Three statements are given followed by three conclusions numbered I, II and III. Assuming the statements to be true, even if they seem to be at variance with commonly known facts, decide which of the conclusions logically follow(s) from the statements. Statements: All apples are pineapples. Some apples are mangoes. No mango is a grape. Conclusions: I. Some pineapples are not grapes. II. No pineapple is a grape. III. Some apples are not grapes.
Only conclusions I and III follow
Understanding Syllogism Statements and Conclusions
This question requires us to analyze three given statements and determine which of the three conclusions logically follow from these statements, assuming the statements are true.
Analyzing the Given Syllogism Statements
The statements provided are:
- All apples are pineapples.
- Some apples are mangoes.
- No mango is a grape.
We need to evaluate the given conclusions based on the logical implications of these statements.
Evaluating Syllogism Conclusions using Logical Deduction and Venn Diagrams
Let's break down the statements and examine each conclusion.
From statement 1 ("All apples are pineapples"), we know that the set of apples is completely contained within the set of pineapples.
From statement 2 ("Some apples are mangoes"), there is an overlap between the set of apples and the set of mangoes. This overlap area is also part of the set of pineapples because all apples are pineapples.
From statement 3 ("No mango is a grape"), the set of mangoes and the set of grapes have absolutely no overlap.
Now let's consider the conclusions:
Conclusion I: Some pineapples are not grapes.
We know from statement 2 that "Some apples are mangoes". Let's call this group of apples 'X'. So, X are apples and X are mangoes.
From statement 1, since X are apples, and "All apples are pineapples", it means X are also pineapples.
From statement 3, since X are mangoes, and "No mango is a grape", it means X are not grapes.
So, we have established that there is a group X which are pineapples and X are not grapes. This logically implies that "Some pineapples are not grapes".
Therefore, Conclusion I follows from the statements.
Conclusion II: No pineapple is a grape.
This statement claims that the entire set of pineapples has no overlap with the set of grapes. Let's consider the part of pineapples that are NOT apples (Pineapples - Apples). The given statements do not provide any information about the relationship between this part of pineapples and grapes. It is possible that some pineapples (which are not apples) could be grapes, or none of them could be grapes. Since the statements do not guarantee that NO pineapple is a grape, this conclusion does not logically follow.
For example, if there are pineapples that are not apples, the statements don't tell us anything about their relationship with grapes. A scenario where some non-apple pineapples are grapes is consistent with the statements. Thus, we cannot conclude that "No pineapple is a grape".
Therefore, Conclusion II does not follow from the statements.
Conclusion III: Some apples are not grapes.
From statement 2, we know that "Some apples are mangoes". Let's again call this group of apples 'X'.
From statement 3, we know that "No mango is a grape". Since X are mangoes, it means X are not grapes.
So, we have established that there is a group of apples (X) which are not grapes. This logically implies that "Some apples are not grapes".
Therefore, Conclusion III follows from the statements.
Summary of Conclusions
Let's summarize our findings in a table:
| Conclusion | Logically Follows? | Reasoning |
|---|---|---|
| I. Some pineapples are not grapes. | Yes | Some apples are mangoes (Statement 2), these apples are also pineapples (Statement 1). These mangoes/apples/pineapples are not grapes (Statement 3). Thus, some pineapples are not grapes. |
| II. No pineapple is a grape. | No | While some pineapples (those that are apples and mangoes) are not grapes, the statements don't preclude the possibility that other pineapples (those not included in the 'some apples' group, or pineapples that are not apples at all) could be grapes. |
| III. Some apples are not grapes. | Yes | Some apples are mangoes (Statement 2), and no mango is a grape (Statement 3). Therefore, these 'some apples' are not grapes. |
Based on our analysis, only conclusions I and III logically follow from the given statements.
Revision Table: Syllogism Concepts
| Concept | Description |
|---|---|
| Syllogism | A type of logical argument where a conclusion is derived from two or more propositions (statements). |
| Statements (Premises) | The propositions given as true, from which conclusions are drawn. |
| Conclusion | The proposition that is inferred from the statements. |
| Logical Follows | A conclusion logically follows if it must be true whenever the statements are true. If there's any possibility, consistent with the statements, that the conclusion is false, then it does not logically follow. |
Additional Information: Working with Syllogisms
Solving syllogism problems often involves visualizing the relationships between the categories mentioned in the statements. Venn diagrams are a common tool for this. Each category (like apples, pineapples, mangoes, grapes in this problem) can be represented by a circle, and the relationships between the categories (All, Some, No) dictate how these circles overlap or don't overlap.
