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
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.- Given below are four statements.
Statement 1: All students are inquisitive.
Statement 2: Some students are inquisitive.
Statement 3: No student is inquisitive.
Statement 4: Some students are not inquisitive.
From the given four statements, find the two statements that CANNOT BE TRUE simultaneously, assuming that there is at least one student in the class. - Based only on the truth of the statement ‘Some humans are intelligent', which one of the following options can be logically inferred with certainty?
Given below are two statements and four conclusions drawn based on the statements.
Statement 1: Some bottles are cups.
Statement 2: All cups are knives.
Conclusion I: Some bottles are knives.
Conclusion II: Some knives are cups.
Conclusion III: All cups are bottles.
Conclusion IV: All knives are cups.
Which one of the following options can be logically inferred?
