[This memoir will develop] the doctrine of logical depth and breadth, in a much more thorough working out of my paper of November 1867, and taking into consideration all kinds of terms.

      The question of the nature of the judgment is today more actively debated than any other. It is here that the German logicians are best worthy of attention; and I propose to take occasion to give here an account of modern German logic. Although this seems rather the subject for a book than for a single paper, yet I think, by stretching this memoir, I can bring into it all that is necessary to say about these treatises, which belong to near a dozen distinct schools.

      I shall then show how my own theory follows from attention to the three categories; and shall pass to an elaborate analysis, classification, symbolization, and doctrine of the relations of propositions. This will probably be the longest of all the memoirs, and will balance No. 16, which will be short. I think I shall treat No. 16 as a supplement to No. 15 and divide No. 21 into two parts to be handed in separately.

      No question of logic has of late years received more attention than that of the nature of the judgment. I here propose to show that my method satisfies all conditions. The doctrine is that the proposition, that is, the meaning of the matter of a judgment, is a sign (regarded as identical with any full interpretation of it) which separately designates its object. Thus a portrait with the name of the person portrayed under it expresses a proposition. This proposition need not be asserted. Assertion is a separate act by which a person makes himself responsible for the truth of the proposition. Interrogation, command, etc., equally involve easily defined acts. The question then arises whether a proposition does not involve a peculiar and indefinable element. I show that it does so. But although this element is indefinable, it is easily identified with my second category, and the precise manner in which reaction enters into it is also clear.

      I then examine the principal discussions of the nature of the judgment, and show exactly wherein they are right and wherein wrong.

      I further give a classification of propositions.

      It is necessary to distinguish between a proposition and the assertion of it. To confound those two things is like confounding the writing of one's name idly upon a scrap of paper, perhaps for practice in chirography, with the attachment of one's signature to a binding legal deed. A proposition may be stated without being asserted. I may state it to myself and worry as to whether I shall embrace it or reject it, being dissatisfied with the idea of doing either. In that case, I doubt the proposition. I may state the proposition to you and endeavor to stimulate you to advise me whether to accept or reject it, in which [case] I put it interrogatively. I may state it to myself and be deliberately satisfied to base my action on it whenever occasion may arise, in which case I judge it. I may state it to you and assume a responsibility for it, in which case I assert it. I may impose the responsibility of its agreeing with the truth upon you, in which case I command it. All of these are different moods in which that same proposition may be stated. The German word Urtheil confounds the proposition itself with the psychological act of assenting to it. This confusion is a part of the general refusal of idealism, which still considerably affects almost all German thought, to acknowledge that it is one thing to be and quite another to be represented.

      An assertion is an act which represents that an icon represents the object of an index. Thus, in the assertion, "Mary is red-headed","red-headed" is not an icon itself, it is true, but a symbol. But its interpretant is an icon, a sort of composite photograph of all the red-headed persons one has seen. "Mary" in like manner, is interpreted by a sort of composite memory of all the occasions which forced my attention upon that girl. The putting of these together makes another index which has a force tending to make the icon an index of Mary. This act of force belongs to the second category, and as such, has a degree of intensity. Not that degree in itself belongs to the second category: on the contrary, it belongs to the third. Degree is not a reaction, or effort, but a thought. But degree attaches to every reaction. Consequently, every assertion has a degree of energy.

      I first examine the essential nature of an argument, showing that it is a sign which separately signifies its interpretant. It will be scrutinized under all aspects.