- "All A are B": The circle for A is drawn completely inside the circle for B.
- "Some A are B": The circles for A and B are drawn with an overlapping region, indicating that there are elements common to both categories.
- "No A is B": The circles for A and B are drawn separately, with no overlap, indicating that there are no elements common to both categories.
- "Some A are not B": The circles for A and B are drawn with an overlapping region, but also with a part of A outside of B, indicating that there are elements in A that are not in B.
By drawing diagrams that represent all the statements simultaneously, you can then check if a conclusion must be true in all possible valid diagrams. If a conclusion is true in one diagram but false in another valid diagram, it does not logically follow.
In this specific problem, drawing circles for Apples (A), Pineapples (P), Mangoes (M), and Grapes (G) helps. A inside P. A and M overlapping. M and G separate. This visualization confirms that the A&M overlap area (which is inside P) is not in G, supporting conclusions I and III, while showing a possibility for P (outside A) overlapping with G, which negates conclusion II.
Similar Questions
Two statements are given, followed by three conclusions numbered I, II and III. Assuming the statements to be true, even if they seem to be at variance with commonly known facts, decide which of the conclusions logically follow(s) from the statements.
Statements:
All teachers are engineers.
All clerks are teachers.
Conclusions:
I. Some engineers are clerks.
II. Some teachers are clerks.
III. Some clerks are not engineers.
Two Statements are given followed by Two conclusions numbered I and II. Assuming the statements to be true, even if they seem to be at variance with commonly known facts, decide which of the conclusions logically follow(s) from the statements.
Statements:
All paintings are calligraphies.
All sketching are calligraphies .
Conclusions:
I. At least some calligraphies are paintings.
II. No sketching is a painting.
Two statements are given followed by two conclusions numbered I and II. Assuming the statements to be true, even if they seem to be at variance with commonly known facts, decide which of the conclusions logically follow(s) from the statements.
Statements:
Some vehicles are helicopters.
All helicopters are cars.
Conclusions:
I. All vehicles are cars.
II. Some vehicles are cars.
Given below are two statements and two conclusions. Take the statements to be true even if they are at variance with commonly known facts, and decide whether the conclusion(s) follow(s) the given statements.
Statements:
All eyes are noses.
Some noses are lips.
Conclusions:
I. Some noses are eyes.
II. All lips are noses.
Two statements are given followed by conclusions numbered 1 and 2. Assuming the statements to be true, even if they seem to be at variance with commonly known facts, decide which of the conclusions logically follow(s) from the statements.
Statements:
1. Some mammals are animals.
2. All animals are birds.
Conclusions:
1. All birds are animals.
2. Some mammals are birds.
Two statements are given followed by two conclusions numbered I and II. Assuming the statements to be true, even if they seem to be at variance with commonly known facts, decide which of the conclusions logically follow(s) from the statements.
Statements:
All keys are locks.
All locks are handles.
Conclusions:
I. Some locks are keys.
II. Some handles are keys.
Three Statements are given followed by Three conclusions numbered I, II and III. Assuming the statements to be true, even if they seem to be at variance with commonly known facts, decide which of the conclusions logically follow(s) from the statements.
Statements:
All switches are fans.
No heater is a fan.
Some wires are heaters.
Conclusions:
I. Some wires are fans.
II. No switch is a heater.
III. No wire is a switch.
Three statements are given followed by two conclusions numbered I and II. Assuming the statements to be true, even if they seem to be at variance with commonly known facts, decide which of the conclusions logically follow(s) from the statements.
Statements:
All H are G.
All V are H.
Some V are I.
Conclusions:
I. At least some G are I.
II. All I being H is a possibility.
Three statements are given followed by three conclusions numbered I, II and III. Assuming the statements to be true, even if they seem to be at variance with commonly known facts, decide which of the conclusions logically follow(s) from the statements.
Statements:
No river is a sea.
Some rivers are lakes.
All rivers are ponds.
Conclusions:
I. Some lakes are seas.
II. Some ponds are seas.
III. Some lakes are ponds.
Two statements are given followed by two conclusions numbered I and II. Assuming the statements to be true, even if they seem to be at variance with commonly known facts, decide which of the conclusions logically follow(s) from the statements.
Statements:
All black is white.
Some white is grey.
Conclusions:
I. Some grey is black.
II. Some white is black.
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. 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.
