**Step 1: Introduction of the Concept** In this problem, we are asked to symbolize an English sentence using the provided logical symbols and the domain-specific key. The sentence to symbolize is: "Neither master sewer working for Mackenzie is more industrious than Frances." We must translate this sentence using the logical notation and the key, where \( L(x, y) \) means "x is more lazy than y", so "x is more industrious than y" would be the negation: \( \neg L(x, y) \). --- **Step 2: Understanding the Sentence** The sentence says **neither** master sewer working for Mackenzie is more industrious than Frances. This implies: - For any person, if they are both a master sewer and work for Mackenzie, then they are **not** more industrious than Frances. But "more industrious than Frances" is \( \neg L(x, f) \). So "not more industrious than Frances" is \( L(x, f) \). --- **Step 3: Write the Symbolization** Let’s break it down: - Let \( x \) be any individual in the domain. - \( M(x) \): \( x \) is a master sewer. - \( W(x, m) \): \( x \) works for Mackenzie. - \( L(x, f) \): \( x \) is more lazy than Frances. "Neither master sewer working for Mackenzie is more industrious than Frances" can be written as: > For all \( x \), if \( x \) is a master sewer and \( x \) works for Mackenzie, then \( x \) is **not** more industrious than Frances. So, in logical notation: \[ \forall x \big( (M(x) \land W(x, m)) \rightarrow L(x, f) \big) \] --- **Step 4: Final Answer in a Few Sentences** **Final Answer:** \[ \forall x \big( (M(x) \land W(x, m)) \rightarrow L(x, f) \big) \] --- **Step 5: What We Did in Each Step** 1. **Introduced** the task: symbolizing a natural language sentence using formal logic. 2. **Analyzed** the sentence for meaning and identified the logical structure and negations. 3. **Translated** the sentence using the provided key and symbols. 4. **Presented** the final answer in clear, logical notation. 5. **Summarized** each step to clarify how we arrived at the answer.