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Categorical Syllogisms from this text. It may

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1 Standard-Form Categorical Syllogisms (Belvedere by

M.C. Escher)

2 The Formal Nature of Syllogistic Argument

3 Venn Diagram Technique for Testing Syllogisms

4 Syllogistic Rules and Syllogistic Fallacies

5 Exposition of the Fifteen Valid Forms of the Categorical Syllogism

Appendix: Deduction of the Fifteen Valid Forms

of the Categorical Syllogism

1 Standard-Form Categorical Syllogisms

Categorical propositions can now be used in more extended reasoning. Argu-

ments that rely on A, E, I, and O propositions commonly have two categorical

propositions as premises and one categorical proposition as a conclusion. Such

arguments are called syllogisms; a syllogismis a deductive argument in which a

conclusion is inferred from two premises.

The syllogisms with which we are concerned here are called categorical be-

cause they are arguments based on the relations of classes, or categories—relations

that are expressed by the categorical propositions with which we are familiar.

Syllogism

More formally, we define a categorical syllogismas a deductive argument con-

Anydeductive argument

sisting of three categorical propositions that together contain exactly three terms, in which a conclusion is

each of which occurs in exactly two of the constituent propositions. inferred from two

premises.

Syllogisms are very common, very clear, and readily testable. The system of

categorical syllogisms that we will explore is powerful and deep. The seventeenth- Categorical syllogism

century philosopher and mathematician Gottfried Leibniz said, of the invention A deductive argument

of the form of syllogisms, that it was “one of the most beautiful and also one of consisting of three

categorical propositions

the most important made by the human mind.” Syllogisms are the workhorse

that contain exactly

arguments with which deductive logic, as traditionally practiced, has been made three terms, each of

effective in writing and in controversy. which occurs in exactly

two of the propositions.

It will be convenient to have an example to use as we discuss the parts and

features of the syllogism. Here is a valid standard-form categorical syllogism Standard form

that we shall use as an illustration: Theform in which a

syllogism is said to be

No heroes are cowards. when its premises and

Some soldiers are cowards. conclusion are all

standard-form

Therefore some soldiers are not heroes.

categorical propositions

To analyze such an argument accurately, it needs to be in standard form. Acate- (A, E, I, or O) and are

arranged in standard

gorical syllogismis said to be in standard form(as the above example is) when

order (major premise,

two things are true of it: (1) its premises and its conclusion are all standard-form

then minor premise,

categorical propositions (A, E, I, or O); and (2) those propositions are arranged in then conclusion).

From Chapter 6 of Introduction to Logic, Fourteenth Edition. Irving M. Copi, Carl Cohen, Kenneth McMahon.

Copyright © 2011 by Pearson Education, Inc. Published by Pearson Prentice Hall. All rights reserved.

Categorical Syllogisms

a specified standard order. The importance of this standard form will become evi-

dent when we turn to the task of testing the validity of syllogisms.

To explain the order of the premises that is required to put any syllogism into

standard form, we need the logical namesof the premisesof the syllogism, and the

names of the termsof the syllogism, and we must understand why those names—

very useful and very important—are assigned to them. This is the next essential

step in our analysis of categorical syllogisms. In this chapter, for the sake of brevi-

ty, we will refer to categorical syllogisms simply as “syllogisms,”.

A. Terms of the Syllogism: Major, Minor, and Middle

The three categorical propositions in our example argument above contain exact-

ly three terms: heroes, soldiers, and cowards. To identify the terms by name, we look

to the conclusion of the syllogism, which of course contains exactly two terms.

The conclusion in our sample is an Oproposition, “Some soldiers are not heroes.”

The term that occurs as the predicate of the conclusion (“heroes,” in this case) is

called the major termof the syllogism. The term that occurs as the subjectof the

conclusion (“soldiers,” in this case) is called the minor termof the syllogism. The

third term of the syllogism (“cowards,” in this case), which never occurs in the

Major term conclusion but always appears in both premises, is called the middle term.

