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.
Options:
(A) only (i)
(B) only (iii) and (iv)
(C) only (i) and (ii)
(D) (i), (ii) and (iii)
Official Answer Key:
D
Step-by-Step Solution & Explanation
### **Official Solution & Concept Analysis (GATE BT 2023 Q.6)**
**1. Governing Concept**:
- **Topic**: Categorical Propositions & Square of Opposition
- Given: The proposition *"Some human beings are not cruel creatures"* (Particular Negative, **O** proposition) is **FALSE**.
- By the Classical Square of Opposition:
- If **O** is False, its contradictory **A** (*"All human beings are cruel creatures"*) must be **TRUE**.
- Since **A** is True:
- (i) *"All human beings are cruel creatures"* is **TRUE**.
- (ii) *"Some human beings are cruel creatures"* (**I** proposition, subaltern of A) is **TRUE**.
- (iii) *"Some creatures that are cruel are human beings"* (conversion of I) is **TRUE**.
- (iv) *"No human beings are cruel creatures"* (**E** proposition, contrary of A) is **FALSE**.
- Hence, statements (i), (ii), and (iii) can be inferred with certainty.
**2. Option Analysis**:
- **Option (A) is INCORRECT**: Fails to include the logically entailed subaltern and converted propositions.
- **Option (B) is INCORRECT**: Statement (iv) is demonstrably false.
- **Option (C) is INCORRECT**: Omits valid converted statement (iii).
- **Option (D) [(i), (ii) and (iii)] is CORRECT**: All three statements necessarily hold true.
**3. Final Answer**:
- **Correct Choice**: **D**
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