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 A are S. II. No D is A. Conclusions: I. Some S are A. II. All S are D. III. No A is D.
Both conclusions I and III follows
Understanding Syllogism Statements and Conclusions
This question asks us to analyze given statements and determine which of the provided conclusions logically follow from them. This type of problem falls under the category of logical reasoning, specifically syllogisms or statement-conclusion analysis.
Analyzing the Given Statements
We are given two statements:
- Statement I: All A are S.
- Statement II: No D is A.
We must assume these statements are true, even if they contradict common knowledge. Statement I tells us that the entire set of A is contained within the set of S. Statement II tells us that there is no overlap between the set of D and the set of A. In other words, A and D are mutually exclusive.
We can represent these statements using set notation:
- Statement I: \(A \subseteq S\)
- Statement II: \(D \cap A = \emptyset\) (which is equivalent to \(A \cap D = \emptyset\))
Evaluating the Given Conclusions
Now let's examine each conclusion based on the truth of the statements:
- Conclusion I: Some S are A.
- Conclusion II: All S are D.
- Conclusion III: No A is D.
Evaluation of Conclusion I: Some S are A
Statement I says "All A are S". If all members of set A are also members of set S, then it logically follows that there must be some members of S that are A (specifically, all the members of A are in S). For example, if all dogs are mammals, then some mammals are dogs. Therefore, Conclusion I logically follows from Statement I.
Using set notation: Since \(A \subseteq S\), if A is not empty, then there exist elements in S that are also in A. If A is empty, then the statement "All A are S" is vacuously true, and "Some S are A" would be false unless S is also empty (which is not implied). However, in syllogism, "Some" usually implies existence. A stronger interpretation of "All A are S" combined with "Some S are A" in standard syllogisms is that if "All A are S" is true, and A is non-empty, then "Some S are A" is true. If A is empty, "All A are S" is true, but "Some S are A" is false. However, logical deduction typically means what must be true if the premises are true. Since \(A \subseteq S\), the region representing A is inside S. The elements in the A region are S that are A. So, Conclusion I follows.
Evaluation of Conclusion II: All S are D
Statement I tells us A is inside S. Statement II tells us A and D are separate. These statements do not provide any information about the relationship between the entire set S and the set D. Some parts of S (specifically, the part that is A) are separate from D, but the part of S that is *not* A could potentially overlap with D, be entirely outside D, or even contain D. We cannot definitively conclude that all S are D based on the given information. Therefore, Conclusion II does not logically follow.
Evaluation of Conclusion III: No A is D
Statement II says "No D is A". This is a universal negative statement indicating that the set D and the set A have no elements in common. The statement "No A is D" is simply a rephrasing or equivalent form of "No D is A". If there is no overlap between D and A, there is also no overlap between A and D. Using set notation: \(D \cap A = \emptyset\) is equivalent to \(A \cap D = \emptyset\). Therefore, Conclusion III logically follows directly from Statement II.
Summary of Conclusions
- Conclusion I: Some S are A — Follows.
- Conclusion II: All S are D — Does not follow.
- Conclusion III: No A is D — Follows.
Based on our analysis, both Conclusion I and Conclusion III logically follow from the given statements.
| Conclusion | Follows? | Reasoning |
|---|---|---|
| I. Some S are A. | Yes | If all A are S, then some S must be A (the portion of S that is A). |
| II. All S are D. | No | Statements only define A's relation to S and D; S's relation to D is not fully determined. |
| III. No A is D. | Yes | This is an equivalent statement to "No D is A", which is given as Statement II. |
Therefore, the conclusions that follow are I and III.
Revision Table: Key Concepts in Syllogism
| Term | Explanation |
|---|---|
| Statement (Premise) | A proposition assumed to be true for the purpose of the argument. |
| Conclusion | A proposition that is claimed to follow logically from the premises. |
| Follows Logically | Means the conclusion must be true if the statements are true. |
| Universal Affirmative (All X are Y) | Every member of X is a member of Y. (\(X \subseteq Y\)) |
| Universal Negative (No X is Y) | No member of X is a member of Y. (X and Y are mutually exclusive, \(X \cap Y = \emptyset\)) |
| Particular Affirmative (Some X are Y) | At least one member of X is a member of Y. (\(X \cap Y \neq \emptyset\)) |
| Particular Negative (Some X are not Y) | At least one member of X is not a member of Y. |
Additional Information: Syllogism Validity
A syllogism is considered valid if its conclusion logically follows from its premises. We determined the validity of individual conclusions based on the given statements. In a formal syllogism, there are specific rules and figures that govern validity, but for basic statement-conclusion questions like this, direct logical deduction or using Venn diagrams is a common and effective method.
Venn diagrams are visual tools where circles represent sets (like A, S, D). Overlapping or contained circles show relationships described by the statements. We can draw diagrams for the statements and then check if the conclusions are necessarily true in those diagrams.
For Statement I (All A are S), we draw a circle for A completely inside a circle for S.
For Statement II (No D is A), we draw a circle for D completely separate from the circle for A.
Then we look at the conclusions:
- Conclusion I (Some S are A): In the diagram, the area representing A is inside S, showing some S are A. This holds.
- Conclusion II (All S are D): The diagram shows A is in S and A is separate from D. The part of S outside A could be anywhere relative to D. This does not necessarily hold.
- Conclusion III (No A is D): Since D is separate from A, A is also separate from D. This holds.
This visual method confirms our logical deduction that only Conclusions I and III follow.
Important Questions from Conventional Syllogism
In this question, three statements followed by two conclusions numbered I and II have been given. You have to take the given statements to be true even if they seem to be at variance from the commonly facts and then decide which of the given conclusions logically follows from the given statements disregarding commonly known facts.
Statements: Some flats are apartments.
No apartment is a hall.
Some halls are rooms.
Conclusions: I. At least some rooms are flats.
II. No apartment is a room.
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:
I. Some blue are red.
II. Some green are red.
Conclusions:
I. No blue is green.
II. No red is green.
Given below are two statements, Consider these statements to be true even if they seem factuality absurd, Read the conclusions and then decide which of the given conclusions logically follow(s) from the given statements?
Statements :
1. All vases are flowers.
2. No flowers is a plant.
Conclusions :
1. No vases is a plant.
2. Some plant are vases
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 land are hard.
II. No stone is land.
Conclusions:
I. Some hard are land.
II. Some stone are hard.
In the question two statements are given, followed by three conclusions, I, II and III. You have to consider the statements to be true even if it seems to be at variance from commonly known facts. You have to decide which of the given conclusions, if any, follows from the given statements.
Statement 1 : Some cars are scooters.
Statement 2 : All scooters are buses.
Conclusion I : Some scooters are cars.
Conclusion II : Some buses are cars.
Conclusion III : All cars are buses.
