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 pictures are prizes. Some prizes are erasers. Some erasers are helmets. Conclusions: I. Some helmets are Prizes. II. Some erasers are pictures. III. No helmet is a Prize.
Either conclusion I or III follows
Solving Syllogism Statements and Conclusions
This question requires us to analyze given statements and determine which conclusions logically follow. The statements involve relationships described by "Some", which indicate a partial overlap between categories. Let's break down the statements and conclusions step by step.
Analyzing the Statements
We are given the following statements:
- Some pictures are prizes. (This means there is an overlap between the set of pictures and the set of prizes. Some pictures are also prizes, and consequently, some prizes are also pictures.)
- Some prizes are erasers. (Similarly, there's an overlap between prizes and erasers. Some prizes are erasers, and some erasers are prizes.)
- Some erasers are helmets. (This indicates an overlap between erasers and helmets. Some erasers are helmets, and some helmets are erasers.)
In statements involving "Some", we only know about a partial relationship. We cannot conclude anything definitive about the relationship between elements that are not directly linked or linked only through a single 'Some' connection in a chain. For example, knowing "Some A are B" and "Some B are C" does not guarantee that "Some A are C" or "No A is C". The relationship between A and C is uncertain.
Analyzing the Conclusions
Now let's evaluate each conclusion based on the given statements:
- Conclusion I: Some helmets are Prizes.
The statements provide links from Helmets to Erasers ("Some erasers are helmets") and from Erasers to Prizes ("Some prizes are erasers"). The chain is Helmets <-> Erasers <-> Prizes. Since both connections are "Some", we cannot definitively conclude that there is any overlap between Helmets and Prizes. It's possible that the overlapping part of Erasers (which are Helmets) and the overlapping part of Erasers (which are Prizes) are different parts of the Erasers set, resulting in no direct overlap between Helmets and Prizes. However, it is also possible that there is an overlap. Thus, this conclusion does not necessarily follow. This conclusion is a possibility, but not a certainty.
- Conclusion II: Some erasers are pictures.
The statements link Erasers to Prizes ("Some prizes are erasers") and Prizes to Pictures ("Some pictures are prizes"). The chain is Erasers <-> Prizes <-> Pictures. Again, both connections are "Some". Similar to Conclusion I, we cannot definitively conclude a relationship between Erasers and Pictures. The overlapping part of Prizes (which are Erasers) and the overlapping part of Prizes (which are Pictures) might not coincide. This conclusion does not necessarily follow. It is a possibility, but not a certainty.
- Conclusion III: No helmet is a Prize.
This conclusion is a negative statement claiming there is absolutely no overlap between Helmets and Prizes. As discussed in Conclusion I, based on the "Some" statements linking Helmets, Erasers, and Prizes, we cannot definitively say whether there is an overlap or not. Conclusion I stated "Some helmets are Prizes" (implying overlap is possible). Conclusion III states "No helmet is a Prize" (implying no overlap is possible). Since we couldn't definitively prove "Some helmets are Prizes" from the statements, we also cannot definitively prove "No helmet is a Prize". This conclusion does not necessarily follow. It is a possibility, but not a certainty.
Identifying the Either-Or Case
Let's look closely at Conclusions I and III:
- Conclusion I: Some helmets are Prizes.
- Conclusion III: No helmet is a Prize.
These two conclusions form a complementary pair (also known as a contradictory pair). One asserts that there is at least some overlap ("Some A are B"), while the other asserts there is absolutely no overlap ("No A is B"). For any two categories, either there is *some* overlap (or total overlap), or there is *no* overlap. There is no third possibility. Since our analysis showed that we cannot definitively conclude *either* Conclusion I or Conclusion III from the statements (both are possibilities but not certainties), one of them *must* be true. This is the 'either-or' condition.
Conclusion II ("Some erasers are pictures") is also a possibility but not a certainty. However, it does not form a complementary pair with any other given conclusion (we don't have "No eraser is a picture"). Therefore, Conclusion II stands alone as a conclusion that might or might not follow.
