Logic of Officer Response

The main statement provided is: "The officer will respond if you write complain officially." This is a conditional statement, indicating a dependency where one event must occur for another to potentially happen.

To analyze this logically, we can represent the statements using propositional logic symbols:

• Let $ P $ represent the statement: "You write complain officially."
• Let $ Q $ represent the statement: "The officer responds."

The main statement "The officer will respond if you write complain officially" can be formally written as:

This logical form means that if $ P $ (writing the complaint officially) is true, then $ Q $ (the officer responds) must also be true. If $ P $ is false, the truth value of $ Q $ does not affect the validity of the statement $ P \rightarrow Q $.

Complain Official Propositions

The question presents four specific propositions related to the scenario:

• Statement A: You wrote the complain officially. (This corresponds to $ P $)
• Statement B: You did not write the complain officially. (This corresponds to $ \neg P $)
• Statement C: The officer responded. (This corresponds to $ Q $)
• Statement D: The officer did not respond. (This corresponds to $ \neg Q $)

The task is to identify an ordered pair of statements (let's call them Statement 1 and Statement 2) that meet two criteria:

1. Statement 1 must logically imply Statement 2.
2. Both Statement 1 and Statement 2 must be logically consistent with the main statement ($ P \rightarrow Q $).

We need to evaluate the given options, which are pairs of these statements (A, B, C, D).

Statement Implication Analysis

Let's systematically examine the logical implications and consistency for each potential pair:

DB Statement Logic Evaluation

Let's take a closer look at the pair DB:

• Statement D: "The officer did not respond." ( $ \neg Q $)
• Statement B: "You did not write the complain officially." ( $ \neg P $)

1. Implication Check ($ \neg Q \rightarrow \neg P $):

The primary statement is $ P \rightarrow Q $. Its contrapositive is $ \neg Q \rightarrow \neg P $. Since a statement and its contrapositive are logically equivalent, the implication $ \neg Q \rightarrow \neg P $ is valid. This means if the officer did not respond (D), it logically follows that you did not write the complaint officially (B), given the initial rule.

2. Consistency Check:

If Statement D ($ \neg Q $) is true, then for the rule $ P \rightarrow Q $ to hold, Statement B ($ \neg P $) must necessarily be true. If Statement B ($ \neg P $) is true, the rule $ P \rightarrow Q $ remains valid irrespective of whether the officer responded or not. Thus, both statements D and B are fully consistent with the main conditional statement.

Because the pair DB satisfies both the implication requirement and the consistency requirement, it is the correct choice.

Statement Pair Consistency Check

The other options fail to meet the criteria:

• BD: While B ($ \neg P $) and D ($ \neg Q $) might occur together, $ \neg P $ does not logically guarantee $ \neg Q $.
• DA: D ($ \neg Q $) implies $ \neg P $, which directly contradicts A ($ P $). Therefore, they cannot form a consistent pair where D implies A.
• BC: B ($ \neg P $) does not logically guarantee C ($ Q $). The officer might respond for reasons other than an official complaint, or might not respond at all if no complaint is filed.

Only the ordered pair DB demonstrates a valid logical implication (the contrapositive) that is also consistent with the original conditional statement.