The definition of the Port Royalists, therefore, makes comprehension include all necessary marks, whether essential or proper.
The Port Royalists attribute comprehension immediately to any ideas. Very many logicians attribute it immediately only to concepts. Now a concept, as defined by them, is strictly only the essence of an idea; they ought therefore to include in the comprehension only the essential marks of a term. These logicians, however, abstract so entirely from the real world, that it is difficult to see why these essential marks are not at the same time all the marks of the object as they suppose it.
There can, I think, be no doubt that such writers as Gerlach and Sigwart make comprehension include all marks, necessary or accidental, which are universally predicable of the object of the concept.
Again, most German writers regard the comprehension as a sum either of concepts (Drobisch, Bachmann, etc.) or of elements of intuition (Trendelenburg). But many English writers regard it as the sum of real external attributes (Shedden, Spalding, Devey, De Morgan, Jevons, McGregor, Fowler).
According to most writers, comprehension consists of the (necessary) attributes thought as common to the objects. Shedden defines it as consisting of all the attributes common to the things denoted.
Again, most logicians consider as marks only such as are virtually14 predicated; a few, perhaps, only such as are actually thought, and still fewer include those which are habitually thought. Here and there is found an author who makes comprehension include all true attributes, whether thought or not.
There is also a difference in the mode of reckoning up the marks. Most writers count all distinguishable marks, while a few consider coextensive marks as the same.
In the use of the term “extension” the want of a definite convention is still more marked. The Port Royalists define it as “those subjects to which the idea applies.” It would appear, therefore, that it might include mere fictions.
Others limit the term to real species, and at the same time extend it to single beings. This is the case with Watts, and also with Friedrich Fischer.
Others are most emphatic in declaring that they mean by it things, and not species, real or imaginary. This is the case with Bachmann, Esser, and Schulze.
Others make it include neither concepts nor things, but singular representations. This is the case with the strict Kantian.
The following table exhibits this diversity:—
Extension embraces
Again, logicians differ as to whether by extension they mean the concepts, species, things, or representations to which the term is habitually applied in the judgment, or all to which it is truly applicable. The latter position is held by Herbart, Kiesewetter, etc.; the former by Duncan, Spalding, Vorländer, Überweg, etc.
Some logicians include only actual things, representations, etc., under extension (Bachmann, Fries, Herbart); others extend it to such as are merely possible (Esser, Ritter, Gerlach).
Finally, some few logicians speak of the two quantities as numerical, while most writers regard them as mere aggregates of diverse objects or marks.
§4. Denials of the Inverse Proportionality of the two Quantities, and Suggestions of a third Quantity
Until lately the law of the inverse proportionality of extension and comprehension was universally admitted. It is now questioned on various grounds.
Drobisch says that the comprehension varies arithmetically, while the extension varies geometrically. This is true, in one sense.
Lotze, after remarking that the only conception of a universal which we can have is the power of imagining singulars under it, urges that the possibility of determining a concept in a way corresponding to each particular under it is a mark of that concept, and that therefore the narrower concepts have as many marks as the wider ones. But, I reply, these marks belong to the concept in its second intention, and are not common marks of those things to which it applies, and are therefore no part of the comprehension. They are, in fact, the very marks which constitute the extension. No one ever denied that extension is a mark of a concept; only it is a certain mark of second intention.
Vorländer’s objection is much more to the purpose. It is that if from any determinate notion, as that of Napoleon, we abstract all marks, all determination, what remains is merely the conception something, which has no more extension than Napoleon. “Something” has an uncertain sphere, meaning either this thing or that or the other, but has no general extension, since it means one thing only. Thus, before a race, we can say that some horse will win, meaning this one, that one, or that one; but by some horse we mean but one, and it therefore has no more extension than would a term definitely indicating which,—although this latter would be more determinate, that is, would have more comprehension. I am not aware that those who adhere to Kant’s unmodified doctrine have succeeded in answering this objection.
Überweg has the following remarks.15
To the higher representation, since conformably to its definition it contains only the common elements of content of several lower representations, belongs in comparison to each of the lower a more limited content, but a wider circuit. The lower representation, on the contrary, has a richer content but narrower circuit. Yet by no means by every diminution or increase of a given content does the circuit increase or diminish, nor by every increase or diminution of a given circuit does the content diminish or increase.
I am surprised that he does not explain himself further upon this point, which it is the principal object of this paper to develop.
De Morgan says:16
According to such statements as I have seen, “man residing in Europe, drawing breath north of the equator, seeing the sun rise before those in America,” would be a more intensively quantified notion than “man residing in Europe”; but certainly not less extensive, for the third and fourth elements of the notion must belong to those men to whom the first and second belong.
Mr. De Morgan adopts the definitions of extension and comprehension given by the Port Royalists. According to those definitions, if the third and fourth elements necessarily belong to the notion to which the first and second belong, they are parts of the comprehension of that second notion which is composed of the first and second elements, and therefore the two notions are equal in comprehension; but if this is not the case, then the second notion can be predicated of subjects of which the first cannot, for example, of “man residing in Europe drawing breath south of the Equator”; for that there is really no such man will not affect the truth of the proposition, and therefore the second notion is more extensive than the first.
Two logicians, only, as far as I remember, Archbishop Thomson17 and Dr. W. D. Wilson,18 while apparently admitting Kant’s law, wish to establish a third quantity of concepts. Neither gentleman has defined his third quantity, nor has stated what its relations to the other two are. Thomson calls his Denomination. It seems to be the same as Extension regarded in a particular way. Dr. Wilson terms his new quantity Protension; it has something to do with time, and appears to be generally independent of the other two. It is plain, indeed, that as long as Kant’s law holds, and as long as logical quantities can only be compared as being more or less and not directly measured, and as long as the different kinds of quantity cannot be compared at all, a third quantity must be directly proportional to one or other of the known quantities, and therefore must measure the same thing, or else must be independent of the other two, and be quite unconnected with them.
§5. Three Principal Senses in which Comprehension