Question

Human beings are one among many creatures that inhabit an imagined world. In this imagined world, some creatures are cruel. If in this imagined world, it is given that the statement "Some human beings are not cruel creatures" is FALSE, then which of the following set of statement(s) can be logically inferred with certainty?
(i) All human beings are cruel creatures.
(ii) Some human beings are cruel creatures.
(iii) Some creatures that are cruel are human beings.
(iv) No human beings are cruel creatures.

• A. only (i)
• B. only (iii) and (iv)
• C. only (i) and (ii)
• D. (i), (ii) and (iii)

Hint:

Remember the "Square of Opposition" in logic. The statement "Some A are not B" (Particular Negative) and "All A are B" (Universal Affirmative) are contradictories. This means if one is false, the other must be true, and vice-versa.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This question deals with categorical propositions in logic. We are given a statement that is false and asked to determine what other statements must be true. The key is to understand the logical negation of the given statement.
Step 2: Detailed Explanation:
The given statement is: "Some human beings are not cruel creatures". This statement is of the form "Some S are not P".
We are told that this statement is FALSE.
In formal logic, the negation of a statement has the opposite truth value. So, the negation of "Some S are not P" must be TRUE.
The logical negation of "Some S are not P" is "All S are P".
Therefore, the statement "All human beings are cruel creatures" must be TRUE. This corresponds to statement (i).
Now, let's evaluate the other statements based on the fact that "(i) All human beings are cruel creatures" is TRUE.
- (ii) Some human beings are cruel creatures.
In classical logic, if a universal affirmative statement ("All S are P") is true, and the subject class (S, human beings) is not empty, then the particular affirmative statement ("Some S are P") is also considered true. Since "All" human beings are cruel, it certainly follows that "Some" (at least one) are. So, (ii) is TRUE.
- (iii) Some creatures that are cruel are human beings.
This is another way of saying "Some cruel creatures are human beings". If all human beings are cruel creatures, it logically follows that the set of cruel creatures includes all human beings. Therefore, some of the cruel creatures are indeed human beings. So, (iii) is TRUE.
- (iv) No human beings are cruel creatures.
This is the direct contradictory of statement (i). Since we have established that (i) is TRUE, statement (iv) must be FALSE.
Thus, the statements that can be inferred with certainty are (i), (ii), and (iii).
Step 3: Why This is Correct:
The falsity of "Some human beings are not cruel creatures" logically implies the truth of its negation, "All human beings are cruel creatures." This, in turn, implies the truth of the particular statements (ii) and (iii). Therefore, (i), (ii), and (iii) are all certain inferences.