Effects of Definitions of Logical Implication on Unit Structure of Logic in Textbooks: Praxeological Analysis of a High School Mathematics Textbook Focusing on Indirect Proof

Abstract

　　The purpose of this paper is to elaborate effects of definitions of logical implication on the unit structure of logic in mathematics textbooks.  Especially, this study focuses on the effects of their definitions on indirect proof. For this purpose, we carried out praxeological analysis of a Japanese high school mathematics textbook within Anthropological Theory of the Didactic.  The analysis consists of two parts.  First, we briefly review a definition of implication by inclusive relationships of sets.  Second, we identify what types of tasks appear how many tasks each type has respectively.  As a result, we found that the definition of logical implication justified solutions for almost all of tasks.  Especially, we can explain and justify the validity of proof by contrapositive by using the definition.  On the other hand, we showed that the definition did not validate of the method of proof by contradiction.  Furthermore, we suggested that above differences were caused by the following three reasons: 1) The current conception of implication based on the concept of set in Japanese school mathematics does not subsume the conception of non-implicational proposition, that is, singular proposition; 2) As long as following the description of the textbook, we cannot define negation for any propositions (we can define negation only for open sentences); nevertheless, 3) The textbook does not explicitly describe some other concepts, such as a logical consequence, required for explaining the validity of proof by contradiction.