I'll do my best, Samantha! I may need further clarification from you - are we talking about negating conditional claims for the purpose of the Assumption Negation Technique, or are we looking only at negating claims in conditional statements to create either Contrapositives or else Mistaken Negations? That chapter isn't dealing with Assumption questions, so I am assuming the latter is what you mean. If this response doesn't address what you need, please follow up with additional info.
Let's take a hypothetical conditional relationship, not from any particular LR question but from my imagination:
If I have a second helping of lasagna, I won't have room for dessert.
(This is pure nonsense - I always have room for dessert - but let's pretend it's a real thing.)
The claim starts off with the sufficient condition indicator "if", and that tells me that having seconds on lasagna is my Sufficient Condition. When that condition occurs, the necessary condition is, well, necessary - it has to happen. In this case, the necessary condition is the other part of my claim: I won't have room for dessert. I might diagram this claim this way:
2HL
RD
(2HL means second helping of lasagna, while RD means room for dessert, and I have struck that term out to indicate that it is negated - I will NOT have room).
The rule for any conditional claim, assuming that it is a valid claim (and we are going to pretend it is, even though I will find a way to get that tiramisu down my gullet one way or another), is that the contrapositive of that claim must also be valid. To create the contrapositive, we reverse the order of the terms, moving them from one side of the arrow to the other, and we also negate them, which means cross out the ones that are not crossed out and remove the strikethrough from any that are already crossed out. The contrapositive here, then, is as follows:
RD
2HL
I would translate this back to English by saying "if I have room for dessert, then I must not have had a second helping of lasagna". I have reversed the order of the terms and negated them both. If the first claim is true, the second must also be true.
Now, one common mistake that we see on the LSAT, made by the authors and also by students, is what we call a Mistaken Negation. That happens when we cross out the terms but leave them in the original order. That would look like this, in our hypo:
2HL RD
That reads as "I did not have a second helping of lasagna, therefore I must have room for dessert".
The thing about Mistaken Negations is that, while they
might be true (maybe I didn't have a second helping and I have room), but they don't
have to be true. Perhaps I refrained from a second helping of lasagna, but I'm full anyway? Maybe because I had more ceasar salad, or extra meatballs, or just because my eyes were bigger than my stomach and my first helping of lasagna was gigantic? The point is that the truth of the original claim (a second helping
guarantees no room for dessert) doesn't prove the negation of that claim (not having a second helping doesn't guarantee that I will have room for dessert). Mistaken Negations are not sound logic, because the first claim doesn't prove the second one.
So, to sum up, when you have a conditional premise and you accept it as true (which you usually will do when evaluating conditional arguments), you CAN confidently claim that the contrapositive is true (reverse the terms and negate them), but you CANNOT prove that the Mistaken Negation is true (just negate them and leave them in the original order).
Negation of conditional claims comes up in other ways, too, such as in a Cannot Be True question, where the thing that cannot be true is that the sufficient condition occurs but the necessary condition does not occur. In our example, it cannot be true that I have a second helping of lasagna and still have room for dessert. That's impossible, because the original claim, which we accepted as true, means that a second helping absolutely guarantees, without exception, that I will NOT have room for dessert.
Let us know if that did the trick, Samantha! If you need help with negating conditional claims for use with the Assumption Negation technique, that's a whole other discussion that we would be happy to have. Meanwhile, keep at it and good luck!