The term that occurs as The premises of a syllogism also have names. Each premise is named after

the predicate term of the the term that appears both in it and in the conclusion. The major term and the

conclusion in a

minor term must each occur in a different premise. The premise containing the

standard-form

categorical syllogism. major term is called the major premise. In the example, “heroes” is the major

term, so the premise containing “heroes”—“No heroes are cowards”—is the

Minor term

major premise. It is the major premise not because it appears first, but only be-

The term that occurs as

cause it is the premise that contains the major term; it would be the major

the subject term of the

conclusion in a premise no matter in what order the premises were written.

standard-form The premise containing the minor term is called the minor premise. In the

categorical syllogism.

example, “soldiers” is the minor term, so the premise containing “soldiers”—

Middle term

“Some soldiers are cowards”—is the minor premise. It is the minor premise not

In a standard-form

because of its position, but because it is the premise that contains the minor term.

categorical syllogism

(which must contain

exactly three terms), the

term that appears in

overview

both premises but does

not appear in the

conclusion. The Parts of a Standard-Form Categorical Syllogism

Major premise Major Term The predicate term of the conclusion.

In a standard-form

categorical syllogism, Minor Term The subject term of the conclusion.

the premise that

Middle Term The term that appears in both premises but not in the

contains the major term.

conclusion.

Minor premise

In a standard-form Major Premise The premise containing the major term.

categorical syllogism,

Minor Premise The premise containing the minor term.

the premise that

contains the minor term.

(cid:21)(cid:20)(cid:21)

Categorical Syllogisms

Asyllogism is in standard form, we said, when its premises are arranged in a

specified standard order. Now we can state that order: In a standard-form syllo-

gism, the major premise is always stated first, the minor premise second, and the conclu-

sion last. The reason for the importance of this order will soon become clear.

B. The Mood of the Syllogism

Every syllogism has a mood. The moodof a syllogism is determined by the types

(A, E, I, or O) of standard-form categorical propositions it contains. The mood of

the syllogism is therefore represented by three letters, and those three letters are

always given in standard-form order. That is, the first letter names the type of the

syllogism’s major premise; the second letter names the type of the syllogism’s

minor premise; the third letter names the type of the syllogism’s conclusion. In

our example syllogism, the major premise (“No heroes are cowards”) is an E

proposition; the minor premise (“Some soldiers are cowards”) is an Iproposition;

the conclusion (“Some soldiers are not heroes”) is an O proposition. Therefore

the mood of this syllogism is EIO.

C. The Figure of the Syllogism

The mood of a standard-form syllogism is not enough, by itself, to characterize

its logical form. This can be shown by comparing two syllogisms, Aand B, with

the same mood, which are logically very different.

Major Term Middle Term

A. All great scientists are college graduates.

Minor Term Middle Term

Some professional athletes are college graduates.

Minor Term Major Term

Therefore some professional athletes are great scientists.

Middle Term Major Term

B. All artists are egotists. Mood

Middle Term Minor Term A characterization of

Some artists are paupers. categorical syllogisms,

determined by the forms

Minor Term Major Term

of the standard-form

Therefore some paupers are egotists.

categorical propositions it

contains. Since there are

just four forms of

Both of these are of mood AII, but one of them is valid and the other is not.

propositions, A,E,I, and

The difference in their forms can be shown most clearly if we display their log- O,and each syllogism

ical “skeletons” by abbreviating the minor terms as S(subject of the conclusion), contains exactly three

the major terms as P(predicate of the conclusion), and the middle terms as M. such propositions, there

are exactly 64 moods,

Using the three-dot symbol “‹” for “therefore,” we get these skeletons:

each mood identified by

the three letters of its

A. All Pis M. B. All Mis P.

constituent propositions,

Some S is M. Some M is S. AAA,AAI,AAE,and so

‹ Some Sis P. ‹ Some Sis P. on, to OOO.

(cid:21)(cid:20)(cid:22)

Categorical Syllogisms

These are very different. In the one labeled A, the middle term, M, is the predi-

cate term of both premises; but in the one labeled B, the middle term, M, is the

subject term of both premises. Syllogism B will be seen to be a valid argument;

syllogism A, on the other hand, is invalid.

