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 standard-form 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 AEE-2. 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: AII-3

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.