Question

Human beings are one among many creatures that inhabit an imagined world. In this world, some creatures are cruel. If it is given that the statement "Some human beings are not cruel creatures" is FALSE, then which of the following statements 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:

The negation of "Some \(A\) are not \(B\)" is "All \(A\) are \(B\)." From universals plus existence, you can immediately infer the corresponding existential statements.

Answer 1

The Correct Option is D

Step 1: Negate the given false statement correctly.
The statement "Some humans are not cruel" has logical form \(\exists x\,(\text{Human}(x)\wedge \neg \text{Cruel}(x))\).
Its negation (which must be true) is \(\forall x\,(\text{Human}(x)\Rightarrow \text{Cruel}(x))\) i.e., all humans are cruel. Hence (i) is true.

Step 2: Use existence of humans.
Humans are said to be "one among many creatures," so humans exist. From (i), if humans exist and all of them are cruel, then some humans are cruel. Thus (ii) is true.

Step 3: Translate to "some cruel are humans."
Since at least one human exists and every human is cruel, there exists at least one creature that is both cruel and human. Therefore (iii) is true.

Step 4: Reject the contradictory option.
(iv) "No human beings are cruel" contradicts (i), hence it is false.

Final Answer:
\[ \boxed{\text{(D) } (i), (ii) \text{ and } (iii)} \]