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. Statement : All shoes are slippers. All slippers are socks. All socks are sandals Conclusions: (I) Some sandals are slippers (II) Some socks are shoes. (III) All sandals are shoes.
Both Conclusion I and II follow.
Analyzing Syllogism Statements and Conclusions
This problem asks us to analyze logical conclusions drawn from given statements, a common task in syllogism questions. We assume the statements are true and determine which of the provided conclusions logically follow.
Given Statements:
- All shoes are slippers.
- All slippers are socks.
- All socks are sandals.
Given Conclusions:
- (I) Some sandals are slippers.
- (II) Some socks are shoes.
- (III) All sandals are shoes.
Solving Syllogism using Venn Diagrams
One effective way to solve syllogism problems is by using Venn diagrams. We represent each category (shoes, slippers, socks, sandals) as circles and show the relationships described in the statements.
Let's visualize the statements:
- All shoes are slippers: Draw a circle representing 'shoes' completely inside a larger circle representing 'slippers'.
- All slippers are socks: Draw the 'slippers' circle completely inside an even larger circle representing 'socks'.
- All socks are sandals: Draw the 'socks' circle completely inside the largest circle representing 'sandals'.
The resulting diagram will show concentric circles, with 'shoes' as the innermost, followed by 'slippers', then 'socks', and finally 'sandals' as the outermost circle.
Diagrammatic representation:
Sandals > Socks > Slippers > Shoes (where '>' means 'contains')
Evaluating the Conclusions based on the Diagram:
Conclusion (I): Some sandals are slippers.
Look at the Venn diagram. The circle for 'slippers' is located inside the circle for 'sandals'. If all slippers are within the sandals category, it means that the area covered by slippers is also part of the area covered by sandals. Therefore, there is an overlap, and specifically, all slippers are sandals. If all slippers are sandals, then it is definitely true that some sandals are slippers. This conclusion logically follows from the statements.
Conclusion (II): Some socks are shoes.
Look at the Venn diagram again. The circle for 'shoes' is located inside the circle for 'socks'. This means that all shoes fall within the category of socks. If all shoes are socks, then the area covered by shoes is part of the area covered by socks. This implies that there is an overlap, and specifically, all shoes are socks. If all shoes are socks, then it is definitely true that some socks are shoes. This conclusion logically follows from the statements.
Conclusion (III): All sandals are shoes.
Examine the Venn diagram. The circle for 'sandals' is the outermost circle, and the circle for 'shoes' is the innermost circle. The sandals circle is much larger than the shoes circle, and only a part of the sandals circle is occupied by shoes (specifically, the part that contains socks, slippers, and shoes). It is not true that the entire 'sandals' circle is inside the 'shoes' circle. Therefore, the statement "All sandals are shoes" does not logically follow from the given statements. This conclusion does not follow.
Summary of Conclusions:
| Conclusion | Follows? | Reasoning |
|---|---|---|
| (I) Some sandals are slippers. | Yes | Since all slippers are socks and all socks are sandals, all slippers are also sandals. If all slippers are sandals, then some sandals must be slippers. |
| (II) Some socks are shoes. | Yes | Since all shoes are slippers and all slippers are socks, all shoes are also socks. If all shoes are socks, then some socks must be shoes. |
| (III) All sandals are shoes. | No | The diagram shows sandals contain socks, which contain slippers, which contain shoes. It is not necessary for all sandals to be shoes. Only the sandals that are also socks (and thus slippers and shoes) are shoes. |
Based on our analysis, both Conclusion I and Conclusion II logically follow from the given statements, while Conclusion III does not.
Revision Table: Syllogism Basics
| Term | Explanation | Example Relationship |
|---|---|---|
| Statement | A proposition providing information about the relationship between two categories. | All A are B, Some A are B, No A are B, Some A are not B. |
| Conclusion | A proposition that is claimed to logically follow from the given statements. | Must be necessarily true if the statements are true. |
| Syllogism | A form of reasoning where a conclusion is drawn from two or more premises (statements). | Statements + Conclusion = Syllogism structure. |
Additional Information: Syllogism Rules and Types
Syllogisms are part of deductive reasoning. When solving syllogism problems, remember:
- Assume statements are true, even if they contradict real-world facts.
- The conclusion must be necessarily true based only on the statements.
- If there is any possibility, however small, for the conclusion to be false based on the statements, then it does not follow.
Common types of relationships in syllogisms:
- Universal Affirmative (All A are B): A is completely included in B. Implies Some A are B and Some B are A (if A is not empty).
- Universal Negative (No A are B): A and B are completely separate. Implies No B are A, Some A are not B, and Some B are not A.
- Particular Affirmative (Some A are B): There is at least one element common to A and B. Implies Some B are A.
- Particular Negative (Some A are not B): There is at least one element in A that is not in B. Does not imply Some B are not A.
In this specific problem, all statements were of the "All A are B" type, leading to nested inclusions in the Venn diagram.
Important Questions from 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:
All dancers are talented.
Some girls are dancers.
Conclusions:
I. Some girls are talented.
II. All talented are girls.
III. All girls are talented.
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 directors are actors.
No actor is a producer.
All choreographers are directors.
Conclusions:
I. No choreographer is producer.
II. Some actors are choreographers.
III. No director is a producer.
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 follows from the statements.
Statements:
All lemons are plums.
All plums are dates.
Some dates are mangoes.
Conclusions:
I. Some lemons are mangoes.
II. Some mangoes are plums.
III. All lemons are dates.
IV. Some mangoes are dates.
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 cards are postcards.
Some cards are envelopes.
All envelopes are copies.
Conclusions:
I. Some copies are envelopes.
II. Some postcards are copies.
III. Some cards are copies.
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 employees are tax-payers.
Some employees are farmers.
Some farmers are doctors.
Conclusions:
I. No farmer is a tax-payer.
II. Some farmers are tax-payers.
