Since we have been working on some logical reasoning questions I'm going to try to collect the principles I'm following here for everybody's reference. Please feel free to discuss and add more.

If X then Y
This is the equivalent of: If non Y then non X.
Example: If it rains, then I will take an umbrella with me. I don't have a umbrella with me. That must mean it is not raining.

This is NOT equivalent to: If Y then X, or If Y then non X, or if non Y then X. In fact, if we know "If X then Y" and Y occurred, X may or may not happen.
Example. If it rains, then I will definitely take an umbrella with me. I have a umbrella with me today. Is it raining? It may or may not be raining. I said if it rains I will take an umbralla with me. But I could also take an umbralla with me just for the sake of it, even if it doesn't rain. By the same token, if it is not raining, do I have an umbralla with me? I may or may not have.

Using symbals:
X->Y<nonY>nonX
These two below are the same thing:
nonX->Y<nonY>X
X->nonY <Y>non X

Y if and only if X
This is the equivalent of: If X then Y, AND if Y then X. Also, if non X then non Y. If non Y then non X.

Example:
I will take an umbralla with me if and only if it rains. If it rains, then I have the umbralla with me. If I have the umbralla, then it must be raining. If I don't have the umbralla, then it mustn't be raining. If it isn't raining, then I don't have the umbralla with me.

Y unless X
This is the equivalent of: If non X then Y. Also, if non Y then X.

Example:
I will take an umbralla with me unless it is sunny. If it is not sunny, I will take an umbralla with me. If I don't have an umbralla with me, it must mean that it is sunny. However, if it is sunny, I may or may not take an umbralla with me. If I have my umbralla with me, it may or may not be sunny.

Good job....to put this in our heads mathematically, u can deduce these.


----> just like we inverse inequality sign when we multiply -ve, we can perhaps do the same for X---> Y

1) For X--> Y......-ve on both sides will give.....(non Y ----> non X)...note the relation is reversed i.e. Y to X now

2) For (non X ----> Y)......multiply-ve on both sides.....(non Y ----> X)

3) For (X----> non Y).....multiply-ve on both sides.....(Y-----> non X)

Just a thought on mathematical approach for easier recollection.

Great post Honghu, it is often not easy to verbalize such abstract concepts
_________________

Best Regards,

Paul

This is seriously an awesome post, our forum here is worth more than any kaplan course!:read
You guys are awesome!

We have to be careful about sometimes, always, and never.

If A = always doing something
then non A = not doing something sometimes

If A = never doing something
then non A = doing something sometimes

If A = doing something sometimes
then non A = never doing something

If A = not doing something sometimes
then non A = always doing something

For example:

Birds sing sometimes. A never sings. Therefore A is not a bird.
Birds don't sing sometimes. A always sings. Therefore A is not a bird.

Compare to:

Birds always sing. A doesn't sing sometimes. Therefore A is not a bird.
Birds never sing. A sings sometimes. Therefore A is not a bird.

Compare to:
Birds sing sometimes. A sings sometimes. Is A a bird? We don't know. A may be a person who sings sometimes.
Birds sing sometimes. A doesn't sing sometimes. Is A not a bird? We don't know. A maybe a bird who sings sometimes and doesn't sing the other times.

Oh this is a gr8 post
Thx Hong...

Yes, that's very good. If raining is the necessary condition of me bringing the umbrella, then if I have the umbrella you know that it must be raining. However, if it is raining, I may or may not take the umbrella with me.

Hi Honghu,
for the if x then y section, you have this sentence: "This is NOT equivalent to: If Y then X, or If Y then non X, or if non Y then X. In fact, if we know "If X then Y" and Y, X may or may not happen. "

I think you meant: "In fact, if we know "If X then Y", then X may not or may not happen if Y occurs (y->not sure)"

Is that right ?

1. My Dog Dylan loves being brushed. At the moment, he is not a happy doggie, so I can't have just brushed him.

If X, then Y => Not Y, then Not X
if I brush my Dog, he will love it => He is not happy, I can't have brushed him
CORRECT LOGIC.

2. Dylan barks loudly when he is alarmed or frightened. One night I woke up when he barked fiercely. I concluded that he must have been either alarmed or frightened, so I tiptoed down the stairs expecting to find a burglar in the house. Was my conclusion a logical one?

When Dylan is alarmed or frightened ==> Dylan barks loudly
Dylan barked loudly ==> alarmed or frightened.

Wrong Logic

Correct logic is :
If X, Then Y ==>
a) X then Y or
b) Not Y, Not X

Please correct me if I am wrong.

I didn't Understand this.
First, You said that 'if it rains, then I will take an umbrella with me.' The 'If/then' here is a condition that implies that you will take an umbrella "only" if it is raining. Does GMAT have such devious logic?