This lesson is from Bachhuber, Andrew H., S.J. 1957. "Chapter 3: The Attributive Proposition," An Introduction to Logic. New York: Appleton-Century-Crofts, Inc. Pp. 34-8.
The rules governing the quantity, or extension, of the predicate are of very great importance both theoretically and practically. A thorough mastery of them is an absolutely necessary prerequisite to an understanding of the conversion of propositions, which we shall treat in Chapter 5, and of the categorical syllogism, which we shall treat in Chapter 7.
We must prefix our treatment of these rules with two very important cautions:
The first caution is a warning never to confuse the quantity of the predicate with the quantity of the proposition itself. The quantity of a proposition, as we saw earlier, is determined solely by the quantity of the subject term; and the quantity of the predicate is irrelevant to the quantity of the proposition. A proposition is singular if the subject term is singular, particular if the subject term is particular, and universal if the subject term is universal. In other words, a proposition is singular, particular, or universal, depending on whether something is affirmed or denied of a definitely designated individual or group, of an indeterminately designated portion of the subject’s absolute extension, or of each individual included in the subject’s absolute extension.
The second caution is a reminder that the predicate does not have quantity, or extension, in the same way as the subject. The subject term has quantity directly and by its very nature as subject. The subject, according to Aristotelian terminology, indicates “matter,” and the predicate expresses a “form” that is received into this subject as into matter (or else denied of it). Now matter, not form, is the basis of quantity; and, until we reflect on the relationship of the subject and predicate, we do not think of the predicate from the point of view of quantity at all, but only from the point of view of comprehension. Suppose, for instance, that you do not know what a platyhelminth is and are then told, “A platyhelminth is a worm.” When you grasp the meaning of this proposition, the form “worm” is, as it were, received into the “matter” indicated by “platyhelminth” —the comprehension of “worm” is drawn into the comprehension of “platyhelminth,” and the nature of a platyhelminth is revealed to you insofar as a platyhelminth is whatever is signified by the predicate (that is, insofar as it is a worm). Only later, when you have reflected on the subject-predicate relationship, do you see that the extension of the subject is drawn into the extension of the predicate, and so forth. The fact that quantifiers (“every,” “some,” and so on) are attached to the subject but not to the predicate likewise reveals a difference in the way in which each of them has quantity.
This second caution will save us from the error of thinking that an attributive proposition is nothing more than an assertion of quantitative relationships of terms. This caution will also help us understand both the value and the limitations of diagrams in which, for instance, a circle is enclosed within another circle. These diagrams are visual aids to grasping quantitative relationships of subject and predicate, and only that. Now that we have given these two cautions, we are ready to treat of the rules governing the quantity, or extension, of the predicate.
Sometimes the predicate of an attributive proposition is singular. It is singular if it stands for one individual or group and likewise designates this individual or group definitely, as in the following examples:
The first man to make a solo non-stop flight across the Atlantic was Lindbergh.
John is not the tallest boy in the room.
He is not the first to do that.
Notice that this rule holds for both affirmative and negative propositions. Notice, too, that a singular term is singular independently of its function in a proposition.
If the predicate is not singular, the following rules (which are based on the function of the predicate and on an analysis of the subject-predicate relationship) are applicable:
a. Rule for the Affirmative Proposition
The predicate of an affirmative proposition is particular or undistributed (unless it is singular).
This diagram displays the most common relationship in extension of the subject and predicate of an affirmative proposition. The large circle represents the extension of "animal." Each of the dotted circles represents the indeterminate part of the extension of “animal” embraced by “dog,” “cats,” and “pig,” respectively. The words “bird,” “sheep,” “elephant,” “man,” and so on, show that there are, or at least might be, other animals besides “dog,” “cats,” and pig. Now, when we say that a dog is an animal, we do not mean that a dog is every animal, or this or that animal, but that a dog is some animal: we mean that “dog” is identical with an indeterminately designated portion of the extension of “animal.” Thus, we see that the relationship of the subject and predicate in extension is the reverse of their relationship in comprehension: the comprehension of the predicate, as we have seen, is drawn into the subject, with the predicate expressing one of the innumerable attributes of the subject; but, from the point of view of extension, the subject is drawn into the extension of the predicate and embraces an inde¬ terminately designated part thereof.
[Note: The fact that a certain form is in one subject does not exclude its presence in other subjects. For instance, the fact that the form “whiteness” is in this paper does not exclude “whiteness” from a white shirt. Thus, in the proposition “This paper is white,” the predicate “white” is particular, since this paper is only one of the innumerable subjects that can have whiteness.b. Rule for the Negative Proposition
The predicate of a negative proposition is universal or distributed (unless it is singular).
The subject of a negative proposition is completely excluded from the extension of the predicate, and the predicate is completely excluded from the extension of the subject. Consider the example “No dog is a cat” and the accompanying diagram.
You can go through the entire extension of “cat” without finding a single dog, and through the entire extension of “dog” without finding a single cat. Even the predicate of a singular or particular negative proposition is always universal (unless it is singular), for you can go through the entire extension of the predicate without finding an instance of the subject. For instance, if you are not a Hottentot, we can look at all the Hottentots without finding you; and if some cats are not black, we can look at all black things without finding those cats of which to be black has been denied, as a glance at the diagram will make clear.