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.
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.
