LRB page 169 #6 mini-drill

[Stephanie Oswalt]

We recently received the following question from a student. An instructor will respond below. Thank you!

I was hoping you could clarify my confusion on question #6 page 169 on the valid and invalid statement recognition mini-drill (edition 2017). I am unsure if my issue stems from misunderstanding the relationship behind conditional reasoning and negation. The book states, "in considering the form of the statements, the position of the slash is irrelevant when determining if you are looking at a repeat, contrapositive, mistaken reversal or negation." After reviewing the two examples it summarizes this concept by stating," the form determines the result." So, applying this idea to question #6:

Original Statement: K L
Attempted Inference: L K

I thought the answer was a Mistaken Negation, but the correct answer it is a Mistaken Reversal. By ignoring the slashes and focusing on the reversed positions I can see how it is a mistaken reversal. I am confused because I interpreted the attempted inference as the positions reversed as an attempt to infer a contrapositive, but the slashes were carried over onto the wrong positions. What am I missing?

Thank you!

Sincerely,
SA

[Jonathan Evans]

Hi, SA,

Good question. As you noticed, we didn't quite make it to a contapositive here. You correctly observed that we have flipped the positions of the terms but have not negated the statements.

I am confused because I interpreted the attempted inference as the positions reversed as an attempt to infer a contrapositive, but the slashes were carried over onto the wrong positions.

This is correct, except that what you have described is not a contrapositive but a Mistaken Reversal™!

When you reverse the positions but do not negate the statements, that is the definition of a Mistaken Reversal™. The contrapositive requires you both to (1) reverse the sufficient and necessary statements and (2) negate each statement. When the attempted inference involves a Mistaken Reversal or a Mistaken Negation, the author has done one of those two steps but not both. The point of this exercise is to recognize when the author has (1) repeated the original conditional (valid) (2) made the contrapositive (valid) (3) made a Mistaken Negation (invalid) or (4) made a Mistaken Reversal (invalid).

We'd like to think the author is always on his or her way to making a contrapositive, but if you do only one but not both steps, it's not a contrapositive, it's either a Mistaken Negation or a Mistaken Reversal. For example:

• If I skip lunch, I'll be hungry. SL H
• SL H, Repeat
• SL H, Mistaken Negation
• H SL, Mistaken Reversal
• H SL, Contrapositive

It is not possible for something to be both a Mistaken Negation/Reversal and a contrapositive at the same time. The contrapositive is a valid logical inference; the Mistaken Negation and Reversal are not.

Does this make sense?