form of the verb "to be." And, fortunately, although there
are unsettled disputes about the proper interpretation of the
relationship asserted, there is one mode of interpretation which
will always enable us to avoid error in syllogisms containing
such propositions. This is to consider the relation one of in-
clusion. That is, to say that most business men are progres-
sive is to say that most business men are included in the class
of individuals to whom the term "progressive" applies. Simi-
larly, to affirm that no airship is safe is to deny the relation
of inclusion and to assert that of exclusion between airships
and safe things; the class of objects described by the term
"airship" and the class described by the term "safe" have no
objects in common. In general, the student will find that where !
no other relation is verbally affirmed in any proposition, the
relation in which it is to be construed is that of inclusion or
its opposite. The special problems of inference involving such
relations will occupy us below. They are important because
the relation is a very common one in our reasoning.

tion of

terms and relations

depends

In the third place, it is well to note that while many proposi- Detertions are so simple that their analysis into terms and relations minacan be confidently performed without knowing how they are to be used in a syllogism, this is not always the case. The meaning of a proposition is its meaning in the syllogism in which it is to be used, and in complex propositions this may on how not be apparent, so far as determining the terms and relations they are is concerned, unless the rest of the argument is known. For used example, if we were given the proposition: "The cost per mile of a subway is very high," and knew nothing else about the reasoning in which it was to be used, it would be quite impossible to say with assurance what the precise terms and relation are. But the other premise (or the conclusion) gives us our common term and enables us to construe the proposition. "Chicago will build a subway." Hence "subway" is the common term. The proposition can then be formulated in this fashion: "A subway is a means of transportation whose cost per mile is very high." Here "subway" is one term; "means of transportation very high" is the other, and the relation is one of inclusion.

a relation

a term

Another illustration thus appears of the fact that the real unit of deduction is the syllogism, and that the meaning of its various elements is to be determined by their function in the syllogism. That we face no such problem in the case of propositions like "Omaha is west of Chicago," or "James is father of Sally," is simply due to the fact that in statements of this sort there is no possible ambiguity about either the terms or the relation.

A given In certain cases, moreover, it is impossible to tell clearly, withword may out the remainder of the argument, whether a certain word in be part of a proposition is a part of the relation or a part of one of the or part of terms. Suppose, for example, we were given the proposition: "Mr. A. is an inhabitant of Australia." At first sight we should naturally assume that "Mr. A.” and “Australia" were the terms, and "inhabitant of" the relation. This would be correct if "Australia" were one of the terms in the other premise, making it the common element between them, as in "Australia is the smallest continent." Then we could deduce the conclusion, "Mr. A. is an inhabitant of the smallest continent." But suppose the other premise were the following: "All the people of Australia are enterprising." This shows that the common element of the two premises is not "Australia," but "people" or "inhabitants of Australia." Accordingly, "inhabitant of" becomes a part of one of the terms instead of the relation, and the relation must be construed as that of inclusion. It is meant that Mr. A. is included within the class of inhabitants of Australia.

The student will find practice desirable in discerning terms and relations in situations of this kind. When a single proposition is given to be dissected, it is usually best to assume that as large a part of the proposition as possible is to be assigned to the relation. This practice has the additional advantage of affording familiarity with a larger number of relations than we should otherwise be apt to consider.

EXERCISE.-Pick out terms and relations in the following pairs of premises. Draw the conclusion permitted in each case.

1. Roosevelt was a hunter of big game.

Roosevelt was a president of the United States.

2. Yuma is named after the Yuma Indians.

The Yuma Indians inhabited the lower valley of the Colorado River.

3. John struck his little brother.

Anybody who strikes his little brother is punished.

4. John struck his little brother.

His little brother had torn his book.

tions

In the fourth place, while the propositions which have been Complex heretofore used for illustration are all propositions containing proposibut two terms and one relation, it is to be noticed that a proposition is not necessarily of this simple structure. There may be three, four, or more terms, and a corresponding complexity of relationship. Such propositions are not very common in deduction, and they interpose no special difficulties in the study of the principles of correct inference, but for the sake of completeness of analysis the fact must be mentioned. Consider the following syllogism:

Father told Robert to bring the paper.

