Chapter VI: The Indirect Method of Inference (2)

The operator will readily collect the various conclusions in the manner described in previous pages, as, for instance that A is always C, that not-C is not-B and not-A; and not-B is not-A but either C or not-C. The results are thus to be read off exactly as in the case of the Logical Slate, or the Logical Abacus.

Disjunctive propositions are to be treated in an exactly similar manner. Thus, to work the premises

A = AB ꖌ AC
B ꖌ C = BD ꖌ CD,

it is only necessary to press in succession the keys

A (left), copula, A (right), B, ꖌ, A, C, full stop.
B (left), ꖌ, C, copula, B (right), D, ꖌ, C, D, full stop.

The combinations then remaining will be as follows

ABCD *a*BCD *abc*D
AB*c*D *a*B*c*D *abcd.*
A*c*CD *ab*CD

On pressing the left-hand key A, all the possible combinations which do not contain A will disappear, and the description of A may be gathered from what remain, namely that it is always D. The full-stop key restores all combinations consistent with the premises and any other selection may be made, as say not-D, which will be found to be always not-A, not-B, and not-C.

At the end of every problem, when no further questions need be addressed to the machine, we press the Finis key, which has the effect of bringing into view the whole of the conceivable combinations of the alphabet. This key in fact obliterates the conditions impressed upon the machine by moving back into their ordinary places those combinations which had been rejected as inconsistent with the premises. Before beginning any new problem it is requisite to observe that the whole sixteen combinations are visible. After the Finis key has been used the machine represents a mind endowed with powers of thought, but wholly devoid of knowledge. It would not in that condition give any answer but such as would consist in the primary laws of thought themselves. But when any proposition is worked upon the keys, the machine analyses and digests the meaning of it and becomes charged with the knowledge embodied in that proposition. Accordingly it is able to return as an answer any description of a term or class so far as furnished by that proposition in accordance with the Laws of Thought. The machine is thus the embodiment of a true logical system. The combinations are classified, selected or rejected, just as they should be by a reasoning mind, so that at each step in a problem, the Logical Alphabet represents the proper condition of a mind exempt from mistake. It cannot be asserted indeed that the machine entirely supersedes the agency of conscious thought; mental labour is required in interpreting the meaning of grammatical expressions, and in correctly impressing that meaning on the machine; it is further required in gathering the conclusion from the remaining combinations. Nevertheless the true process of logical inference is really accomplished in a purely mechanical manner.

It is worthy of remark that the machine can detect any self-contradiction existing between the premises presented to it; should the premises be self-contradictory it will be found that one or more of the letter-terms disappears entirely from the Logical Alphabet. Thus if we work the two propositions, A is B, and A is not-B, and then inquire for a description of A, the machine will refuse to give it by exhibiting no combination at all containing A. This result is in agreement with the law, which I have explained, that every term must have its negative (p. 74). Accordingly, whenever any one of the letters A, B, C, D, *a*, *b*, *c*, *d*, wholly disappears from the alphabet, it may be safely inferred that some act of self-contradiction has been committed.

It ought to be carefully observed that the logical machine cannot receive a simple identity of the form A = B except in the double form of A = B and B = A. To work the proposition A = B, it is therefore necessary to press the keys--

A (left), copula, B (right), full stop;
B (left), copula, A (right), full stop.

The same double operation will be necessary whenever the proposition is not of the kind called a partial identity (p. 40). Thus AB = CD, AB = AC, A = B ꖌ C, A ꖌ B = C ꖌ D, all require to be read from both ends separately.

The proper rule for using the machine may in fact be given in the following way:--(1) *Read each proposition as it stands, and play the corresponding keys*: (2) *Convert the proposition and read and play the keys again in the transposed order of the terms.* So long as this rule is observed the true result must always be obtained. There can be no mistake. But it will be found that in the case of partial identities, and some other similar forms of propositions, the transposed reading has no effect upon the combinations of the Logical Alphabet. One reading is in such cases all that is practically needful. After some experience has been gained in the use of the machine, the worker naturally saves himself the trouble of the second reading when possible.

