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 N are K.
II. All K are P.
Conclusion:
I. Some P are not K.
II. No P is N.
Neither conclusion follows
Analyzing Logical Conclusions from Syllogism Statements
The question asks us to analyze two statements and determine which of the given conclusions logically follow based on these statements. We must assume the statements are true, even if they contradict common knowledge.
The given statements are:
- Statement I: All N are K.
- Statement II: All K are P.
The given conclusions are:
- Conclusion I: Some P are not K.
- Conclusion II: No P is N.
Step-by-Step Logical Analysis
Let's analyze the relationship between N, K, and P based on the given statements.
- Statement I says, "All N are K." This means that the set of N is completely contained within the set of K. Every element that is in N is also in K.
- Statement II says, "All K are P." This means that the set of K is completely contained within the set of P. Every element that is in K is also in P.
Combining these two statements, if all N are K, and all K are P, then it logically follows that all N must also be P. We can represent this relationship as a nested structure: N ⊂ K ⊂ P. The set N is inside K, and the set K is inside P.
Evaluating Conclusion I: Some P are not K
Conclusion I states, "Some P are not K." This conclusion claims that there exists at least one element in the set P that is not in the set K.
From the statements, we know that "All K are P." This means K is a subset of P. While this relationship allows for the possibility that P might be larger than K (meaning there could be some P outside of K), it does not guarantee it.
Consider a scenario where the set K is identical to the set P (K=P). In this case, "All K are P" is true (since all of K is indeed P). However, the conclusion "Some P are not K" would be false because every element in P is also in K. Since there is a possible scenario (consistent with the statements) where Conclusion I is false, it does not logically follow from the statements.
Evaluating Conclusion II: No P is N
Conclusion II states, "No P is N." This conclusion claims that there is no overlap between the set P and the set N. In other words, no element that is in P is also in N.
From our analysis of the statements, we deduced that "All N are P." This means that the set N is completely contained within the set P. Every element in N is also in P. This implies that there is a definite overlap between P and N; specifically, the entire set N is part of P.
The conclusion "No P is N" directly contradicts the logical deduction that "All N are P." Therefore, Conclusion II does not logically follow from the statements; in fact, it is false based on the given statements.
Summary of Conclusions Analysis
Based on the logical analysis of the statements:
- Conclusion I ("Some P are not K") is not necessarily true because it is possible for K and P to be the same set while still satisfying the statements.
- Conclusion II ("No P is N") is false because the statements logically imply that "All N are P".
Since neither conclusion can be definitively proven to be true based on the given statements, neither conclusion logically follows.
Revision Table: Syllogism Statements and Conclusions
| Statement/Conclusion | Details | Follows Logically? |
|---|---|---|
| Statement I: All N are K | N is a subset of K (N ⊂ K) | Given as True |
| Statement II: All K are P | K is a subset of P (K ⊂ P) | Given as True |
| Combined Logic | N is inside K, and K is inside P. Thus, N is inside P (N ⊂ P). It follows that All N are P. | Derived |
| Conclusion I: Some P are not K | Claims existence of P elements outside K. Not guaranteed by All K are P. Could be false if K=P. | No |
| Conclusion II: No P is N | Claims no overlap between P and N. Contradicts the derived logic All N are P. | No |
Additional Information on Syllogism and Logic
Syllogism is a form of logical reasoning where a conclusion is derived from two or more premises (statements). Categorical syllogisms, like this one, use statements about categories or sets (e.g., All N are K, Some P are not K).
Key types of categorical statements include:
- Universal Affirmative (A): All S are P (e.g., All dogs are animals)
- Universal Negative (E): No S is P (e.g., No dog is a cat)
- Particular Affirmative (I): Some S are P (e.g., Some dogs are friendly)
- Particular Negative (O): Some S are not P (e.g., Some dogs are not brown)
To determine if a conclusion logically follows, we check if the conclusion must be true whenever the premises are true. If there is any possibility, consistent with the premises, where the conclusion is false, then the conclusion does not logically follow.
In this problem, the statements are of type A. The conclusions are of type O and E. By using techniques like Venn diagrams (mentally or on paper) or set theory representation, we can visualize the relationships and test the validity of the conclusions.
Similar Questions
- Read the following statement carefully and identify the conclusion that follows.
Statement:
All poets are writers. Some writers are critics. No critic is a fool.
Conclusions:
I. Some poets are critics.
II. Some writers are not fools. - Statements: All dancers are artists. No artist is lazy.
Conclusions: I. No dancer is lazy. II. Some lazy are dancers. - Read the following statement carefully and identify the conclusion that follows.
Statement:
All oranges are fruits. No fruit is a vegetable. All vegetables are green.
Conclusions:
I. No orange is green.
II. No orange is a vegetable. - Read the following statement carefully and identify the conclusion that follows.
Statement:
All kids are students. Some students are athletes. No athlete is a scholar.
Conclusions:
I. Some kids may be athletes.
II. No kid is a scholar. - Statements: Some keys are locks. Some locks are doors.
Conclusions: I. Some keys are doors. II. All locks being keys is a possibility. - Read the following statement carefully and identify the conclusion that follows.
Statement:
All musicians are artists. Some artists are dancers. No dancer is blind.
Conclusions:
I. Some artists are not blind.
II. All blind are not dancers. Statements: Some pens are papers. All papers are books. Conclusions: I. Some pens are books. II. All books are pens.
Read the following statement carefully and identify the conclusion that follows.
Statement:
All birds are animals. Some animals are tigers. No tiger is a bird.
Conclusions:
I. Some tigers are birds.
II. Some animals are not birds.
- Statements: All dogs are cats. Some cats are rats. Conclusions: I. Some rats are dogs. II. All dogs are rats.
Read the following statement carefully and identify the conclusion that follows.
Statement:
All pens are blue things. Some blue things are plastic. No plastic is recyclable.
Conclusions:
I. Some pens are plastic.
II. Some blue things are not recyclable.
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:
Some students are players.
All players are male.
Conclusions:
I. All males are players.
II. Some males are students.
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:
No creative is an employer.
All experts are creative.
All workers are experts.
Conclusions:
I. No employer is an expert.
II. No worker is an employer.
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 actors are choreographers.
All choreographers are producers.
Not a single producer is a director.
Conclusions:
I. Some actors are directors.
II. Not a single actor is a director.
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 :
1. All flowers are tulips.
2. No tulips are whites.
Conclusions :
I. All tulips are flowers.
II. Some tulips are whites.
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 rats are dogs.
Some rats are hens.
Conclusions:
I. Some rats are dogs.
II. Some hens are rats.
