Unit 2: Immediate Inference and Square of Opposition

Immediate Inference

Immediate inference is a form of deduction where the conclusion is drawn directly from a single premise without the need for a middle term. This process relies on changing the form of the proposition while attempting to preserve its truth value.

1. Conversion

Conversion is the process of interchanging the subject and predicate terms of a categorical proposition.

• A Proposition (All S is P): Converts to Some P is S (Conversion by Limitation).
• E Proposition (No S is P): Converts to No P is S (Simple Conversion).
• I Proposition (Some S is P): Converts to Some P is S (Simple Conversion).
• O Proposition (Some S is not P): Cannot be validly converted.

2. Obversion

Obversion involves changing the quality of the proposition (affirmative to negative or vice versa) and replacing the predicate term with its complement (contradictory term).

• A: No S is non-P.
• E: All S is non-P.
• I: Some S is not non-P.
• O: Some S is non-P.

3. Contraposition

Contraposition involves replacing the subject with the complement of the predicate and the predicate with the complement of the subject.

• A: All non-P is non-S.
• E: Some non-P is not non-S (Contraposition by Limitation).
• O: Some non-P is not non-S.
• I: Cannot be validly contraposed.

Square of Opposition

The Square of Opposition is a diagrammatic representation of the logical relationships between the four standard-form categorical propositions (A, E, I, O) when they share the same subject and predicate.

1. Traditional (Aristotelian) Square

This traditional model assumes Existential Import—meaning it assumes the subjects of universal propositions actually exist.

• Contradictories (A & O; E & I): One must be true and the other must be false.
• Contraries (A & E): Both cannot be true, but both can be false.
• Subcontraries (I & O): Both cannot be false, but both can be true.
• Subalternation (A to I; E to O): If the universal is true, the particular must be true.

2. Modern (Boolean) Square

George Boole rejected the assumption of existential import for universal propositions. In the Boolean view, universal propositions (A and E) are hypothetical.

• The only relationship that remains valid in the Boolean Square is Contradiction.
• Contraries, Subcontraries, and Subalternation are considered logically invalid in modern formal logic.