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 S are B. II. Some B are C. Conclusions: I. All S are C. II. No C is B. III. All B are S.
Neither conclusion follows
Understanding Logic Statements and Conclusions
This question is based on the topic of Logic, specifically Syllogism, where we are given statements and asked to determine which conclusions logically follow from these statements. We must assume the given statements are true, even if they contradict common knowledge.
Analyzing the Given Statements
We have two statements:
- Statement I: All S are B.
- Statement II: Some B are C.
Let's break down what these statements mean:
- "All S are B" means that every element of set S is also an element of set B. In a Venn diagram, the circle representing S would be entirely inside the circle representing B.
- "Some B are C" means there is at least one element that is in set B and is also in set C. In a Venn diagram, the circles representing B and C overlap. The overlap area represents the elements that are both B and C.
Examining the Conclusions
Now let's look at each conclusion and see if it must be true based on the statements.
- Conclusion I: All S are C.
- Conclusion II: No C is B.
- Conclusion III: All B are S.
Step-by-Step Analysis of Each Conclusion
Let's analyze Conclusion I: All S are C.
- From Statement I, we know S is inside B.
- From Statement II, we know B and C overlap.
- Consider a scenario where the overlap between B and C is only in the part of B that is *outside* of S. In this case, even though some B are C, none of the S elements might be C.
- Therefore, "All S are C" does not necessarily follow from the statements. It might be true in some cases (e.g., if S is entirely within the B and C overlap), but it is not *always* true. For a conclusion to be logically valid, it must be true in *all* possible scenarios based on the statements.
Let's analyze Conclusion II: No C is B.
- From Statement II, we are explicitly told that "Some B are C".
- "Some B are C" means there is an intersection between sets B and C.
- "No C is B" means there is no intersection between sets C and B (they are mutually exclusive).
- These two ideas ("Some B are C" and "No C is B") directly contradict each other. Since Statement II is given as true, Conclusion II must be false.
- Therefore, "No C is B" does not follow from the statements.
Let's analyze Conclusion III: All B are S.
- From Statement I, we know "All S are B". This means S is a subset of B.
- This statement does not imply that B is a subset of S. There can be elements in B that are not in S.
- For example, if S = {1, 2} and B = {1, 2, 3}, then "All S are B" is true. But "All B are S" ({1, 2, 3} are {1, 2}) is false because 3 is in B but not in S.
- Therefore, "All B are S" does not necessarily follow from the statements.
Summary of Conclusion Analysis
Based on our analysis:
- Conclusion I ("All S are C") does not necessarily follow.
- Conclusion II ("No C is B") contradicts Statement II and therefore does not follow.
- Conclusion III ("All B are S") does not necessarily follow.
Since none of the given conclusions logically follow from the statements, the correct option is the one indicating that neither conclusion follows.
| Conclusion | Analysis Based on Statements | Logically Follows? |
|---|---|---|
| I. All S are C. | S is in B. Some B are C. S might be in the part of B not overlapping C. | No |
| II. No C is B. | Statement II says "Some B are C", directly contradicting this. | No |
| III. All B are S. | Statement I says All S are B (S <subset> B). Doesn't mean B <subset> S. | No |
Conclusion
None of the conclusions I, II, or III can be definitively proven to be true based solely on the given statements. Therefore, neither conclusion follows.
Revision Table: Key Concepts in Syllogism
| Term | Description | Example |
|---|---|---|
| Statement | A premise assumed to be true. | All dogs are mammals. |
| Conclusion | A judgment drawn from statements. | Some mammals are pets. > Conclusion: Some dogs are pets (not always true). |
| Syllogism | A logical argument where a conclusion is inferred from two or more statements. | Statements > Conclusion. |
| Categorical Proposition | A statement that relates two categories (or terms). | All S are P, No S is P, Some S are P, Some S are not P. |
Additional Information: Types of Logic Statements
In Syllogism, statements relate two categories using terms like 'All', 'No', and 'Some'. Understanding these types is crucial:
- Universal Affirmative (A): All S are P. (Every member of S is a member of P).
- Universal Negative (E): No S is P. (No member of S is a member of P).
- Particular Affirmative (I): Some S are P. (At least one member of S is a member of P).
- Particular Negative (O): Some S are not P. (At least one member of S is not a member of P).
Our given statements are of types A ("All S are B") and I ("Some B are C"). To determine valid conclusions, one often uses Venn diagrams or rules of inference to visualize or formally test the relationships between the categories mentioned in the statements.
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.
