This episode begins the discussion of the style of proof known as Natural Deduction, invented by Gerhard Gentzen, a student of Hermann Weyl, himself a student of David Hilbert (sorry, I said incorrectly that Gentzen was Hilbert's own student).  Each logical connective (like OR, AND, IMPLIES, etc.) has introduction rules that let you prove formulas built with that connective; and elimination rules that let you deduce consequences from a proven formula built with that connective.