Square of Opposition: Categorical Propositions

Square of Opposition

• Analysis of relationships between categorical propositions

Four Types of Propositions

• Categorical propositions have a quantity (universal or particular) and a quality (affirmative or negative).

A Proposition (Universal Affirmative)

• An "A" proposition has universal quantity and is affirmative (AUaf).
• Example: "All dogs are animals."
• "All" determines the quantity as universal while "are" affirms the claim about the subject.

E Proposition (Universal Negative)

• An "E" proposition has universal quantity and is negative (EUne).
• Example: "All dogs are not animals."
• The phrase 'are not' denies that "all dogs" belong to the "idea of animals".

I Proposition (Particular Affirmative)

• An "I" proposition has particular quantity and is affirmative (IPaf).
• Example: "Some dogs are animals."
• The phrase 'some dog' indicates only a certain number are, and "are" affirms the claim.

O Proposition (Particular Negative)

• An "O" proposition has particular quantity and is negative (OPne).
• Example: "Some dogs are not animals."
• The phrase 'some dog' indicates only a certain number are, and "are not" denies the claim.

Relations of Propositions

• Combining propositions creates relationships affecting truth claims.
• Four Relationships arise: Contrary, Subcontrary, Subaltern, Contradiction.

Contrary Relationship (A-E)

• It exists between Universal Affirmative (A) and Universal Negative (E) propositions.
• Only one pair of contraries exists, differing only in quality.
• Both are universal propositions, but one affirms while the other denies.

Contrary Rule

• If one contrary is true, the other is false.
• If one contrary is false, the other's truth value is doubtful.

Contrary Truth Table

• If A is true, E is false; if E is false, A is doubtful.
• Truth value depends on what is initially given.

Subcontrary Relationship (I-O)

• It occurs between particular affirmative (I) and particular negative (O) propositions.
• Only one pair of subcontraries exists, differing only in quality, according to the square of oppositions.

Subcontrary Rule

• If one subcontrary is false, the other is true.
• If one subcontrary is true, the other is doubtful.

Subcontrary Truth Table

• If I is false, O is true; if I is true, O is doubtful.

Subaltern Relationship (A-I and E-O)

• The relation between Universal and Particular propositions sharing the same quality.
• Two pairs exist: (A) to (I), and (E) to (O).

Subaltern Rule

• If the universal is true, the particular is true.
• If the universal is false, the particular is doubtful.
• If the particular is true, the universal is doubtful.
• If the particular is false, the universal is false

Subaltern Truth Table (Universal to Particular)

• If A is true, I is true; if A is false, I is doubtful.

Subaltern Truth Table (Particular to Universal)

• If I is true, A is doubtful; if I is false, A is false.

Contradictory Relationship (A-O and E-I)

• The relation between universal and particular propositions having different qualities and quantity.
• Two pairs exist: Universal Affirmative (A) to Particular Negative (O), and Universal Negative (E) to Particular Affirmative (I).

Contradictory Rule

• One member of each pair denies the other.
• If the universal is true, the particular is false, and vice versa.

Contradictory Truth Table

• If A is true, O is false; if E is false, I is true.
• If U is true, P is false and and If P is false and U is true.

Exercises

• Practice determining the proposition, its relationship, and its truth value.