Aristotle on the Logical Relation between the Principles of Non-Contradiction and Excluded Middle in Metaphysics Γ.4

Christopher Izgin

Abstract:

At Metaphysics Γ.4, 1008a2–7, Aristotle argues that a denial of the principle of non-contradiction (PNC) entails a denial of the principle of excluded middle (PEM). What do we learn from this and other passages about the logical relation between the two principles? I argue that PEM implies PNC, and not vice versa. Aristotle treats PNC as the principle of all other axioms. Since PEM is an axiom, PNC is the principle of PEM. The sense in which it is the principle of PEM is that it is the starting-point from which PEM becomes known: one cannot grasp PEM with out already having grasped PNC. PNC enjoys logical priority because it implicitly occurs in PEM, and not vice versa. This is guaranteed if the disjunction in PEM is exclusive. It follows that PNC does not imply PEM.