Bibliography

Carnegie Institution Correspondence

But in my paper on probable inference in the Johns Hopkins “Studies in Logic”, owing to the excessive weight I at that time placed on formalistic considerations, I fell into the error of attaching a name the synonym I then used for Abduction, to a probable inference which I correctly described, forgetting that according to my own earlier and correct account of it, abduction is not of the number of probable inferences. It is singular that I should have done that, when in the very same paper I mention the existence of the mode of inference which is true abduction. Thus, the only error that paper contains is the designation as abduction of a mode of induction somewhat resembling abduction, which may properly be called abductive induction.

Methodeutic has a special interest in Abduction, or the inference which starts a scientific hypothesis. For it is not sufficient that a hypothesis should be a justifiable one. Any hypothesis which explains the facts is justified critically. But among justifiable hypotheses we have to select that one which is suitable for being tested by experiment. There is no such need of a subsequent choice after drawing deductive and inductive conclusions.

Abduction is reasoning which professes to be such that in case there is any ascertainable truth concerning the matter in hand, the general method of this reasoning, though not necessarily each special application of it, must eventually approximate to the truth.

Of these three classes of reasonings Abduction is the lowest. So long as it is sincere, and if it be not, it does not deserve to be called reasoning, Abduction cannot be absolutely bad. For sincere efforts to reach the truth, no matter in how wrong a way they may be commenced, cannot fail ultimately to attain any truth that is attainable. Consequently, there is only a relative preference between different abductions; and the ground of such preference must be economical. That is to say, the better abduction is the one which is likely to lead to the truth with the lesser expenditure of time, vitality, etc.

… Besides these three types of reasoning there is a fourth, Analogy, which combines the characters of the three, yet cannot be adequately represented as composite. …

I use the word belief to express any kind of holding for true or acceptance of a proposition. Belief, in this sense, is a composite thing. Its principal element is not a matter of consciousness at all; but is a habit established in the believer’s nature, in consequence of which he would act, should occasion present itself, in ceratin ways. However, not every habit is a belief. A belief is a habit with which the believer is deliberately satisfied. This implies that he is aware of it, and being aware of it does not struggle against it. A third important characteristic of belief is that while other habits are contracted by repeatedly performing the act under the conditions, belief may be, and commonly if not invariably is contracted, by merely imagining the situation and imagining what would be our experience and what our conduct in such a situation; and this mere imagination at once establishes such a habit that if the imagined case were realized we should really behave in that way.

My first real discovery about mathematical procedure was that there are two kinds of necessary reasoning, which I call the Corollarial and the Theorematic, because the corollaries affixed to the propositions of Euclid are usually arguments of one kind, while the more important theorems are of the other.

Deduction is divisible into sub-classes in various ways; of which the most important is into Corollarial and Theorematic. Corollarial deduction is where it is only necessary to imagine any case in which the premisses are true in order to perceive immediately that the conclusion holds in that case. All ordinary syllogisms and some deductions in the logic of relatives belong to this class. Theorematic deduction is deduction in which it is necessary to experiment in the imagination upon the image of the premiss in order from the result of such experiment to make corollarial deductions to the truth of the conclusion. The subdivisions of theorematic deduction are of very high theoretical importance.

A belief is chiefly an affair of the soul, not of the consciousness; a doubt, on the contrary, is chiefly an affair of consciousness. It is an uneasy feeling, a special condition of irritation, in which the idea of two incompatible modes of conduct [is] before the doubter’s imagination, and nothing determines him, indeed he feels himself forbidden, to adopt either and reject the other. Of course it is not necessary that the degrees of dissatisfaction with the opposite alternatives should be equal. Like irritations generally, doubt sets up a reaction which does not cease until the irritation is removed. If we accept this account of the matter, doubt is not the direct negation or contrary of belief; for the two mainly affect different parts of the man. Speaking physiologically, belief is a state of the connections between different parts of the brain, doubt an excitation of brain-cells. Doubt acts quite promptly to destroy belief. Its first effect is to destroy the state of satisfaction. Yet the belief-habit may still subsist. But imagination so readily affects this habit, that the former believer will soon begin to act in a half-hearted manner, and before long the habit will be destroyed. The most important character of doubt is that no sooner does a believer learn that another man equally well- informed and equally competent doubts what he has believed, than he begins by doubting it himself. Probably the first symptom of this state of irritation will be anger at the other man. Such anger is a virtual acknowledgment of one’s own doubt; that is to say, not a genuine doubt, or feeling of uneasiness, but a sense that it is possible we may come to doubt it. Such doubt, at first of a purely external nature, sets up as reaction an effort to enter into the doubt and to comprehend it. Indeed, it is not necessary that one should actually meet with a man who doubts; for such is the influence of imagination in such matters that as soon as a believer can imagine that a man, equally well-informed and equally competent with himself, should doubt, doubt actually begins to set in, in his own state of feeling. From this follows the important corollary that if a man does not himself really doubt a given proposition he cannot imagine how it can be doubted, and therefore cannot produce any argument tending to allay such doubt. It thus appears that it is one thing to question a proposition and quite another to doubt it. We can throw any proposition into the interrogative mood at will; but we can no more call up doubt than we can call up the feeling of hunger at will. What one does not doubt one cannot doubt, and it is only accidentally that attention can be drawn to it in a manner which suggests the idea that there might be a doubt. Thence comes a critical attitude, and finally, perhaps, a genuine doubt may arise.

