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.
Analyze the Given Premise
The core information provided is that the statement “Some human beings are not cruel creatures” is FALSE.
Derive the Certain Truth
In formal logic, when a statement of the form "Some A are not B" is determined to be false, its negation must be true. The logical negation of "Some A are not B" is "All A are B".
- Let 'A' represent the category 'human beings'.
- Let 'B' represent the category 'cruel creatures'.
Given that "Some human beings are not cruel creatures" (Some A are not B) is FALSE, we can logically conclude with certainty that "All human beings are cruel creatures" (All A are B) is TRUE. This forms our primary inference.
Evaluate Each Statement for Certainty
Statement (i): All human beings are cruel creatures.
This statement is the direct logical consequence derived from the given premise. Therefore, statement (i) is TRUE and can be inferred with certainty.
Statement (ii): Some human beings are cruel creatures.
If the statement "All human beings are cruel creatures" is TRUE, it logically follows that "Some human beings are cruel creatures" must also be TRUE. This inference assumes the existence of at least one human being, which is standard in such logic problems.
Therefore, statement (ii) is TRUE and can be inferred with certainty.
Statement (iii): Some creatures that are cruel are human beings.
This statement asserts that the intersection between the set of 'cruel creatures' and the set of 'human beings' is non-empty. This is logically equivalent to statement (ii). If "Some human beings are cruel creatures" is true, then "Some creatures that are cruel are human beings" must also be true.
Therefore, statement (iii) is TRUE and can be inferred with certainty.
Statement (iv): No human beings are cruel creatures.
This statement directly contradicts statement (i) ("All human beings are cruel creatures"), which we have already established as TRUE. A statement and its contradiction cannot both be true.
Therefore, statement (iv) is FALSE and cannot be inferred with certainty.
Final Conclusion
Based on the logical analysis, the statements that can be inferred with certainty are (i), (ii), and (iii).
Important Questions from Syllogism
Six statements, S1 to S6, are given.
S1: All GATE candidates are engineers.
S2: Some GATE candidates are CAT candidates.
S3: All GRE candidates are GATE candidates.
S4: All engineers are GRE candidates.
S5: Some CAT candidates are GRE candidates.
S6: Some CAT candidates are engineers.
Based on S1-S6, following options provide a set of three statements, followed by a fourth statement. Identify the option(s) where the fourth statement can be necessarily inferred from the set of three statements.
Statements :
All pens are pencils.
No pencil is cap.
Conclusions:
(I) All caps are pencils.
(II) No pen is a cap.
Statements:
All huts are mansions.
All mansions are temples.
Conclusions :
(I) Some temples are huts.
(II) Some temples are mansions.
Given below are two statements 1 and 2, and two conclusions I and II.
Statement 1: All entrepreneurs are wealthy.
Statement 2: All wealthy are risk seekers.
Conclusion I: All risk seekers are wealthy.
Conclusion II: Only some entrepreneurs are risk seekers.
Based on the above statements and conclusions, which one of the following options is CORRECT?
Statements:
1. All heroes are winners.
2. All winners are lucky people.
Inferences:
I. All lucky people are heroes.
II. Some lucky people are heroes.
III. Some winners are heroes.
Which of the above inferences can be logically deduced from statements 1 and 2?
