Read the given statements and conclusions carefully. Assuming that the information given in the statements is true, even if it appears to be at variance with commonly known facts, decide which of the given conclusions logically follow(s) from the statements. Statements: All dinosaurs are extinct. No tigers are extinct. Some tigers are omnivores. Conclusions: I. Some omnivores are dinosaurs. II. No tigers are dinosaurs. III. Some tigers are dinosaurs.
Only conclusion II follows
Understanding Statements and Conclusions in Logical Reasoning
This question requires us to carefully read the provided statements and determine which of the given conclusions logically follow based *only* on the information presented in the statements. We must assume the statements are true, even if they contradict real-world knowledge.
Analyzing the Given Statements
Let's break down the statements one by one:
- Statement 1: All dinosaurs are extinct. This means the set of 'Dinosaurs' is completely contained within the set of 'Extinct' things.
- Statement 2: No tigers are extinct. This means the set of 'Tigers' has no overlap whatsoever with the set of 'Extinct' things. The intersection of 'Tigers' and 'Extinct' is empty.
- Statement 3: Some tigers are omnivores. This means there is at least one tiger that is also an omnivore. The set of 'Tigers' and the set of 'Omnivores' have some overlap.
Evaluating the Conclusions
Now, let's examine each conclusion based on the logical relationships established by the statements.
Conclusion I: Some omnivores are dinosaurs.
- Statement 3 tells us some tigers are omnivores. Let's call this group 'Omnivore Tigers'.
- Statement 2 tells us no tigers are extinct. So, these 'Omnivore Tigers' are not extinct.
- Statement 1 tells us all dinosaurs are extinct. This means anything that is a dinosaur must be extinct.
- Since the 'Omnivore Tigers' are not extinct, they cannot be dinosaurs.
- Do the statements provide any information about other omnivores (those that are not tigers) and dinosaurs? No. We know some omnivores are tigers (from S3), and those specific omnivores cannot be dinosaurs (from S2 and S1). The statements do not give us any information about other omnivores being dinosaurs. Therefore, we cannot conclude that 'Some omnivores are dinosaurs'. This conclusion does not logically follow.
Conclusion II: No tigers are dinosaurs.
- From Statement 1, we know: If something is a dinosaur, then it is extinct. (\( \text{Dinosaur} \implies \text{Extinct} \))
- From Statement 2, we know: If something is a tiger, then it is not extinct. (\( \text{Tiger} \implies \text{Not Extinct} \))
- These two statements together create a clear separation. If an animal is a tiger, it is not extinct. If an animal is a dinosaur, it is extinct. An animal cannot be both extinct and not extinct at the same time.
- Therefore, nothing can be both a tiger and a dinosaur. The set of 'Tigers' and the set of 'Dinosaurs' have no common elements. This conclusion logically follows from Statements 1 and 2. Statement 3 is not needed to reach this conclusion.
Conclusion III: Some tigers are dinosaurs.
- This conclusion states that there is at least one tiger that is also a dinosaur.
- From our analysis of Conclusion II, we determined that 'No tigers are dinosaurs' logically follows from the statements.
- If it is true that 'No tigers are dinosaurs', then it is impossible for the conclusion 'Some tigers are dinosaurs' to also be true.
- Therefore, this conclusion does not logically follow. In fact, it is contradicted by a conclusion that *does* follow from the statements.
Summary of Conclusions
Based on our analysis:
- Conclusion I: Some omnivores are dinosaurs. (Does NOT follow)
- Conclusion II: No tigers are dinosaurs. (FOLLOWS)
- Conclusion III: Some tigers are dinosaurs. (Does NOT follow)
Only Conclusion II logically follows from the given statements.
| Statement | Logical Representation |
|---|---|
| All dinosaurs are extinct. | \( \text{Dinosaurs} \subseteq \text{Extinct} \) |
| No tigers are extinct. | \( \text{Tigers} \cap \text{Extinct} = \emptyset \) |
| Some tigers are omnivores. | \( \text{Tigers} \cap \text{Omnivores} \ne \emptyset \) |
| Conclusion | Follows? | Reasoning |
|---|---|---|
| I. Some omnivores are dinosaurs. | No | Statements don't link non-tiger omnivores to dinosaurs. Tiger omnivores are not extinct, thus not dinosaurs. |
| II. No tigers are dinosaurs. | Yes | Tigers are not extinct, dinosaurs are extinct. No overlap possible. |
| III. Some tigers are dinosaurs. | No | Contradicted by Conclusion II, which follows. |
Revision Table: Key Concepts in Logical Reasoning
| Term | Explanation |
|---|---|
| Statement | A premise assumed to be true for the purpose of deduction. |
| Conclusion | A proposition that is claimed to follow logically from the statements. |
| Logically Follows | A conclusion logically follows if it must be true whenever the statements are true. |
| Syllogism | A form of logical argument where a conclusion is derived from two or more premises. |
Additional Information: Deductive vs. Inductive Reasoning
Logical reasoning problems like this one rely on deductive reasoning. Here's a brief look at the difference:
- Deductive Reasoning: Starts with general statements (premises) and arrives at a specific, certain conclusion. If the premises are true and the logic is valid, the conclusion *must* be true. This is what we used here.
- Inductive Reasoning: Starts with specific observations and arrives at a general conclusion or prediction. The conclusion is probable, but not certain. For example, observing that all swans you've ever seen are white might lead to the inductive conclusion that all swans are white (which is not universally true).
In statements and conclusions questions, you are always using deductive reasoning based *strictly* on the provided statements.
Important Questions from Conventional Syllogism
Read the given statements and conclusions carefully. Assuming that the information given in the statements is true, even if it appears to be at variance with commonly known facts, decide which of the given conclusions logically follow(s) from the statements.
Statements:
No bank is an office.
All offices are stalls.
Conclusions:
I. No bank is a stall.
II. No stall is a bank.
III. Some stalls are offices.
IV. All the stalls are offices
Read the given statements and conclusions carefully. Assuming that the information given in the statements is true, even if it appears to be at variance with commonly known facts, decide which of the given conclusions logically follow(s) from the statements.
Statements:
All flowers are beautiful.
Vaidehi is beautiful.
Conclusions:
I. Vaidehi is a flower.
II. Some beautiful are flowers.
Read the given statements and conclusions carefully. Assuming that the information given in the statements is true, even if it appears to be at variance with commonly known facts, decide which of the given conclusions logically follow(s) from the statements.
Statements:
1. All rugs are blankets.
2. All blankets are pillows.
3. Some blankets are frames.
Conclusions:
I. All pillows are rugs.
II. Some pillows are rugs.
III. All rugs are frames
Read the given statements and conclusions carefully. Assuming that the information given in the statements is true, even if it appears to be at variance with commonly known facts, decide which of the given conclusions logically follow(s) from the statements.
Statements:
Some fingers are toes.
Some toes are rings.
Some rings are hands.
Conclusions:
I. Some hands are toes.
II. Some rings are fingers.
III. Some hands are fingers.
V. Some fingers are rings.
Read the given statements and conclusions carefully. Assuming that the information given in the statements is true, even if it appears to be at variance with commonly known facts, decide which of the given conclusions logically follow(s) from the statements.
Statements:
All polygons are angles.
All angles are diagonals.
All cones are cubes.
All cubes are decagons.
No diagonal is a cube.
Conclusions:
I. Some diagonals are polygons.
II. All diagonals are decagons.
III. No polygon is a cone.
IV. Some cubes are angles.
