Two statements are given followed by two conclusions numbered I and II. Assuming the statements to be true, even if they seem to be at variance with commonly known facts, decide which of the conclusions logically follow(s) from the statements. Statements: All keys are locks. All locks are handles. Conclusions: I. All locks are keys. II. All handles are keys.
Neither conclusion I nor II follows
Understanding Syllogism Statements and Conclusions
This question asks us to analyze two statements and determine which of the given conclusions logically follow from them. We must assume the statements are true, even if they contradict common knowledge. This type of logical reasoning problem is known as a syllogism.
Analyzing the Given Statements
We have two statements:
- All keys are locks.
- All locks are handles.
Let's represent the categories:
- 'Keys' (K)
- 'Locks' (L)
- 'Handles' (H)
The first statement, "All keys are locks," means that the set of 'Keys' is entirely contained within the set of 'Locks'. In set notation, this can be represented as K $\subseteq$ L.
The second statement, "All locks are handles," means that the set of 'Locks' is entirely contained within the set of 'Handles'. In set notation, this is L $\subseteq$ H.
Combining these two statements, if all Keys are Locks and all Locks are Handles, it logically follows that all Keys are also Handles. This gives us the relationship K $\subseteq$ L $\subseteq$ H.
Evaluating the Conclusions Based on the Statements
Now let's examine each conclusion:
Conclusion I: All locks are keys.
From the statements, we know K $\subseteq$ L (All keys are locks). This means that every key is a lock. However, it does not necessarily mean that every lock is a key. The set of Locks (L) could be larger than the set of Keys (K), containing some items that are locks but not keys. For example, if there are 10 keys and 15 locks, where the 10 keys are among the 15 locks, then "All keys are locks" is true, but "All locks are keys" is false. Therefore, Conclusion I does not logically follow from the statements.
Conclusion II: All handles are keys.
From the combined statements, we established that K $\subseteq$ H (All keys are handles). This means that every key is a handle. However, this does not imply that every handle is a key. The set of Handles (H) could be much larger than the set of Keys (K), containing many items that are handles but not keys. For instance, door knobs and levers are types of handles, but they are not typically keys. Based on the statements, H is a superset of K. H could be equal to K, but it could also be much larger. Therefore, Conclusion II does not logically follow from the statements.
Final Decision on Logical Conclusions
Based on the analysis of both conclusions, neither Conclusion I ("All locks are keys") nor Conclusion II ("All handles are keys") can be logically deduced as true from the given statements ("All keys are locks" and "All locks are handles").
Let's summarize the relationships:
| Statement/Relationship | Meaning | Logical Follows? |
|---|---|---|
| Statement 1: All keys are locks | Set of Keys is inside Set of Locks (K $\subseteq$ L) | Given as true |
| Statement 2: All locks are handles | Set of Locks is inside Set of Handles (L $\subseteq$ H) | Given as true |
| Combined: All keys are handles | Set of Keys is inside Set of Handles (K $\subseteq$ H) | Yes, logically follows |
| Conclusion I: All locks are keys | Set of Locks is inside Set of Keys (L $\subseteq$ K) | No, not necessarily from K $\subseteq$ L |
| Conclusion II: All handles are keys | Set of Handles is inside Set of Keys (H $\subseteq$ K) | No, not necessarily from K $\subseteq$ H |
Therefore, neither conclusion follows.
Syllogism Revision Table
Reviewing key concepts for syllogism problems:
- Statements: Given propositions assumed to be true.
- Conclusions: Propositions to be tested for logical validity based on statements.
- Logical Deduction: The process of arriving at a conclusion that must be true if the statements are true.
- "All A are B": Means the set of A is a subset of the set of B. It does NOT mean "All B are A".
- "Some A are B": Means there is at least one element common to both set A and set B.
- "No A are B": Means the set of A and the set of B are disjoint (have no elements in common).
Additional Information on Logical Reasoning
Syllogisms are a fundamental part of deductive reasoning. They involve drawing conclusions from premises (statements). Validity in syllogisms depends on the logical structure, not the truthfulness of the statements in the real world. When solving syllogism questions, it is often helpful to visualize the relationships using Venn diagrams or think in terms of sets and subsets. Always follow the exact wording and scope of the statements provided. Do not introduce outside information or assumptions about the real-world relationship between the terms like 'keys', 'locks', and 'handles'. The logic must flow solely from the given 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.
