Three statements are given followed by three conclusions numbered 1, 2 and 3. 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 shoes are slippers. Some slippers are socks. All socks are sandals. Conclusions: 1. Some sandals are shoes. 2. Some sandals are slippers. 3. Some socks are shoes.
Only conclusion 2 follows
Analyzing Syllogism Statements and Conclusions
This question asks us to determine which of the given conclusions logically follow from the provided statements, assuming the statements are true. This type of problem is a common part of logical reasoning and is often solved using techniques like Venn diagrams or rules of syllogism.
Given Statements:
- Some shoes are slippers.
- Some slippers are socks.
- All socks are sandals.
Given Conclusions:
- Some sandals are shoes.
- Some sandals are slippers.
- Some socks are shoes.
Step-by-Step Analysis using Venn Diagrams
We can visualize the relationships described in the statements using Venn diagrams. Let's represent each category (Shoes, Slippers, Socks, Sandals) with a circle.
Statement 1: Some shoes are slippers.
This means there is an overlap between the circle representing 'Shoes' and the circle representing 'Slippers'.
Statement 2: Some slippers are socks.
This means there is an overlap between the circle representing 'Slippers' and the circle representing 'Socks'.
Statement 3: All socks are sandals.
This means the entire circle representing 'Socks' is contained within the circle representing 'Sandals'.
Now, let's evaluate each conclusion based on these relationships:
Evaluating Conclusion 1: Some sandals are shoes.
From the statements, we know 'Some shoes are slippers' and 'Some slippers are socks', and 'All socks are sandals'. The overlap between 'Shoes' and 'Slippers' does not necessarily mean 'Shoes' overlap with 'Socks' or 'Sandals'. While it's possible for the 'Shoes' circle to overlap with the 'Sandals' circle (via the parts that are 'Socks' or 'Slippers'), it is not guaranteed. We can draw a scenario where 'Shoes' and 'Sandals' have no direct overlap while still satisfying all three statements. Therefore, this conclusion does not logically follow.
Evaluating Conclusion 2: Some sandals are slippers.
We have the statements 'Some slippers are socks' and 'All socks are sandals'. If all socks are inside the group of sandals, and some slippers are part of the group of socks, then those specific slippers must also be part of the group of sandals. This directly implies that some slippers are sandals, or conversely, some sandals are slippers. This conclusion logically follows from statements 2 and 3.
Evaluating Conclusion 3: Some socks are shoes.
We have 'Some shoes are slippers' and 'Some slippers are socks'. This describes a relationship between Shoes and Socks mediated by Slippers. However, a 'Some A are B' and 'Some B are C' structure does not guarantee a 'Some A are C' or 'Some C are A' relationship. We can draw the circles such that the overlap between 'Slippers' and 'Shoes' is separate from the overlap between 'Slippers' and 'Socks'. Thus, 'Socks' and 'Shoes' might not overlap at all while satisfying the statements. Therefore, this conclusion does not logically follow.
Summary of Conclusion Analysis
| Conclusion | Statements Involved | Logically Follows? | Reasoning |
|---|---|---|---|
| 1. Some sandals are shoes. | 1, 2, 3 | No | Not a guaranteed overlap between Shoes and Sandals based on given statements. |
| 2. Some sandals are slippers. | 2, 3 | Yes | If some slippers are socks and all socks are sandals, then some slippers must be sandals (and thus, some sandals are slippers). |
| 3. Some socks are shoes. | 1, 2 | No | 'Some A are B' and 'Some B are C' does not guarantee 'Some A are C'. |
Based on the analysis, only Conclusion 2 logically follows from the given statements.
Revision Table: Key Syllogism Concepts
| Term | Definition | Example Statement Type |
|---|---|---|
| Statement | A premise assumed to be true in a syllogism. | "All A are B", "Some A are B", "No A are B", "Some A are not B" |
| Conclusion | A judgment or decision reached by reasoning from the statements. | Derived from the combination of two or more statements. |
| Middle Term | The term that appears in both statements but not in the conclusion. | In "All A are B" and "All B are C", 'B' is the middle term. |
| Syllogism | A form of logical reasoning where a conclusion is derived from two or more statements. | The entire structure of statements and conclusion. |
Additional Information: Types of Statements in Syllogism
Syllogism problems typically use four main types of statements, often referred to as Categorical Propositions:
- Universal Affirmative (A-type): All S are P. (e.g., All dogs are mammals)
- Universal Negative (E-type): No S are P. (e.g., No cats are dogs)
- Particular Affirmative (I-type): Some S are P. (e.g., Some students are intelligent)
- Particular Negative (O-type): Some S are not P. (e.g., Some birds are not black)
In the given problem, we encountered I-type statements ('Some shoes are slippers', 'Some slippers are socks') and an A-type statement ('All socks are sandals'). The logical combination rules and Venn diagrams help determine what conclusions can be validly drawn from these combinations.
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.
