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+\chapter{From the founding situations of the SIA to its formalization}
+\label{intro} % Always give a unique label
+% use \chaptermark{}
+% to alter or adjust the chapter heading in the running head
+
+
+
+\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}.
+
+
+
