Counter-intuitive contrapositives: Explain? Forum

[conticuere]

Hi everyone,

I of course understand the contrapositive and its essential importance. Sometimes, though, I find a particular instance of it counter-intuitive. Please help me think through this. I'm dumb.

Let ~ = not.

~P ---> ~Q

cp:

Q ---- > P.

not having P guarantees not having Q. But for some reason for me it seems strange to think (in some cases) that having Q guarantees P.

Here's a case in which it makes perfect sense:

If I am not voting in the general election, I am not voting for Clinton. (not A guarantees not B)
If I vote for Clinton, I am voting in the general election. (B guarantees A)

I thought of some reasonable (I thought) real world examples:

If I don't have AIDS, I don't have the HIV virus. (~A >> ~H) We can agree this is true.
If I have the HIV virus, I have AIDS. (H>>>A). We know this is not true.

If I don't buy a dog, I will most certainly not buy a cat. Let's say this is a person's conviction.

If I buy a cat, I will buy a dog. Why would C guarantee D?

WHY AM I DUMB? Are my examples ill-formulated?

[Lawfayette]

Try using "necessary" to identify the necessary condition and I think it makes more sense.

"If I do not buy a dog, then it is NECESSARY that I do not buy a cat"

So if you are not buying a dog, it has to be the case that you are not buying a cat. Therefore, if you buy a cat, there's no way you can not not buy a dog (confusing language...), so you must have bought a dog.

Does that make sense?

[RoryC]

I believe you're thinking too much. It can get confusing if you try to think of LSAT statements in a real-world sense - as you've demonstrated with your examples. If I were you (and what I do) is to just reverse statements mindlessly, for lack of a better word. Don't read too much into what they're saying if it confuses you.

[Sprout]

I believe you're thinking too much. It can get confusing if you try to think of LSAT statements in a real-world sense - as you've demonstrated with your examples. If I were you (and what I do) is to just reverse statements mindlessly, for lack of a better word. Don't read too much into what they're saying if it confuses you.

This. In formal logic to properly illustrate contrapositives you need to make sure whatever you're plugging in will actually make sense.
Whatever you asign your A & B's or P's & Q's to should be as simple as possible in order to understand how contrapositives work. I think of them as little grouping subsets. This is stupid but roll with it:

L1: If I am a lady, I am a human.
(If p --> q)

L2: If I am not a human, then I am not a lady.
(If ~q --> ~p)

Notice how you literally cannot be a lady in the first Line & not be a human. It is logically impossible because for every X that is a lady, X is a human. In the exact same way (contrapositive) if I am not a human, I cannot be a lady. Being human is a requirement for being a lady.

For me, I like to think of the p's and q's as groups. Picture a little circle of all things that are P. Then draw a bigger circle around it called Q. All P's are inside of Q. Therefore, if something is NOT Q, it cannot be P. Make sense?

[conticuere]

I believe you're thinking too much. It can get confusing if you try to think of LSAT statements in a real-world sense - as you've demonstrated with your examples. If I were you (and what I do) is to just reverse statements mindlessly, for lack of a better word. Don't read too much into what they're saying if it confuses you.

This. In formal logic to properly illustrate contrapositives you need to make sure whatever you're plugging in will actually make sense.
Whatever you asign your A & B's or P's & Q's to should be as simple as possible in order to understand how contrapositives work. I think of them as little grouping subsets. This is stupid but roll with it:

L1: If I am a lady, I am a human.
(If p --> q)

L2: If I am not a human, then I am not a lady.
(If ~q --> ~p)

Notice how you literally cannot be a lady in the first Line & not be a human. It is logically impossible because for every X that is a lady, X is a human. In the exact same way (contrapositive) if I am not a human, I cannot be a lady. Being human is a requirement for being a lady.

For me, I like to think of the p's and q's as groups. Picture a little circle of all things that are P. Then draw a bigger circle around it called Q. All P's are inside of Q. Therefore, if something is NOT Q, it cannot be P. Make sense?

Yes it makes sense, with your example as with mine about the Clinton/voting thing. In 9/10 cases it makes sense.

