-symmetric
rules and these relationships can often be expressed in terms of
causality, we do not claim, as Régis said earlier, that they are
deterministic, but simply that they have the ability to allow
quantifiable assumptions to be made about their predictability.


\section{A software for SIA}


When the number of subjects and variables increases, it is difficult
to perform the associated calculations and especially to provide the
two representations: graph and hierarchy.
At first, both tasks were done by hand. Régis did it for his thesis
with obstinacy in order to represent the links between 40 variables.
Then, he wrote a first program in the Basic language that performed
the calculations and built the {\bf hierarchy}.
Imagine the excitement of the novice programmer that Régis was to see
a hierarchy develop before his eyes, where the levels did not
overlap.
A doctoral student from I.C. Lerman -H.Rostam- in his turn built the
{\bf implicative graph} using a more sophisticated program whose algorithm
Régis had developed.
Then, two of my doctoral students, Saddo Ag Almouloud~\cite{AgAlmouloud} and Harrisson
Ratsimba-Rajohn~\cite{Ratsimba}, integrated the whole into a single software that we
called {\bf Implicative Cohesive Hierarchy Classification} under the
sympathetic acronym "{\bf C.H.I.C.}", called CHIC in the following. The acronym was chosen from the French
meaning, in English the meaning is more debatable.
Finally, since the late 1990s, Raphaël Couturier~\cite{Couturiera} has unified the
complete processing of calculations and representations while
continuously improving it due to the development of the underlying
theory and the variety of applications.
In particular, a CHIC option makes it possible to show the different
possible predictors and descendants of a variable, but also to
determine the optimal conjunction of these in the sense of their
originality.
The possibility of modifying the construction threshold of the
implicit graph highlights the plasticity of the structure of all the
variables, while dialectically reconciling the instability of a
dynamic system with its structural stability.
A first version written in R was presented by Raphaël Couturier at the
ASI 8 symposium (Radès, Tunisia, November 2015), then in a more
advanced state at the ASI 9 symposium (Belfort, France, October 2017).


This software is operational throughout the world since 26 countries,
mostly Greek-Latin, own and use it\footnote{ Currently, Pablo Gregori (Castellon) and Ruben Rodriguez (Riobamba) are working with Raphaël Couturier to translate CHIC into an open computer version under R.}.
For research, as we can understand, it plays the dual role of
{\bf developer and analyzer}.
We will come back to these properties later.
As a result, questions about the possibilities and limitations of ASI
are emerging through general but also specific questions due to
different traditions and cultures.
In addition to the letter-writing link, they are strengthened through
our international meetings on ASI, from ASI 1 (Caen) to ASI 7 (Sao
Paulo) in 2013 and recently in 2015 in Tunisia and 2017 in Belfort.
These are the questions that drive the theory and its tool towards
continuous developments as we will see.
To illustrate this, Régis will quote Anne Lauvergeon~\cite{Lauvergeon} "... when you produce,
you also end up conceiving".
Or as H. Atlan~\cite{Atlanb} (p. 110) writes  "...using beliefs first as
practices can be more effective than focusing immediately on
theoretical beliefs, explained in statements, which are already
supposed to take the place of certain knowledge...".


\section{Extension of the method to other variables}

It is therefore through the different situations encountered that the
limitation to binary variables, which served to give a set meaning to
involvement, appeared constraining.
The one who fired the first was Marc Bailleul~\cite{Bailleula} who wanted to look for
ordered preference relationships between teachers' assertions.
{\bf Modal variables} ("a little", "a lot", "not at all",...) had to be
processed.
He proposed a first index satisfying the expression measure of the
type: "{\bf if for a the modality "a little" is chosen then generally for b
a modality greater than or equal to this one is chosen}".
In his thesis, Marc Bailleul obtained excellent results, some of which
were unpredictable, with regard to 4 conceptions of mathematics
teaching as seen by professors.
We have used his study in several papers and in some chapters of the
Proceedings of our various symposia on ASI.


In order to proceed, with regard to the same problem, by extending the
intensity of involvement between Boolean variables, with
J.-B. Lagrange~\cite{Lagrange}, we then defined a new measure relating to {\bf
  numerical variables} making it possible to assign a value to
statements such as: "{\bf if $a = \alpha$ then generally $b \ge
  \beta$}". Jean-Claude Régnier~\cite{Regniera}, for his part, reduced the problem of
the search for relationships between preferences to that of {\bf rank
  variables}, which provided another extension of the SIA.

