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 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.
Consider the given statement(s) to be true and decide which of the given conclusions/assumptions can definitely be drawn from the given statement.
Statement 1: All elephants are men.
Statement 2: All men are socks.
Conclusion I: Some socks are elephants.
Conclusion II: All elephants are socks.In the question, a word is represented by only one set of numbers as given in any one of the alternatives. The sets of numbers given in the question are represented by two classes of alphabets in two matrices given below. The columns and rows of Matrix 1 are numbered from 0 to 4 and that of Matrix II are numbered from 5 to 9. A letter from these matrices can be represented first by its row and next by its column, e.g., 'A' can be represented by 04, 12, etc., and 'U' can be represented by 59, 66, etc. Similarly, you have to identify the set for the word 'ROOT'.
MATRIX I
0 1 2 3 4 0 B C D E A 1 D E A B C 2 C D E A B 3 A B C D E 4 E A B C D MATRIX II
5 6 7 8 9 5 R O N T U 6 T U R O N 7 O N T U R 8 U R O N T 9 N T U R O Consider the given statement/s to be true and decide which of the given conclusions/assumptions can definitely be drawn from the given statement.
Statements: 1. All fish are tortoise.
2. No tortoise is a crocodile.
Conclusion: 1. No crocodile is a fish.
II. No fish is a crocodile.
Consider the given statements to be true and decide which of the given conclusions/assumptions can definitely be drawn from the given statement.
Statements: 1. All books are novels.
2. Some novels are poems.
Conclusions: I. Some books are poems.
II. Some poems are novels.Consider the statement to be true and decide which of the conclusions logically follow:
Statement:
All dogs are animals.
All animals are living beings.
Conclusion
I:Dog is not an animal.
II. Dog is an animal.In the question, a word is represented by only one set of numbers as given in any one of the alternatives. The sets of numbers given in the matrices are represented by two classes of alphabets as in two matrices given below. The columns and rows of Matrix I are numbered from 0 to 4 and that of Matrix II are numbered from 5 to 9. A letter from these matrices can be represented first by its row and next by its column, e.g., 'A' can be represented by 01, 13, etc., and 'S' can be represented by 55, 66, etc. Similarly, you have to identify the set for the word 'BOTH'.
MATRIX I
0 1 2 3 4 0 F A N O I 1 I O F A N 2 A N O I F 3 O F I N A 4 N I A F O MATRIX II
5 6 7 8 9 5 S E H B T 6 H S E T B 7 B T S E H 8 E H T B S 9 T S E H B Consider the given statement/s to be true and decide which of the given conclusions/assumptions can definitely be drawn from the given statement.
Statement I: All astronomers are scientists.
Statement II: Some scientists are shopkeepers.
Conclusion I: All astronomers are shopkeepers.
Conclusion II: All astronomers are scientists.
Conclusion III: Some shopkeepers are astronomers.
Conclusion IV: All scientists are astronomers.A word is represented by only one set of numbers as given in any one of the options. The sets of numbers are formed using two matrices below. The rows and columns of Matrix I are numbered 0 to 4 and Matrix II are numbered 5 to 9. A letter is represented by its row and column numbers. For example, ‘G’ can be coded as 12 or 21, and ‘L’ as 76 or 67.
Find the correct code for the word ‘LIFE’.
Consider the given statement/s to be true and decide which of the given conclusions/assumptions can definitely be drawn from the given statement:
Statements: Some cars are caves.
Conclusion: I. Some cars are not buses
II. Some caves are buses
III. No cave is a bus
IV. Some caves are cars
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.
