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.
Similar Questions
Two statements are given, followed by three conclusions numbered I, II and III. 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 teachers are engineers.
All clerks are teachers.
Conclusions:
I. Some engineers are clerks.
II. Some teachers are clerks.
III. Some clerks are not engineers.
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 paintings are calligraphies.
All sketching are calligraphies .
Conclusions:
I. At least some calligraphies are paintings.
II. No sketching is a painting.
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:
Some vehicles are helicopters.
All helicopters are cars.
Conclusions:
I. All vehicles are cars.
II. Some vehicles are cars.
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.
Two statements are given followed by conclusions numbered 1 and 2. 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:
1. Some mammals are animals.
2. All animals are birds.
Conclusions:
1. All birds are animals.
2. Some mammals are birds.
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. Some locks are keys.
II. Some handles are keys.
Three Statements are given followed by Three conclusions numbered I, II and III. 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 switches are fans.
No heater is a fan.
Some wires are heaters.
Conclusions:
I. Some wires are fans.
II. No switch is a heater.
III. No wire is a switch.
Three 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 H are G.
All V are H.
Some V are I.
Conclusions:
I. At least some G are I.
II. All I being H is a possibility.
Three statements are given followed by three conclusions numbered I, II and III. 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 apples are pineapples.
Some apples are mangoes.
No mango is a grape.
Conclusions:
I. Some pineapples are not grapes.
II. No pineapple is a grape.
III. Some apples are not grapes.
Three statements are given followed by three conclusions numbered I, II and III. 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:
No river is a sea.
Some rivers are lakes.
All rivers are ponds.
Conclusions:
I. Some lakes are seas.
II. Some ponds are seas.
III. Some lakes are ponds.
Important Questions from Conventional Syllogism
In this question, three statements followed by two conclusions numbered I and II have been given. You have to take the given statements to be true even if they seem to be at variance from the commonly facts and then decide which of the given conclusions logically follows from the given statements disregarding commonly known facts.
Statements: Some flats are apartments.
No apartment is a hall.
Some halls are rooms.
Conclusions: I. At least some rooms are flats.
II. No apartment is a room.
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:
I. Some blue are red.
II. Some green are red.
Conclusions:
I. No blue is green.
II. No red is green.
Given below are two statements, Consider these statements to be true even if they seem factuality absurd, Read the conclusions and then decide which of the given conclusions logically follow(s) from the given statements?
Statements :
1. All vases are flowers.
2. No flowers is a plant.
Conclusions :
1. No vases is a plant.
2. Some plant are vases
In the following question below are given some statements followed by some conclusions based on those statements. Taking the given statements to be true even if they seem to be at variance from commonly known facts. Read all the conclusions and then decide which of the given conclusion logically follows the given statements.
Statements:
I. All A are S.
II. No D is A.
Conclusions:
I. Some S are A.
II. All S are D.
III. No A is D.
In the following question below are given some statements followed by some conclusions based on those statements. Taking the given statements to be true even if they seem to be at variance from commonly known facts. Read all the conclusions and then decide which of the given conclusion logically follows the given statements.
Statements:
I. Some land are hard.
II. No stone is land.
Conclusions:
I. Some hard are land.
II. Some stone are hard.
