The semantic perspective in logic appears in Aristotle, and is developed in the late nineteenth and early twentieth, mainly by Frege (Begrischift, 1882), Wittgenstein (Tractatus logico philosophicus, 1921), Tarski (The concept of truth in the language of deductive sciences, 1933, The semantic conception of truth, 1944)) and Quine (Methods of logic, 1950). In particular, Tarski (1936, 1944) provides a semantic definition of truth "formally correct and materially adequate", through the crucial notion of satisfaction of an open sentence by an object, and developed a model theoretic point of view, in which semantics is at the very core. Semantics is to put in relation with syntax on the one hand, and pragmatic on the other hand. Semantics concerns the relation between signs and objects they refer to; syntax concerns the rules of integration of signs in a given system, and pragmatics the relationship between subjects and signs (Morris, Foundations of the Theory of Signs, 1938; Eco, Il segno, 1971).

According with Da Costa (Logiques classiques et non classiques, 1997), it is necessary to take in account these three aspects for a right understanding of logical mathematical fields. In mathematics education, the importance of considering syntax, semantic and has already been support by many authors, especially in the field of numbers and algebra, more often (but not always) with few references to logical philosophy. In this lecture, I aim to show that a model theoretic point of view supports the relevance for mathematical education of considering semantic as a key for conceptualisation; this due to the fact that it offers powerful tools to take in account the articulation between forms and content (Sinaceur, Corps et modèles, 1991), and to distinguish between truth and validity, that are crucial issues in teaching and learning mathematics.

In the first part of this lecture, I will give some insights in epistemological aspects, and will precise and delimitate the meaning of the logical concepts in my research In the second part, I will give some precise examples of the way I use the model theoretic point of view in didactic (logical connectors, numbers construction, equation, calculus, solid’s geometry...) and will open on possible developments.