“Consider the following arguments:
1. If you didn’t pass the test, then you didn’t pass the course.
2. You did pass the test.
3. You did pass the course.
How should I reconstruct/evaluate this argument? The pattern of argument is called “Denying the antecedent” and looks like this:
1. If P then Q.
How do I interpret the negations of “pass the test” and “pass the course”? Can I also interpret it like this:
1. If ¬P then ¬Q.
Yes. It’s the same pattern. P is equivalent to ¬¬P (which is the negation of the antecedent), and Q is equivalent to ¬¬Q which is the negation of the consequent.
You also wrote:
“I have another question concerning validity in this example:
1. All logicians are dull.
2. Irving is not a logician.
3. Irving is dull.
1. All As are Bs
2. x is not an A
3. x is B
Now, does this follow from the premises?”
No. Any valid syllogism with a negative premise must have a negative conclusion so x is B cannot be a valid conclusion from those premises.
I’m not sure what you mean by saying “Shouldn’t it be the invalid pattern:
1. All As are Bs.
2. x is not an A.
3. x is not a B”
There can be more than one invalid conclusion from any given set of premises. However, if what you are suggesting is the “x is not a B” is the more plausible invalid conclusion, then I would agree with you. At least it is negative and so would at least pass rule 4.
Thank you as usual for your questions.