It is no doubt a remarkable fact that a simple identity cannot be impressed upon the machine except in the form of two partial identities, and this may be thought by some logicians to militate against the equational mode of representing propositions.

Before leaving the subject I may remark that these mechanical devices are not likely to possess much practical utility. We do not require in common life to be constantly solving complex logical questions. Even in mathematical calculation the ordinary rules of arithmetic are generally sufficient, and a calculating machine can only be used with advantage in peculiar cases. But the machine and abacus have nevertheless two important uses.

In the first place I hope that the time is not very far distant when the predominance of the ancient Aristotelian Logic will be a matter of history only, and when the teaching of logic will be placed on a footing more worthy of its supreme importance. It will then be found that the solution of logical questions is an exercise of mind at least as valuable and necessary as mathematical calculation. I believe that these mechanical devices, or something of the same kind, will then become useful for exhibiting to a class of students a clear and visible analysis of logical problems of any degree of complexity, the nature of each step being rendered plain to the eyes of the students. I often used the machine or abacus for this purpose in my class lectures while I was Professor of Logic at Owens College.

Secondly, the more immediate importance of the machine seems to consist in the unquestionable proof which it affords that correct views of the fundamental principles of reasoning have now been attained, although they were unknown to Aristotle and his followers. The time must come when the inevitable results of the admirable investigations of the late Dr. Boole must be recognised at their true value, and the plain and palpable form in which the machine presents those results will, I hope, hasten the time. Undoubtedly Boole’s life marks an era in the science of human reason. It may seem strange that it had remained for him first to set forth in its full extent the problem of logic, but I am not aware that anyone before him had treated logic as a symbolic method for evolving from any premises the description of any class whatsoever as defined by those premises. In spite of several serious errors into which he fell, it will probably be allowed that Boole discovered the true and general form of logic, and put the science substantially into the form which it must hold for evermore. He thus effected a reform with which there is hardly anything comparable in the history of logic between his time and the remote age of Aristotle.

Nevertheless, Boole’s quasi-mathematical system could hardly be regarded as a final and unexceptionable solution of the problem. Not only did it require the manipulation of mathematical symbols in a very intricate and perplexing manner, but the results when obtained were devoid of demonstrative force, because they turned upon the employment of unintelligible symbols, acquiring meaning only by analogy. I have also pointed out that he imported into his system a condition concerning the exclusive nature of alternatives (p. 70), which is not necessarily true of logical terms. I shall have to show in the next chapter that logic is really the basis of the whole science of mathematical reasoning, so that Boole inverted the true order of proof when he proposed to infer logical truths by algebraic processes. It is wonderful evidence of his mental power that by methods fundamentally false he should have succeeded in reaching true conclusions and widening the sphere of reason.

The mechanical performance of logical inference affords a demonstration both of the truth of Boole’s results and of the mistaken nature of his mode of deducing them. Conclusions which he could obtain only by pages of intricate calculation, are exhibited by the machine after one or two minutes of manipulation. And not only are those conclusions easily reached, but they are demonstratively true, because every step of the process involves nothing more obscure than the three fundamental Laws of Thought.

*The Order of Premises.*

Before quitting the subject of deductive reasoning, I may remark that the order in which the premises of an argument are placed is a matter of logical indifference. Much discussion has taken place at various times concerning the arrangement of the premises of a syllogism; and it has been generally held, in accordance with the opinion of Aristotle, that the so-called major premise, containing the major term, or the predicate of the conclusion, should stand first. This distinction however falls to the ground in our system, since the proposition is reduced to an identical form, in which there is no distinction of subject and predicate. In a strictly logical point of view the order of statement is wholly devoid of significance. The premises are simultaneously coexistent, and are not related to each other according to the properties of space and time. Just as the qualities of the same object are neither before nor after each other in nature (p. 33), and are only thought of in some one order owing to the limited capacity of mind, so the premises of an argument are neither before nor after each other, and are only thought of in succession because the mind cannot grasp many ideas at once. The combinations of the logical alphabet are exactly the same in whatever order the premises be treated on the logical slate or machine. Some difference may doubtless exist as regards convenience to human memory. The mind may take in the results of an argument more easily in one mode of statement than another, although there is no real difference in the logical results. But in this point of view I think that Aristotle and the old logicians were clearly wrong. It is more easy to gather the conclusion that “all A’s are C’s” from “all A’s are B’s and all B’s are C’s,” than from the same propositions in inverted order, “all B’s are C’s and all A’s are B’s.”

