Abstract

If you think that a proposition can have more than one contradictory, or can have none, then you need read no further. What I will show is that if then It is not obvious that this must be so. If p1 and p2 are distinct but logically equivalent, it might appear that their contradictories, q1 and q2, should also be distinct though logically equivalent. In traditional logic, a pair of propositions which satisfy (3)(i) alone are called contraries, and a pair which satisfy (3)(ii) alone are called sub-contraries.2 The relation of contradictoriness defined by (3) is clearly symmetrical: Let p1 and p2 be two propositions which are logically equivalent, and assume (1). Then p1 has a contradictory, q. We first show that Suppose p2 and q were both true. Then, since p1 and p2 are logically equivalent, p1 and q would both be true. But p1 and q are contradictories, so, by (3)(i) p1 and q cannot both be true. We next show that By (3)(ii) at least one of p1 and q must be true. So, since p1 and p2 are logically equivalent, at least one of p2 and q must be true. From (5) and (6) it follows that p2 and q are contradictories, so by (4) From (7) and (1) we have that p1 and p2 are identical; and since all that was assumed about them was that they are logically equivalent, (2) follows.