It now appears that there are two kinds of deductive reasoning, which might, perhaps, be called explicatory and ampliative. However, the latter term might be misunderstood; for no mathematical reasoning is what would be commonly understood by ampliative, although much of it is not what is commonly understood as explicative. It is better to resort to new words to express new ideas. All readers of mathematics must have felt the great diffrence between corollaries and major theorems, although these words are not sharply distinguished. It is needless to say that the words come to us, not from Euclid, but from the editions of Euclid’s elements. The great body of the propositions called corollaries (all but 27 in the whole 13 books) are due to commentators, and are of an obvious kind. Kant’s characterization of all deductive reasoning is true of them: they are mere explications of what is implied in previous results. The same is true of a good many of Euclid’s own theorems; probably the numerical majority of the whole 369 of them are of this character. But many are of a different nature. We may call the two kinds of deduction corollarial and theorematic.
This appears to be in harmony with Kant’s view of deduction, namely, that it merely explicates what is implicitly asserted in the premisses. This is what is called a half-truth. Deductions are of two kinds, which I call corollarial and theorematic. The corollarial are those reasonings by which all corollaries and the majority of what are called theorems are deduced; the theorematic are those by which the major theorems are deduced. If you take the thesis of a corollary, – i.e. the proposition to be proved, and carefully analyze its meaning, by substituting for each term its definition, you will find that its truth follows, in a straightforward manner, from previous propositions similarly analyzed. But when it comes to proving a major theorem, you will very often find you have need of a lemma, which is a demonstrable proposition about something outside the subject of inquiry; and even if a lemma does not have to be demonstrated, it is necessary to introduce the definition of something which the thesis of the theorem does not contemplate. In the most remarkable cases, this is some abstraction; that is to say, a subject whose existence consists in some fact about other things. Such, for example, are operations considered as in themselves subject to operation; lines, which are nothing but descriptions of the motion of a particle, considered as being themselves movable; collections; numbers; and the like. When the reform of mathematical reasoning now going on is complete, it will be seen that every such supposition ought to be supported by a proper postulate. At any rate Kant himself ought to admit, and would admit if he were alive today, that the conclusion of reasoning of this kind, although it is strictly deductive, does not flow from definitions alone, but that postulates are requisite for it.
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.
How it can be that, although the reasoning is based upon the study of an individual schema, it is nevertheless necessary, that is, applicable, to all possible cases, is one of the questions we shall have to consider. Just now, I wish to point out that after the schema has been constructed according to the precept virtually contained in the thesis, the assertion of the theorem is not evidently true, even for the individual schema; nor will any amount of hard thinking of the philosophers’ corollarial kind ever render it evident. Thinking in general terms is not enough. It is necessary that something should be DONE. In geometry, subsidiary lines are drawn. In algebra permissible transformations are made. Thereupon, the faculty of observation is called into play. Some relation between the parts of the schema is remarked. But would this relation subsist in every possible case? Mere corollarial reasoning will sometimes assure us of this. But, generally speaking, it may be necessary to draw distinct schemata to represent alternative possibilities. Theorematic reasoning invariably depends upon experimentation with individual schemata. We shall find that, in the last analysis, the same thing is true of the corollarial reasoning, too; even the Aristotelian “demonstration why.” Only in this case, the very words serve as schemata. Accordingly, we may say that corollarial, or “philosophical” reasoning is reasoning with words; while theorematic, or mathematical reasoning proper, is reasoning with specially constructed schemata.
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 Deduction is an argument whose Interpretant represents that it belongs to a general class of possible arguments precisely analogous which are such that in the long run of experience the greater part of those whose premisses are true will have true conclusions. Deductions are either Necessary or Probable. Necessary Deductions are those which have nothing to do with any ratio of frequency, but profess (or their interpretants profess for them) that from true premisses they must invariably produce true conclusions. A Necessary Deduction is a method of producing Dicent Symbols by the study of a diagram. It is either Corollarial or Theorematic. A Corollarial Deduction is one which represents the conditions of the conclusion in a diagram and finds from the observation of this diagram, as it is, the truth of the conclusion. A Theorematic Deduction is one which, having represented the conditions of the conclusion in a diagram, performs an ingenious experiment upon the diagram, and by the observation of the diagram, so modified, ascertains the truth of the conclusion.
Deduction has two parts. [—] Explication is followed by Demonstration, or Deductive Argumentation. [—] It invariably requires something of the nature of a diagram; that is, an “Icon,” or Sign that represents its Object in resembling it. It usually, too, needs “Indices,” or Signs that represent their Objects by being actually connected with them. But it is mainly composed of “Symbols,” or Signs that represent their Objects essentially because they will be so interpreted. Demonstration should be Corollarial when it can. An accurate definition of Corollarial Demonstration would require a long explanation; but it will suffice to say that it limits itself to considerations already introduced or else involved in the Explication of its conclusion; while Theorematic Demonstration resorts to a more complicated process of thought.
There are two kinds of Deduction; and it is truly significant that it should have been left for me to discover this. I first found, and subsequently proved, that every Deduction involves the observation of a Diagram (whether Optical, Tactical, or Acoustic) and having drawn the diagram (for I myself always work with Optical Diagrams) one finds the conclusion to be represented by it. Of course, a diagram is required to comprehend any assertion. My two genera of Deductions are first those in which any Diagram of a state of things in which the premisses are true represents the conclusion to be true and such reasoning I call Corollarial because all the corollaries that different editors have added to Euclid’s Elements are of this nature. Second kind. To the Diagram of the truth of the Premisses something else has to be added, which is usually a mere May-be, and then the conclusion appears. I call this Theorematic reasoning because all the most important theorems are of this nature.