I think I figured out the problem with my HIV/AIDS example (sorry, it just came to mind). I wrote above:

If I don't have AIDS, I don't have the HIV virus. (~A >> ~H) We can agree this is true.
If I have the HIV virus, I have AIDS. (H>>>A). We know this is not true.

I think my first premise is false on second thought, for the very reason that the contrapositive is false (even if you don't have AIDS you could still have HIV).

The cat/dog example is still a little abstract for me, but I'll go with it.

[Clearly]

I believe you're thinking too much. It can get confusing if you try to think of LSAT statements in a real-world sense - as you've demonstrated with your examples. If I were you (and what I do) is to just reverse statements mindlessly, for lack of a better word. Don't read too much into what they're saying if it confuses you.

This. In formal logic to properly illustrate contrapositives you need to make sure whatever you're plugging in will actually make sense.
Whatever you asign your A & B's or P's & Q's to should be as simple as possible in order to understand how contrapositives work. I think of them as little grouping subsets. This is stupid but roll with it:

L1: If I am a lady, I am a human.
(If p --> q)

L2: If I am not a human, then I am not a lady.
(If ~q --> ~p)

Notice how you literally cannot be a lady in the first Line & not be a human. It is logically impossible because for every X that is a lady, X is a human. In the exact same way (contrapositive) if I am not a human, I cannot be a lady. Being human is a requirement for being a lady.

For me, I like to think of the p's and q's as groups. Picture a little circle of all things that are P. Then draw a bigger circle around it called Q. All P's are inside of Q. Therefore, if something is NOT Q, it cannot be P. Make sense?

Yes it makes sense, with your example as with mine about the Clinton/voting thing. In 9/10 cases it makes sense.

I think I figured out the problem with my HIV/AIDS example (sorry, it just came to mind). I wrote above:

If I don't have AIDS, I don't have the HIV virus. (~A >> ~H) We can agree this is true.
If I have the HIV virus, I have AIDS. (H>>>A). We know this is not true.

I think my first premise is false on second thought, for the very reason that the contrapositive is false (even if you don't have AIDS you could still have HIV).

The cat/dog example is still a little abstract for me, but I'll go with it.

Noap, it makes sense 10/10 times, you're just confusing yourself. If the contra doesn't make sense it's because the original didn't make sense, it was just formulated in a way that you thought made sense.

[conticuere]

I believe you're thinking too much. It can get confusing if you try to think of LSAT statements in a real-world sense - as you've demonstrated with your examples. If I were you (and what I do) is to just reverse statements mindlessly, for lack of a better word. Don't read too much into what they're saying if it confuses you.

This. In formal logic to properly illustrate contrapositives you need to make sure whatever you're plugging in will actually make sense.
Whatever you asign your A & B's or P's & Q's to should be as simple as possible in order to understand how contrapositives work. I think of them as little grouping subsets. This is stupid but roll with it:

L1: If I am a lady, I am a human.
(If p --> q)

L2: If I am not a human, then I am not a lady.
(If ~q --> ~p)

Notice how you literally cannot be a lady in the first Line & not be a human. It is logically impossible because for every X that is a lady, X is a human. In the exact same way (contrapositive) if I am not a human, I cannot be a lady. Being human is a requirement for being a lady.

For me, I like to think of the p's and q's as groups. Picture a little circle of all things that are P. Then draw a bigger circle around it called Q. All P's are inside of Q. Therefore, if something is NOT Q, it cannot be P. Make sense?

Yes it makes sense, with your example as with mine about the Clinton/voting thing. In 9/10 cases it makes sense.

I think I figured out the problem with my HIV/AIDS example (sorry, it just came to mind). I wrote above:

If I don't have AIDS, I don't have the HIV virus. (~A >> ~H) We can agree this is true.
If I have the HIV virus, I have AIDS. (H>>>A). We know this is not true.

I think my first premise is false on second thought, for the very reason that the contrapositive is false (even if you don't have AIDS you could still have HIV).

The cat/dog example is still a little abstract for me, but I'll go with it.

Noap, it makes sense 10/10 times, you're just confusing yourself. If the contra doesn't make sense it's because the original didn't make sense, it was just formulated in a way that you thought made sense.

That's what I guessed, that I confused myself with my own examples, which are ill-formed.