These examples show that although the form of a syllogism is partially de-

scribed by its mood (AII in both of these cases), syllogisms that have the same

mood may differ importantly in their forms, depending on the relative positions

of their middle terms. To describe the form of a syllogism completely we must

state its mood(the three letters of its three propositions) andits figure—where by

figurewe mean the position of the middle term in its premises.

Syllogisms can have four—and only four—possible different figures:

1. The middle term may be the subject term of the major premise and the

predicate term of the minor premise; or

2. The middle term may be the predicate term of both premises; or

3. The middle term may be the subject term of both premises; or

4. The middle term may be the predicate term of the major premise and

the subject term of the minor premise.

These different possible positions of the middle term constitute the first,

second, third, and fourth figures, respectively. Every syllogism must have one

or another of these four figures. The characters of these figures may be visual-

ized more readily when the figures are schematized as in the following array,

in which reference to mood is suppressed and the quantifiers and copulas are

not shown—but the relative positions of the terms of the syllogism are

brought out:

M — P P — M M — P P — M

S — M S — M M — S M — S

∴S — P ∴S — P ∴S — P ∴S — P

First Figure Second Figure Third Figure Fourth Figure

Any standard-form syllogism is completely described when we specify its

mood and its figure. The syllogism we have been using as an example is in the

second figure; “cowards,” the middle term, is the predicate term of both premis-

es. Its mood, as we pointed out, is EIO. So it is completely described as being a

syllogism of the form EIO–2. It is a valid syllogism, as we noted; every valid syl-

logistic form, as we shall see, has it own name. The name of this form, EIO–2, is

Festino. We say of this syllogism that it is “in Festino.”

Figure

The position of the Here is another example:

middle term in the

No Mis P.

premises of a standard-

form categorical All S is M.

syllogism. ‹ No Sis P.

(cid:21)(cid:20)(cid:23)

Categorical Syllogisms

This syllogism is in the first figure (its middle term is the subject of the major

premise and the predicate of the minor premise); its mood is EAE. So we may

characterize it completely as EAE–1, a form whose unique name is Celarent. Any

syllogism of this form is “in Celarent,” just as any syllogism of the earlier form is

“in Festino.” Because Celarent(EAE–1) and Festino(EIO–2) are known to be valid

forms, we may conclude that whenever we encounter an argument in one of

these forms, it too is valid.

With these analytical tools we can identify every possible categorical syllo-

gism by mood and figure. If we were to list all the possible moods, beginning

with AAA, AAE, AAI, AAO, AEA, AEE, . . ., and so on, continuing until every

possibility had been named, we would eventually (upon reaching OOO) have

enumerated sixty-four possible moods. Each mood can occur in each of the four

figures; 4 × 64 = 256. It is certain, therefore, that there are exactly 256 distinct

forms that standard-form syllogisms may assume.

Of these 256 possible forms, as we shall see, only a few are valid forms. Each

of those valid forms has a unique name, as will be explained.

EXERCISES

Rewrite each of the following syllogisms in standard form, and name its mood

and figure. (Procedure:first, identify the conclusion; second, note its predicate

term, which is the major term of the syllogism; third, identify the major prem-

ise, which is the premise containing the major term; fourth, verify that the other

premise is the minor premise by checking to see that it contains the minor term,

which is the subject term of the conclusion; fifth, rewrite the argument in stan-

dard form—major premise first, minor premise second, conclusion last; sixth,

name the mood and figure of the syllogism.)

EXAMPLE

1. No nuclear-powered submarines are commercial vessels, so no war-

ships are commercial vessels, because all nuclear-powered submarines

are warships.

SOLUTION

Step 1. The conclusion is “No warships are commercial vessels.”

Step 2. “Commercial vessels” is the predicate term of this conclusion

and is therefore the major term of the syllogism.

Step 3. The major premise, the premise that contains this term, is “No

nuclear-powered submarines are commercial vessels.”

Step 4. The remaining premise, “All nuclear-powered submarines are

warships,” is indeed the minor premise, because it does contain

the subject term of the conclusion, “warships.”

(cid:21)(cid:20)(cid:24)