A Cluster Model for Indirect Proof

Abstract

　　The purpose of this paper is to problematize the characterization of indirect proof in previous research and to propose a cluster model for indirect proof in order to capture a student’s overall process of developing the concept of indirect proof. We review and analyze polysemous characterizations of “indirect proof” in research and practice from a theoretical perspective of “cluster models.” As a result, we find that we should consider the concept of indirect proof as essentially polysemous. It has dual aspects of a method and its product, and has at least the following five meanings: A) proof based on logical laws such as law of excluded middle and double negation elimination; B) an approach to prove an equivalent different statement, especially with a meta-theorem in logic; C) a heuristic device through the introduction of an assumption “if …”; D) any method for proving in which (1) we prove an equivalent statement with the given statement or (2) we disprove a contradictory of the given statement; and E) a collective term for proof by contradiction, by contraposition, and so on. In cognitive linguistic term, the concept of indirect proof can be understood as a cluster concept of the above four meanings from A) to D).

　　Based on this result, we draw an implication for more constructive research on indirect proof. We need to flexibly design a research plan and to explore the relationships between different meanings of indirect proof. In addition, we also need to explore new potential meanings of indirect proof based on the meaning E) as an extensional definition and to continue to update the cluster model of indirect proof.