Aristotle’s Categories: Chapter 3a — Dictum de omni et nullo

When one thing is predicated of another, as of a subject, whatever things are said of the predicate, may be also said of the subject (1b10-1b12)

This principle allows us, when given two premises, to infer some conclusion. Specifically, if in premise 1 we say something like “mortal” of the predicate “man,” and in premise 2 we predicate “man” of the subject “Socrates,” we can conclude that “Socrates” is “mortal.” This general structure of reasoning is known as a syllogism, a core component of Aristotle’s logic which is developed throughout the Organon. This principle, known in a slightly different form as dictum de omni et nullo (the maxim of all and none), serves as an axiom from which all valid syllogistic forms can be derived. The standard version of dictum de omni et nullo is established in The Prior Analytics, chapter 1 (24b27-24b30):

But for one thing to be in the whole of another, and for one thing to be predicated of the whole of another, are the same thing, and we say it is predicated of the whole, when nothing can be assumed of the subject, of which the other may not be asserted, and as regards being predicated of nothing, in like manner. (24b27-24b30)

This is a bit harder to parse, but the first part, “But for one thing to be in the whole of another, and for one thing to be predicated of the whole of another, are the same thing” is just a statement asserting that logically one thing containing the “whole of another” is equivalent to that thing being predicated of the “whole of another”. The “whole of another” encompasses essentially all things ($X$) which the “other” ($Y$) can be predicated of. Therefore, when we say that a thing is predicated of the “whole of another”, we are saying that some thing $Z$ is predicated of all the things $X$, which another thing $Y$ is predicated of. We may equate this to the more vague “whatever things are said of the predicate” above (things that are said of a predicate are things ($Z$) predicated of the predicate ($Y$), which is, of course, predicated of some thing ($X$)). The text continues, establishing when something is “predicated of the whole”. Note that according to the analysis above, the things $Z$ are the things “predicated of the whole”, while the “subject” are the things $X$, which $Y$ can be predicated of. Therefore, according to the text, when there is no subject $X$ which $Y$ can not be asserted of (equivalently, when $Y$ can be asserted (or predicated) of all subjects $X$), $Z$ can necessarily be predicated of all $X$. It is necessary, then, that if something is predicated of the “other” (or “whole”), it can also be predicated of anything of which the “other” can be predicated. Clearly, this is semantically equivalent to (1b10-1b12).

(1b10-1b12) also suggests an inherent hierarchical structure of things, which through the use of this axiom of logic is made useful. For example, we can establish a hierarchical structure, such as Socrates being a particular instance of man, and man being a particular instance of things which are mortal. Aristotle’s principle here gives this hierarchical structure utility in discourse, in that invoking this structure allows us to conclude that Socrates is mortal. When establishing such a hierarchical structure, we are fundamentally performing the process of abstraction over particular things. From Wikipedia:

Abstraction is a process where general rules and concepts are derived from the use and classifying of specific examples, literal (real or concrete) signifiers, first principles, or other methods. (source)

For example, given a collection of individuals, say Socrates, Plato, and Aristotle, we can observe specific similarities among them and abstract the idea “man”. We can then observe similarities among “man” and other living beings, such as “ox”, and abstract the similarity termed “mortal”. As established above, Aristotle’s principle (and so syllogistic logic itself) ensures the utility of hierarchical structures. Since such hierarchical structures are established via the process of abstraction, this ensures the utility of abstraction itself, wherever it is applied.

Next, Aristotle provides an example of his own:

as “the man” is predicated of “some certain man,” but “the animal” is predicated of “the man,” wherefore “the animal” will be predicated of “some certain man,” since “the certain man” is both “man” and “animal.” (1b13-1b15)

That is, some certain man is a man, and any man is an animal. Since we can say of a subject any thing which is said of the thing which is predicated of that subject, we can say anything of some certain man that we can say of man in general. This implies that since we can say of man that man is animal, we can say that some certain man is animal.