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.
Similar Questions
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.
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.
Statements:
Some spices are medicines.
All medicines are chemicals.
No chemical is cheap.
Conclusions:
I. Some medicines are cheap.
II. Some chemicals are spices.
III. No medicine is cheap.
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.
Statements:
All oranges are apples.
Some oranges are bananas.
All bananas are fruits.
Conclusions:
I. Some fruits are apples.
II. Some apples are bananas.
III. Some oranges are fruits.
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. Some Z are X.
II. Some W are Z.
Conclusions:
I. Some X are not W.
II. Some X are not Z.
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.
Statements:
Some mobiles are flat.
All flats are TVs.
Some TVs are plastic.
Conclusions:
I. Some mobiles are TVs.
II. Some flats are plastic.
III. All plastic are TVs.
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.
Statements:
No cup is a plate.
All cups are utensils.
Some bottles are cups.
Conclusions:
I. Some utensils are not plates.
II. Some bottles are not plates.
III. No bottle is a plate.
Three statements are followed by three conclusions numbered I, II and III. You have to consider these statements to be true, even if they seem to be at variance with commonly known facts. Decide which of the given conclusions logically follow/s from the given statements.
Statements:
Some watches are projectors.
All projectors are lamps.
Some heaters are watches.
Conclusions:
(I) Some heaters are projectors.
(II) Some lamps are heaters.
(III) All watches are lamps
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. No A is B.
II. No C is B.
Conclusion:
I. Some C are not A.
II. Some A are B.
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 benches are tables.
Some tables are chairs.
Some chairs are pencils.
Conclusions:
I. Some benches are chairs.
II. Some tables are pencils.
III. Some pencils are chairs.
IV. Some pencils are tables.
Important Questions from Syllogism
The statements below are followed by conclusions labeled I, II and III. Assuming that the information in the statements is true, even if it appears to be at variance with generally established facts, decide which conclusion(s) logically and definitely follow(s) from the information given in the statements.
Statements:
All twenty are thirty.
All thirty are forty.
All forty are sixty.
All sixty are seventy.
Conclusions:
I. Some forty are thirty.
II. Some seventy are sixty.
III. No thirty is twenty.The statements below are followed by two conclusions labelled I and II. Assuming that the information in the statements is true, even if it appears to be at variance with generally established facts, decide which conclusion(s) logically and definitely follow(s) from the information given in the statements.
Statements:
All Strong are animals.
Some animals are Tigers.
All Tigers are Sharp.
Conclusions:
I. Some Strong are Sharp.
II. No Strong is Sharp.The statements below are followed by conclusions labelled I, II and III. Assuming that the information in the statements is true, even if it appears to be at variance with generally established facts, decide which conclusion(s) logically and definitely follow(s) from the information given in the statements.Statements:
Some women are weak.
Some weaks are female.
All female are iron.
All iron are gold.Conclusions:
I. Some weaks are iron.
II. Some gold are weaks.
III. Some women are female.
The statements below are followed by two conclusions labeled I and II. Assuming that the information in the statements is true, even if it appears to be at variance with generally established facts, decide which conclusion(s) logically and definitely follow(s) from the information given in the statements.Statements:
Some teaspoons are glasses.
All teddies are teaspoons.
Conclusions:
I. Some teddies are glasses.
II. Some glasses are teddies.
The statements below are followed by two conclusions labeled I and II. Assuming that the information in the statements is true, even if it appears to be at variance with generally Established facts, decide which conclusion(s) logically and definitely follow(s) from the Information, Given in the Statements.
Statements:
All bangles are rings.
Some rings are toys.
Some toys are dolls.
Conclusions:
I. Some rings are bangles.
II. Some dolls are rings.
