The Equivalence of Propositions.
One great advantage which arises from the study of this Indirect Method of Inference consists in the clear notion which we gain of the Equivalence of Propositions. The older logicians showed how from certain simple premises we might draw an inference, but they failed to point out whether that inference contained the whole, or only a part, of the information embodied in the premises. Any one proposition or group of propositions may be classed with respect to another proposition or group of propositions, as
1. Equivalent,
2. Inferrible,
3. Consistent,
4. Contradictory.
Taking the proposition “All men are mortals” as the original, then “All immortals are not men” is its equivalent; “Some mortals are men” is inferrible, or capable of inference, but is not equivalent; “All not-men are not mortals” cannot be inferred, but is consistent, that is, may be true at the same time; “All men are immortals” is of course contradictory.
One sufficient test of equivalence is capability of mutual inference. Thus from
All electrics = all non-conductors,
I can infer
All non-electrics = all conductors,
and vice versâ from the latter I can pass back to the former. In short, A = B is equivalent to a = b. Again, from the union of the two propositions, A = AB and B = AB, I get A = B, and from this I might as easily deduce the two with which I started. In this case one proposition is equivalent to two other propositions. There are in fact no less than four modes in which we may express the identity of two classes A and B, namely,
| FIRST MODE. | SECOND MODE. | THIRD MODE. | FOURTH MODE. | ||
| A = B | a = b | A = AB |  | a = ab | |
B = AB | b = ab | ||||