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 propositions.
To give in this place illustrations of all the four kinds of relation would require much space: as the examples given in previous sections or chapters may serve more or less to explain the relations of inference, consistency, and contradiction, I will only add a few instances of equivalent propositions or groups.
In the following list each proposition or group of propositions is exactly equivalent in meaning to the corresponding one in the other column, and the truth of this statement may be tested by working out the combinations of the alphabet, which ought to be found exactly the same in the case of each pair of equivalents.
A = | Ab | B = | aB | ||||
A = | b | a = | B | ||||
A = | BC | a = | b ꖌ c | ||||
A = | AB ꖌ AC | b = | ab ꖌ AbC | ||||
A ꖌ B = | B ꖌ d | ab = | cd | ||||
A ꖌ c = | B ꖌ d | aC = | bD | ||||
| A = | ABc ꖌ AbC |  | A = | AB ꖌ AC | |||
AB = | ABc | ||||||
A = | B |  | | A = | B | ||
B = | C | A = | C | ||||
A = | AB | | | A = | AC | ||
B = | BC | B = | A ꖌ aBC | ||||
Although in these and many other cases the equivalents of certain propositions can readily be given, yet I believe that no uniform and infallible process can be pointed out by which the exact equivalents of premises can be ascertained. Ordinary deductive inference usually gives us only a portion of the contained information. It is true that the combinations consistent with a set of premises may always be thrown into the form of a proposition which must be logically equivalent to those premises; but the difficulty consists in detecting the other forms of propositions which will be equivalent to the premises. The task is here of a different character from any which we have yet attempted. It is in reality an inverse process, and is just as much more troublesome and uncertain than the direct process, as seeking is compared with hiding. Not only may several different answers equally apply, but there is no method of discovering any of those answers except by repeated trial. The problem which we have here met is really that of induction, the inverse of deduction; and, as I shall soon show, induction is always tentative, and, unless conducted with peculiar skill and insight, must be exceedingly laborious in cases of complexity.
De Morgan was unfortunately led by this equivalence of propositions into the most serious error of his ingenious system of Logic. He held that because the proposition “All A’s are all B’s,” is but another expression for the two propositions “All A’s are B’s” and “All B’s are A’s,” it must be a composite and not really an elementary form of proposition.[82] But on taking a general view of the equivalence of propositions such an objection seems to have no weight. Logicians have, with few exceptions, persistently upheld the original error of Aristotle in rejecting from their science the one simple relation of identity on which all more complex logical relations must really rest.
The Nature of Inference.
The question, What is Inference? is involved, even to the present day, in as much uncertainty as that ancient question, What is Truth? I shall in more than one part of this work endeavour to show that inference never does more than explicate, unfold, or develop the information contained in certain premises or facts. Neither in deductive nor inductive reasoning can we add a tittle to our implicit knowledge, which is like that contained in an unread book or a sealed letter. Sir W. Hamilton has well said, “Reasoning is the showing out explicitly that a proposition not granted or supposed, is implicitly contained in something different, which is granted or supposed.”[83]
Professor Bowen has explained[84] with much clearness that the conclusion of an argument states explicitly what is virtually or implicitly thought. “The process of reasoning is not so much a mode of evolving a new truth, as it is of establishing or proving an old one, by showing how much was admitted in the concession of the two premises taken together.” It is true that the whole meaning of these statements rests upon that of such words as “explicit,” “implicit,” “virtual.” That is implicit which is wrapped up, and we render it explicit when we unfold it. Just as the conception of a circle involves a hundred important geometrical properties, all following from what we know, if we have acuteness to unfold the results, so every fact and statement involves more meaning than seems at first sight. Reasoning explicates or brings to conscious possession what was before unconscious. It does not create, nor does it destroy, but it transmutes and throws the same matter into a new form.