From there, following a question from E. Diday placed within the
framework of {\bf symbolic variables} and by an extension of numerical
variables, we assigned values to expressions "{\bf if $a$ takes its
  values in the interval $I_a$ then generally $b$ takes its values in
  $I_b$}".
The main idea was to partition the range of values taken by each
variable into sub-intervals maximizing their inter-class variance.
The interest of this new category of variables, called {\bf
  intervals}, was manifested in teaching, in comparing performance
hierarchically in different disciplines and in seeking to transfer
specific skills to other skills.


We then see that of these {\bf interval variables} treated in
collaboration with E. Diday and Pascale Kuntz~\cite{Grasj}, we naturally arrive at
{\bf fuzzy variables} as we find them in the study of relationships of
the type: "if the tension $a$ of a subject is rather high then
generally his heart rate $b$ is too".
We anticipate the practical applications that can result from this in
construction problems or failure detection.
Thanks to the collaboration of Fabrice Guillet, Maurice Bernardet,
Raphaël Couturier, Filippo Spagnolo, this allowed Régis to model the
notion of fuzzy variable in the context of interval variables and to
make a presentation at a conference on fuzzy logic~\cite{Spagnolo}.

A problem occurred one day in the implication processing of too large
a set of variables to the point of making the resulting graphs
unreadable. With Fabrice Guillet and Robin Gras~\cite{Grask}, we defined an
equivalence relationship between the variables, based on their
neighbouring implicit behaviours, which led to the substitution of a
leading representative of this package for a package of variables.
This {\bf reduction} has proved effective in many other situations, for
example in Laurence Ndong's thesis~\cite{Ndonga}.

A little stop to talk about the collaborations in the DUKe team, to
 which Régis belongs in the Laboratory of Digital Sciences of Nantes
 (LS2N), collaborations always based on epistemological choices in
 response to semantic expectations.
The explosion of a multitude of application or theoretical questions,
still not symmetrical, has led to joint work undertaken since 1990
with some members of the DUKe team: Régis will mention for example,
with Pascale Kuntz~\cite{Graso} on {\bf rule hierarchies} (ah! the demonstration of the
ultrametric of the hierarchy! the algebraic model of hierarchy !),
on {\bf exception rules} with Einoshin Suzuki~\cite{Grasr}, {\bf rule redundancy} still with
Pascale, with Julien Blanchard on {\bf entropic analysis}, {\bf temporal
variables, gene expression in bioinformatics} with Gérard Ramstein~\cite{Ramstein},
etc...

For example, how did the need for time variables become apparent?
Well, to account for the changing relationships between economic,
social or cognitive variables.
Indeed, can we explain in teaching, the involvement between variables
when we proceed to interventions during the year on learning or in
socio-psychology by successive interventions, by interviews or by
internships (in collaboration with D. Pasquier)?
We then formalized these time-indexed variables into {\bf vector variables},
where a variable is modeled by a time-set vector.
Then, Julien Blanchard, in collaboration with Fabrice Guillet and
Régis, defined {\bf sequential variables} modelled by a Poisson process in a
different way.
So many new concepts and new fields of application born of various
problems and expected extensions.

{\bf Generalized extensions} of the SIA have recently been introduced:
\begin{itemize}
  \item
On the one hand, to a continuous set of subjects, for example colours,
opinions, with a given distribution law, generalization since until
then the model was limited to discrete and finite sets;
\item
On the other hand, to all the values of different types of variables, taken in continuous spaces, for example fields, with given laws;
\item
  Finally, in this book, we present an extension to continuous variables, which represents a major theoretical advance in data analysis.
\end{itemize}

Régis insists on a remark likely to satisfy any Cartesian mind: during
each extension (nature of the variables, structure of the subject and
variable spaces), we have tried to prove and we believe that {\bf the
restriction to the fundamental case of binary variables and discrete
spaces was always satisfied}.
The mathematical interlocking is original and rigorous.


\section{Explanatory role of additional variables}

Another question bothered Régis very early on.
Is there an internal structure in the population of subjects involved
in a study that would explain the variable structure revealed by the
ASI?
Assuming an implicit variable structure obtained by a graph or
hierarchy, is it possible to designate the subjects and categories of
subjects more or less responsible for the elements of these
structures?
For example, if we observe that in all school classes a certain
conception of geometric notions leads to certain response behaviours,
to which type of pupils, to which type of didactic variable, to which
context can it be more specifically attributed?
Conversely, which students would be resistant to it?
With the theses of Harrisson Ratsimba-Rajohn and Marc Bailleul, we
have formalized and exploited, in the SIA, the notion of the
additional variable (or {\bf exogenous} as opposed to the {\bf endogenous}
variables analysed) and its {\bf contribution} (we also speak of {\bf typicality})
to structural elements, {\bf networks} or hierarchical classes of rules for
example, semantically explained during a previous analysis.
Even better, we were able to retroactively define a {\bf dual topological
structure} on all subjects.
Thus, for example, an "emblematic conception of the republican school"
is reinforced in a privileged school environment where the distance
inferred between pupils by an element of the rule structures is the
lowest (work with Dominique Lahanier-Reuter. 