Induction is the highest and most typical form of reasoning. In my essay of 1883, I only recognized two closely allied logical forms of pure induction, one of which is undoubtedly the highest. I have since discovered eight other forms which include those almost exclusively used by reasoners who are not adepts in logic.

27. Of Methodeutic. The first business of this memoir is to develope a precise conception of the nature of methodeutical logic. In methodeutic, it is assumed that the signs considered will conform to the conditions of critic, and be true. But just as critical logic inquires whether and how a sign corresponds to its intended ultimate object, the reality; so methodeutic looks to the purposed ultimate interpretant and inquires what conditions a sign must conform to, in order to be pertinent to the purpose. Methodeutic has a special interest in Abduction, or the inference which starts a scientific hypothesis. For it is not sufficient that a hypothesis should be a justifiable one. Any hypothesis which explains the facts is justified critically. But among justifiable hypotheses we have to select that one which is suitable for being tested by experiment. There is no such need of a subsequent choice after drawing deductive and inductive conclusions. Yet although methodeutic has not the same special concern with them, it has to develop the principles which are to guide us in the investigation of proofs, those which are to govern the general course of an investigation, and those which determine what problems shall engage our energies. It is, therefore, throughout of an economic character. Two other problems of methodeutic which the old logics usually made almost its only business are, first, the principles of definition, and of rendering ideas clear; and second, the principles of classification.

No. 27 of Methodeutic. The first business of this memoir is to show the precise nature of methodeutic; how it differs from critic; how, although it considers not what is admissable but what is advantageous, it is nevertheless a purely theoretical study, and not an art; how it is from the most strictly theoretical point of view, an absolutely essential and distinct department of logical inquiry; and how upon the other hand, it is readily made useful to a researcher into any science, even mathematics. It strongly resembles the purely mathematical part of political economy, which is also a theoretical study of advantages. Of the different classes of arguments, abductions are the only ones in which after they have been admitted to be just, it still remains to inquire whether they are advantageous. But since the whole business of heuretic, so far as its theory goes, falls under methodeutic, there is no kind of argumentation that methodeutic can pass over without notice. Nor is methodeutic confined to the consideration of arguments. On the contrary, its special subjects have always been understood to be the definition and division of terms. The formation of systems of propositions, although it has been neglected, should also evidently be included in methodeutic. In its method, methodeutic is less strict than critic.

The word criticism carries a meaning in philosophy which has so little resemblance to the criticism of literature that the latter meaning throws no light on the former. Philosophical criticism is applied to an idea which we have already adopted, but which we remark that we have not deliberately adopted. The mere fact that it has been adopted, as if hastily, that is, without deliberation, though it does not necessarily create a doubt, suggests the idea that perhaps a doubt might arise. The critical attitude consists in reviewing the matter to see in what manner corrections shall be made. This is what one does when one reads over a letter one has written to see whether some unintended meaning is suggested. The criticism is always of a process, the process which led to the acceptance of the ideas. It supposes that this process is subject to the control of the will; for its whole purpose is correction, and one cannot correct what one cannot control.

…a sign is something, A, which brings something, B, its interpretant sign determined or created by it, into the same sort of correspondence with something, C, its object, as that in which itself stands to C.

A sign is something, A, which brings something, B, its interpretant sign, determined or created by it, into the same sort of correspondence (or a lower implied sort) with something, C, its object, as that in which itself stands to C.

Deduction is divisible into sub-classes in various ways; of which the most important is into Corollarial and Theorematic. Corollarial deduction is where it is only necessary to imagine any case in which the premisses are true in order to perceive immediately that the conclusion holds in that case. All ordinary syllogisms and some deductions in the logic of relatives belong to this class. Theorematic deduction is deduction in which it is necessary to experiment in the imagination upon the image of the premiss in order from the result of such experiment to make corollarial deductions to the truth of the conclusion. The subdivisions of theorematic deduction are of very high theoretical importance.

My first real discovery about mathematical procedure was that there are two kinds of necessary reasoning, which I call the Corollarial and the Theorematic, because the corollaries affixed to the propositions of Euclid are usually arguments of one kind, while the more important theorems are of the other. The peculiarity of theorematic reasoning is that it considers something not implied at all in the conceptions so far gained, which neither the definition of the object of research nor anything yet known about could of themselves suggest, although they give room for it. Euclid, for example, will add lines to his diagram which are not at all required of suggested by any previous proposition, and which the conclusion that he reaches by this means says nothing about. I show that no considerable advance can be made in thought of any kind without theorematic reasoning. When we come to consider the heuretic part of mathematical procedure, the question how such suggestions are obtained will be the central point of the discussion.

[—] it is proper to divide all Theorematic reasoning into the Non-abstractional and the abstractional.

Imagine, for example, an endless succession of objects. [—] Yet this proof will rest on some proposition which is simply self evident. But as long as one only has the idea of simple endless series, one may think forever, and not discover the theorem, until something suggests that other idea to the mind. What I call the theorematic reasoning in mathematics consists in so introducing a foreign idea, using it, and finally deducing a conclusion from which it is eliminated.