Chapter 6 - Conditional Reasoning

[jamiesam26]

Hi All, I have a few questions about chapter 6 that I would love further insight into.

1. It is correct to assume that Mistaken Reversal and Mistaken Negation answer choices are never going to be the correct one? The only answer choices that should be correct if the question has a conditional statement is the Repeat or the Contrapositive answer forms?

2. When diagramming, should I diagram like this:

Sufficient Necessary

A -------> B

Basically, setting up the sufficient part first, no matter what part of the sentence it appeared in first, and the necessary portion second, following the same logic, no matter what part of the sentence it was identified at?

That is the only thing I really seem to be hung up on for now. Thanks!

[Jonathan Evans]

Hi Jamiesam,

Welcome to the forums! Good questions.

1. It is correct to assume that Mistaken Reversal and Mistaken Negation answer choices are never going to be the correct one? The only answer choices that should be correct if the question has a conditional statement is the Repeat or the Contrapositive answer forms?

If we are referring specifically to Must Be True questions (ones in which we must make a valid inference based on the premises), the answer is yes. If you have correctly analyzed the conditional reasoning in the stimulus and find an answer choice that represents a Mistaken Negation™ or Mistaken Reversal™ of this conditional statement, then this answer choice is definitely incorrect. A necessary condition is not in and of itself sufficient to guarantee the condition for which it is necessary.

• Ex. To get a car loan, you must produce a (1) pay stub, (2) a credit report, and (3) a down payment. Here, there are three conditions necessary to get a car loan, but none of them by themselves are sufficient to guarantee that you get the car loan. Further, even if you meet all three conditions, you still might not get the car loan! You might also be required to pay off your old car, for instance.

If you are asked what we can infer based on the information above, could we properly infer that if I produce a credit report, I will be given the auto loan? No, this would be a Mistaken Reversal™. Can we conclude that if I didn't get the auto loan, then I didn't produce a pay stub? No, again, this is fallacious, a Mistaken Negation™.

Likewise, a condition that is sufficient to guarantee the truth of another condition is not in and of itself necessary for that condition to be true.

• Ex. If you buy me a vanilla ice cream cone, I'll pay for dinner tonight. Here, buying me a vanilla ice cream cone will guarantee that I pay for dinner, but it is not necessarily required. I might pay for dinner anyways, or maybe I'll pay for dinner if you buy me a chocolate ice cream cone. Who knows?

If we are asked what we can infer from this conditional, could we conclude that if you don't buy me a vanilla ice cream cone, then I won't pay for dinner? No, as stated, I might pay for dinner anyways. This is the Mistaken Negation™. By the same logic, can we conclude that if I do buy dinner, then you did buy me a vanilla ice cream cone? Again, no. This doesn't work. This is the Mistaken Reversal™.

It bears noting that if we are faced with a "Double Arrow" situation (material equivalence, conditional that goes both directions), then the conditions involved are both sufficient and necessary for one another. To spin off a PowerScore example:

• Jackie will go to the party if and only if Billy goes too.
  
  J B
  B J
  
  J B

In this case, there are neither Mistaken Negations nor Reversals because both conditions are necessary and sufficient for one another.

When diagramming, should I diagram like this: [...]
Basically, setting up the sufficient part first, no matter what part of the sentence it appeared in first, and the necessary portion second, following the same logic, no matter what part of the sentence it was identified at?

I want to be certain that I am clear about your question here. If I understand the question correctly, you wish to know what to do in a situation such as the following:

• Thelma will drive over the cliff if Louise will too.

In this case, you might wonder, should we diagram this statement thus:

• TC LC

In general, I advice students not to use this syntax. There is nothing ipso facto "wrong" with writing a conditional this way. However, if we pair it with another conditional:

• Louise will surrender to the authorities only if Thema does not drive over the cliff.
  
  LS TC

Then we start to get into some confusing syntax.

Consider the two ways we could write these statements together:

• TC LC
  LS TC
  
  OR
  
  LC TC
  LS TC

Note that in the second case, it might be more apparent that LC TC has the contrapositive TC LC.

• LS TC
  TC LC

Thus, we can infer LS LC. If Louise surrenders, then Thelma won't drive off the cliff. If Thelma doesn't drive off the cliff, then Louise won't either. Thus, if Louise surrenders, Louise won't drive off the cliff.

This inference is far clearer to me keeping the sufficient conditions on the left and the necessary conditions on the right. I often encourage students to stack them on top of each other if faced with multiple conditionals. That way they can read from left to right and repeat (see whether a true necessary condition reoccurs as a sufficient condition).

I hope this helps!

[Dave Killoran]

Hi Jamiesam,

Please let me take a moment to add a thought or two to Jonathan's excellent and comprehensive answer.

With question #2, you have the diagram correct, although I'll note that you don't need to write out "sufficient" and "necessary." I suspect you did that for purposes of clarity, but it's worth making sure Once you are familiar enough with these arrows, you'll know what you are looking at, and it won't be an issue of course.

As for the direction, you are correct that the sufficient condition appears at the "start" of the arrow. As you will see later in the book, I talk about how the left-to-right direction of the arrow means very little, and what is more important is where the arrow starts and what it points to. For example, each of these four arrow representations are identical in meaning:

• A B
  
  B A
  
  A
  |
  |
  v
  B
  
  B
  ^
  |
  |
  A

Those last two diagrams are meant to have arrows between A and B; I did my best with the tools available here! Either way, this ability for the direction of the arrow to move around is very helpful when you run into Formal Logic in Chapter 13. that's a really tough chapter in general, but you are already ahead of the game by understanding some of these ideas now.

Jonathan covered everything very well, but if you have any questions, please let us know. Thanks!