 Boethius here draws a distinction between affirmative and negative conditionals and explains that negative conditionals have the form ‘if a, then not b’ and ‘if not a, then not b.’ This statement is quite different from the reading offered by Kneale and Kneale. Note that AT follows from BT′ and AT′ follows from BT in logics in which A → A is theorem and modus ponens is an admissible inference rule.