Negating Answer Choices for Assumption Questions

[jessicamorehead]

Emily, thank you for your response!! I think I was having a problem with only thinking in terms of polar opposition, rather than logical. Thanks for clearing that up!

[jessicamorehead]

Emily,

I redid some homework problems (pg 6-34) and I am still a little confused.

Question 1: "Most drivers are not good drivers." I negated this to "Most drivers may be good drivers" since that is "softer" than the logical opposite of just removing the "not". However, the book lists "Most drivers are good drivers" as the correct answer. Am I wrong? When is it okay to just remove/add a "not"?

Question 7: "An increase in our company budget could lead to record growth." I negated this to "An increase in our company budget may not lead to record growth." The book negated it to "An increase in our company budget cannot lead to record growth."

Question 8: "Unless we protect our rights, we will lose them." I simplified this statement into the following using the unless questions: NOT lose rights protect our rights. I know for conditional statements, you negate them by showing that the necessary isn't actually necessary. So, I negated this into "We will not lose our rights even if we do not protect them." The book lists the correct answer as, "Even if we protect our rights, we will lose them." Are these two statements equivalent, since the book's answer just removes the two negatives in my answer?

Question 12: "The university cannot give a substantial contribution without imperiling its own endowment." I negated this to "The university can give a substantial contribution even if it does not imperil its own endowment."

I guess I am confused on when can negates to may not vs cannot?

[Francis O'Rourke]

Hi Jessica,

As Emily said in her earlier post, generally we should provide a negation that states that "something doesn't HAVE to happen."
Applying that rule to question number one, we should write: "it is not necessarily true that most drivers are not good drivers."

That construction is unfortunately a bit of a mouthful, and the answers on pages 6-92 to 6-93 try to balance the strictly correct negations with ease of understanding. So, you may see some answers that simply add or drop a "not" to negate the original statement, but as Emily said earlier this will get you to the right answer the vast majority of times.


Number eight is a more complicated example since it is a conditional statement. It states that if we don't lose our rights (i.e. if we keep our rights), then we must protect them. We can also phrase this in the contrapositive by saying 'if we don't protect our right, we will lose them.'

To negate a conditional statement, we need to express that the sufficient condition will not always lead to the necessary condition. The answer on page 6-93 may be phrased a bit too strongly than what is needed. the negation of this statement may be better expressed as "Even if we do not protect our rights, we may or may not lose our rights."


Number seven is interesting because the original statement only speaks of possibility. That is, it states that something could be true. You may describe this as a "soft" statement. The negation in this case will lead us to a stronger statement.


As for number 12, what you wrote is logically identical to the answer provided in the book. The only difference is that you constructed your negation without using the word "without."

The original statement informed us that giving a substantial contribution would necessarily imperil the endowment. The negation of this is simply that giving a substantial contribution does not necessarily imperil the endowment. Your negation communicates this idea as well as the answer on page 6-93.


Let us know if you have any additional questions, and good luck!

[jessicamorehead]

Francis,

Thank you for your response, yet again! It sounds like, with practice, the negation technique will become easier. I just need to stop thinking about it with such a rigid mindset. I'll update you if I encounter any issues and need further clarification!

Jessica

[rabia.osman]

Hi,

Quick Question:

For the Statement Negation Drill: Unless we protect our rights, we will lose them. Can i also read it as : If we did not lose our rights then we can protect our rights ?

PR : Protect rights; and L: lose

I am diagramming that NOT L --> PR

Now, would the opposite of that be PR --> L? and is "Even if we protect our rights we will lose them" diagrammed as PR --> L

[Adam Tyson]

Your understanding of the original claim looks perfect to me, rabia.osman! The only place I see you going astray is at the end, where you said:

Now, would the opposite of that be PR --> L? and is "Even if we protect our rights we will lose them" diagrammed as PR --> L

First, the logical opposite would be anything that shows a sufficient condition happening and a necessary condition NOT happening. That would mean the so-called necessary condition isn't really necessary! Not that it CANNOT occur, but that it NEED NOT occur.

So, I would negate your statement as "Even if we don't protect our rights, we might not lose them."

Finally, we don't diagram the negation of a conditional statement, because the negation is saying that a relationship is NOT conditional. Instead of diagramming it, just put it in those "even if" terms - the sufficient can happen "even if" the necessary does not. Again, not that the necessary condition cannot happen, but that it doesn't have to.

Good work, keep that up!

[Katya W]

Hi PS, I am re-posting this because I posted a couple of weeks ago and did not receive a reply, so I figure it got lost in the mix! Also, my apologies if this thread is not the appropriate one for this topic. Sometimes I have a hard time finding the appropriate thread for a random, non-prep test related question.

Anyway, I came across the below on a practice drill I was doing for diagramming conditional/formal logic, and I don’t understand how the question was diagrammed in the answer. And no real explanation was provided. So I’m hoping you all can help me since you are my main source of all support and answers!

Q: No one who really cares would ever just give to a charity without understanding how the charity operates.

The answer diagrammed it like this:

If really cares Does NOT give without understanding
Contrapositive: If gives without understanding does NOT really care

How was this answered drawn? It seems to me like the sufficient condition was all cut up, and that “without“ was incorrectly inserted as part of the necessary condition instead of becoming the necessary condition. Oh, and the negate the sufficient clause principle I am used to implementing when I see unless, without, etc, was not used here. I was honestly confused on how to diagram this at all in the first place. And then once I saw their diagram I became even more confused.

Do you have any idea what happened here? Thank you! Katya

[Christen Hammock]

Hi Katya!

This drill applies the Unless Equation, since "without" is one of our trigger words for that kind of conditional statement. More about the Unless Equation can be found in this helpful blog post!

There are two conditions here: really caring, and giving without understanding how the charity operates.

The first step in the Unless Equation is to take the term modified by "unless" (or, in this case, "without") and treat that as our necessary condition:

No one . . . would ever just give to a charity without understanding how the charity operates. "No one would ever" provides the "NOT" here, so the statement becomes Does NOT give without understanding.

The second step of the Unless Equation is to 1) take the other term, 2) negate it. This becomes our sufficient condition.

When negated, "no one who really cares" becomes "someone who really cares." That's where If really cares comes from.

Thanks for reaching back out!

[Katya W]

Thank you Christen for getting back to me! What's tripping me up here is that don't we usually not repeat the ”without(or unless)” in the diagram? Also, why is the ’NOt’ being applied to both the necessary AND the sufficient condition in your explanation? Sorry! I'm still confused

Thank you!

[Adam Tyson]

Hey Katya, I think your original approach here is perfect! There is a more complex way of doing this, involving a "nested conditional," and that would look something like this:

Care (Give Understand)

You might read this as "If you care, then if you give to a charity, you understand how that charity operates."

Now since this question is in a thread about negating answer choices, just in case you wanted to know how to negate this, we want to say it isn't true. That means the necessary condition may not be necessary. Here's how I would do that:

"Someone who cares might give to a charity even if they don't understand how it operates."

That denies the truth of the original claim, and that is what negation is all about!