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.
Analyzing Syllogism Statements and Conclusions
This question requires us to analyze the logical relationship between two statements and two conclusions. We must assume the given statements are true, even if they contradict common knowledge, and determine which conclusions logically follow.
Understanding the Given Statements
Let's break down the provided statements:
- Statement I: No A is B.
- This statement tells us that there is absolutely no overlap between the set of A and the set of B. They are completely separate or disjoint.
- In terms of relationships, if something belongs to A, it cannot belong to B, and vice versa.
- Statement II: No C is B.
- This statement tells us that there is absolutely no overlap between the set of C and the set of B. They are also completely separate or disjoint.
- If something belongs to C, it cannot belong to B, and vice versa.
Both statements establish a relationship of mutual exclusion with the set B, but they do not directly tell us anything about the relationship between A and C.
Evaluating the Conclusions
A conclusion logically follows from the statements only if it is true in *every possible scenario* where the statements are true. If we can find even one scenario where the statements are true but the conclusion is false, then the conclusion does not logically follow.
Conclusion I: Some C are not A.
This conclusion claims that there are some elements in C that do not belong to A. Let's consider the relationship between A and C based on the given statements:
We know A is disjoint from B ($\text{A} \cap \text{B} = \emptyset$), and C is disjoint from B ($\text{C} \cap \text{B} = \emptyset$). These statements restrict A and C only in relation to B. They don't put any constraints on the relationship between A and C.
Let's visualize possible relationships between A and C (while keeping A and C disjoint from B):
- Scenario 1: A and C are completely separate (disjoint).
- If No A is C, then it is true that Some C are not A (in fact, all C are not A, and 'all' implies 'some'). In this scenario, the conclusion is true.
- Scenario 2: A and C overlap.
- If Some A are C and Some C are A, then there are parts of C that are not in A. In this scenario, the conclusion "Some C are not A" is true.
- Scenario 3: C is a subset of A.
- If All C are A, then every element of C is also an element of A. In this scenario, there are no elements in C that are not in A. The conclusion "Some C are not A" is false.
- Scenario 4: A is a subset of C.
- If All A are C, then there are elements in C that are not in A (unless C is empty, which is usually not assumed in syllogisms unless explicitly stated or implied). In this scenario, the conclusion is true.
Since we found a scenario (Scenario 3, where C is a subset of A) where the statements are true but Conclusion I ("Some C are not A") is false, Conclusion I does not logically follow from the statements.
Conclusion II: Some A are B.
This conclusion claims that there is at least one element that belongs to both A and B.
Look back at Statement I: "No A is B". This statement explicitly says that there are *no* elements in common between A and B. Conclusion II ("Some A are B") is the direct contradiction (or rather, the existential affirmation contradicting the universal negation) of Statement I.
If Statement I is true (which we must assume), then Conclusion II must be false. Therefore, Conclusion II does not logically follow from the statements.
Final Decision
Based on our analysis, neither Conclusion I nor Conclusion II logically follows from the given statements because:
- Conclusion I is not necessarily true in all possible scenarios where the statements hold.
- Conclusion II directly contradicts Statement I.
Therefore, neither conclusion follows.
Revision Table: Syllogism Analysis Summary
| Statement/Conclusion | Description | Follows? | Reasoning |
|---|---|---|---|
| Statement I | No A is B ($\text{A} \cap \text{B} = \emptyset$) | Given | Assumed true as per question instructions. |
| Statement II | No C is B ($\text{C} \cap \text{B} = \emptyset$) | Given | Assumed true as per question instructions. |
| Conclusion I | Some C are not A ($\exists \text{x} : \text{x} \in \text{C} \land \text{x} \notin \text{A}$) | No | Can be false if C is a subset of A. The statements provide no direct link between A and C. |
| Conclusion II | Some A are B ($\exists \text{x} : \text{x} \in \text{A} \land \text{x} \in \text{B}$) | No | Directly contradicts Statement I ("No A is B"). |
Additional Information on Logical Syllogisms
Syllogisms are a type of logical argument that applies deductive reasoning to arrive at a conclusion based on two or more propositions that are asserted or assumed to be true. Here are some key concepts related to this problem:
- Statements (Premises): These are the propositions given as true. In this case, "No A is B" and "No C is B".
- Conclusions: These are propositions that are supposed to follow logically from the statements.
- Logical Consequence: A conclusion follows logically from statements if and only if it is impossible for the statements to be true and the conclusion false simultaneously. If there's even one possibility where statements are true but the conclusion is false, the conclusion does not follow.
- Disjoint Sets: The statement "No X is Y" means the sets X and Y have no members in common. Their intersection is empty ($\text{X} \cap \text{Y} = \emptyset$). Both statements in this problem describe disjoint relationships with the set B.
- Relationship between Universal Negation and Existential Affirmation:
- "No A is B" is a universal negation (applies to all A and all B regarding their intersection).
- "Some A are B" is an existential affirmation (claims existence of at least one element in the intersection).
- These two statements are contradictory. If one is true, the other must be false.
- Undefined Relationships: If statements do not provide a direct link between two terms (like A and C in this case), then conclusions involving only those two terms usually do not follow unless they are derivable through an intermediate term (which isn't the case here in a way that necessitates the conclusion).
Understanding these principles is crucial for solving syllogism problems correctly. We must rely only on the information given in the statements and the rules of logic.
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
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.
