(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.
Assume that p = Nicholas Thompson teaches this course. Assume q = I will get a passing grade.
The symbolic form is ~q->~p
The truth table is:
p. ~p q. ~q. ~q->~p
T. F T. F. T
T. F F. T. F
F. T T. F. T
F. T F. T. T
So the conclusion is valid.
f t t f f
To translate the argument into symbolic form, we can assign variables to represent the premises and conclusion:
P: Nicholas Thompson teaches this course.
Q: I will get a passing grade.
The argument can be stated as follows:
If P, then Q.
Not Q.
Therefore, not P.
In symbolic form:
P -> Q
~Q
∴ ~P
To determine if the argument is valid or invalid, we can compare it to a standard form or use a truth table.
Standard Form:
In standard form, the argument would look like this:
1. P -> Q
2. ~Q
∴ ~P
This argument follows the form Modus Tollens, which states that if the conditional statement P -> Q is true and its consequent (Q) is false, then the antecedent (~P) must also be true. Therefore, the argument is valid.
Truth Table:
A truth table is another method to test the validity of an argument. We can construct a truth table for all possible combinations of P and Q to determine if there is any case where the premises are all true while the conclusion is false.
P | Q | P -> Q | ~Q | ~P
---------------------------
T | T | T | F | F
T | F | F | T | F
F | T | T | F | T
F | F | T | T | T
As we can see, there is no situation where the premises are all true while the conclusion is false. Therefore, the argument is valid.