A
categorical syllogism consists of three categorical propositions—two premises
and a conclusion. To be tested for validity, it must be stated in standard
form. Standard-form categorical syllogisms can be sorted in terms of mood and
figure into 256 possible arrangements, only some of which pass muster as valid
in the system. To be valid (that is, for its conclusion to “follow” from its
premises), it must satisfy certain formal restrictions on the number of terms that
may occur in it; on the positions in which those terms may occur; on the
“distribution” of the middle, major, and minor terms; and on the occurrence of
negative statements

**I**

A
categorical syllogism consists of three standard-form categorical statements:
two serving as premises and the third as the conclusion. Each of the individual statements must be
unambiguous. Example: “All
undergraduates aren’t philosophy majors” might be either a universal negative
claim that “No undergraduate students are philosophy majors” or a much less
sweeping particular negative that “Some undergraduates are not philosophy
majors.”

To be in
standard form, a syllogism must have exactly three terms, each of which occurs
in two of its three propositions.

1. The predicate term of the conclusion (called the major term) will also occur as either the subject or predicate
of the premise that is stated first (hence, the major premise).

2. The subject term of the conclusion (called the minor term) will also occur as either the subject or predicate
of the premise that is stated second (hence, the minor premise).

3. The third term (called the middle term) will occur in both of the premises (it can be the
subject or the predicate term of either one) and will not occur in the
conclusion.

Major
premise: middle term,
major term (in either order)

Minor
premise: middle term,
minor term (in either order)

Conclusion: minor
term, major term (in this order only)

Some
nonstandard-form syllogisms can be reduced to standard form by reducing their
number of terms to three—for example, if the syllogism seems to have more than
three terms because of synonymy or because of the use of complementary terms—by
manipulating their constituent propositions by means of immediate inferences,
or by placing their premises and conclusion in proper order.

Example: The argument “All Athenians are
mortal because they are all Greeks and no Greeks are immortal” can be reduced
to standard form by obverting “No Greeks are immortal” to “All Greeks are
mortal,” by specifying the reference of “they,” and by placing the statements
in proper order (major premise, minor premise, conclusion):

All Greeks are mortal.

All Athenians are Greeks.

Therefore, all Athenians are mortal.

Note: A syllogism with irreducibly more than three
terms is not valid in this system.

**II**

For a
standard-form categorical syllogism to be valid, it must comply with rules that
restrict its structure in terms of (a) the “distribution” of the middle, major,
and minor terms and (b) the occurrence of negative statements.

Restrictions
on distribution: The distribution of a term has to do with whether or not the
proposition in which it occurs conveys some information about all, or only part
of, the class it names. No term can be distributed
in the conclusion that is not distributed in the premise in which it occurs;
that is, a conclusion cannot say more than the premises support.

1. If the
major term is distributed in the conclusion but not in the major premise, the
argument fails due to “illicit process of the major term.” If the minor term is
distributed in the conclusion but not in the minor premise, the argument fails
due to “illicit process of the minor term.”

2. The
middle term must be distributed in at least one of the two premises. If it is
not, the argument fails due to “undistributed middle.”

There are
also restrictions on negative propositions.

1.
If one of the premises is negative, the conclusion must be negative.

2.
If both of the premises are negative, no valid conclusion can be drawn.

Again,
there must be exactly three terms and exactly three propositions, not four or
more.

**III**

Standard-form
categorical syllogisms display both mood and figure. The mood of a syllogism is captured by listing
the quality/quantity of each of its propositions in order (AAA, AEO, EIA, and
so on).

The figure
of a syllogism depends on where its middle term resides.

Figure One: The middle term (M) is
the subject of the major premise and predicate of the minor. Figure Two: The middle term (M) is the
predicate of both premises.

Figure Three: The middle term (M) is
the subject of both premises.

Figure Four: The middle term (M) is
the predicate of the major premise and subject of the minor.

In all four
figures, the subject of the conclusion (S) appears in the minor premise (the
second premise) of the syllogism and is known as the minor term. The predicate
of the conclusion (P) appears in the major premise (the first premise) of the
syllogism and is known as the major term. C. There are 256 possible moods and
figures for standard form syllogisms, from AAA-1 to OOO-4. Very few are valid.

Example:

AAA-1
(Valid).

All Greeks
are mortal.

All
Athenians are Greeks

Therefore,
all Athenians are mortal.

Example:

AAA-2
(Invalid, undistributed middle).

All
Rastafarians are bearded.

All billy
goats are bearded.

Therefore, all billy goats are Rastafarians.

By
Aristotelian standards, 24 of the 256 possible categorical syllogisms are
valid.

Problems
with null classes remain, as will be seen.

**[Courtesy: Professor James Hall]**