# Denying the antecedent

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• #19196

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.

I have another question concering 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? Shouldn’t it be the invalid pattern:

1. All As are Bs.
2. x is not an A.
3. x is not a B.

#19197

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

#19198

I have now confirmed my valid interpretation of the first argument. I am slowly learning to identify patterns where the premises have switched places and where the negations have changed places, compared to the pre-given patterns of arguments I have studied before.

As for the second one, I was looking for a specific invalid pattern. I had only seen two patterns of invalid arguments in predicate logic prior to seeing the argument above, and I was looking for one of them to fit the argument I translated. It would be helpful to have some more examples of invalid arguments just in case I stumble upon these types of arguments in the future.

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