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 holders are hangers. All hangers are hammers. Some hammers are axes. Conclusions: I. Some axes are hangers. II. Some hammers are holders. III. Some hangers are holders.
Only conclusions II and III follow
Solving Syllogism Problems: Step-by-Step Analysis
Syllogism questions test your ability to draw logical conclusions from given statements. You must assume the statements are true, even if they contradict common knowledge, and determine which conclusions logically follow.
Analyzing the Statements
We are given three statements:
- Statement 1: All holders are hangers.
- Statement 2: All hangers are hammers.
- Statement 3: Some hammers are axes.
Let's represent these relationships. From Statement 1 and Statement 2, we can deduce a relationship between holders and hammers:
- If All A are B (holders are hangers) and All B are C (hangers are hammers), then All A are C (holders are hammers).
So, from Statements 1 and 2, we know:
- All holders are hammers.
Statement 3 introduces a 'some' relationship between hammers and axes. This means there is an overlap between the group of hammers and the group of axes, but not necessarily covering all of either group.
Evaluating the Conclusions
Now let's examine each conclusion based on the given statements and our deduction:
Conclusion I: Some axes are hangers.
We know All hangers are hammers, and Some hammers are axes. A Venn diagram can help visualize this. The set of hangers is entirely inside the set of hammers. The set of axes overlaps with the set of hammers. However, the overlap between axes and hammers could be entirely within the part of hammers that is NOT hangers. Therefore, we cannot definitively conclude that some axes are hangers. This conclusion does not logically follow.
Consider this illustration:
| Relationship | Example |
|---|---|
| All Hangers are Hammers | If all dogs are mammals. |
| Some Hammers are Axes | If some mammals are swimmers. |
| Conclusion I: Some Axes are Hangers? | Can we conclude some swimmers are dogs? No. Mammals can be swimmers, but those swimmers might not be dogs (they could be whales, seals, etc.). Similarly, the hammers that are axes might not be the hammers that are hangers. |
Conclusion I does not follow.
Conclusion II: Some hammers are holders.
From the statements, we deduced that All holders are hammers. A fundamental rule in syllogism is that if "All A are B" is true, then "Some B are A" is also true. If every single holder is included in the group of hammers, then there must be at least some hammers that are holders (specifically, all the ones that are holders). Therefore, Conclusion II logically follows from the statements.
Rule: All A are B → Some B are A
Applying this to "All holders are hammers": All (holders) are (hammers) → Some (hammers) are (holders).
Conclusion II follows.
Conclusion III: Some hangers are holders.
Look at Statement 1: All holders are hangers. Using the same rule as above, if "All A are B" is true, then "Some B are A" is true. If every single holder is included in the group of hangers, then there must be at least some hangers that are holders (specifically, all the ones that are holders).
Rule: All A are B → Some B are A
Applying this to "All holders are hangers": All (holders) are (hangers) → Some (hangers) are (holders).
Conclusion III follows.
Summary of Conclusions
- Conclusion I: Some axes are hangers - Does not follow.
- Conclusion II: Some hammers are holders - Follows.
- Conclusion III: Some hangers are holders - Follows.
Based on our analysis, only conclusions II and III logically follow from the given statements.
Final Answer
Only conclusions II and III follow.
Syllogism Problem Revision Table
| Statement / Conclusion | Relationship | Evaluation (Follows/Does Not Follow) | Reasoning |
|---|---|---|---|
| Statement 1 | All holders are hangers | Given | Basis for deduction |
| Statement 2 | All hangers are hammers | Given | Basis for deduction |
| Statement 3 | Some hammers are axes | Given | Basis for deduction |
| Deduction (from 1 & 2) | All holders are hammers | Logically derived | Transitive property (All A=B, All B=C → All A=C) |
| Conclusion I | Some axes are hangers | Does Not Follow | Overlap between hammers and axes doesn't guarantee overlap with the subset of hammers that are hangers. |
| Conclusion II | Some hammers are holders | Follows | From "All holders are hammers", the converse "Some hammers are holders" is a valid immediate inference. |
| Conclusion III | Some hangers are holders | Follows | From "All holders are hangers", the converse "Some hangers are holders" is a valid immediate inference. |
Additional Information on Syllogism
Syllogism is a form of logical reasoning where a conclusion is drawn from two or more premises (statements). Here are some key concepts:
- Statements: These are the given premises that you must accept as true. They typically involve relationships like 'All', 'Some', 'No', 'Some Not'.
