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\chapter{From the founding situations of the SIA to its formalization}
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\abstract{
Starting from mathematical didactic situations, the implicitative
statistical analysis method develops as problems are encountered and
questions are asked.
Its main objective is to structure data crossing subjects and
variables, to extract inductive rules between variables and, based on
the contingency of these rules, to explain and therefore forecast in
various fields: psychology, sociology, biology, etc.
It is for this purpose that the concepts of intensity of implication,
class cohesion, implication-inclusion, significance of hierarchical
levels, contribution of additional variables, etc., are based.
Similarly, the processing of binary variables (e.g., descriptors) is
gradually being supplemented by the processing of modal, frequency
and, recently, interval and fuzzy variables.
}

\section{Preamble}

Human operative knowledge is mainly composed of two components: that
of facts and that of rules between facts or between rules themselves.
It is his learning that, through his culture and his personal
experiences, allows him to gradually develop these forms of knowledge,
despite the regressions, the questioning, the ruptures that arise at
the turn of decisive information.
However, we know that these dialectically contribute to ensuring a
balanced operation.
However, the rules are inductively formed in a relatively stable way
as soon as the number of successes, in terms of their explanatory or
anticipatory quality, reaches a certain level (of confidence) from
which they are likely to be implemented.
On the other hand, if this (subjective) level is not reached, the
individual's economy will make him resist, in the first instance, his
abandonment or criticism.
Indeed, it is costly to replace the initial rule with another rule
when a small number of infirmations appear, since it would have been
reinforced by a large number of confirmations.
An increase in this number of negative instances, depending on the
robustness of the level of confidence in the rule, may lead to its
readjustment or even abandonment.
Laurent Fleury~\cite{Fleury}, in his thesis, correctly cites the
example - which Régis repeats - of the highly admissible rule: "all
Ferraris are red".
This very robust rule will not be abandoned when observing a single or
two counter-examples.
Especially since it would not fail to be quickly
re-comforted.

Thus, contrary to what is legitimate in mathematics, where not all
rules (theorem) suffer from exception, where determinism is total,
rules in the human sciences, more generally in the so-called "soft"
sciences, are acceptable and therefore operative as long as the number
of counter-examples remains "bearable" in view of the frequency of
situations where they will be positive and effective.
The problem in data analysis is then to establish a relatively
consensual numerical criterion to define the notion of a level of
confidence that can be adjusted to the level of requirement of the
rule user.
The fact that it is based on statistics is not surprising.
That it has a property of non-linear resistance to noise (weakness of
the first counter-example(s)) may also seem natural, in line with the
"economic" meaning mentioned above.
That it collapses if counter-examples are repeated also seems to have
to guide our choice in the modeling of the desired criterion.
This text presents the epistemological choice we have made.
As such it is therefore refutable, but the number of situations and
applications where it has proved relevant and fruitful leads us to
reproduce its genesis here.

\section{Introduction}

Different theoretical approaches have been adopted to model the
extraction and representation of imprecise (or partial) inference
rules between binary variables (or attributes or characters)
describing a population of individuals (or subjects or objects).
But the initial situations and the nature of the data do not change
the initial problem.
It is a question of discovering non-symmetrical inductive rules to
model relationships of the type "if a then almost b".
This is, for example, the option of Bayesian networks~\cite{Amarger}
or Galois lattices~\cite{Simon}.
But more often than not, however, since the correlation and the
${\chi}^2$ test are unsuitable because of their symmetric nature,
conditional probability~\cite{Loevinger, Agrawal,Grasn}  remains the
driving force behind the definition of the association, even when the
index of this selected association is multivariate~\cite{Bernard}.



Moreover, to our knowledge, on the one hand, most often the different
and interesting developments focus on proposals for a partial
implication index for binary data~\cite{Lermana} or \cite{Lallich}, on
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). 
