Directions: Define the necessary symbols, rewrite the argument in symbolic form, and us a truth table to determine whether the argument is valid. If the argument is invalid, interpret the specific circumstances that cause the argument to be invalid.

Problem:
1. If you argue with a police officer, you get a ticket.
2. If you do not break the speed limit, you do not get a ticket.
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Therefore, if you break the speed limit, you argue with a police a police officer.

To solve this problem using symbolic logic, we need to assign appropriate symbols to the statements given. Let's define the following symbols:

- P: You argue with a police officer.
- Q: You get a ticket.
- R: You break the speed limit.

Now, let's rewrite the given statements in symbolic form:
1. If P, then Q: P → Q
2. If ¬R, then ¬Q: ¬R → ¬Q

Finally, we need to determine whether the argument is valid using a truth table. A truth table displays all possible combinations of truth values for the given statements to determine the truth value of the conclusion.

Here's the truth table:

| P | Q | R | ¬R | ¬Q | P → Q | ¬R → ¬Q | Conclusion: R → P |
|---|---|---|----|----|-------|---------|------------------|
| T | T | T | F | F | T | F | T |
| T | T | F | T | F | T | F | T |
| T | F | T | F | T | F | T | T |
| T | F | F | T | T | F | T | T |
| F | T | T | F | F | T | F | T |
| F | T | F | T | F | T | T | T |
| F | F | T | F | T | T | F | T |
| F | F | F | T | T | T | T | T |

In the conclusion column (R → P), we can see that for all truth value combinations, the conclusion is true. Hence, the argument is valid.

Interpreting the specific circumstances that cause the argument to be invalid is not applicable in this case since the argument is valid.