ABSTRACT
The deductive closure of knowledge looks forward from knowledge of the premise to knowledge of the conclusion: If Sherlock Holmes knows a premise entails a conclusion while also knowing the premise, then he thereby knows the conclusion. Counter-closure looks backward from knowledge of the conclusion to knowledge of the premise: If Holmes knows a conclusion by virtue of deducing it from a premise, then he knew that premise throughout the inferential process. Counter-closure explains how Holmes managed to know more than Watson without any relevant difference in testimony, perceptions, or other evidence. This mode of explanation comes on-line at about age six (when children realize there are inferences and purposeful thinking). Socrates presupposes counter-closure when arguing that Meno's slave boy must have innate knowledge of geometrical premises. Philosophers rarely notice the presupposed rule of inference because counter-closure is even more intuitive than closure. Nevertheless, developmental psychology reinforces Peter Klein's ‘useful false belief’ counter-examples to counter-closure. But empirical research on reasoning also suggests that there are ‘useful invalidities’ in which one learns from invalid inferences. Indeed, developmental psychology supports the irony that counter-closure is itself a ‘useful invalidity’.
