Consider the following three statements:
(i) Some roses are red.
(ii) All red flowers fade quickly.
(iii) Some roses fade quickly.
Which of the following statements can be logically inferred from the above statements?
Logical Inference Analysis
This question requires analyzing the logical connection between three statements to determine when the third statement can be definitively inferred.
Statement Breakdown
- Statement (i): Some roses are red. This is a particular affirmative statement. It means there exists at least one rose that is red. Let R be the set of roses and D be the set of red things. Then, $R \cap D \neq \emptyset$.
- Statement (ii): All red flowers fade quickly. This is a universal affirmative statement. Assuming 'red' implies 'red flower' in this context, it means everything that is red also fades quickly. Let F be the set of things that fade quickly. Then, $D \subseteq F$.
- Statement (iii): Some roses fade quickly. This is a particular affirmative statement, suggesting $R \cap F \neq \emptyset$.
Deriving the Conclusion
We need to find the condition under which statement (iii) is a logical consequence of statements (i) and (ii).
- Assume statements (i) and (ii) are true.
- From statement (i), we know there is at least one entity that is both a rose and red. Let's call this entity 'x'. So, $x$ is a rose AND $x$ is red.
- From statement (ii), we know that if something is red, it fades quickly.
- Since 'x' is red (from step 2), it must also fade quickly (based on step 3).
- Combining these facts, 'x' is a rose (from step 2) and 'x' fades quickly (from step 4).
- This directly implies that there exists at least one rose that fades quickly, which is statement (iii).
Therefore, if statements (i) and (ii) are true, statement (iii) must logically follow. This corresponds to the condition described in Option 3.
Evaluating Other Options
The validity of statement (iii) depends entirely on the truthfulness of statements (i) and (ii). Scenarios where either (i) or (ii) (or both) are false do not impact the logical inference derived from them being true. Option 3 correctly captures the condition under which the inference holds.
Important Questions from Syllogism
- Consider the following sentences:
All benches are beds .No bed is a bulb. Some bulbs are lamps.
Which of the following can be inferred?
i. Some beds are lamps.
ii. Some lamps are beds. Given below are two statements followed by two conclusions. Assuming these statements to be true, decide which one logically follows.
Statements:
I. No manager is a leader.
II. All leaders are executives.
Conclusions:
I. No manager is an executive.
II. No executive is a manager.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?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.