*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* }

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 = A*b* . . . . . . . B = *a*B
A = *b* . . . . . . . . *a* = B
A = BC . . . . . . . . *a* = *b* ꖌ *c*
A = AB ꖌ AC . . . . . . *b* = *ab* ꖌ A*b*C
A ꖌB = C ꖌ D . . . . . . . *ab* = *cd*
A ꖌ *c* = B ꖌ *d* . . . . . . *a*C = *b*D
A = AB*c* ꖌ A*b*C . . .{ A = AB ꖌ AC
{ AB = AB*c*

A = B } { A = B
B = C } . . . . . . . . . { A = C

A = AB } { A = AC
B = BC }. . . . . . . . . { B = A ꖌ *a*BC

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.

[82] *Syllabus of a proposed system of Logic*, §§ 57, 121, &c.
*Formal Logic*, p. 66.

*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]

[83] Lectures on Metaphysics, vol. iv. p. 369.

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.

[84] Bowen, *Treatise on Logic*, Cambridge, U.S., 1866; p. 362.

The difficult question still remains, Where does novelty of form begin? Is it a case of inference when we pass from “Sincerity is the parent of truth” to “The parent of truth is sincerity?” The old logicians would have called this change *conversion*, one case of immediate inference. But as all identity is necessarily reciprocal, and the very meaning of such a proposition is that the two terms are identical in their signification, I fail to see any difference between the statements whatever. As well might we say that *x* = *y* and *y* = *x* are different equations.

Another point of difficulty is to decide when a change is merely grammatical and when it involves a real logical transformation. Between a *table of wood* and a *wooden table* there is no logical difference (p. 31), the adjective being merely a convenient substitute for the prepositional phrase. But it is uncertain to my mind whether the change from “All men are mortal” to “No men are not mortal” is purely grammatical. Logical change may perhaps be best described as consisting in the determination of a relation between certain classes of objects from a relation between certain other classes. Thus I consider it a truly logical inference when we pass from “All men are mortal” to “All immortals are not-men,” because the classes *immortals* and *not-men* are different from *mortals* and *men*, and yet the propositions contain at the bottom the very same truth, as shown in the combinations of the Logical Alphabet.

The passage from the qualitative to the quantitative mode of expressing a proposition is another kind of change which we must discriminate from true logical inference. We state the same truth when we say that “mortality belongs to all men,” as when we assert that “all men are mortals.” Here we do not pass from class to class, but from one kind of term, the abstract, to another kind, the concrete. But inference probably enters when we pass from either of the above propositions to the assertion that the class of immortal men is zero, or contains no objects.

It is of course a question of words to what processes we shall or shall not apply the name “inference,” and I have no wish to continue the trifling discussions which have already taken place upon the subject. What we need to do is to define accurately the sense in which we use the word “inference,” and to distinguish the relation of inferrible propositions from other possible relations. It seems to be sufficient to recognise four modes in which two apparently different propositions may be related. Thus two propositions may be--

1. *Tautologous* or *identical*, involving the same relation between the same terms and classes, and only differing in the order of statement; thus “Victoria is the Queen of England” is tautologous with “The Queen of England is Victoria.”

2. *Grammatically related*, when the classes or objects are the same and similarly related, and the only difference is in the words; thus “Victoria is the Queen of England” is grammatically equivalent to “Victoria is England’s Queen.”

3. *Equivalents* in qualitative and quantitative form, the classes being the same, but viewed in a different manner.

4. *Logically inferrible*, one from the other, or it may be *equivalent*, when the classes and relations are different, but involve the same knowledge of the possible combinations.