Adapting Formal Logic for Everyday Mathematics

Abstract

Although logic is considered central to mathematics and computer science, there is evidence that teaching logic has not been a great success. We identify three issues where what is typically taught conflicts with what is needed by those who are supposed to apply logic. First, what is taught about the notion of implication often disagrees with human intuition. We argue that in some cases human intuition is wrong, and in some others teaching is to blame. Second, the formal concepts of logical consequence, logical equivalence and tautology are not the similar concepts that everyday mathematicians and computer scientists need. The difference is small enough to go unnoticed but big enough to cause confusion. Third, how to deal with undefined operations such as division by zero is left informal and perhaps fuzzy. These problems also harm development of computer tools for education. We present suggestions about how to address them in teaching.