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 stars are moons. Some moons are suns. All suns are satellites. Conclusions : I. Some satellites are moons. II. Some satellites are stars. III. Some moons are stars.
Both conclusions I and III follow.
Analyzing the Logic Syllogism: Stars, Moons, Suns, and Satellites
This problem requires us to analyze logical statements and determine which conclusions can be drawn from them, regardless of whether the statements align with real-world facts. We are given three statements and three conclusions. Our task is to assess the validity of each conclusion based solely on the given statements.
Understanding the Statements
Let's list the provided statements clearly:
- Statement 1: All stars are moons.
- Statement 2: Some moons are suns.
- Statement 3: All suns are satellites.
Evaluating the Conclusions
Now, we will examine each conclusion one by one to see if it logically follows from the statements.
Conclusion I: Some satellites are moons.
To evaluate this, let's look at the statements involving 'satellites' and 'moons'. Statements 2 and 3 are relevant here:
- Statement 2: Some moons are suns. (This tells us there's an overlap between the set of moons and the set of suns).
- Statement 3: All suns are satellites. (This tells us that the entire set of suns is inside the set of satellites).
If there are some moons that are also suns (from Statement 2), and all suns belong to the category of satellites (from Statement 3), then those specific moons which are suns must also be satellites. Therefore, there must be some entities that are both moons and satellites. This means "Some satellites are moons" is a valid conclusion.
We can represent this relationship:
- Moons $\cap$ Suns $\ne \emptyset$ (from Statement 2)
- Suns $\subseteq$ Satellites (from Statement 3)
Combining these, if something is in Moons $\cap$ Suns, it must also be in Moons and in Suns. Since Suns $\subseteq$ Satellites, anything in Suns is also in Satellites. So, anything in Moons $\cap$ Suns is also in Moons $\cap$ Satellites. Since Moons $\cap$ Suns is not empty, Moons $\cap$ Satellites is also not empty.
Thus, Conclusion I logically follows.
Conclusion II: Some satellites are stars.
Let's connect 'satellites' and 'stars'. We have statements involving stars (Statement 1) and satellites (Statement 3), with 'moons' and 'suns' acting as intermediaries.
- Statement 1: All stars are moons. (Stars $\subseteq$ Moons)
- Statement 2: Some moons are suns. (Moons $\cap$ Suns $\ne \emptyset$)
- Statement 3: All suns are satellites. (Suns $\subseteq$ Satellites)
From Statement 1, all stars are inside the set of moons. From Statements 2 and 3, we concluded that some moons are satellites (as shown for Conclusion I). However, the part of 'moons' that overlaps with 'suns' (and thus with 'satellites') might be entirely outside the set of 'stars' within the 'moons' set.
Consider a scenario: Let Moons be represented by a large circle. Stars are a smaller circle completely inside the Moons circle. Suns overlap with the Moons circle, but this overlap region might be in the part of the Moons circle that is outside the Stars circle. Since Suns are inside Satellites, this overlap region is also inside the Satellites circle. There is no guarantee that the overlap between Moons and Suns/Satellites involves any element that is also a Star.
Therefore, Conclusion II does not logically follow from the statements. It might be true in some cases, but it is not necessarily true in all cases based *only* on the given statements.
Conclusion III: Some moons are stars.
Let's look at the statements involving 'moons' and 'stars'. Statement 1 is directly relevant:
- Statement 1: All stars are moons. (Stars $\subseteq$ Moons)
This statement means that every single entity that is a 'star' is also a 'moon'. If we assume that the category 'stars' is not empty (which is standard practice in these types of problems unless specified), then there must exist at least one entity that is a 'star'. Since this entity is a 'star', according to Statement 1, it must also be a 'moon'. Therefore, there exists at least one entity that is both a 'moon' and a 'star'. This is exactly what "Some moons are stars" means.
This conclusion is derived through the logical conversion of an 'All' statement. The conversion of "All A are B" is "Some B are A". Applying this to "All stars are moons", we get "Some moons are stars".
Thus, Conclusion III logically follows.
Summary of Conclusions
- Conclusion I: Some satellites are moons. - Follows.
