Explain Square of oppositions with some examples

The Square of Opposition is a diagram representing the different ways in which propositions can be opposed to one another in terms of their truth values.

It originated in Aristotle’s works on logic and was later developed by medieval logicians. The square is composed of four types of categorical propositions: A (universal affirmative), E (universal negative), I (particular affirmative), and O (particular negative). These propositions are related to each other in specific ways: contradiction, contrariety, subcontrariety, and subalternation.

Types of Propositions

1. A (Universal Affirmative): All S are P.
2. E (Universal Negative): No S are P.
3. I (Particular Affirmative): Some S are P.
4. O (Particular Negative): Some S are not P.

Relationships in the Square of Opposition

1. Contradiction: Propositions are contradictories if they cannot both be true and cannot both be false. (A and O, E and I)
2. Contrariety: Propositions are contraries if they cannot both be true but can both be false. (A and E)
3. Subcontrariety: Propositions are subcontraries if they cannot both be false but can both be true. (I and O)
4. Subalternation: This refers to the relationship between a universal proposition and its corresponding particular proposition. If the universal proposition is true, the particular proposition must also be true. (A to I, E to O)

The Square of Opposition Diagram

       A (All S are P)      ----contrariety----       E (No S are P)
          |                                            |
     subalternation                             subalternation
          |                                            |
          I (Some S are P)  ----subcontrariety----      O (Some S are not P)

Examples

Let’s take an example with the subject “dogs” (S) and the predicate “mammals” (P):

1. A: All dogs are mammals.
2. E: No dogs are mammals.
3. I: Some dogs are mammals.
4. O: Some dogs are not mammals.

Contradiction

• A and O: “All dogs are mammals” (A) contradicts “Some dogs are not mammals” (O). Both cannot be true at the same time, and both cannot be false at the same time.
• E and I: “No dogs are mammals” (E) contradicts “Some dogs are mammals” (I). Both cannot be true at the same time, and both cannot be false at the same time.

Contrariety

• A and E: “All dogs are mammals” (A) and “No dogs are mammals” (E) cannot both be true, but they can both be false. If one is true, the other must be false.

Subcontrariety

• I and O: “Some dogs are mammals” (I) and “Some dogs are not mammals” (O) cannot both be false. At least one of them must be true. Both can be true simultaneously.

Subalternation

• A to I: If “All dogs are mammals” (A) is true, then “Some dogs are mammals” (I) must also be true.
• E to O: If “No dogs are mammals” (E) is true, then “Some dogs are not mammals” (O) must also be true.
• Conversely, if “Some dogs are mammals” (I) is false, then “All dogs are mammals” (A) must also be false.
• If “Some dogs are not mammals” (O) is false, then “No dogs are mammals” (E) must also be false.

Practical Application

The Square of Opposition can be used to analyze arguments and assess the logical relationships between different propositions. For instance, if you know that a universal statement (A or E) is true, you can deduce the truth of the corresponding particular statement (I or O) using subalternation. If you know two statements are contradictory, you can determine that if one is true, the other must be false.

In summary, the Square of Opposition is a useful tool for understanding the logical relationships between different types of propositions, enabling clearer and more precise reasoning.