The conditional, "if … then …" (→) has been a contentious topic throughout the history of Western logic, and numerous accounts of its behavior have been proposed. One recurrent account (usually called the material conditional) is that A→B is true just if the antecedent, A, is false or the consequent, B, is true. This account was built into the logic of Frege and Russell, and so came to assume orthodoxy throughout much of the twentieth century (at least where there are no subjunctive moods in the antecedent or consequent). The account has obvious problems, however. It entails, for example, that both of the following are true—which they do not appear to be: "If Melbourne is the capital of Australia, it is in China" (false antecedent), "If there is life on another planet, Canberra is the capital of Australia" (true consequent).

It is natural to suppose...