it. Now we always find that the sphere of one concept has something in common with the sphere of other concepts. That is to say, part of what is thought under one concept is the same as what is thought under other concepts; and conversely, part of what is thought under these concepts is the same as what is thought under the first; although, if they are really different concepts, each of them, or at least one of them, contains something which the other does not contain; this is the relation in which every subject stands to its predicate. The recognition of this relation is called judgment. The representation of these spheres by means of figures in space, is an exceedingly happy idea. It first occurred to Gottfried Plouquet, who used squares for the purpose. Lambert, although later than him, used only lines, which he placed under each other. Euler carried out the idea completely with circles. Upon what this complete analogy between the relations of concepts, and those of figures in space, ultimately rests, I am unable to say. It is, however, a very fortunate circumstance for logic that all the relations of concepts, according to their possibility, i.e., a priori, may be made plain in perception by the use of such figures, in the following way:—
(1.) The spheres of two concepts coincide: for example the concept of necessity and the concept of following from given grounds, in the same way the concepts of Ruminantia and Bisulca (ruminating and cloven-hoofed animals), also those of vertebrate and red-blooded animals (although there might be some doubt about this on account of the annelida): they are convertible concepts. Such concepts are represented by a single circle which stands for either of them.
(2.) The sphere of one concept includes that of the other.
(3.) A sphere includes two or more spheres which exclude each other and fill it.
(4.) Two spheres include each a part of the other.
(5.) Two spheres lie in a third, but do not fill it.
This last case applies to all concepts whose spheres have nothing immediately in common, for there is always a third sphere, often a much wider one, which includes both.
To these cases all combinations of concepts may be referred, and from them the entire doctrine of the judgment, its conversion, contraposition, equipollence, disjunction (this according to the third figure) may be deduced. From these also may be derived the properties of the judgment, upon which Kant based his pretended categories of the understanding, with the exception however of the hypothetical form, which is not a combination of concepts, but of judgments. A full account is given in the Appendix of “Modality,” and indeed of every property of judgments on which the categories are founded.
With regard to the possible combinations of concepts which we have given, it has only further to be remarked that they may also be combined with each other in many ways. For example, the fourth figure with the second. Only if one sphere, which partly or wholly contains another, is itself contained in a third sphere, do these together exemplify the syllogism in the first figure, i.e., that combination of judgments, by means of which it is known that a concept which is partly or wholly contained in another concept, is also contained in a third concept, which again contains the first: and also, conversely, the negation; the pictorial representation of which can, of course, only be two connected spheres which do not lie within a third sphere. If many spheres are brought together in this way we get a long train of syllogisms. This schematism of concepts, which has already been fairly well explained in more than one textbook, may be used as the foundation of the doctrine of the judgment, and indeed of the whole syllogistic theory, and in this way the treatment of both becomes very easy and simple. Because, through it, all syllogistic rules may be seen in their origin, and may be deduced and explained. It is not necessary, however, to load the memory with these rules, as logic is never of practical use, but has only a theoretical interest for philosophy. For although it may be said that logic is related to rational thinking as thorough-bass is to music, or less exactly, as ethics is to virtue, or æsthetics to art; we must yet remember that no one ever became an artist by the study of æsthetics; that a noble character was never formed by the study of ethics; that long before Rameau, men composed correctly and beautifully, and that we do not need to know thorough-bass in order to detect discords: and just as little do we need to know logic in order to avoid being misled by fallacies. Yet it must be conceded that thorough-bass is of the greatest use in the practice of musical composition, although it may not be necessary for the understanding of it; and indeed æsthetics and even ethics, though in a much less degree, and for the most part negatively, may be of some use in practice, so that we cannot deny them all practical worth, but of logic even this much cannot be conceded. It is nothing more than the knowledge in the abstract of what every one knows in the concrete. Therefore we call in the aid of logical rules, just as little to enable us to construct a correct argument as to prevent us from consenting to a false one, and the most learned logician lays aside the rules of logic altogether in his actual thought. This may be explained in the following way. Every science is a system of general and therefore abstract truths, laws, and rules with reference to a special class of objects. The individual case coming under these laws is determined in accordance with this general knowledge, which is valid once for all; because such application of the general principle is far easier than the exhaustive investigation of the particular case; for the general abstract knowledge which has once been obtained is always more within our reach than the empirical investigation of the particular case. With logic, however, it is just the other way. It is the general knowledge of the mode of procedure of the reason expressed in the form of rules. It is reached by the introspection of reason, and by abstraction from all content. But this mode of procedure is necessary and essential to reason, so that it will never depart from it if left to itself. It is, therefore, easier and surer to let it proceed itself according to its nature in each particular case, than to present to it the knowledge abstracted from this procedure in the form of a foreign and externally given law. It is easier, because, while in the case of all other sciences, the general rule is more within our reach than the investigation of the particular case taken by itself; with the use of reason, on the contrary, its necessary procedure in a given case is always more within our reach than the general rule abstracted from it; for that which thinks in us is reason itself. It is surer, because a mistake may more easily occur in such abstract knowledge, or in its application, than that a process of reason should take place which would run contrary to its essence and nature. Hence arises the remarkable fact, that while in other sciences the particular case is always proved by the rule, in logic, on the contrary, the rule must always be proved from the particular case; and even the most practised logician, if he remark that in some particular case he concludes otherwise than the rule prescribes, will always expect to find a mistake in the rule rather than in his own conclusion. To desire to make practical use of logic means, therefore, to desire to derive with unspeakable trouble, from general rules, that which is immediately known with the greatest certainty in the particular case. It is just as if a man were to consult mechanics as to the motion of his body, and physiology as to his digestion; and whoever has learnt logic for practical purposes is like him who would teach a beaver to make its own dam. Logic is, therefore, without practical utility; but it must nevertheless be retained, because it has philosophical interest as the special knowledge of the organisation and action of reason. It is rightly regarded as a definite, self-subsisting, self-contained, complete, and thoroughly safe discipline; to be treated scientifically for itself alone and independently of everything else, and therefore to be studied at the universities. But it has its real value, in relation