But since Régis have cited throughout the pages the contributions and
the role of collaborators as spurs for the development of SIA, he must
highlight the critical, scratchy but constructive role that
Jean-Claude Régnier plays both with regard to theory and its
applications and with regard to dissemination, the popularization of
SIA since he has taken over the presidency of the international SIA
events.
How many slips of the tongue, mistakes, clumsiness, excessive haste on
my part, Pascale Kuntz and Jean-Claude Régnier have straightened it
out! Nor do Régis forget the faithful dedication and computer skills
of Raphaël Couturier, without whom the SIA, deprived of CHIC, and now
of RCHIC, would only be a theoretical and perhaps speculative
construction to contemplate.
He has overcome, with CHIC, Régis' handicap in computer science as
Pascale Kuntz has kindly done for English.
I am also concerned that this last failure has cost the ASI a better
international audience.
However, other international teams are also involved in the
development and application work of the SIA using CHIC software.
Régis quote, in particular, given their regularity of participation:
\begin{itemize}
\item 
actively led by Pilar Orus (University of Castellon in Spain), a
diversity of Hispanic nuclei in Spain (Pablo Gregori, Eduardo Lacasta,
Miguel Wilhelmi, ...), Cuba (Larisa Zamora,...), Chile
(Guzman-Retamal,...); Argentina (Pablo Carranza,...), Ecuador (Ruben
Pazmiño) have created permanent links;
\item in Italy, the recently deceased Filippo Spagnolo had created a
  team that is now closing in on Benedetto Di Paola;
\item in Brazil, around Saddo Ag Almouloud, Vladimir Andrade and
    thanks to the strengthening provided by agreements with
    Jean-Claude Régnier and Lyon II University;
\item in Cyprus around Athanasios Gagatsis, in Slovakia with Lucia
  Rumanova,
\item more episodically, researchers from Belgium, Switzerland,
  Germany, Greece, Romania, Czech Republic, Japan, Vietnam, Canada,
  Mexico, Argentina, Gabon, Tunisia, Algeria; and Régis
  forget, use SIA's method and tool.
\end{itemize}


\section{Specificities of the SIA and conclusion}
SIA compared and measured with other data analysis methods, has
important original characteristics. Régis summarize them:  
\begin{itemize}
\item
 the {\bf successive models} of variables responding to explicit
 epistemological constraints compatible with the semantics of the
 situations to be modeled;
\item
  the {\bf asymmetry} of the method;
\item
the progressive {\bf extension} of the nature of the treated variables while
maintaining the extension properties;
\item
the {\bf pedagogical and ergonomic capacities} of the representations of two
structures on all variables: a graph for the rules, a hierarchy for
the generalized rules;
\item
the {\bf structural duality} of the two spaces at stake: subjects and
variables with the notions of contribution and typicality to the
structures;
\item 
the {\bf extension of the finite discrete to continuous} for both variables
and subjects;
\item
the {\bf originality of the dialectical reasoning} underlying the definition
of simple and generalized rules;
\item 
the {\bf simplicity of the underlying mathematical model} ensuring
accessibility, plasticity and fertility useful to meet application
expectations in a wide range of fields.
\end{itemize}

Some may say that SIA, by the amplitude of its fields of application,
by the homogeneity of its analytical and graphic properties, presents
an original {\bf paradigmatic nature}.
Should we recognize him? Djamel Zighed, Director of Digital at the
Agence Universitaire de la Francophonie, expresses this idea as
follows: "The SIA is not a method but a broad theoretical framework in
which modern problems of knowledge extraction from data are
addressed. It is a general theory in the field of causality because it
responds to weaknesses in other theories, it provides formal tools and
practical problem-solving methods. Its applications are multiple... ».
Régis would add that SIA represents a true philosophical mode of
thought since, as its title says, it draws its source from the
improbability of errors, the improbability of the false, to establish
the bridge that allows us to overcome contingency and lead to
necessity.