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
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
I can infer
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,
The Indirect Method of Inference furnishes a universal and clear criterion as to the relationship of propositions. The import of a statement is always to be measured by the combinations of terms which it destroys. Hence two propositions are equivalent when they remove the same combinations from the Logical Alphabet, and neither more nor less. A proposition is inferrible but not equivalent to another when it removes some but not all the combinations which the other removes, and none except what this other removes. Again, propositions are consistent provided that they jointly allow each term and the negative of each term to remain somewhere in the Logical Alphabet. If after all the combinations inconsistent with two propositions are struck out, there still appears each of the letters A, a, B, b, C, c, D, d, which were there before, then no inconsistency between the propositions exists, although they may not be equivalent or even inferrible. Finally, contradictory propositions are those which taken together remove any one or more letter-terms from the Logical Alphabet.
What is true of single propositions applies also to groups of propositions, however large or complicated; that is to say, one group may be equivalent, inferrible, consistent, or contradictory as regards another, and we may similarly compare one proposition with a group of