Arnold Cusmariu

Gilbert Ryle has argued that Plato's Theory of Forms is a "logically vicious" doctrine because it's fundamental concept of exemplification leads to a vicious infinite regress. David Armstrong and Alan Donagan have agreed with Ryle. After making Ryle's argument logically explicit, I show the exemplification regress is illusory. Exemplification is a genuine universal alongside other relations; there is nothing paradoxical in its being exemplified over and over and over ... Platonism can define logical properties of this relation but not the relation itself, however.

Mathematics textbooks teach logical reasoning by example, a practice started by Euclid; while logic textbooks treat logic as a subject in its own right without practical application to mathematics. Stuck in the middle are students seeking mathematical proficiency and educators seeking to provide it. To assist them, the article explains in practical detail how to teach logic-based skills such as: making mathematical reasoning fully explicit; moving from step to step in a mathematical proof in logically correct ways; and checking to make sure inferences are logically correct. The methodology can easily be extended beyond the four examples analyzed.