- Conclusion II: Some satellites are stars. - Does not follow.
- Conclusion III: Some moons are stars. - Follows.
Based on our analysis, both Conclusion I and Conclusion III logically follow from the given statements.
Revision Table: Syllogism Analysis Key Points
| Statement Type | Example | Conversion | Implication |
|---|---|---|---|
| All A are B (Universal Affirmative) | All stars are moons. | Some B are A (Some moons are stars). | A $\subseteq$ B |
| Some A are B (Particular Affirmative) | Some moons are suns. | Some B are A (Some suns are moons). | A $\cap$ B $\ne \emptyset$ |
| No A are B (Universal Negative) | (Not present) | No B are A. | A $\cap$ B = $\emptyset$ |
| Some A are not B (Particular Negative) | (Not present) | No simple conversion. | Exists x such that x $\in$ A and x $\notin$ B |
Additional Information: Basics of Logic Syllogism
A syllogism is a form of logical reasoning where a conclusion is drawn from two or more statements (premises). In these types of questions, you must assume the statements are true and use only logical rules to determine if a conclusion is necessary. Commonly, Venn diagrams are used to visually represent the sets and their relationships described in the statements.
Key concepts in syllogism:
- Statements/Premises: The initial propositions given as true. They usually describe relationships between categories (like 'stars', 'moons').
- Conclusion: The statement that you need to verify if it necessarily follows from the premises.
- Validity: A conclusion is valid if it must be true whenever the premises are true. It does not depend on the real-world truth of the statements or conclusion.
- Middle Term: A term that appears in both premises but not in the conclusion (e.g., 'moons' in combining Statement 1 & 2).
- Distribution: Refers to whether a term in a statement covers all members of the class it represents. For example, in "All stars are moons", 'stars' is distributed, but 'moons' is not (because there might be moons that are not stars).
When combining statements, pay attention to the types of statements (All, Some, No, Some Not) and the distribution of terms to correctly deduce conclusions. A conclusion cannot be universal if both premises are particular, and a conclusion cannot be affirmative if one premise is negative (and vice versa, with exceptions).
Similar Questions
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 follows from the statements.
Statements:
All lemons are plums.
All plums are dates.
Some dates are mangoes.
Conclusions:
I. Some lemons are mangoes.
II. Some mangoes are plums.
III. All lemons are dates.
IV. Some mangoes are dates.
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 cards are postcards.
Some cards are envelopes.
All envelopes are copies.
Conclusions:
I. Some copies are envelopes.
II. Some postcards are copies.
III. Some cards are copies.
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 pens are newspapers.
Some newspapers are novels.
No novel is a book.
Conclusions:
I. Some books are pens.
II. Some novels are pens.
III. No book is a pen.
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:
Some spices are medicines.
All medicines are chemicals.
No chemical is cheap.
Conclusions:
I. Some medicines are cheap.
II. Some chemicals are spices.
III. No medicine is cheap.
In this question, 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 follows/follow from the statements.
Statements:
I. No streak is a line.
II. Some lines are bands.
III. All bands are flashes.
Conclusions:
I. Some flashes are streaks.
II. No flash is a line.
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 oranges are apples.
Some oranges are bananas.
All bananas are fruits.
Conclusions:
I. Some fruits are apples.
II. Some apples are bananas.
III. Some oranges are fruits.
Select the Venn diagram that best illustrates the relationship between the following classes.
Swimming, Running, Sports
Select the Venn diagram that best illustrates the relationship between the following classes.
Dialect, Language, Communication
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:
Some seeds are fruits.
Some fruits are jams.
All jams are desserts.
Conclusions:
I. Some jams are seeds.
II. Some desserts are jams.
III. Some fruits are desserts.
Three statements are followed by three conclusions numbered I, II and III. You have to consider these statements to be true, even if they seem to be at variance with commonly known facts. Decide which of the given conclusions logically follow(s) from the given statements.
Statements:
All tools are spanners.
Some spanners are utensils.
No spanner is an iron.
Conclusions:
(I) Some tools are irons.
(II) Some utensils are tools.
(III) Some utensils are not iron.
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.
