Logical Reasoning: Square of Opposition

Square of Opposition Relationships

The square of opposition visually represents the logical connections between four types of categorical propositions:

• A: Universal Affirmative (All S are P)
• E: Universal Negative (No S are P)
• I: Particular Affirmative (Some S are P)
• O: Particular Negative (Some S are not P)

Key Relationships & Conditions

• Contradictory:
  • Statements: A and O, E and I
  • Condition: If one is true, the other is always false. If one is false, the other is always true.
• Contrary:
  • Statements: A and E
  • Condition: Both cannot be true at the same time. However, both can be false.
• Subcontrary:
  • Statements: I and O
  • Condition: Both cannot be false at the same time. However, both can be true.
• Subalternation:
  • Statements: A and I, E and O
  • Condition:
    • If A is true, then I is also true.
    • If E is true, then O is also true.
    • If I is false, then A is also false.
    • If O is false, then E is also false.

Examples:

Let’s consider the following statements about “students” and “athletes”

• A: All students are athletes.
• E: No students are athletes.
• I: Some students are athletes.
• O: Some students are not athletes.

Examples

1. Contradictory (A and O)

• If “All students are athletes” (A) is true, then “Some students are not athletes” (O) is false.
• If “Some students are not athletes” (O) is true, then “All students are athletes” (A) is false.

2. Contradictory (E and I)

• If “No students are athletes” (E) is true, then “Some students are athletes” (I) is false.
• If “Some students are athletes” (I) is true, then “No students are athletes” (E) is false.

3. Contrary (A and E)

• “All students are athletes” (A) and “No students are athletes” (E) cannot both be true.
• It is possible for both A and E to be false (e.g., if some students are athletes and some are not).

4. Subcontrary (I and O)

• “Some students are athletes” (I) and “Some students are not athletes” (O) cannot both be false.
• It is possible for both I and O to be true.

5. Subalternation (A and I)

• If “All students are athletes” (A) is true, then “Some students are athletes” (I) is also true.
• If “Some students are athletes” (I) is false, then “All students are athletes” (A) is also false.

6. Subalternation (E and O)

• If “No students are athletes” (E) is true, then “Some students are not athletes” (O) is also true.
• If “Some students are not athletes” (O) is false, then “No students are athletes” (E) is also false.