Worksheet on logical reasoning and the law of syllogism with multiple-choice questions.

Problem Analysis:

The task involves analyzing statements and determining whether conclusions can be logically drawn based on the given premises. This is a classic exercise in Deductive Reasoning using the Law of Detachment or the Law of Syllogism.

#### Key Concepts:
1. Law of Detachment:
If you have a conditional statement \( P \rightarrow Q \) (if \( P \), then \( Q \)) and you know that \( P \) is true, then you can conclude that \( Q \) is true.

2. Law of Syllogism:
If you have two conditional statements \( P \rightarrow Q \) and \( Q \rightarrow R \), then you can infer \( P \rightarrow R \).

Let's solve each problem step by step.

---

Problem 1:

- Premise 1: If a figure is a square, then it is a rectangle.
- Premise 2: If a figure is a rectangle, then it has 4 right angles.
- Question: What conclusion can you make? Which law applies?

#### Solution:
1. Identify the structure:
- Premise 1: \( \text{Square} \rightarrow \text{Rectangle} \)
- Premise 2: \( \text{Rectangle} \rightarrow \text{4 right angles} \)

2. Apply the Law of Syllogism:
- Since \( \text{Square} \rightarrow \text{Rectangle} \) and \( \text{Rectangle} \rightarrow \text{4 right angles} \), we can infer:
\[
\text{Square} \rightarrow \text{4 right angles}
\]

3. Conclusion:
- If a figure is a square, then it has 4 right angles.

4. Which law applies?:
- The Law of Syllogism.

#### Final Answer for Problem 1:
\[
\boxed{\text{If a figure is a square, then it has 4 right angles. Law of Syllogism.}}
\]

---

Problem 2:

- Premise: If I miss my bus, then I'll be late for school.
- Given: I missed my bus.
- Question: What conclusion can you make? Which law applies?

#### Solution:
1. Identify the structure:
- Premise: \( \text{Miss bus} \rightarrow \text{Late for school} \)
- Given: \( \text{Miss bus} \) is true.

2. Apply the Law of Detachment:
- Since \( \text{Miss bus} \rightarrow \text{Late for school} \) and \( \text{Miss bus} \) is true, we can conclude:
\[
\text{Late for school}
\]

3. Conclusion:
- I will be late for school.

4. Which law applies?:
- The Law of Detachment.

#### Final Answer for Problem 2:
\[
\boxed{\text{I will be late for school. Law of Detachment.}}
\]

---

Problem 3:

- Premise: If we live some trees, then we have some shade.
- Given: We have some trees.
- Question: What conclusion can you make? Which law applies?

#### Solution:
1. Identify the structure:
- Premise: \( \text{Have trees} \rightarrow \text{Have shade} \)
- Given: \( \text{Have trees} \) is true.

2. Apply the Law of Detachment:
- Since \( \text{Have trees} \rightarrow \text{Have shade} \) and \( \text{Have trees} \) is true, we can conclude:
\[
\text{Have shade}
\]

3. Conclusion:
- We have some shade.

4. Which law applies?:
- The Law of Detachment.

#### Final Answer for Problem 3:
\[
\boxed{\text{We have some shade. Law of Detachment.}}
\]

---

Problem 4:

- Premise: All accountants enjoy mathematics.
- Given: Stephanie is an accountant.
- Question: What conclusion can you make? Which law applies?

#### Solution:
1. Identify the structure:
- Premise: \( \text{Accountant} \rightarrow \text{Enjoy mathematics} \)
- Given: Stephanie is an accountant.

2. Apply the Law of Detachment:
- Since \( \text{Accountant} \rightarrow \text{Enjoy mathematics} \) and Stephanie is an accountant, we can conclude:
\[
\text{Stephanie enjoys mathematics.}
\]

3. Conclusion:
- Stephanie enjoys mathematics.

4. Which law applies?:
- The Law of Detachment.

#### Final Answer for Problem 4:
\[
\boxed{\text{Stephanie enjoys mathematics. Law of Detachment.}}
\]

---

Problem 5:

- Premise: If a person is a librarian, then they read books.
- Given: If a person is a friend of Bill’s, then they read books.
- Question: What conclusion can you make? Which law applies?

#### Solution:
1. Identify the structure:
- Premise 1: \( \text{Librarian} \rightarrow \text{Read books} \)
- Premise 2: \( \text{Friend of Bill’s} \rightarrow \text{Read books} \)

2. Analyze the premises:
- Both premises lead to the same conclusion (\( \text{Read books} \)), but there is no direct logical connection between "librarian" and "friend of Bill’s."
- No valid conclusion can be drawn using either the Law of Detachment or the Law of Syllogism because the premises do not form a chain or directly relate to each other.

