LR And/Or Statements

[ChicaRosa]

Why is it that when you do the contrapositive of an "and" statement Ex: A and B C , it changes to an "or" with the rest being negated C Aor B, and vice versa if the statement began with an "or"?

Why can't the contrapositive be C A and B and vice versa if the statement began with an "or" statement: C A or B?

Thanks!

[Dave Killoran]

Hi Chica,

This is a good question, but let me start by asking you: Why does it have to be the way that it is?

In other words, we know that's how it works, so can you come up with a counter example that shows where it wouldn't work, and why your method would cause issues? If it helps, when thinking about this try to take it out of the abstract A, B, C realm and into a real world example so we can more easily see what is going on. For example, what if the statement was: if you have a 4.0 GPA and a 180 LSAT score, then you will be accepted to Harvard Law School" ? If you take the contrapositive, what happens to the "and," and if you keep it as "and" does the statement then make sense?

Give it a shot and see how you do!

[ChicaRosa]

Hi Chica,

This is a good question, but let me start by asking you: Why does it have to be the way that it is?

In other words, we know that's how it works, so can you come up with a counter example that shows where it wouldn't work, and why your method would cause issues? If it helps, when thinking about this try to take it out of the abstract A, B, C realm and into a real world example so we can more easily see what is going on. For example, what if the statement was: if you have a 4.0 GPA and a 180 LSAT score, then you will be accepted to Harvard Law School" ? If you take the contrapositive, what happens to the "and," and if you keep it as "and" does the statement then make sense?

Give it a shot and see how you do!

Hi Mr.Killoran,

Let's see if the contrapositive is 4.0 GPA and 180 LSAT score accepted at Harvard Law School
then it would mean that you didn't meet the two conditions to be accepted at Harvard. I think the problem is that it's not necessary to meet the two conditions to not be accepted at Harvard. I mean a person could have a 4.0 GPA and have an LSAT score that is lower than 180 and not be accepted or a person has a 180 LSAT score and they don't meet the 4.0 GPA requirement. So with the contrapositive being "or" it fulfills at least one of the possibilities for not being accepted at Harvard.

Am I right?

Thank You!

[Nikki Siclunov]

Hi ChicaRosa,

Let me jump in real quick and add my 2c.

I'm afraid you misdiagrammed the contrapositive of Dave's example. Your version seems like the Mistaken Negation of it

Here's what Dave provided:

• 4.0 GPA
  
  and Harvard Law
  
  180 LSAT

In his example (which is not the real world, mind you!), having a perfect GPA and LSAT score is sufficient to guarantee admission. To arrive at the contrapositive, you need to ask yourself what must be true if you didn't get into Harvard Law? Does your failure to get admitted also establish that you failed to meet both sufficient conditions? Of course not. It's entirely possible you failed to meet just one of them, though it's also possible that you failed to meet either of them. Thus, we arrive at the following contrapositive:

4.0 GPA

Harvard Law OR

180 LSAT

Note that the "or" in logic is an inclusive conjunction: it's possible that both propositions united by it actually occur. So, if I tell you that I'll either go to law school or to business school, I'm not excluding the possibility of attending both.

Let me know if this helps!

Thanks,

[ChicaRosa]

Hi ChicaRosa,

Let me jump in real quick and add my 2c.

I'm afraid you misdiagrammed the contrapositive of Dave's example. Your version seems like the Mistaken Negation of it

Here's what Dave provided:

• 4.0 GPA
  
  and Harvard Law
  
  180 LSAT

In his example (which is not the real world, mind you!), having a perfect GPA and LSAT score is sufficient to guarantee admission. To arrive at the contrapositive, you need to ask yourself what must be true if you didn't get into Harvard Law? Does your failure to get admitted also establish that you failed to meet both sufficient conditions? Of course not. It's entirely possible you failed to meet just one of them, though it's also possible that you failed to meet either of them. Thus, we arrive at the following contrapositive:

4.0 GPA

Harvard Law OR

180 LSAT

Note that the "or" in logic is an inclusive conjunction: it's possible that both propositions united by it actually occur. So, if I tell you that I'll either go to law school or to business school, I'm not excluding the possibility of attending both.

Let me know if this helps!

Thanks,

Hi Nikki,

Thanks for your explanation! I thought it was very helpful After reading your response I realized I mistakenly did a mistaken reversal with David's example when I was doing the contrapositive in my head while replying to his post at the same time