Dear Daniel,

You wrote:

“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.

2. ¬P.

3. ¬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.

2. P.

3. 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.

Pattern:

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.

Best wishes,

GC