Valid and Invalid Statement Recognition Mini-Drill Answer Key #1

[Tommy2456]

Hi there guys! I am doing the drill on page 169. Can you explain why # 1/6 is a Mistaken Reversal and maybe tell the main points that I should remember from the Contrapositive, MR, and MN. I think I may be confusing myself. Thanks.

[Adam Tyson]

Thanks for the question, Tommy! Conditional reasoning causes a lot of folks trouble at first, and it's a major concept in both LR and LG, so it's really important to get it right. We're happy to spend a little time on such a crucial part of the test.

In any conditional relationship, there is a sufficient condition (the "if" part of the relationship) and a necessary condition (the "then" part). Whenever that sufficient condition happens, the necessary must happen, because it's, well, necessary.

When presented with a conditional claim, if we accept it as true, we must also accept the inverse relationship, what is called the contrapositive. If we believe that A is sufficient for B (that is, if A happens, B must also happen), then we also must accept that if B does not happen, A cannot happen. We reverse the order of the variables and negate them both. Those two claims - the original claim and its contrapositive - are equally true, and they are logically equivalent to each other. In other words, they mean the exact same thing and are interchangeable.

Two common mistakes happen in conditional reasoning, and those are Mistaken Reversals and Mistaken Negations. A Mistake Reversal occurs when you reverse the order of the variables, and that's all you do. A Mistaken Negation occurs when you negate the terms but leave them in their original order. That's what happens in #1 and #6 in that drill.

Here's an example to help you:

If you jump our of an airplane, you will fall towards the earth (original claim)

Now, if you do not jump out of an airplane (the sufficient condition does not occur), can I prove that you do not fall towards the earth? Nope - you could have been pushed out, or jumped from a helicopter, or off the edge of a tall building. Maybe you are falling toward the earth because your rollercoaster went off the tracks. The absence of the sufficient condition (you didn't jump out of a plane) does not prove the absence of the necessary condition (you didn't fall). To conclude otherwise is to make a mistaken negation.

What if you are falling toward the earth? Does that prove you jumped from a plane? Again, nope - just see all the alternative explanations above, and any others you can think of. The presence of the necessary condition (you fell) proves nothing about the presence of the sufficient condition (jumping). To conclude that it does is to make a mistaken reversal.

Finally, what if I say you did not fall towards the earth? Now what can I prove? I can prove you did not jump out of an airplane. The absence of the necessary condition DOES prove the absence of the sufficient condition. In fact, saying "if you don't fall, you didn't jump" is exactly the same as saying if you jump, you fall. They are two sides of the same coin. That's the contrapositve, the thing you can prove.

Incidentally, and not wanting to confuse you here, mistaken reversals and mistaken negations are the contrapositve of each other. They are logically equivalent to each other, and are thus equally unprovable and equally unsound logic.

If it helps, keep it simple and mechanical - reverse the order and negate the terms, and you're good. If you change the order and don't negate, or negate and don't change, you've made a mistake. Also remember that two negatives make a positive, so if you start off with a negative term, negating it makes it positive. Thus:

If you do not rest, you will not be healthy: R -> H
Contrapositve: H -> R

Let us know if that helps! Keep at it, it will click for you, and when it does you will see big improvements in your performance on both LR and games.

[Tommy2456]

Wow! This was just what I needed in a response. Thank you so much!!!