Laboratoire Leibniz IMAG, Grenoble, France

The implication is an omnipresent concept in mathematics, constitutive of proofs. However, the implication, often identified with the object of natural logic, is almost never taught as a mathematical object. It appears as a clear and easy object whereas many students have difficulties related to this concept until the end of the university. For this study we tried to answer the following questions:

- What is the mathematical object "implication"?

- What is its "life" in french teaching?

- How can we build a didactical situation that question the implication?

To answer the first question, we present, in chapter 1, an epistemological and didactical analysis of the mathematical implication from three points of view: formal logic, sets theory and deductive reasoning. To answer the second one, we studied, in chapter 2, the "life" of the implication, seen from these three points of view, in some schoolbooks from high school to university. In chapter 3, we present our first results, particularly, what we call the "causal-temporal conception" of the implication. Following these results, we formulated our thesis:

*"It is necessary and sufficient to know and establish links between these three points of view on the implication for a good apprehension and a correct use of it."*

To support our thesis we built a didactical engineering, presented in the second part, which allows to question the implication by an "interplay" between these three points of view. This engineering shows, particularly, the relevance of the sets point of view to study the implication.