40 The Undistributed Middle Term

The scheme given at the left on page 141 is typical of the syllogism. For variety let us take a slightly different format of a valid syllogism, illustrate, schematize, and diagram it in circles:

so

No M is P Some S is M Some S is not P

No raw fish is edible Some Japanese dishes are raw fish   So some Japanese dishes are not edible

It will be seen that in this traditional form of argument the conclusion is entailed by two premises taken together. When the classes of raw fish, edible things, and Japanese dishes are related as the premises require, the conclusion is inescapable. Any argument so constructed is valid. The circles show this clearly. The conclusion "Some Japanese dishes are not edible" is true if the premises given are true.

Note that both premises relate to the same subject matter. This means only that there is a concept common to both, M in the formula (raw fish in the example), which, once it has related the other two classes of the premises, disappears -- it does not occur in the conclusion. This is called the "middle term" (hence "M"). The most important rule of the syllogism is that the middle term must be distributed -- important because frequently violated. In the invalid examples about the freshmen and the American Only League, the middle term was not distributed; in letters, every relevant case of P and of S is accounted for, but not every case of M. We must make clear what it means to say that a term is distributed.

In the formulas on page 141 both premises and the conclusion are "All" propositions. These are called universal propositions, since they refer to the whole universe under discussion: "All men must die. All dictators are men. Therefore, all dictators (thank God!) must die." Since such propositions give information about every case, the terms ("men," "dictators") are said to be distributed. As in this example, the formula shows another characteristic. The propositions are affirmative; they make positive statements, or, better, they do not contain the words "no," "not," or their equivalents.

It is permissible to use negative premises, provided that only one occurs in any given syllogism. "No good Christian is selfish" distributes both terms, since it states something that every last good Christian is not. It is clearly universal. There is another class of proposition, the non-universal or particular proposition. "Some men are not self-seeking" is evidently both particular and negative. Particular premises do not distribute their first terms; simply from the truth of "Some men are not self-seeking" we cannot know whether or not "Some men are self-seeking" is also true: we can know that "Some good Christians are selfish" is false merely from the truth of "No good Christian is selfish." Like the case of the negative premise, only one particular premise can occur in a given syllogism. It follows that if this particular premise is also negative, then the remaining premise must be both universal and affirmative. "All aggressive types (P) are self-seeking (M); some men (S) are not self-seeking (M); therefore, some men (S) are not aggressive types (P)." Let us look at this last example.

We see

1. there are two premises and a conclusion. This arrangement of sentences combines three terms in the following way:
   1. the subject ("S") of the conclusion "men" occurs in the minor premise, and its predicate ("P") "aggressive types" in the major premise. (This is all that is meant by calling these two propositions "major" and "minor" premises, that is, in order, the premises containing the predicate and the subject of the conclusion. Which premise comes first is immaterial.)
   2. the middle term ("M") "self-seeking" is distributed once: we know that no matter who the self-seeking are they are not some men;
   3. there is no term distributed in the conclusion which is not distributed in the premises: "Aggressive types" is the only term distributed in the conclusion (by the negative, as above). Since the subject of the conclusion "men" is not distributed in the premises -- it is particular -- it may not be, and is not, distributed in the conclusion; the conclusion must be particular since the minor premise is particular.
2. We also see that at least one of the premises is affirmative, and
3. that since there is one negative premise, the conclusion is negative.

These observations can be briefed as rules. All valid syllogisms observe them. If a syllogism is invalid, it has violated one or more of them. They constitute the necessary and sufficient conditions for a valid syllogism.

1. The two premises and the conclusion must each contain only two terms, and the syllogism altogether three and only three terms. These three terms are so combined that no term will be distributed in the conclusion which is not distributed in the premises and that the middle term will be distributed at least once.
2. At least one of the premises must be affirmative and one universal.
3. If one of the premises is negative, the conclusion must be negative, if particular, then particular.

A syllogistic argument which violates any of these rules is invalid. We shall now try to see intuitively, with the help of examples, why this is so.

EXAMPLE	COMMENT
Peter argues. "All weapons ought to be abolished. All propaganda ought to be abolished, too. You can see that propaganda is a weapon."	Paul takes a sheet of paper and works out the argument, as follows: S = propaganda, P = weapons, M = things that ought to be abolished. Thus, so All P is M All S is M All S is P           Paul says, "Your argument, Peter, is simply another example of the undistributed middle as found and diagramed on p. 142."
Peter tries again, "Well, I mean that propaganda is always a weapon and that some weapons are not safe in anybody's hands. So propaganda is not safe, either."	Paul applies the formulas again. In this argument, they are: S (subject of the conclusion) = propaganda; and P (predicate of the conclusion) = things safe in anybody's hands; M (middle term) = weapon (s). so All S is M Some M is not P Some S is not P           Paul shakes his head, "Just another undistributed middle."
"No proof exists that smoking causes tuberculosis. Some smokers do not get tuberculosis. So some smokers get tuberculosis from other causes."	This example is more complicated than the usual artificial examples employed in explanation. Peter's first premise seems to contain three terms, and so does the conclusion. But the classes might be stated in such a way that one of the terms would get absorbed. Even if this can be done, the argument is invalid because both premises are negative. Can the student see how other rules are violated?
"Some people get on my nerves; so some people better watch out!"	This is a truncated argument, called an enthymeme (see p. 3). It will depend on what proposition is supplied whether or not a valid argument results: "Some people get on my nerves; all those who get on my nerves had better watch out; so some people had better watch out!" Perfectly valid, if a little vapid. Note here that the major and minor premises are reversed. How is this demonstrated? (See p. 144.) Can the student invent any plausible major premise which would make the argument invalid?
That man Jones was a member of Stone Throwers, Inc., and that organization is a branch of the Hit-and-Runners. It's plain as the nose on my face that old Jones is a Hit-and-Runner.	Too bad, but valid. The ambiguity of the term "branch" and of the "er" formations (in Hit-and-Runner), like some "-ite" and "-ist" formations, are not problems related to formal logic (see #10).
Peter says, "I've heard that Jones voted for Schmidt. He must be a Democrat."	Another typical enthymeme. To make this enthymeme into a valid argument one would have to supply an unlikely major premise: "AH those who voted for Schmidt were Democrats."
Scientist Smith belongs to the A.S.C., a scientific organization. Communists belong to the A.S.C. Therefore, Smith is a communist.	This is guilt by association. The argument seldom occurs so baldly. One begins by proving what perhaps is never denied, that Scientist Smith is a member of the A.S.C. Then one proves that Jones, Robinson and some other members of the A.S.C. are "communists." Finally, in a triumphant enthymeme, the speaker makes a gesture with his hands and leaves the audience to draw the conclusion. As the audience goes out, members can be heard to say to one another: "Think of an important scientist like Smith being a communist!" This conclusion is warranted only if association with communists makes Smith a communist himself.