Notes: Square of Opposition | Logical Reasoning for UGC NET PDF Download

What is Square of Opposition? 

The square of opposition is a conceptual tool used in categorical logic to visually represent the logical relationships between certain types of propositions based on their form. 

This framework helps in understanding how different propositions relate to each other in terms of truth and falsehood.

Basic Propositions

The square of opposition is built around four basic types of propositions, each occupying a corner of the square:

1. A Propositions (Universal Affirmatives):

   • Form: All S are P.
   • Example: "All dogs are mammals."

2. E Propositions (Universal Negatives):

   • Form: No S are P.
   • Example: "No dogs are reptiles."

3. I Propositions (Particular Affirmatives):

   • Form: Some S are P.
   • Example: "Some dogs are friendly."

4. O Propositions (Particular Negatives):

   • Form: Some S are not P.
   • Example: "Some dogs are not friendly."

Relationships in the Square of Opposition

1. Contradictories:

   • Definition: Two propositions are contradictories if they cannot both be true or both be false at the same time. If one is true, the other must be false.
   • Examples:
     • A and O propositions: "All dogs are mammals" (A) and "Some dogs are not mammals" (O).
     • E and I propositions: "No dogs are reptiles" (E) and "Some dogs are reptiles" (I).

2. Contraries:

   • Definition: Two propositions are contraries if they cannot both be true at the same time, but they can both be false.
   • Examples:
     • A and E propositions: "All dogs are mammals" (A) and "No dogs are mammals" (E). Both cannot be true simultaneously, but both can be false if some dogs are and some dogs are not mammals.

3. Sub-Contraries:

   • Definition: Two propositions are sub-contraries if they cannot both be false at the same time, but they can both be true.
   • Examples:
     • I and O propositions: "Some dogs are friendly" (I) and "Some dogs are not friendly" (O). Both statements can be true, but they cannot both be false simultaneously.

4. Sub-Alternation:

   • Definition: This relationship involves a universal proposition (A or E) and its corresponding particular proposition (I or O) with the same subject and predicate. The truth of the universal proposition implies the truth of the particular proposition.
   • Examples:
     • From A to I: If "All dogs are mammals" (A) is true, then "Some dogs are mammals" (I) must also be true.
     • From E to O: If "No dogs are reptiles" (E) is true, then "Some dogs are not reptiles" (O) must also be true.

Immediate Inferences

Immediate inferences can be drawn directly from one proposition to another based on the square of opposition. Here are some examples:

1. From A Proposition (All S are P):

   • If true:
     • E (No S are P) is false.
     • I (Some S are P) is true.
     • O (Some S are not P) is false.
   • If false:
     • O (Some S are not P) is true.
     • E and I are undetermined.

2. From E Proposition (No S are P):

   • If true:
     • A (All S are P) is false.
     • I (Some S are P) is false.
     • O (Some S are not P) is true.
   • If false:
     • I (Some S are P) is true.
     • A and O are undetermined.

3. From I Proposition (Some S are P):

   • If true:
     • E (No S are P) is false.
     • A and O are undetermined.
   • If false:
     • A (All S are P) is false.
     • E (No S are P) is true.
     • O (Some S are not P) is true.

4. From O Proposition (Some S are not P):

   • If true:
     • A (All S are P) is false.
     • E and I are undetermined.
   • If false:
     • A (All S are P) is true.
     • E (No S are P) is false.
     • I (Some S are P) is true.

The square of opposition is a vital tool in classical logic, allowing for a clear understanding of how different types of categorical propositions relate to one another. By recognizing the relationships of contraries, contradictories, sub-contraries, and sub-alternations, one can determine the truth or falsehood of related propositions, facilitating logical reasoning and argumentation.