Robert is my older brother.

Therefore, father told my older brother to bring the paper.

By a violent transformation of the first premise—Robert is one told by father to bring the paper-it may be construed as containing only two terms in a single relation. But such twisting is obviously unnatural. As it stands, the premise contains three terms and two relations, and there is no difficulty in advancing by the other premise to the proper conclusion without any change in the form of statement at all. And if we add to this premise the phrase "on the hall table," we introduce another term and another relation without adding any strain to the process of reasoning necessary to complete the deduction. EXERCISE A.-Analyze the following into terms and relations, assigning as large a part of the proposition to the relation as possible.

1. Helen is prettier than Caroline.

2. The waves of the sea are deep blue.

3. My copy of Omar Khayyám is under the shelf.

4. The world is too much with us.

modifies the phrase "with us".

5. This knife is just as sharp as that.

6. Amundsen discovered the South Pole.

7. Daniel Boone fought the Indians.

(Observe that "too much"

What the

does

B. Arrange the following syllogisms in the form necessary to identify terms and relations in each proposition:

1. I shall not all die, for love and reason are immortal.

2. Deep lies the snow in the forest, hence it will not melt quickly. 3. All trespassers will be prosecuted. Prosecution is an annoyance. Therefore all trespassers will suffer annoyance.

Section 4. THE FUNDAMENTAL STRUCTURE of the SYLLOGISM

Returning now to the syllogism as a whole in the light of syllogism the analysis of its elements, we are in a position to complete our really study of its fundamental structure. We observed that one term is common to the two premises, and that this term is the one omitted in the conclusion. That is, the other terms become. related in a certain way in the conclusion by having been successively related to this (now omitted) term in the premises. The term which thus binds the premises and enables them to yield a conclusion about the other terms is called for this reason the middle term; the others are the subject term or the predicate term, according to their position in the conclusion. The subject term is sometimes also called the minor term and the predicate. the major term, No special terminology has become technical for cases where the conclusion contains three or more terms.

But the central question which still needs to be asked is, what happens to the relations asserted in the premises as we pass to the conclusion? The answer is, of course: The relations are combined in accordance with their meanings to form a single relation in the conclusion. This is clear from the illustrative syllogisms with which we are now familiar. But in order to discover more clearly the laws according to which relations are thus combined the student must experiment with syllogisms in which the terms have been replaced by neutral symbols. Thus we avoid distraction by the terms and the rela

tions are brought into full prominence. The following will exemplify the method suggested:

If A is greater than B, and B greater than C, then A is greater than C.

If A is in tune with B, and B is in tune with C, then A is in tune with C.

If A is north of B, and B is west of C, then A is northwest of C.
If A is brother of B, and B is brother of C, then A is brother of C.
If A is father of B, and B is father of C, then A is grandfather of C.
If A is as old as B, and B is older than C, then A is older than C.
If all A is B, and all B is C, then all A is C.

If no A is B, and some C is A, then no B is C.

If all A is B, and some C is A, then some C is B.

If A explored B, and B is (identity) C, then A explored C.

If A is descendant of B, and B is descendant of C, then A is descend-
ant of C.

If A is writer of B, and B is (inclusion) C, then A is writer of
one (or some) C.

If A happened earlier than B, and B happened earlier than C, then
A happened earlier than C.

If A is B, and B is C, then A is C.

If more than half A is B, and more than half A is C, then some

B is C.

Generalizing from such a study, we may formulate the struc- Symbolic ture of the syllogism in a symbolic scheme as follows:

A (R1) B
B (R2) C

Therefore, A (RıR1) C.1

That is, B is the middle term, and since A stands to it in the

relation R1 in one premise, and B itself stands in the relation R2 to C in the other, the conclusion asserts that A stands in the relation (RR) to C, the symbol of relation here expressing the organic unity of R1 and R2 in a single relation in accordance with their meaning. Every valid syllogism of three terms must conform to this essential structure. The student will find it interesting to work out for himself a symbolism for cases where one premise contains more than two terms.

'I owe this formula to my former teacher, Professor Montague of Columbia University.

representation of

the structure of

the syl

logism