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.
Neither conclusion follows
Let's analyze the given statements and conclusions based on the principles of logical reasoning and syllogisms. We will assume the statements are true, even if they contradict common knowledge, and determine which conclusions logically follow.
Syllogism Statements and Conclusions
We are given two statements:
- Statement I: Some Z are X.
- Statement II: Some W are Z.
And two conclusions:
- Conclusion I: Some X are not W.
- Conclusion II: Some X are not Z.
Analyzing Statement I: Some Z are X
This statement tells us that there is at least one element that belongs to both the set Z and the set X. It establishes an overlap between Z and X. However, this statement does not provide any information about:
- Whether all Z are X.
- Whether all X are Z.
- The relationship between the elements of X that are not Z.
For example, if Z represents 'Animals' and X represents 'Dogs', the statement "Some Animals are Dogs" is true. In this case, all Dogs (X) are Animals (Z). So, there would be no Dogs (X) that are not Animals (Z).
Analyzing Statement II: Some W are Z
This statement tells us that there is at least one element that belongs to both the set W and the set Z. It establishes an overlap between W and Z. Similar to Statement I, it doesn't specify if all W are Z or if all Z are W, or the nature of elements in W that are not Z.
Combining Statements and Analyzing Conclusions
We have overlaps: (Z and X) and (W and Z). The term 'Z' is common to both statements, acting as a middle term. We need to see if these overlaps necessitate a specific relationship between W and X.
Evaluating Conclusion I: Some X are not W
Does the fact that Some Z are X and Some W are Z force a situation where Some X must not be W?
Let's consider possible scenarios using Venn diagrams or simple examples:
Scenario 1: Maximum overlap
Imagine that the sets W, Z, and X significantly overlap, or even that a portion of Z that is X is also part of W. It's possible that all the elements that are both Z and X are also part of W. In fact, it is even possible that all X are W. For example, let Z = Students, X = Those who scored > 90%, W = Those who received a scholarship. Statement I: Some Students are those who scored > 90%. (Some Z are X) - True. Statement II: Some who received a scholarship are Students. (Some W are Z) - True. Is Conclusion I necessarily true: Some who scored > 90% are not those who received a scholarship? (Some X are not W). Not necessarily. It could be that every student who scored > 90% also received a scholarship. In this specific scenario, "Some X are not W" would be false.
Since we found a possible scenario where Conclusion I is false while the statements are true, Conclusion I does not logically follow from the statements.
Evaluating Conclusion II: Some X are not Z
Does the fact that Some Z are X necessarily mean Some X are not Z?
Let's revisit Statement I: Some Z are X.
This statement confirms the existence of elements that are in the intersection of Z and X. It says nothing about the elements in X that are outside of this intersection. Consider the example used before: Z = Animals, X = Dogs. Statement I: Some Animals are Dogs. (Some Z are X) - True. Conclusion II: Some Dogs are not Animals. (Some X are not Z) - False, because all Dogs are Animals.
Since it is possible for Statement I to be true while Conclusion II is false (as shown in the Animals/Dogs example where all X are Z), Conclusion II does not logically follow from Statement I. Statement II (Some W are Z) is irrelevant to the relationship between X and Z in this context.
Summary of Conclusions
Based on our analysis:
- Conclusion I (Some X are not W) does not logically follow because we can construct scenarios where the statements are true but this conclusion is false.
- Conclusion II (Some X are not Z) does not logically follow because Statement I (Some Z are X) does not preclude the possibility that all X are Z.
Therefore, neither of the given conclusions logically follows from the given statements.
| Statement Type | Relationship |
|---|---|
| Some A are B | Partial overlap between A and B. Does not imply anything about 'Some A are not B' or 'Some B are not A', nor does it imply 'All A are B' or 'All B are A'. |
Revision Table: Syllogism Rules
When analyzing syllogisms with 'Some' statements:
- Two particular premises (like 'Some... are...' and 'Some... are...') do not yield any valid universal conclusions (like 'All...' or 'No...').
- Two particular affirmative premises ('Some... are...' and 'Some... are...') do not yield any valid particular negative conclusions (like 'Some... are not...').
- In standard syllogisms, two 'Some' premises do not guarantee a conclusion about the relationship between the extreme terms.
Additional Information: Understanding Syllogisms and Validity
A syllogism is a form of deductive reasoning where a conclusion is drawn from two given premises. The validity of a syllogism depends on its logical form, not on the truthfulness of the statements in the real world. A conclusion is valid if it must be true whenever the premises are true. If it is possible for the premises to be true and the conclusion to be false, then the conclusion is not valid.
Statements involving "Some" (like "Some A are B") indicate at least one, and potentially all. This ambiguity means we must consider all possible interpretations consistent with the statement when testing conclusions. If a conclusion is false in even one such interpretation where the premises are true, the conclusion is not logically valid.
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.
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.
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 tyre are white.
II. All grey are white.
Conclusions:
I. No white is tyre.
II. No tyre is grey.
III. Some white are not grey.
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.
