Which notion of implication is the right one? From logical considerations to a didactic perspective

Abstract

Implication is at the very heart of mathematical reasoning. As many authors have shown, pupils and students experience serious difficulties in using it in a suitable manner. In this paper, we support the thesis that these difficulties are closely related with the complexity of this notion. In order to study this complexity,we refer to Tarski's semantic truth theory,which contributes to clarifying the different aspects of implication: propositional connective, logically valid conditional, generalized conditional,inference rules. We will show that for this purpose, it is necessary to extend the classical definition of implication as a relation between propositions to a relation between open sentences with at least one free variable. This permits to become aware of the fact that, in some cases, the truth-value of a given mathematical statement is not constrained by the situation, contrary to the common standpoint that, in mathematics, a statement is either true or false. In the present paper, the didactic relevance of this theoretical stance will be illustrated by an analysis of two problematic situations and the presentation of some experimental results from our research on first-year university students' understanding of implication.