Some tables are shelves. Some shelves are chairs. All chairs are benches. Which of the following conclusions can be deduced from the preceding sentences?
i. At least one bench is a table
ii. At least one shelf is a bench
iii. At least one chair is a table
iv. All benches are chairs
Analyze the Premises:
- Premise 1: Some tables are shelves. (Let T = Tables, S = Shelves. This means T and S have a non-empty intersection: T <0xC2><0x8A> S <0xE2><0x89><0xA0> ∅)
- Premise 2: Some shelves are chairs. (Let C = Chairs. This means S and C have a non-empty intersection: S <0xC2><0x8A> C <0xE2><0x89><0xA0> ∅)
- Premise 3: All chairs are benches. (Let B = Benches. This means the set C is a subset of the set B: C <0xE2><0x89><0xA4> B)
Evaluating Conclusions
Conclusion ii: At least one shelf is a bench
- From Premise 2, we know there exists at least one item that is both a shelf (S) and a chair (C). Let's call this item 'x'. So, 'x' is in S and 'x' is in C.
- From Premise 3, we know that everything that is a chair (C) is also a bench (B).
- Since 'x' is a chair (from Premise 2), 'x' must also be a bench (based on Premise 3).
- Therefore, 'x' is both a shelf (S) and a bench (B).
- This confirms that at least one shelf is a bench (S <0xC2><0x8A> B <0xE2><0x89><0xA0> ∅). Conclusion ii is valid.
Conclusion i: At least one bench is a table
- We know Some T are S, and Some S are C, and All C are B.
- The shelves that are tables might be completely different from the shelves that are chairs.
- Example: {Tables: T1, T2}, {Shelves: T1, S1}, {Chairs: S1, C1}, {Benches: S1, C1, B1}. Here, T1 is a Table and Shelf. S1 is a Shelf and Chair. C1 is a Chair and Bench. B1 is just a Bench. All Chairs are Benches.
- In this example, no bench is a table. Therefore, this conclusion cannot be reliably deduced. Conclusion i is invalid.
Conclusion iii: At least one chair is a table
- Similar to conclusion i, the overlap between Tables and Shelves (T <0xC2><0x8A> S) does not necessarily overlap with the overlap between Shelves and Chairs (S <0xC2><0x8A> C).
- Using the same example: {Chairs: S1, C1}, {Tables: T1, T2}. No chair is a table. Conclusion iii is invalid.
Conclusion iv: All benches are chairs
- Premise 3 states "All chairs are benches" (C <0xE2><0x89><0xA4> B). This means the set of chairs is contained within the set of benches.
- This does *not* imply the reverse: that all benches are chairs (B <0xE2><0x89><0xA4> C). There could be benches that are not chairs.
- Using the example again: {Benches: S1, C1, B1}, {Chairs: S1, C1}. B1 is a bench but not a chair. Conclusion iv is invalid.
Final Deduction
Only conclusion ii, "At least one shelf is a bench," can be logically deduced from the given premises.
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.
