Understanding the Square of Opposition and Logical Inferences

The question asks us to use the principles of the Square of Opposition to determine which statements can be immediately inferred to be false, given that the statement "All apes are mammals" is true. This statement is a Universal Affirmative (A) statement of the form "All S are P", where S is "apes" and P is "mammals".

Categorical Statements and the Square of Opposition

In logic, categorical statements relate two categories (or classes). There are four standard types:

• A (Universal Affirmative): All S are P (e.g., All dogs are mammals). Represented as $\forall x (Sx \rightarrow Px)$.
• E (Universal Negative): No S are P (e.g., No dogs are cats). Represented as $\forall x (Sx \rightarrow \neg Px)$.
• I (Particular Affirmative): Some S are P (e.g., Some dogs are black). Represented as $\exists x (Sx \land Px)$.
• O (Particular Negative): Some S are not P (e.g., Some dogs are not black). Represented as $\exists x (Sx \land \neg Px)$.

The Square of Opposition illustrates the logical relationships between these four types of statements when they share the same subject (S) and predicate (P) terms.

Analyzing the Given True Statement

We are given that the statement "All apes are mammals" (A statement) is True.

Using the Square of Opposition, we can infer the truth values of the corresponding E, I, and O statements:

• A is True: "All apes are mammals".
• Relationship with E (Contrary): If A is true, then E ("No apes are mammals") must be False. They cannot both be true.
• Relationship with I (Subalternation): If A is true, then I ("Some apes are mammals") must be True. Truth flows from A to I.
• Relationship with O (Contradictory): If A is true, then O ("Some apes are not mammals") must be False. They have opposite truth values.

Evaluating the Options

Now let's look at the given options and their truth values based on our inferences:

• A. "No apes are mammals." This is an E statement. We inferred that if A is true, the corresponding E statement is False.
• B. "Some apes are mammals." This is an I statement. We inferred that if A is true, the corresponding I statement is True.
• C. "Some mammals are apes." This statement has the subject and predicate swapped compared to the original. This is the converse of "All apes are mammals". The conversion of an A statement ("All S are P") is an I statement ("Some P are S"). If the original A statement ("All apes are mammals") is true, its converse I statement ("Some mammals are apes") is also generally considered True (assuming the categories S and P are not empty).
• D. "Some apes are not mammals." This is an O statement. We inferred that if A is true, the corresponding O statement is False.

Identifying False Statements

Based on our analysis using the Square of Opposition and conversion rules, the statements that can be immediately inferred to be false are:

• A. "No apes are mammals" (E) - Inferred as False.
• D. "Some apes are not mammals" (O) - Inferred as False.

Statement B ("Some apes are mammals" - I) is true, and Statement C ("Some mammals are apes" - Converse of A) is also true.

Therefore, the statements inferred to be false are A and D.

Conclusion on Inferences

Given that "All apes are mammals" is true:

• "No apes are mammals" (E) is False (Contrary).
• "Some apes are mammals" (I) is True (Subalternation).
• "Some mammals are apes" (I, converse of A) is True (Conversion).
• "Some apes are not mammals" (O) is False (Contradictory).

The statements immediately inferred to be false are A and D.

Revision Table: Truth Inferences from A being True

Additional Information: Conversion of Categorical Statements

Conversion is a type of immediate inference where the subject and predicate terms of a statement are interchanged. The validity of conversion depends on the type of statement:

• E statements (No S are P): Validly convert to E statements (No P are S). E.g., "No dogs are cats" converts to "No cats are dogs". If "No S are P" is true, then "No P are S" is true.
• I statements (Some S are P): Validly convert to I statements (Some P are S). E.g., "Some students are artists" converts to "Some artists are students". If "Some S are P" is true, then "Some P are S" is true.
• A statements (All S are P): Do not validly convert to A statements (All P are S). E.g., "All dogs are mammals" is true, but "All mammals are dogs" is false. However, A statements can validly convert by limitation (or per accidens) to I statements (Some P are S), provided the subject class is not empty. If "All S are P" is true, then "Some P are S" is true.
• O statements (Some S are not P): Do not validly convert. E.g., "Some animals are not dogs" is true, but "Some dogs are not animals" is false.

In the context of option C, "Some mammals are apes" is the converse of the given A statement "All apes are mammals". Since "All apes are mammals" is true, its converse "Some mammals are apes" (an I statement) is also true by conversion per accidens (assuming there are apes).