Based on the rules of syllogism and the analysis of "Some" statements, when a positive ('Some') and a negative ('No') conclusion regarding the same two categories are both individually uncertain (cannot be definitely concluded), then the 'either-or' condition applies to that pair.
In this case, Conclusions I (Some helmets are Prizes) and III (No helmet is a Prize) are individually uncertain, and they form a complementary pair. Therefore, either Conclusion I or Conclusion III must follow.
Conclusion II is uncertain and does not form a complementary pair with Conclusion I or III. Therefore, Conclusion II does not necessarily follow on its own or as part of an either-or case with the other two conclusions.
Thus, the logical inference is that either Conclusion I or Conclusion III follows from the statements.
| Conclusion | Relationship | Analysis from Statements | Follows? |
|---|---|---|---|
| I. Some helmets are Prizes. | Helmets <-> Erasers <-> Prizes | Indirect link via 'Some' statements. Cannot be certain. | Cannot be Determined (CBD) |
| II. Some erasers are pictures. | Erasers <-> Prizes <-> Pictures | Indirect link via 'Some' statements. Cannot be certain. | Cannot be Determined (CBD) |
| III. No helmet is a Prize. | Helmets <-> Erasers <-> Prizes | Negation of Conclusion I. Cannot be certain. | Cannot be Determined (CBD) |
Since Conclusions I and III are CBD and form a complementary pair ('Some' and 'No' between the same two terms), the correct inference is that either I or III follows.
Revision Table: Key Syllogism Concepts
| Statement Type | Meaning | Implication |
|---|---|---|
| All A are B | Every A is included in B. | If A is B, some A is B, some B is A is possible (if B exists). |
| No A is B | There is no overlap between A and B. | If No A is B, then No B is A. |
| Some A are B | At least one A is a B. There is overlap. | Does not guarantee 'All A are B' or 'No A is B'. Does not imply any relationship between A/B and a third term C based on 'Some B are C'. |
| Some A are not B | At least one A is not a B. | Does not guarantee 'No A is B' or 'All A are B'. |
Additional Information: Either-Or Cases in Syllogism
An 'either-or' case, also called a complementary pair or contradictory pair, arises when two conclusions are made about the same subject and predicate, and they represent mutually exclusive and exhaustive possibilities. The most common pairs are:
- "Some A are B" and "No A is B"
- "All A are B" and "Some A are not B"
If, after analyzing the statements, you find that a conclusion of the form "Some A are B" is possible but not certain (CBD), and a conclusion of the form "No A is B" is also possible but not certain (CBD), then you can logically conclude that *either* "Some A are B" *or* "No A is B" follows. This is because these two statements cover all possible relationships between A and B – either there is some overlap or there is none.
The same logic applies to the "All A are B" and "Some A are not B" pair. If both are CBD, then either "All A are B" or "Some A are not B" follows.
In this problem, Conclusions I ("Some helmets are Prizes") and III ("No helmet is a Prize") form the first type of complementary pair. Since both were found to be CBD, the correct logical inference is "Either conclusion I or III follows". Conclusion II does not form such a pair with either I or III, and since it is also CBD, it doesn't follow on its own.
Similar Questions
Read the given statements and conclusions carefully. Assuming that the information given in the statements are 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:
1. All colours are reds.
2. Some reds are greens.
3. All greens are dark.
Conclusions:
I. All colours being dark is a possibility
II. Some reds are dark.
III. Some reds are colours.
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.
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:
Some tents are cones.
All the cones are motels.
Conclusions:
All the motels are cones.
Some tents are motels.The statements below are followed by four conclusions labelled I, II, III and IV. 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:
No laptop is notebook.
All notebooks are tablets.
Conclusions:
I. No laptop is tablets.
II. No tablets is laptop.
III. Some tablets are notebooks.
IV. All the tablets are notebooks.
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 cats are hens.
All hens are goat.
Conclusions:
Conclusion I: Some hens are cat.
Conclusion II: Some goats are hens.
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.
