Unit 4: Mediate Inference (Syllogism)

Table of Contents

Mediate Inference: Categorical Syllogism

As defined in Unit 3, a Mediate Inference is one where the conclusion is drawn from two or more premises.
The most famous type is the Categorical Syllogism.

A Categorical Syllogism is a deductive argument consisting of three categorical propositions (two premises and one conclusion) that together contain exactly three terms.

This is the classic "All men are mortal..." argument structure. Its purpose is to link two terms (S and P) by "mediating" them through a third, common term (M).

Structure of a Syllogism

Let's analyze a standard-form example:

  1. All mammals are animals.
  2. All dogs are mammals.
  3. Therefore, all dogs are animals.

The Three Terms

Each of the three terms appears exactly twice in the syllogism.

  1. Major Term (P): The predicate term of the conclusion. (In example: "animals")
  2. Minor Term (S): The subject term of the conclusion. (In example: "dogs")
  3. Middle Term (M): The term that appears in both premises but not in the conclusion. It is the "mediating" term. (In example: "mammals")

The Three Propositions

Standard Form

A syllogism is in "Standard Form" when:

  1. The Major Premise is listed first.
  2. The Minor Premise is listed second.
  3. The Conclusion is listed last.

Example (Standard Form):

Mood of Syllogism

The Mood of a syllogism is a statement of the proposition types (A, E, I, O) of its major premise, minor premise, and conclusion, in that order.

Example 1:

The Mood of this syllogism is A-A-A.

Example 2:

The Mood of this syllogism is E-I-O.

There are 64 possible moods (4 x 4 x 4), but only a few are valid.

Figure of Syllogism

The Figure of a syllogism is determined by the position of the Middle Term (M) in the two premises.

There are only four possible figures. You must memorize these patterns.

Figure 1 Figure 2 Figure 3 Figure 4
M - P
S - M
-------
S - P 		P - M
S - M
-------
S - P 		M - P
M - S
-------
S - P 		P - M
M - S
-------
S - P
(M is Subject in major, Predicate in minor) (M is Predicate in both) (M is Subject in both) (M is Predicate in major, Subject in minor)
[Diagram Placeholder: The Four Figures]

A visual diagram showing the 'Z', 'backward C', 'C', and 'backward Z' patterns of the Middle Term (M).

Full Description: The "full name" of a syllogism combines its Mood and Figure.
Our first example: "All M are P; All S are M; So, All S are P" has the mood A-A-A. The middle term M is in the position of Figure 1.
Therefore, its full logical name is AAA-1.

Our second example: "No M are P; Some S are M; So, Some S are not P" has the mood E-I-O. The middle term M is in Figure 1.
Therefore, its name is EIO-1.

Valid Moods

There are 256 possible syllogistic forms (64 moods x 4 figures). However, only a very small number are valid.

The 15 Unconditionally Valid Forms (valid from both Aristotelian and modern Boolean perspectives) are:

The 9 Conditionally Valid Forms (valid only from the Aristotelian perspective, which assumes existential import) are:

This gives a total of 24 valid forms in traditional logic.

Mnemonic for Valid Moods (Figure 1):

Medieval scholars created a poem to memorize the valid moods. The names for Figure 1 are the most famous:

BARBARA, CELARENT, DARII, FERIO

This tells you the valid moods for Figure 1 are: AAA, EAE, AII, EIO.
(You are not required to memorize all 24, but knowing the Figure 1 moods is very helpful.)