Understanding the Syllogism Question

This question asks us to analyze logical statements and determine which conclusions necessarily follow. We are given three statements about the relationships between planets, asteroids, orbits, and stars, and two potential conclusions. We must assume the statements are true, even if they contradict common knowledge, and use only the information provided in the statements to evaluate the conclusions.

The statements are:

• Statement I: All planets are asteroids.
• Statement II: All orbits are asteroids.
• Statement III: No star is an asteroid.

The conclusions are:

• Conclusion I: No planet is a star.
• Conclusion II: No star is an orbit.

Analyzing the Statements with Set Theory

We can represent these statements using set theory concepts or Venn diagrams. Let's consider the categories as sets:

• Planets (P)
• Asteroids (A)
• Orbits (O)
• Stars (S)

Based on the statements:

• Statement I: All planets are asteroids. This means the set of Planets (P) is a subset of the set of Asteroids (A). Mathematically, \(P \subseteq A\). Every element in the set P is also an element in the set A.
• Statement II: All orbits are asteroids. This means the set of Orbits (O) is a subset of the set of Asteroids (A). Mathematically, \(O \subseteq A\). Every element in the set O is also an element in the set A. Note that Statements I and II tell us that both Planets and Orbits are contained within Asteroids. They don't tell us anything about the relationship between Planets and Orbits.
• Statement III: No star is an asteroid. This means the set of Stars (S) has no common elements with the set of Asteroids (A). Their intersection is empty. Mathematically, \(S \cap A = \emptyset\). This implies that anything that is a Star cannot be an Asteroid, and anything that is an Asteroid cannot be a Star.

Evaluating Conclusion I: No planet is a star

Let's see if this conclusion follows from the statements.

We know from Statement I that All planets are asteroids (\(P \subseteq A\)).

We also know from Statement III that No star is an asteroid (\(S \cap A = \emptyset\)).

Consider if Conclusion I were false. If Conclusion I were false, it would mean that 'Some planet is a star' is true. If there is an entity that is both a planet and a star, let's call it 'x'.

• According to Statement I (All planets are asteroids), if 'x' is a planet, then 'x' must also be an asteroid. So, 'x' is an asteroid.
• According to Statement III (No star is an asteroid), if 'x' is a star, then 'x' cannot be an asteroid. So, 'x' is not an asteroid.

This leads to a contradiction: 'x' must be an asteroid and 'x' cannot be an asteroid. This contradiction arises from assuming that 'Some planet is a star' is true.

Therefore, the assumption must be false, and 'No planet is a star' must be true.

Conclusion I logically follows from the statements.

Evaluating Conclusion II: No star is an orbit

Now let's evaluate the second conclusion.

We know from Statement II that All orbits are asteroids (\(O \subseteq A\)).

We also know from Statement III that No star is an asteroid (\(S \cap A = \emptyset\)).

Consider if Conclusion II were false. If Conclusion II were false, it would mean that 'Some star is an orbit' is true. If there is an entity that is both a star and an orbit, let's call it 'y'.

• According to Statement II (All orbits are asteroids), if 'y' is an orbit, then 'y' must also be an asteroid. So, 'y' is an asteroid.
• According to Statement III (No star is an asteroid), if 'y' is a star, then 'y' cannot be an asteroid. So, 'y' is not an asteroid.

Again, this leads to a contradiction: 'y' must be an asteroid and 'y' cannot be an asteroid. This contradiction arises from assuming that 'Some star is an orbit' is true.

Therefore, the assumption must be false, and 'No star is an orbit' must be true.

Conclusion II logically follows from the statements.

Summary of Conclusions

Based on our analysis:

• Conclusion I (No planet is a star) follows from the statements.
• Conclusion II (No star is an orbit) follows from the statements.

Therefore, both conclusions I and II logically follow from the given statements.

Final Answer Determination

Both Conclusion I and Conclusion II are found to logically follow from the given statements. Therefore, the correct option is the one stating that both conclusions I and II follow.

Revision Table: Syllogism Practice

Here is a quick look at the problem structure and result:

Additional Information: Syllogism Basics

Syllogism questions test your ability to draw logical inferences from given statements. Key principles include:

• Assume statements are true: Regardless of real-world facts, treat the given statements as absolute truth within the context of the problem.
• Do not add outside information: Base your conclusions solely on the relationships described in the statements.
• Analyze relationships: Identify how the different categories (Planets, Asteroids, Stars, Orbits in this case) relate to each other (e.g., "all A are B", "some A are B", "no A is B").
• Test conclusions: Evaluate each conclusion to see if it is necessarily true given the relationships established by the statements. If a conclusion could be false in any possible scenario consistent with the statements, it does not follow. If it must be true in all consistent scenarios, it follows.

In this specific problem, the structure is: If A ⊂ C and B ⊂ C, and C ∩ D = ∅, then A ∩ D = ∅ and B ∩ D = ∅. Here, A = Planets, B = Orbits, C = Asteroids, D = Stars.

Since Planets are inside Asteroids, and Stars are completely outside Asteroids, there is no way for Planets and Stars to overlap. Hence, "No Planet is a Star".

Similarly, since Orbits are inside Asteroids, and Stars are completely outside Asteroids, there is no way for Orbits and Stars to overlap. Hence, "No Star is an Orbit".