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 conclusion logically follow(s) from the statements. Statements: No photographer is an architect. All engineers are photographers. All postmen are architects. Conclusions: I. No engineer is an architect. II. No engineer is a postman. III. No photographer is a postman.
All conclusions follow
Solving the Syllogism Problem
This problem involves analyzing logical statements and determining which conclusions can be drawn from them. We are given three statements and three conclusions. We must assume the statements are true and check which conclusions logically follow.
Understanding the Statements
Let's represent the categories using letters:
- P: Photographers
- A: Architects
- E: Engineers
- M: Postmen
The statements can be written as:
- No photographer is an architect. (P and A are disjoint sets: \(P \cap A = \emptyset\))
- All engineers are photographers. (E is a subset of P: \(E \subseteq P\))
- All postmen are architects. (M is a subset of A: \(M \subseteq A\))
Analyzing the Conclusions
Let's examine each conclusion based on the given statements.
Conclusion I: No engineer is an architect.
- Statement 2 tells us that all engineers are photographers (\(E \subseteq P\)).
- Statement 1 tells us that no photographer is an architect (\(P \cap A = \emptyset\)).
- If all engineers are a part of the group 'photographers', and the group 'photographers' has no members in common with the group 'architects', then the group 'engineers' can also have no members in common with the group 'architects'.
- Therefore, no engineer is an architect logically follows.
Conclusion II: No engineer is a postman.
- From Conclusion I, we established that no engineer is an architect (\(E \cap A = \emptyset\)).
- Statement 3 tells us that all postmen are architects (\(M \subseteq A\)).
- If no engineer is an architect, and all postmen are a part of the group 'architects', then no engineer can be a postman.
- Therefore, no engineer is a postman logically follows.
Conclusion III: No photographer is a postman.
- Statement 1 tells us that no photographer is an architect (\(P \cap A = \emptyset\)).
- Statement 3 tells us that all postmen are architects (\(M \subseteq A\)).
- If no photographer is an architect, and all postmen are a part of the group 'architects', then no photographer can be a postman.
- Therefore, no photographer is a postman logically follows.
Summary of Conclusions
| Conclusion | Logical Follows? | Reasoning |
|---|---|---|
| I. No engineer is an architect. | Yes | Engineers are Photographers, Photographers are not Architects. |
| II. No engineer is a postman. | Yes | Engineers are not Architects, Postmen are Architects. |
| III. No photographer is a postman. | Yes | Photographers are not Architects, Postmen are Architects. |
Based on the analysis of each conclusion, all three conclusions logically follow from the given statements.
Revision Table: Syllogism Basics
| Term | Explanation | Example (from this problem) |
|---|---|---|
| Statement | A premise assumed to be true for the purpose of logic. | "All engineers are photographers." |
| Conclusion | A judgment or decision reached by reasoning. | "No engineer is an architect." |
| Syllogism | A form of reasoning where a conclusion is drawn from two or more premises. | The entire problem structure. |
| Disjoint Sets | Sets that have no elements in common. | Photographers and Architects (Statement 1). |
| Subset | A set where every element is also an element of a larger set. | Engineers are a subset of Photographers (Statement 2). |
Additional Information: Solving Syllogisms
Solving syllogism problems requires careful attention to the logical relationships between different categories mentioned in the statements. Two common methods are:
- Venn Diagrams: Drawing overlapping or separate circles to represent the sets (categories) and shading regions based on the statements. This provides a visual way to see the relationships and check if a conclusion is supported by the diagram.
- Logical Deduction: Using rules of logic to combine statements and derive new valid conclusions. This involves understanding terms like "all," "some," "no," and how they affect set relationships (subset, intersection, disjoint).
It is crucial to remember that in syllogisms, you must accept the statements as true, even if they contradict real-world knowledge. The validity of a conclusion depends solely on whether it logically follows from the given statements, not on its factual correctness in reality.
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.