- Conclusions: These are inferences that you must evaluate to see if they are logically supported by the statements.
- Types of Statements:
- Universal Affirmative (A): All S are P.
- Universal Negative (E): No S are P.
- Particular Affirmative (I): Some S are P.
- Particular Negative (O): Some S are not P.
- Immediate Inference: Drawing a conclusion directly from a single statement. For example, from "All dogs are mammals", you can immediately infer "Some mammals are dogs".
- Mediate Inference: Drawing a conclusion from two or more statements, like in the problem above where we combined two statements.
- Venn Diagrams: A useful tool to visually represent the relationships between categories in statements. Overlapping circles show shared elements.
Remember, the key is to follow the logic strictly based on the statements, ignoring any outside information or commonly known facts.
Similar Questions
Three Statements are given followed by Four conclusions numbered I, II, III and IV. 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 curtains are pillows.
No pillow is a bedsheet.
Some curtains are bedsheets.
Conclusions:
I. No curtain is a bedsheet.
II. No pillow is a curtain.
III. Some bedsheets are curtains.
IV. Some pillows are curtains.
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 dates are cashews.
All cashews are raisins.
Some raisins are almonds.
Conclusions:
I. All almonds being dates is a possibility.
II. All dates are raisins.
III. No cashew is an almond.
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.
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:
Some plants are flowers.
Some flowers are fruits.
All flowers are gardens.
Conclusions:
I. Some gardens are plants.
II. Some fruits are gardens.
III. Some gardens are flowers.
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.
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 classrooms are schools.
No school is a park.
All houses are parks.
Conclusions:
I. No house is a classroom.
II. No park is a classroom.
III. No school is a house.
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 fans are hammers.
All hammers are fishes.
Conclusions:
I. No fans are fishes.
II. Some fans are fishes.
Read the given statements and conclusions carefully. Assuming that the information given in the statement 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 fans are coolers.
All coolers are equipment.
Some equipment are mobiles.
Conclusions:
I. Some equipments are fans.
II. Some mobiles are fans.
III. No equipment is fan and cooler both
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 inkpots are kettles.
All utensils are candies.
No kettle is utensil.
Conclusions:
1. Some inkpots are candies.
2. Some kettles are inkpots.
3. Some candies are inkpots.
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 blankets are quilts.
Some quilts are bedsheets.
Conclusions:
I. All quilts are blankets.
II. All blankets are bedsheets.
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:
1. Some flowers are chemicals.
2. Some chemicals are herbs.
Conclusions:
I. No flower is a herb.
II. Some flowers are herbs.
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 students are teachers.
All teachers are professors.
Conclusions:
1. All students are professors.
2. All professors are students.
3. No teacher is a professor.
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 dentists are surgeons.
II. All the surgeons are doctors.
Conclusions:
I. Some dentists are doctors.
II. No surgeons are dentists.
The statements below are followed by two conclusions labelled I and II. Assuming that the information in the statements is true, even if it appears at variance with generally established facts. decide which conclusion(s) logically and definitely follow(s) from the information given in the statements.
Statements:
1) Some angers are griefs.
2) Some griefs are loves.
Conclusion:
I. Some griefs are angers.
II. Some loves are griefsThe statements below are followed by two conclusions labeled I and II. Assuming that the information in the statements is true, even if it appears to be at variance with generally established facts. decide which conclusion(s) logically and definitely follow(s) from the information given in the statements.
Statements:
Some cows are deer.
Some deer are fishes.
Conclusion:
I. Some cows are fishes.
II. Some fishes are cows.
