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group that the circles stand for, or something that does not have their qualities. *
* If we wished to distinguish in our diagrams between syllogisms in which the major premise describes objects as they are in themselves and those in which it tells of some causal or other relation between objects, we could indicate the latter in some such way as the following, using the arrow to indicate the relation :
That is to say, The Slavs, including all the Russians, hate the Semites, including all the Jews.
Every Slav hates every Semite, and some Russians are Slavs and
some Jews are Semites
In the text I have treated universal propositions as though they were nothing more nor less than statements about every member of a certain
group of objects, and have omitted any reference to the causal relation which they so often imply. (See above, p. 109.)
The principle and cautions can be easily restated so as to take account of this causal relation, as follows:
Any object or relation (S) which possesses or involves a given relation (M) possesses or involves every other relation (P) which that relation (M) involves, provided that M really (i.c., necessarily or always) involves it.
The fact that an object or relation (S) does not possess or involve a relation (M) which involves another relation (P) is no evidence that the object or relation (S) does not possess or involve this other relation (P).
The first paragraph covers the rule and the more important part of the first caution ; the second covers the second caution. To cover the rest of the first caution we must add that a statement about some undesignated member or members of a group of objects (S) will not enable us to say anything about any particular one of them.
Here is an example of the sort of causal connections I speak of :
Going to war involves distress for the soldier's family. Each of the diagrams in the text is intended to represent at least one sample of every object mentioned in either premise, and to preserve the distinction between those that are described in a given respect and those that are not. I have tried to make the illustrations as concrete as possible by using closed figures-circles or dots—to represent things, and lines drawn through them to indicate attributes. When the sense is once understood these circles are not so good as a matter of practical convenience as simple letters written one after the other with the understanding that the first letter stands for a thing and those that come after it for attributes. AB would thus mean that all A's have the attribute B (i.e., are B's). A, AB would mean that some A's have the attribute B. A, ABC, BC, would mean that some A's are B and all B's are C, and that therefore some A's are C. When these letters are used, a stroke over a letter indicates the absence of the attribute in question. ABC, BC means that all A's are non- -B, and all non-B's are C, and therefore all A's are C.
When we use these letters it is easy to get from pictorial representations to algebraic. A = AB means that every A has the attribute B, i.e., that every A is also B.
The pictorial representations most used are Euler's. These do not attempt to represent individual objects or to preserve the distinction between things and attributes ; but deal with the mutual relations of classes. As the diagrams are drawn on precisely the same principle
for all the figures (including the fourth) there is no denying their convenience for any one who has to work out a set of problems. Here are some of the diagrams for the first figure.
The dotted lines are used to indicate doubt as to where the circle or the part of the circle in question should fall.
THE SECOND FIGURE OF THE SYLLOGISM.
In this figure each premise describes an object or set of
objects, and from the nature of the two descripFunction and general tions the conclusion tells whether or not the cautions.
objects are identical with each other. The man that came to my house was tall and thin; The man that went to your house was short and fat
(i.e., not tall and thin); ... The man that came to my house is not the man that went to your
house. Crows do not sing;,
This bird sings; .:. This bird is not a crow.
Whales suckle their young;
Fishes do not; .:. Whales are not fishes.
The second and third of these examples differ from the first in this respect: In the first, two given objects are compared, and we conclude that they are not identical; in the second a given object is compared with a class of objects, and we conclude that it does not belong to the class—that it is not the kind of thing to which the class-name applies; in the third two classes of objects are compared. In the first case we are concerned with identification in the narrowest sense of the term ; in the others with classification. In the general description of the figure I have mentioned only the identification because it is the more fundamental
and the classification really depends upon it. If we could not distinguish between individual objects, we could not distinguish between classes. To say that whales are not fishes is to say that there is not any whale which is identical with any fish. To classify is thus merely to distinguish between individuals in groups, and the principles by which we distinguish classes are nothing more than those by which we distinguish individuals.
In this figure, therefore, both premises are concerned with descriptive relations, and the conclusion with a relation of identity. *
The next thing to be noticed about the figure is that the two premises must describe their objects in the same respect. If I describe the man I saw as tall, and you describe the one you saw as agreeable, the descriptions indicate absolutely nothing about the identity of the men.
But even when both premises describe their objects in the same respect a conclusion is not always possible. If each of two persons had met a tall man named Smith, they would not necessarily have met the same man. Two Dromios or two atoms of hydrogen might have innumerable points of
* For the purposes of this figure propositions which in themselves are not strictly descriptive are treated as such. When we conclude, for example, that Newhaven and New Haven are different cities because one is on the road from Paris to London, and the other on the road from New York to Boston, the geographical or spatial relations of each of them to the adjacent cities are practically regarded as a part of the city itself. The distance between Newhaven and London or Paris is a spatial relation and belongs as much to London or Paris as to Newhaven. But when it serves to distinguish Newhaven from New Haven it is treated as though it belonged, like its size or its history, to Newhaven itself. For this reason we make the word Newhaven the subject of the sentence in which the facts are expressed. We do not say “ London is a usual terminus for persons traveling from Paris and Newhaven", or "A good way to reach London from Paris is through Newhaven ", or " London is so far from Newhaven and so much farther from Paris by way of Newhaven”. What the syllogism involves is not the distance of London or Paris as such, but all of them in so far as they characterize Newhaven.