Given below are two statements and two conclusions. Take the statements to be true even if they are at variance with commonly known facts, and decide whether the conclusion(s) follow(s) the given statements. Statements: All eyes are noses. Some noses are lips. Conclusions: I. Some noses are eyes. II. All lips are noses.
Only conclusion I follows
Understanding Syllogism and Drawing Conclusions
This question asks us to analyze two statements and determine which of the given conclusions logically follow from them. We must accept the statements as true, even if they contradict real-world knowledge. This is a standard type of question in logical reasoning, specifically dealing with syllogisms.
Analyzing the Syllogism Statements
Let's break down the given statements:
- Statement 1: All eyes are noses.
- Statement 2: Some noses are lips.
We can think of these statements in terms of categories or sets. Let 'Eyes' be set E, 'Noses' be set N, and 'Lips' be set L.
- Statement 1 (All E are N): This means that the set of Eyes (E) is completely contained within the set of Noses (N). In other words, if something is an eye, it must also be a nose.
- Statement 2 (Some N are L): This means that there is at least one element that is both in the set of Noses (N) and the set of Lips (L). There is an overlap or intersection between the sets N and L. However, it does not mean that all elements in N are in L, or that all elements in L are in N.
Evaluating the Conclusions
Now, let's examine each conclusion based on the statements:
Conclusion I: Some noses are eyes.
Statement 1 says "All eyes are noses". If every single eye is also a nose, then it logically follows that there must be some noses that are eyes (specifically, all the noses that happen to be eyes). Consider this: if you have a group of 10 items, and all 10 are red, then it's true that some of those 10 items are red. The statement "All A are B" always implies "Some B are A" (assuming A is not an empty set, which is typical in syllogism problems).
Therefore, Conclusion I directly follows from Statement 1.
Conclusion II: All lips are noses.
Statement 2 says "Some noses are lips". This means that the intersection between the set of noses and the set of lips is not empty. There could be several possible scenarios that satisfy "Some noses are lips":
- Only a few noses are lips, and the rest are not lips.
- Only a few lips are noses, and the rest are not noses.
- All lips are noses (This is one possibility, but not the only one guaranteed by the statement).
- All noses are lips (This is also a possibility, but not the only one).
The statement "Some N are L" only guarantees an overlap. It does not guarantee that the entire set of Lips (L) is contained within the set of Noses (N). For example, if there are 10 noses and 10 lips, and 3 items are both noses and lips, the statement is true. But there could still be 7 lips that are not noses. Therefore, we cannot definitively conclude that "All lips are noses".
Therefore, Conclusion II does not follow from the statements.
Final Decision
Based on our analysis:
- Conclusion I: Some noses are eyes - Follows.
- Conclusion II: All lips are noses - Does not follow.
Thus, only conclusion I follows from the given statements.
| Statement/Conclusion | Relationship | Follows? |
|---|---|---|
| Statement 1 | All Eyes → Noses | Given as true |
| Statement 2 | Some Noses ↔ Some Lips (Overlap) | Given as true |
| Conclusion I | Some Noses → Eyes? | Yes (From Statement 1) |
| Conclusion II | All Lips → Noses? | No (Statement 2 only guarantees overlap) |
Revision Table: Syllogism Rules
| Statement Type | Implied Conversions | Meaning (Sets A, B) |
|---|---|---|
| All A are B | Some B are A | Set A is a subset of Set B ($A \subseteq B$) |
| No A are B | No B are A | Sets A and B have no common elements ($A \cap B = \emptyset$) |
| Some A are B | Some B are A | Sets A and B have at least one common element ($A \cap B \neq \emptyset$) |
| Some A are not B | No valid simple conversion | There is at least one element in Set A that is not in Set B |
Additional Information: Syllogism Basics
Syllogism is a type of logical argument that applies deductive reasoning to arrive at a conclusion based on two or more propositions that are asserted or assumed to be true. In these problems, the structure is typically two statements (premises) and one or more conclusions. You must strictly adhere to the information given in the statements and not bring in any outside knowledge.
Key points for solving syllogism problems:
- Draw diagrams (like Venn diagrams) to visualize the relationships described in the statements.
- Analyze each conclusion independently based on whether it is necessarily true given the statements.
- Be careful with "some" statements. "Some" means "at least one", but can also include "all". However, "Some A are B" does NOT imply "All A are B" or "All B are A".
- The conclusion must follow logically and inevitably from the statements. If there is any possibility, however remote in the real world, for the conclusion to be false while the statements are true, then the conclusion does not follow.
This question tests your understanding of the implications of "All" and "Some" relationships in logical reasoning.
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.
