Understanding mathematical conditionals: An educational perspective informed by philosophy, linguistics and psychology

Conditionals - sentences of the form 'if A then B' - are ubiquitous in mathematics, where they are treated as true unless A is true and B is false. Conditionals are ubiquitous in everyday life, too, but there interpretations vary. This creates a challenge for students, who must learn an interpretation that might feel unnatural. How can we help them toward mathematically valid reasoning? In this theoretical paper, I argue that a sensible answer should build on work in philosophy, linguistics and psychology. I apply work from these fields to mathematical learning, especially at the transition to proof. I argue that day-to-day use of mathematical conditionals reflects the common inferential reading of everyday conditionals, so that an effective explanation of mathematical conditionals might: discuss the peculiarities of the material conditional, with reference to truthfunctionality; observe that universal mathematical conditionals are sensibly subject to an inferential reading; and entrench habitual counterexample search.