In the following question below are given some statements followed by some conclusions. 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 pen are black. II. Some pen are blue. III. No black is red. Conclusions: I. No pen is red. II. Some blue are black.
Only III
Analyzing Syllogism Statements and Conclusions
This question asks us to analyze a set of statements and determine which of the given conclusions logically follow from these statements, assuming the statements are true.
Understanding the Given Statements
We have three statements:
- Statement I: Some pen are black. This tells us there's an overlap between the set of 'pen' and the set of 'black' things.
- Statement II: Some pen are blue. This indicates an overlap between the set of 'pen' and the set of 'blue' things.
- Statement III: No black is red. This means the set of 'black' things and the set of 'red' things are completely separate; they have no elements in common.
Evaluating the Conclusions
Now let's examine each conclusion based on the provided statements:
Conclusion I: No pen is red.
Let's consider this conclusion. From Statement I, we know some pens are black. From Statement III, we know no black thing is red. This means the pens that are black cannot be red. However, the statements do not give us any information about the pens that are *not* black. These other pens (which could be blue, as per Statement II, or some other color not mentioned) could potentially be red. Since the statements do not rule out the possibility of some pens being red (specifically, the pens that are not black), the conclusion "No pen is red" does not logically follow from the given statements.
Conclusion II: Some blue are black.
Let's look at this conclusion. Statement I says "Some pen are black," and Statement II says "Some pen are blue." Both statements involve 'pen' as a common element. However, the fact that some pens are black and some pens are blue does not necessarily mean that some blue things are black. It's possible that the set of pens that are black is completely separate from the set of pens that are blue. For example, some pens could be *only* black, and some pens could be *only* blue, with no overlap between the black pens and the blue pens. Therefore, the conclusion "Some blue are black" does not logically follow from these statements.
Conclusion III: Some black are pens.
Let's analyze this conclusion. Statement I is "Some pen are black." This establishes a relationship where there is at least one element (or entity) that is both a 'pen' and 'black'. The structure "Some A are B" logically implies its converse, "Some B are A". If some members of the 'pen' category are also in the 'black' category, then it must be true that some members of the 'black' category are also in the 'pen' category. Therefore, the conclusion "Some black are pens" logically follows directly from Statement I.
Summary of Conclusions
- Conclusion I: Does not follow.
- Conclusion II: Does not follow.
- Conclusion III: Logically follows from Statement I.
Based on our analysis, only Conclusion III logically follows from the given statements.
| Conclusion | Logically Follows? | Reasoning |
|---|---|---|
| I. No pen is red. | No | Statements don't preclude non-black pens from being red. |
| II. Some blue are black. | No | Statements show overlap with 'pen', but not necessarily between 'blue' and 'black' directly. |
| III. Some black are pens. | Yes | Directly follows from the converse of Statement I ("Some pen are black"). |
Conclusion
Only Conclusion III is valid based on the provided statements. Therefore, the option indicating "Only III" is the correct choice.
Revision Table: Key Syllogism Rules
| Statement Type | Relation | Converse | Example (A=Cats, B=Mammals) |
|---|---|---|---|
| All A are B | A <span style="font-size: smaller;">⊂</span> B | Some B are A | All Cats are Mammals → Some Mammals are Cats |
| No A is B | A ∩ B = ∅ | No B is A | No Cats are Dogs → No Dogs are Cats |
| Some A are B | A ∩ B ≠ ∅ | Some B are A | Some Cats are Black → Some Black are Cats |
| Some A are not B | A ∖ B ≠ ∅ | (Not valid) | Some Cats are not Black → Cannot conclude Some Black are not Cats |
Additional Information: Syllogism Fundamentals
Syllogism is a form of logical reasoning where a conclusion is drawn from two or more statements (premises). In these types of problems:
- Assume the statements are absolutely true, regardless of real-world facts.
- The conclusion must *necessarily* follow from the statements; it cannot just be a possibility.
- Venn diagrams or other graphical methods can be helpful tools to visualize the relationships described in the statements and check the validity of conclusions.
- Pay close attention to quantifiers like "All," "No," and "Some." These words define the extent of the relationship between categories.
Understanding the valid conversions of statements (like "Some A are B" → "Some B are A") is crucial for solving syllogism problems efficiently and accurately.
Similar Questions
Read the given statements and conclusions carefully. Assuming that the information given in the statement 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 chairs are cupboards.
Some chairs are radios.
No chair is piano.
Conclusions:
I. Some radios are cupboards.
II. Some cupboards are not piano.
III. Some radios are not piano.
Read the given statements and conclusions carefully. Assuming that the information given in the statements is ture, 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 aero-planes are cars.
All helicopters are cars.
All cars are scooters.
Conclusions:
1. All helicopters are scooters.
2. Some aero-planes are helicopters.
3. No aero-plane is helicopter.
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 books are grapes.
Some grapes are fruits.
Conclusions:
I. No grape is a book.
II. Some grapes are books.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 fans are radio.
All fans are light.
Conclusion I: Some radios are light.
Conclusion II: Some lights are radio.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 scissors are spoons.
All spoons are plates.
Conclusions:
1. Some plates are spoons.
2. Some spoons are scissors.
3. Some plates are scissors.
Read the given statements and conclusions carefully. Assuming that the information given in the statement 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:
No boat is ship.
Some ships are rods.
Conclusions:
I. No boat is rod.
II. Some rods are ships.
III. Some rods are not boats.
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 bottles are glasses.
All glasses are jugs.
Conclusions:
I. All bottles are jugs.
II. All jugs are bottles.
III. Some glasses are bottles.
In the following question below are given some statements followed by some conclusions. 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 cups are plates.
II. All plates are red.
Conclusions:
I. All cups are red.
II. All plates are cups.
III. All red are plates.In the following question below are given some statements followed by some conclusions. 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 sky are blue.
II. Some blue are soft.
Conclusions:
I. Some sky are soft.
II. Some soft are sky.What will come in the place of the question mark (?) in the following equation if '+' and '-' are interchanged and 'x' and '÷' are interchanged?
108 × 2 − 14 ÷ 6 + 27 = ?
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.
