Distribution - Amateur Logician

of Categorical Propositions

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Distribution of Categorical Propositions

A Rough Diagrammatic Expression of Categorical Propositions…

These circles represent the set of objects that are members of a given term.
One circle represents the “set” S (subject) and the other the “set” P (predicate).
*S represents at least one member of the “set” S.

Some Quirks about Particular Propositions. . .

In ordinary conversation, it would be odd to claim “some men are mortal.” We would rather say, “all men are mortal.” The former — “some men are mortal” — might seem to imply that some men are not in fact mortal. With categorical propositions in logic, however, nothing of that sort is implied. So, don’t think that!

The truth of an I proposition does NOT rule out the truth of the corresponding A proposition. In other words, the I proposition “some men are mortal” does NOT rule out its corresponding A proposition “all men are mortal.” Both can be true. In fact, both categorical propositions are true in this example: it is true that “some men are mortal” and it is true that “all men are mortal.”

Please recall that we have defined a particular proposition as referencing at least one member of the subject in question. Thus, because there is at least one man who is mortal, it is true that “some men are mortal.”

Distribution of Categorical Propositions. . .

We’ve already seen that quality and quantity are features of categorical propositions.

Distribution deals with the individual terms in a proposition. That is, it deals with the subject as a term and the predicate as a term. (Though remember that terms can be composed of more than one word. So a so-called “term” can be a long phrase.)

Distributed Term:
— A distributed term refers to all members that belong to it.
— That is, there is reference to its full “extension.”

Undistributed Term:
— An undistributed term doesn’t refer to each member that belongs to it.
— That is, there is no reference to its full “extension.”

Example: “All men are mortal.”
“Men” is clearly a distributed term. The proposition is about all men.
“Mortal” is an undistributed term. There’s no reference to every mortal being.

The quantifier “all” reveals that we are referring to the entire “extension” of men. The A type proposition references each and every member of the subject. Thus, the subject is distributed. It makes a claim about all men (about Socrates, you, me, etc.).

Conversely, the proposition only reveals that morality is a predicate of men; it tells us no more than that. There’s no reference to everything that has this predicate. Thus, the predicate is undistributed.

In general, only the subject will be distributed in an A proposition.

We can similarly analyze I, E, and O propositions.

When you do so, you’ll discover a pattern.

Universal propositions, by their very definition, will always have distributed subjects. Every member belonging to the subject is being referenced. Hence, all of the extension is being referenced of the subject in A and E propositions.

Negative propositions, as the subject is being excluded from the entire extension of the predicate, will always have distributed predicates. There is no way otherwise to deny a predicate of a subject. Hence, predicates will be distributed in E and O propositions.

Let’s return to some examples.

Example: “Some dogs are well-trained.”
“Dogs” is undistributed; after all, there’s the quantifier “some.”
“Well-trained” is undistributed; after all, there’s no reference to all that is “well-trained.”

No term is distributed in an I proposition. The subject isn’t being referenced in its entire extension. Nor is the predicate.

Example: “No Euclidean triangle is a figure with 360 degree angles.”
“Euclidean triangle” is distributed.
“Figure with 360 degree angles” is distributed.

Both subject and predicate are distributed in an E proposition. Pictures are worth thousands of words. Review the “Rough Diagrammatic Expression of Categorical Propositions.” The circles are entirely separate, since not one member of the subject has the given predicate. Declaring this necessitates reference to the entire extension of both the subject and the predicate.

To declare that “no” Euclidean triangle has some predicate can only be done when referencing every Euclidean triangle (for otherwise: there then could be at least one Euclidean triangle having the predicate, and thus it would be impossible that none of them have it). Equally, the predicate must be referenced in its entire extension, since the subject is excluded from the whole extension of the class “figure with 360 degree angles” (for otherwise: there could be no denial whatsoever).

Example: “Some men are not virtuous.”
“Men” is undistributed.
“Virtuous” is distributed; after all, the entire extension of “virtuous” is being denied.

Only the predicate is distributed in an O proposition. There’s only reference to “some of” the members of the subject. Thus, the subject cannot be distributed. Here the predicate must be distributed, since that’s the only way to deny the predicate as belonging to some members of the subject. Indeed, if the whole extension weren’t referenced, then no absolute denial of that predicate could take place.