(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.
If a cow, then mammal - hypothesis
If a mammal then a cow - converse, invalid
If not a cow , then not a mammal - inverse, invalid
If not a mammal, then not a cow - contrapositive - true
Here:
If NT teaches, then pass - hypothesis
If pass, then NT - converse, invalid
If not NT, then not pass - inverse, invalid
If not pass, then not NT - contrapositive, true (this is your case)
(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 I can get my child to preschool by 9:45AM, then I can take the 9:00AM class.
If I can take the 9AM class, then I can be done by 2PM
If I can get my child to preschool by 8:45AM then I can be done by 2PM
To translate the argument into symbolic form, we can assign variables to different components of the argument:
Let P represent "Nicholas Thompson teaches this course."
Let Q represent "I will get a passing grade."
The argument can then be translated as:
If P, then Q.
Not Q.
Therefore, not P.
In logical symbols, the argument can be represented as:
P → Q
¬Q
∴ ¬P
Now, to determine if the argument is valid or invalid, we can compare it to a standard form or use a truth table.
In this case, we can use a truth table with two variables (P and Q). The truth table will display all possible combinations of true (T) and false (F) for P and Q and evaluate the logical expressions accordingly.
Truth Table:
| 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 |
The truth table reveals that there is at least one row where all premises are true (the first row), but the conclusion is false. Therefore, the argument is invalid since it is possible for all the premises to be true, but the conclusion to be false.
In simpler terms, just because you did not get a passing grade (¬Q), it does not necessarily mean that Nicholas Thompson did not teach the course (¬P).