Which of the following statements is logically equivalent to: "If he studies, he will pass the course."?

Options

A. He passed the course, therefore, he studied.
B. He did not study, therefore, he will not pass the course.
C. He did not pass the course, therefore he did not study.
D. He will pass the course only if he studies.

Explanation

To determine which statement is logically equivalent to "If he studies, he will pass the course," we can analyze the structure of the statement and apply logical reasoning.

### Step-by-Step Solution:

1. **Identify the Statements:**
- Let P be the statement "He studies."
- Let Q be the statement "He will pass the course."
- The original statement can be written as: P → Q (If P, then Q).

2. **Understand Logical Equivalence:**
- A statement P → Q is logically equivalent to its contrapositive, which is ¬Q → ¬P (If not Q, then not P).
- This means if he does not pass the course (¬Q), then he did not study (¬P).

3. **Translate the Contrapositive:**
- The contrapositive ¬Q → ¬P translates to: "If he did not pass the course, then he did not study."

4. **Evaluate the Options:**
- **Option 1:** "He passed the course, therefore he studied." (Not equivalent)
- **Option 2:** "He did not study, therefore he will not pass the course." (Not equivalent)
- **Option 3:** "He did not pass the course, therefore he did not study." (Equivalent)
- **Option 4:** "He will pass the course only if he studies." (Not equivalent)

5. **Conclusion:**
- The statement that is logically equivalent to "If he studies, he will pass the course" is **Option 3:** "He did not pass the course, therefore he did not study."