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.
Neither conclusion I nor II follows
Understanding the Statements in Logic Questions
In logic questions like this, we are given statements and conclusions. Our job is to figure out which conclusions must absolutely be true if we assume the statements are true, even if they contradict things we know in the real world. We need to follow the logic strictly based on the provided statements.
Let's break down the given statements:
- Statement I: Some blue are red. This means there is at least one thing that is both blue and red. It doesn't tell us anything about all blue things or all red things, only about an overlap between the two groups.
- Statement II: Some green are red. This means there is at least one thing that is both green and red. Similar to the first statement, it indicates an overlap between the green group and the red group.
Analyzing the Proposed Conclusions
Now, let's look at the conclusions we need to evaluate:
- Conclusion I: No blue is green. This claims there is absolutely no overlap between the blue group and the green group.
- Conclusion II: No red is green. This claims there is absolutely no overlap between the red group and the green group.
Detailed Logical Deduction Process
To check if a conclusion logically follows, we need to see if it must be true under all possible scenarios allowed by the statements. If we can find even one scenario where the statements are true but the conclusion is false, then the conclusion does not logically follow.
Let's consider the relationship between blue and green based on the statements:
- Statement I tells us about the relationship between blue and red.
- Statement II tells us about the relationship between green and red.
There is no direct statement connecting blue and green. Their relationship is only indirectly mentioned through their connection to red ("Some blue are red", "Some green are red").
Let's test the conclusions:
- Conclusion I: No blue is green. The statements tell us some blue are red, and some green are red. This leaves open the possibility that some of the blue things that are red might also be green, or that some blue things that are *not* red could overlap with green things that are *not* red, or that blue and green could completely overlap, partially overlap, or not overlap at all. Since the statements do not *force* blue and green to have no overlap, this conclusion does not logically follow. For example, it's possible that some items are blue, red, AND green. This would satisfy both statements but make conclusion I false.
- Conclusion II: No red is green. This conclusion states that there is no overlap between red and green. However, Statement II explicitly says, "Some green are red." "Some green are red" means that the intersection of the green set and the red set is not empty. "No red is green" means the intersection of the red set and the green set is empty. These two statements directly contradict each other. Since Statement II is assumed to be true, Conclusion II must be false. Therefore, this conclusion does not logically follow.
Conclusion Assessment
Based on the analysis:
- Conclusion I ("No blue is green") does not logically follow because the statements allow for scenarios where blue and green do overlap.
- Conclusion II ("No red is green") directly contradicts Statement II ("Some green are red") and therefore cannot logically follow if Statement II is true.
Since neither conclusion is guaranteed to be true based solely on the statements, neither conclusion logically follows.
Revision Table: Analyzing Syllogism Conclusions
| Statement / Conclusion | Type | Relation | Logical Status based on Statements |
|---|---|---|---|
| Statement I: Some blue are red. | Particular Affirmative (I) | Blue $\cap$ Red $\neq \emptyset$ | Assumed True |
| Statement II: Some green are red. | Particular Affirmative (I) | Green $\cap$ Red $\neq \emptyset$ | Assumed True |
| Conclusion I: No blue is green. | Universal Negative (E) | Blue $\cap$ Green $= \emptyset$ | Does not follow (Possible but not necessary) |
| Conclusion II: No red is green. | Universal Negative (E) | Red $\cap$ Green $= \emptyset$ | Does not follow (Directly contradicts Statement II) |
Additional Information: Syllogism Basics
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.
- Statements: These provide the premises. They typically involve quantifiers like "All", "Some", and "No", and relate different categories.
- Conclusions: These are inferences drawn from the statements.
- Validity: A conclusion is valid if it must be true whenever the premises (statements) are true. It's about the structure of the argument, not the real-world truth of the statements.
- 'Some' statements: "Some A are B" means there is at least one A that is also B. It does NOT mean "Some but not all". It is compatible with "All A are B".
- 'No' statements: "No A is B" means there is no overlap between A and B.
- Venn Diagrams: A helpful tool to visualize the relationships between categories described in the statements and check if a conclusion holds true in all possible valid diagrams.
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.
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. 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.
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.
