Categorical
Syllogisms
A syllogism is a deductive argument
consisting of two premises and one conclusion. A categorical syllogism
is a syllogism made up of categorical propositions and containing three terms,
each appearing twice in distinct propositions.
All soldiers are patriots
No traitors are patriots
Therefore, no traitors are soldiers
Each of the three terms is named. The major
term is the predicate of the conclusion. The minor term is the
subject of the conclusion. The middle term, which provides the middle ground
or connection between the two premises, occurs once in each premise and not at
all in the conclusion.
Each of the three premises is named. The major
premise is the one that contains the major term. The minor premise
contains the minor term. In a standard form syllogism, the major premise
comes first, the minor premise second, and the conclusion last.
Thus, a syllogism is said to be in standard form when:

1.
All three
statements are standardform categorical propositions. 2.
The two
occurrences of each term are identical. 3.
Each term is
used in the same sense throughout the argument 4.
The major
premise is listed first, the minor second, and the conclusion last. 
Once a categorical syllogism is put into
standard form, its validity or invalidity can be determined through mere
inspection of that form. The form consists of two factors: mood and figure.
The mood of a syllogism consists of the
letter names of the propositions that make it up. The above example's mood is:
AEE. The figure of a syllogism is determined by the placement of the
middle terms. (Note: In the following diagram, M=Middle term, S=Minor term,
P=Major term.)
Figure 1 M P S M S P 
Figure 2 P M S M S P 
Figure 3 M P M S S P 
Figure 4 P M M S S P 
Our original example, then, is AEE2. Note
that there are 256 different possible categorical syllogisms. By Aristotelian standards,
24 are valid. By Boolean standards, 15 are valid.
Unconditionally
Valid Moods and Figures
Figure
1 AAA EAE AII EIO 
Figure
2 EAE AEE EIO AOO 
Figure
3 IAI AII OAO EIO 
Figure
4 AEE IAI EIO 
Conditionally
Valid Moods and Figures
Figure
1 AAI EAO 
Figure
2 AEO EAO 
Figure
3

Figure
4 AEO 
Condition S exists 


AAI EAO 
EAO 
M exists 



AAI 
P exists 
Reconstructing Syllogisms from given Mood
and Figure:
For Example: AII3
First, use the mood to determine the skeleton
of the form:
A: All _______ are ________.
I: Some _______are ________.
I: Some _______are ________.
Then use the figure to insert the middle
terms:
A: All M are ________.
I: Some M are
________.
I: Some _______are ________.
Finally, supply the major and minor terms to
fill in the remainder, using S for the minor term and P for the major term:
A: All M are P.
I: Some M are S.
I: Some S are P.