(a) translate the argument into symbolic form and (b) determine if the argument is valid or invalid. You may compare the argument to a standard form or use a truth table.
If Nicholas Thompson teaches this course, then I will get a passing grade.
I did not get a passing grade.
∴ Nicholas Thompson did not teach the course.
To translate the argument into symbolic form, we can assign the following letters to represent the statements:
P: Nicholas Thompson teaches this course.
Q: I will get a passing grade.
The argument can then be represented as:
If P, then Q.
¬Q.
∴ ¬P.
Now let's determine if the argument is valid or invalid using a truth table.
We have two premises and one conclusion. To evaluate the validity of the argument, we need to check if the conclusion is always true whenever both premises are true.
We can create a truth table and assign truth values (T or F) to P and Q:
| P | Q | If P, then Q. | ¬Q | ∴ ¬P |
|:---:|:---:|:-------------:|:---:|:----:|
| T | T | T | F | F |
| T | F | F | T | F |
| F | T | T | F | T |
| F | F | T | T | T |
We can see that there is no row in which both premises are true and the conclusion is false. In other words, whenever the premises are true, the conclusion is always true. This means that the argument is valid.
Alternatively, we can observe that by contraposition, if the conclusion is false (¬P), then the premise "If P, then Q" is also false (~Q), which contradicts the second premise "¬Q." Therefore, the conclusion must be true to remain consistent with the premises, indicating that the argument is valid.