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 A are M. II. No M is B. Conclusions: I. Some M are A. II. Some B are M. III. Some A are B.
Only conclusion I follows
Understanding Syllogism and Logical Conclusions
This question tests our ability to draw logical conclusions based on given statements. This is a classic problem in logical reasoning, often solved using Venn diagrams or rules of syllogism. We must assume the given statements are true, even if they contradict real-world knowledge.
Analyzing the Given Statements
Let's break down the statements provided:
- Statement I: All A are M. This means that the entire set of 'A' is contained within the set of 'M'.
- Statement II: No M is B. This means that the set of 'M' and the set of 'B' have absolutely no overlap. They are mutually exclusive.
We can visualize these relationships using Venn diagrams:
- Statement I implies a circle representing 'A' is completely inside a circle representing 'M'.
- Statement II implies the circle representing 'M' and a circle representing 'B' are separate and do not intersect.
Putting them together, the 'A' circle is inside the 'M' circle, and the 'M' circle has no connection to the 'B' circle. This implies the 'A' circle also has no connection to the 'B' circle.
Evaluating the Conclusions
Now, let's examine each conclusion based on the truth of the statements:
Conclusion I: Some M are A.
Statement I says "All A are M". If every single element of set A is also an element of set M, then it must logically follow that there are some elements in M that are also in A (at least all the elements that are in A). This conclusion is a direct and valid deduction from Statement I.
Evaluation: This conclusion follows.
Conclusion II: Some B are M.
Statement II says "No M is B". This establishes a complete separation between the set M and the set B. If no M is B, it means there is no overlap between M and B. The conclusion "Some B are M" claims that there is an overlap between B and M, which directly contradicts Statement II.
Evaluation: This conclusion does not follow.
Conclusion III: Some A are B.
From Statement I, we know "All A are M". From Statement II, we know "No M is B". Since all A are contained within M, and M has no relationship with B (specifically, no M is B), it means that nothing that is in M can be in B. As all A are in M, nothing that is in A can be in B. Therefore, "No A is B" is a valid conclusion. The conclusion "Some A are B" claims there is an overlap between A and B, which contradicts the derived conclusion "No A is B".
Evaluation: This conclusion does not follow.
Summary of Conclusion Analysis
Based on our analysis:
- Conclusion I (Some M are A) logically follows from the statements.
- Conclusion II (Some B are M) does not follow from the statements.
- Conclusion III (Some A are B) does not follow from the statements.
Therefore, only Conclusion I is a valid deduction from the given statements.
| Conclusion | Logical Deduction? | Reasoning |
|---|---|---|
| Some M are A | Yes | If All A are M, then Some M must be A. |
| Some B are M | No | Statement says No M is B, meaning no overlap. |
| Some A are B | No | Since All A are M and No M is B, then No A is B. |
Final Answer
Only conclusion I follows from the given statements.
Revision Table: Key Concepts in Syllogism
| Term | Definition | Relationship Example |
|---|---|---|
| Syllogism | A form of logical argument where a conclusion is derived from two premises (statements). | Statements $\implies$ Conclusion |
| Statement | A proposition considered true for the purpose of the argument. | "All A are M", "No M is B" |
| Conclusion | A judgment or decision reached by reasoning. | "Some M are A", "Some B are M" |
| Universal Affirmative (All X are Y) | Every member of the first category is also a member of the second. | All dogs are mammals. |
| Universal Negative (No X is Y) | No member of the first category is a member of the second. | No cat is a dog. |
| Particular Affirmative (Some X are Y) | At least one member of the first category is a member of the second. | Some students are engineers. |
| Particular Negative (Some X are not Y) | At least one member of the first category is not a member of the second. | Some birds are not sparrows. |
Additional Information: Rules of Inference
Understanding basic rules of inference helps solve syllogism problems. A key rule used here is that a universal affirmative statement ("All A are M") implies a particular affirmative statement ("Some M are A"). However, a universal negative statement ("No M is B") implies its converse ("No B is M"), but it does not imply any particular affirmative statement like "Some B are M".
Also, combining a universal affirmative and a universal negative statement requires careful consideration of the middle term (M in this case). If All A are M and No M is B, the connection through the middle term M establishes that there can be no connection between A and B.
Practice with different types of statements and conclusions helps in mastering logical deduction skills required for syllogism questions.
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.