      I shall then come to the important question of the classification of arguments. My paper of April 1867 on this subject divides arguments into deductions, inductions, abductions (my present name, which will be defended), and mixed arguments. I consider this to be the key of logic. In the following month, May 1867, I correctly defined the three kinds of simple arguments in terms of the categories. 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". It was this resemblance which deceived me, and subsequently led me into a further error contrary to my own previous correct statement, namely, to confound abduction and abductive induction. In subsequent reflections upon the rationale of abduction, I was led to see that this rationale was not that which I had in my Johns Hopkins paper given of induction; and in a statement I published in the Monist, I was led to give the correct rationale of abduction as applying to abductive induction and so, in fact, to all induction. All the difficulties with which I labored are now completely disposed of by recognizing that abductive induction is quite a different thing from abduction. It is a very instructive illustration both of the dangers and of the strength of my heuretic method. Similar errors may remain in my system. I shall be very thankful to whomever can detect them. But if its errors are confined to that class, the general fabric of the doctrine is true. I at first saw that there must be three kinds of arguments severally related to the three categories; and I correctly described them. Subsequently studying one of these kinds, I found that besides the typical form, there was another, distinguished from the typical form by being related to that category relation to which distinguishes abduction. I hastily identified it with abduction, not being clear-headed enough to see that, while related to that category, it is not related to it in the precise way in which one of the primary divisions of arguments ought to be, according to the theory of the categories. This is the form of error to which my method of discovery is peculiarly liable. One sees that a form has a relation to a certain category, and one is unable for the time being to attain sufficient clearness of thought to make quite sure that the relation is of the precise nature required. If only one point were obscure, it would soon be cleared up; but the difficulty is at first that one is sailing in a dense fog, through an unknown sea, without a single landmark. I can only say that if others, after me, can find some way of making as important discoveries in logic as I have done while falling into less error, nobody will be more intensely delighted than I shall be. My gratitude to the man who will show me where I am wrong in logic will have no bounds. Thus far, I have had to find out for myself as well as I could. Meantime, be it observed that the kind of error which I have been considering can never amount to anything worse than a faulty classification. All that I asserted about probable inference in my Johns Hopkins paper and in my Monist paper was perfectly true.

      In this paper, besides very important improvements in the subdivision of the three kinds of simple arguments, with several hitherto unrecognized types, and far greater clearness of exposition, I shall have much that is new to say about mixed arguments, which present many points of importance and of interest that have never been remarked. I shall give a new classification of them based, not upon the nature of their elements, but upon their modes of combination. Besides setting forth my own doctrine of the stechiology of argument, I shall examine the most important of those which are opposed to it.

      Most of the German logicians who are free from traditional influences have regarded the judgment as the logical element, because they find in it something sui generis. They find nothing of the sort in the argument. I show just how much truth there is in this, and that there really is a peculiar element in the argument; to wit, the third category. I show what the peculiarity of the German mind which leads to this and a number of other analogous views consists in. I then analyze the nature of the argument, and show among other things that in all reasoning there is a logica utens, or an appeal to a vaguely defined logical doctrine. I show that it follows from the definition of an argument, as a sign which definitely signifies its intended interpretant, that an argument must be a self-conscious sign, and I formally define this self-consciousness, without any resort to psychology or to the peculiar flavor of human self-consciousness. I further show that properly, an argument is self-controlled, although a sign may be quite similar to an argument without being self-controlled. I then show that from the definition of an argument it follows mathematically that every argument is either a deduction, an induction, an abduction, or an argument which mixes these characters. I proceed to show what the chief varieties of these [are], which will be one of the parts of this memoir most emphasized. In particular I distinguish deduction into two types, the corollarial and the theorematic, and induction into three types of widely different natures. This division is specially significant, and has never been published.

      There are three different ways in which a method may be calculated to lead to the truth, these three senses constituting three great classes of reasonings. Deduction is reasoning which professes to pursue such a method that if the premisses are true the conclusion will in every case be true. Probable deduction is, strictly speaking, necessary; only, it is necessary reasoning concerning probabilities. Induction is reasoning which professes to pursue such a method that, being persisted in, each special application of it (when it is applicable) must at least indefinitely approximate to the truth about the subject in hand, in the long run. 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.

      Deduction is only of value in tracing out the consequences of hypotheses, which it regards as pure, or unfounded, hypotheses. 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. 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. But I cannot go into them in this statement.

      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 in undoubtedly the highest. I have since discovered eight other forms which include those almost exclusively used by reasoners who are not adepts in logic. In fact, Norman Lockyer is the only writer I have met with who in his best work, especially his last book, habitually restricts himself to the highest form. Some of his work, however, as for example, that on the orientation of temples, is logically poor.

      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. There are also composite reasonings where an argument of one type is joined to an argument of another type. Such for example, is an induction fortified by the consideration of some known uniformity. Uniformities are of four principal kinds of which Mill distinctly recognizes only a single one.

      Mill's four methods of induction is a heterogeneous division, not at all scientific, and, in part, of very trifling utility. Still, it is better than no classification of inductions at all.