§2. The Plan and Steps of Reasoning
158. We may now profitably ask ourselves what logical goodness is. We have seen that any kind of goodness consists in the adaptation of its subject to its end. One might set this down as a truism. Verily, it is scarcely more, although circumstances may have prevented it being clearly apprehended.
If you call this utilitarianism, I shall not be ashamed of the title. For I do not know what other system of philosophy has wrought so much good in the world as that same utilitarianism. Bentham may be a shallow logician; but such truths as he saw, he saw most nobly. As for the vulgar utilitarian, his fault does not lie in his pressing too much the question of what would be the good of this or that. On the contrary his fault is that he never presses the question half far enough, or rather he never really raises the question at all. He simply rests in his present desires as if desire were beyond all dialectic. He wants, perhaps, to go to heaven. But he forgets to ask what would be the good of his going to heaven. He would be happy, there, he thinks. But that is a mere word. It is no real answer to the question.
159. Our question is, What is the use of thinking? We have already remarked that it is the argument alone which is the primary and direct subject of logical goodness and badness. We have therefore to ask what the end of argumentation is, what it ultimately leads to.
160. The Germans, whose tendency is to look at everything subjectively and to exaggerate the element of Firstness, maintain that the object is simply to satisfy one's logical feeling and that the goodness of reasoning consists in that esthetic satisfaction alone.1) This might do if we were gods and not subject to the force of experience.
Or if the force of experience were mere blind compulsion, and we were utter foreigners in the world, then again we might as well think to please ourselves; because we then never could make our thoughts conform to that mere Secondness.
But the saving truth is that there is a Thirdness in experience, an element of Reasonableness to which we can train our own reason to conform more and more. If this were not the case, there could be no such thing as logical goodness or badness; and therefore we need not wait until it is proved that there is a reason operative in experience to which our own can approximate.2) We should at once hope that it is so, since in that hope lies the only possibility of any knowledge.
161. Reasoning is of three types, Deduction, Induction, and Abduction.3) In deduction, or necessary reasoning, we set out from a hypothetical state of things which we define in certain abstracted respects. Among the characters to which we pay no attention in this mode of argument is whether or not the hypothesis of our premisses conforms more or less to the state of things in the outward world. We consider this hypothetical state of things and are led to conclude that, however it may be with the universe in other respects, wherever and whenever the hypothesis may be realized, something else not explicitly supposed in that hypothesis will be true invariably. Our inference is valid if and only if there really is such a relation between the state of things supposed in the premisses and the state of things stated in the conclusion. Whether this really be so or not is a question of reality, and has nothing at all to do with how we may be inclined to think. If a given person is unable to see the connection, the argument is none the less valid, provided that relation of real facts really subsists. If the entire human race were unable to see the connection, the argument would be none the less sound, although it would not be humanly clear. Let us see precisely how we assure ourselves of the reality of the connection. Here, as everywhere throughout logic, the study of relatives has been of the greatest service. The simple syllogisms, which are alone considered by the old inexact logicians, are such very rudimentary forms that it is practically impossible to discern in them the essential features of deductive inference until our attention has been called to these features in higher forms of deduction.
162. All necessary reasoning without exception is diagrammatic.1) That is, we construct an icon of our hypothetical state of things and proceed to observe it. This observation leads us to suspect that something is true, which we may or may not be able to formulate with precision, and we proceed to inquire whether it is true or not. For this purpose it is necessary to form a plan of investigation and this is the most difficult part of the whole operation. We not only have to select the features of the diagram which it will be pertinent to pay attention to, but it is also of great importance to return again and again to certain features. Otherwise, although our conclusions may be correct, they will not be the particular conclusions at which we are aiming. But the greatest point of art consists in the introduction of suitable abstractions. By this I mean such a transformation of our diagrams that characters of one diagram may appear in another as things. A familiar example is where in analysis we treat operations as themselves the subject of operations. Let me say that it would make a grand life-study to give an account of this operation of planning a mathematical demonstration.2) Sundry sporadic maxims are afloat among mathematicians, and several meritorious books have been written upon the subject, but nothing broad and masterly. With the modern reformed mathematics and with my own and other logical results as a basis, such a theory of the plan of demonstration is no longer a superhuman task.
163. Having thus determined the plan of the reasoning, we proceed to the reasoning itself, and this I have ascertained can be reduced to three kinds of steps.3) The first consists in copulating separate propositions into one compound proposition. The second consists in omitting something from a proposition without possibility of introducing error. The third consists in inserting something into a proposition without introducing error.
164. You can see precisely what these elementary steps of inference are in Baldwin's Dictionary under Symbolic Logic.1) As a specimen of what they are like you may take this:
A is a bay horse,
Therefore, A is a horse.
If one asks oneself how one knows that this is certain, one is likely to reply that one imagines a bay horse and on contemplating the image one sees that it is a horse. But that only applies to the single image. How large a horse did this image represent? Would it be the same with a horse of very different size? How old was the horse represented to be; was his tail docked? Would it be so if he had the blind-staggers, and if so are you sure it would be so whatever of the numerous diseases of the horse afflicted him? We are perfectly certain that none of these circumstances could affect the question in the least. It is easy enough to formulate reasons by the dozen; but the difficulty is that they are one and all far less evident than the original inference. I do not see that the logician can do better than to say that he perceives that when a copulative proposition is given, such as »A is a horse and A has a bay color« any member of the copulation may be omitted without changing the proposition from true to false. In a psychological sense I am willing to take the word of the psychologist if he says that such a general truth cannot be perceived. But what better can we do in logic?
165. Somebody may answer that the copulative proposition contains the conjunction »and« or something equivalent, and that the very meaning of this »and« is that the entire copulation is true if and only if each of the members is singly true; so that it is involved in the very meaning of the copulative proposition that any member may be dropped.
To this I assent with all my heart. But after all, what does it amount to? It is another way of saying that what we call the meaning of a proposition embraces every obvious necessary deduction from it. Considered as the beginning of an analysis of what the meaning of the word »meaning« is, it is a valuable remark. But I ask how it helps us to understand our passing from an accepted judgment A to another judgment C of which we not only feel equally confident but in point of fact are equally sure, barring a possible blunder which could be corrected as soon as attention was called to it, barring another equivalent blunder?
To this the advocate of the explanation by the conception of »meaning« may reply: that is meant which is intended or purposed; that a judgment is a voluntary act, and our intention is not to employ the form of the judgment A, except to the interpretation of images to which judgments, corresponding in form to C, can be applied.
166. Perhaps it may reconcile the psychologist to the admission of perceptual judgments involving generality to be told that they are perceptual judgments concerning our own purposes. I certainly think that the certainty of pure mathematics and of all necessary reasoning is due to the circumstance that it relates to objects which are the creations of our own minds, and that mathematical knowledge is to be classed along with knowledge of our own purposes. When we meet with a surprising result in pure mathematics, as we so often do, because a loose reasoning had led us to suppose it impossible, this is essentially the same sort of phenomenon as when in pursuing a purpose we are led to do something that we are quite surprised to find ourselves doing, as being contrary, or apparently contrary, to some weaker purpose.
But if it is supposed that any such considerations afford any logical justification of primary logical principles I must say that, on the contrary, at the very best they beg the question by assuming premisses far less certain than the conclusion to be established.