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?
i. All human beings are cruel creatures.
ii. Some human beings are cruel creatures.
iii. No human being is a cruel creature.
iv. No human being is not a cruel creature.
A. (i) only
B. (i) and (ii) only
C. (iii) and (iv) only
D. (i), (ii), and (iv) only
GATE 2023 · General Aptitude · Statements Follow · medium
Answer: Statements (i), (ii), and (iv) can be logically inferred (option D).
Negate the given FALSE statement: Given: 'Some humans are NOT cruel' is FALSE. Apply quantifier negation: ~(exists x: Human(x) AND ~Cruel(x)) = forall x: Human(x) -> Cruel(x). Conclusion: ALL humans are cruel creatures.
Evaluate statement (i): All human beings are cruel creatures: This is exactly what we derived in Step 1. Statement (i) is TRUE.
Evaluate statement (ii): Some human beings are cruel creatures: From 'All humans are cruel' and assuming at least one human exists (non-empty domain), 'Some humans are cruel' follows. Statement (ii) is TRUE.
Evaluate statement (iii): No human being is a cruel creature: 'No human is cruel' means forall x: Human(x) -> NOT Cruel(x). This directly contradicts our derived 'All humans are cruel'. Statement (iii) is FALSE.
Evaluate statement (iv): No human being is NOT a cruel creature: 'No human is NOT cruel' = forall x: Human(x) -> NOT(NOT Cruel(x)) = forall x: Human(x) -> Cruel(x) = 'All humans are cruel'. This is the same as statement (i) and is TRUE.
Select the correct option: Statements i, ii, and iv are TRUE. Statement iii is FALSE. This matches option D: (i), (ii), and (iv) only.