The World as Will and Idea, Vol. I — Arthur Schopenhauer, trans. Haldane & Kemp

It follows from what has been said that every concept, just because it is abstract and incapable of presentation in perception, and is therefore not a completely determined idea, has what is called extension or sphere, even in the case in which only one real object exists that corresponds to 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

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really diflerent 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, whiclP he placed under each other. Euler carried_ out the_ idea completely with cirdes. Upon Vf^t 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, te., a priori, may be made plain in perception by the use of such figures, in the following way : —

(i.) 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 Buminantia and Bisulca (ruminating and cloven-hoofed animals), also those of vertebrate and redblooded 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.

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(3.) A sphere includes two or more spheres which exclude each other and lili 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

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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 w’hich 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 tlieory, and in this waj 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 b^.only. a- theofetipaV interest .fgi; philosophy. For although it may be said that logi.c_ js .related to rational thinking as thorough-bass is to music, or less exactly, as ethics is to virtue, or aesthetics to art ; we must yet remember that no one ever became an artist by the study of aesthetics j that a noble character was neverp. 58 fHB WORLD AS IDEA.