3. Conclusion:
- No valid conclusion can be made.

4. Which law applies?:
- None.

#### Final Answer for Problem 5:
\[
\boxed{\text{No valid conclusion.}}
\]

---

Problem 6:

- Premise: If Sam is a pig, then he is a mammal.
- Given: Sam is a pig.
- Question: What conclusion can you make? Which law applies?

#### Solution:
1. Identify the structure:
- Premise: \( \text{Sam is a pig} \rightarrow \text{Sam is a mammal} \)
- Given: \( \text{Sam is a pig} \) is true.

2. Apply the Law of Detachment:
- Since \( \text{Sam is a pig} \rightarrow \text{Sam is a mammal} \) and \( \text{Sam is a pig} \) is true, we can conclude:
\[
\text{Sam is a mammal.}
\]

3. Conclusion:
- Sam is a mammal.

4. Which law applies?:
- The Law of Detachment.

#### Final Answer for Problem 6:
\[
\boxed{\text{Sam is a mammal. Law of Detachment.}}
\]

---

Problem 7:

- Premise: If a quadrilateral ABCD is a square, then it has 4 congruent sides.
- Given: Quadrilateral ABCD is a square.
- Question: What conclusion can you make? Which law applies?

#### Solution:
1. Identify the structure:
- Premise: \( \text{Quadrilateral ABCD is a square} \rightarrow \text{It has 4 congruent sides} \)
- Given: \( \text{Quadrilateral ABCD is a square} \) is true.

2. Apply the Law of Detachment:
- Since \( \text{Quadrilateral ABCD is a square} \rightarrow \text{It has 4 congruent sides} \) and \( \text{Quadrilateral ABCD is a square} \) is true, we can conclude:
\[
\text{It has 4 congruent sides.}
\]

3. Conclusion:
- Quadrilateral ABCD has 4 congruent sides.

4. Which law applies?:
- The Law of Detachment.

#### Final Answer for Problem 7:
\[
\boxed{\text{Quadrilateral ABCD has 4 congruent sides. Law of Detachment.}}
\]

---

Problem 8:

- Premise: If many little dogs are in a movie, then I will be happy.
- Given: Many little dogs are in a movie.
- Question: What conclusion can you make? Which law applies?

#### Solution:
1. Identify the structure:
- Premise: \( \text{Many little dogs in a movie} \rightarrow \text{I will be happy} \)
- Given: \( \text{Many little dogs in a movie} \) is true.

2. Apply the Law of Detachment:
- Since \( \text{Many little dogs in a movie} \rightarrow \text{I will be happy} \) and \( \text{Many little dogs in a movie} \) is true, we can conclude:
\[
\text{I will be happy.}
\]

3. Conclusion:
- I will be happy.

4. Which law applies?:
- The Law of Detachment.

#### Final Answer for Problem 8:
\[
\boxed{\text{I will be happy. Law of Detachment.}}
\]

---

Problem 9:

- Premise: If a man is wealthy, then he buys a mansion.
- Given: If a man is wealthy, then he buys a limousine.
- Question: What conclusion can you make? Which law applies?

#### Solution:
1. Identify the structure:
- Premise 1: \( \text{Wealthy} \rightarrow \text{Buys a mansion} \)
- Premise 2: \( \text{Wealthy} \rightarrow \text{Buys a limousine} \)

2. Analyze the premises:
- Both premises start with the same condition (\( \text{Wealthy} \)), but they lead to different conclusions (\( \text{Buys a mansion} \) and \( \text{Buys a limousine} \)).
- No valid conclusion can be drawn using either the Law of Detachment or the Law of Syllogism because the premises do not form a chain or directly relate to each other.

3. Conclusion:
- No valid conclusion can be made.

4. Which law applies?:
- None.

#### Final Answer for Problem 9:
\[
\boxed{\text{No valid conclusion.}}
\]

---

Final Summary of Answers:

1. \(\boxed{\text{If a figure is a square, then it has 4 right angles. Law of Syllogism.}}\)
2. \(\boxed{\text{I will be late for school. Law of Detachment.}}\)
3. \(\boxed{\text{We have some shade. Law of Detachment.}}\)
4. \(\boxed{\text{Stephanie enjoys mathematics. Law of Detachment.}}\)
5. \(\boxed{\text{No valid conclusion.}}\)
6. \(\boxed{\text{Sam is a mammal. Law of Detachment.}}\)
7. \(\boxed{\text{Quadrilateral ABCD has 4 congruent sides. Law of Detachment.}}\)
8. \(\boxed{\text{I will be happy. Law of Detachment.}}\)
9. \(\boxed{\text{No valid conclusion.}}\)