Unit 3: Mediate Inference and Syllogism

Understanding Mediate Inference

Mediate inference is a type of deductive reasoning where the conclusion is derived from two or more premises. Unlike immediate inference, which moves from one premise directly to a conclusion, mediate inference requires a "middle" ground to bridge the premises. The most common form of mediate inference in classical logic is the Categorical Syllogism.

Categorical Syllogism: Structure

A categorical syllogism is a deductive argument consisting of three categorical propositions (two premises and one conclusion) that together contain exactly three terms, each of which occurs in exactly two of the constituent propositions.

The Three Terms

• Major Term (P): The predicate of the conclusion.
• Minor Term (S): The subject of the conclusion.
• Middle Term (M): The term that appears in both premises but not in the conclusion.

The Two Premises

• Major Premise: The premise containing the Major Term.
• Minor Premise: The premise containing the Minor Term.

Figure and Mood of Syllogisms

The logical form of a syllogism is determined by its Mood and its Figure.

1. Mood

The mood is determined by the types (A, E, I, or O) of categorical propositions it contains. For example, a syllogism with an A-major premise, an E-minor premise, and an E-conclusion has the mood AEE.

2. Figure

The figure is determined by the position of the Middle Term (M) in the premises. There are four possible figures:

• First Figure: M is the subject of the major premise and the predicate of the minor premise.
• Second Figure: M is the predicate of both premises.
• Third Figure: M is the subject of both premises.
• Fourth Figure: M is the predicate of the major premise and the subject of the minor premise.

Copi’s Six Rules for Testing Validity

Irving Copi formulated six essential rules to determine if a categorical syllogism is valid. If a syllogism violates any of these, it is invalid.

1. Rule 1: A valid categorical syllogism must contain exactly three terms, each used in the same sense throughout the argument.
2. Rule 2: In a valid categorical syllogism, the middle term must be distributed in at least one premise.
3. Rule 3: In a valid categorical syllogism, if a term is distributed in the conclusion, it must be distributed in the premises.
4. Rule 4: No categorical syllogism with two negative premises is valid.
5. Rule 5: If either premise is negative, the conclusion must be negative.
6. Rule 6: From two universal premises, no particular conclusion can be drawn (Existential Fallacy in modern logic).

Venn Diagram Techniques

Venn diagrams provide a graphical method to test the validity of syllogisms by representing the relationships between the three classes (S, P, and M).

Representation Steps:

• Draw three overlapping circles representing S, P, and M.
• Diagram the Universal premises first (shading areas).
• Diagram Particular premises (placing an 'X').
• The Test: After diagramming both premises, if the conclusion is already represented, the syllogism is valid. If you must add more to represent the conclusion, it is invalid.