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Does anyone feel "un-intuitive" when we take the contrapositive of conditional statements posited in a way that the sufficient condition is in the negative (e.g.,)

~Dave buys a cat -> Dave buys a dog

~Dave buy a dog -> Dave buys a cat.

How did you get over that "uneased" feeling.

jimmierock wrote:Does anyone feel "un-intuitive" when we take the contrapositive of conditional statements posited in a way that the sufficient condition is in the negative (e.g.,)

~Dave buys a cat -> Dave buys a dog

~Dave buy a dog -> Dave buys a cat.

How did you get over that "uneased" feeling.

You just do? Sadly, the games are easier the less you "think" and the more you just "see".

Noblesse_Oblige wrote:

jimmierock wrote:Does anyone feel "un-intuitive" when we take the contrapositive of conditional statements posited in a way that the sufficient condition is in the negative (e.g.,)

~Dave buys a cat -> Dave buys a dog

~Dave buy a dog -> Dave buys a cat.

How did you get over that "uneased" feeling.

You just do? Sadly, the games are easier the less you "think" and the more you just "see".

I am really beginning thats the case now, it seems like I spend so much time arguing my intuition during practice my head just blows up, I really need to learn to trust the deductions and just go with it.

I'm a bit confused as well. Couldn't it be the case that if Dave buys the dog he didn't have to buy the cat? Maybe I'm just stupid lol.

It might be better to "ease into it" by using a more intuitive example.
If I eat too much ice cream, then I get a tummy ache (expressed as IC -> TA)
So, if I don't have a tummy ache, it isn't possible that I ate too much ice cream (since we know that's what happens if IC obtains). Or, ~TA -> ~IC.
Basically, if A always causes B, we know that if ~B occurs, A couldn't have also occurred (since A causes B, and you can't have both B and ~B). Hence, ~B -> ~A.

JohnV wrote:I'm a bit confused as well. Couldn't it be the case that if Dave buys the dog he didn't have to buy the cat? Maybe I'm just stupid lol.

idk, bro. Just accept it, and let it all soak in.

If not THIS, then THAT

If not THAT, then THIS

~A > B
~B > A

If Im not right, then Im wrong
If Im not wrong, then Im right

I had trouble making it "click" too, but it finally did when I thought of the LG in math terms.

You know how when you have an inequality, if you multiply or divide by a negative number, you have to flip the sign?
Example:
6 > 4.
Multiply it by negative 1.
-6 < -4.

Extend that to making the contrapositive. You're multiplying the statements by negative one, and you have to flip the arrow.
Example:
If A is selected, B is selected.
A --> B
(-1)A <-- (-1)B
Swap the sides so your arrows are facing the same direction, and remove the 1 since it's assumed, and voila...
-B --> -A

And remember, if you have something like "If A is selected, B is not selected," you'd have a negative for the B to start, and when you multiply a negative by a negative, it makes a positive.
Example:
A --> -B
(-1)A <-- (-1)(-B)
B --> -A

I hope this helps. I never actually used the math formulas on the tests, but the logic behind them made SO much more sense when I thought of the rules in those terms.

Just realized you were looking for a negation in the sufficient condition. It doesn't change anything to have a negation as the sufficient condition, but maybe a different example will help.
If I don't take my pills, I feel sick (or, ~P -> S).
Therefore, if I don't feel sick, then I must have taken my pills (~S -> P).

If you can distill it down to symbols/letters, it's probably easier to understand; you don't get bogged down in the soundness of the argument and can concentrate in the validity.

To JohnV: you could be right, but we can't deduce it from the statement, so we can't say for sure. It is also possible that if Dave buys a dog, he buys a cat.

Ok so his original example was wrong, correct?

If Dave buys a cat -> Dave buys a dog.

But if Dave buys a dog, he doesn't necessarily have to buy a cat so (Dave buys a dog -> Dave buys a cat) is the wrong inference.

The correct inference is (If Dave doesn't buy a dog, then Dave doesn't buy a cat).

JohnV wrote:Ok so his original example was wrong, correct?

If Dave buys a cat -> Dave buys a dog.

But if Dave buys a dog, he doesn't necessarily have to buy a cat so (Dave buys a dog -> Dave buys a cat) is the wrong inference.

The correct inference is (If Dave doesn't buy a dog, then Dave doesn't buy a cat).

Nah, you just misread it a little. The OP's conditional was:

~Cat -> Dog (don't forget the tilde!) and for the contrapositive, we swap the sufficient and necessary and negate both

~Dog -> Cat

Another way to look at this is that Dave must buy *at least* 1 of cat/dog. When you have a conditional with mixed "signs", negative/positive means at least one must happen and positive/negative means at most one will happen.

Try thinking in terms of "it's not the case that..."

Thanks everyone, and the math multiplying example was definitely a new approach.

Math multiplying approach was brilliant. Wish I had thought of it that way sooner. SO SIMPLE!

Also helpful:

~A --> B

is equivalent to

A or B (or both)

anon sequitur wrote:Also helpful:

~A --> B

is equivalent to

A or B (or both)

This.

If he doesn't buy A ===> He buys B

If he doesn't buy B ===> He buys A

Overall he is, no matter what, buying one OR the other (whether that is A OR B). However, he also has the option to buy BOTH.
He just can't have "Neither A nor B" or "None"

That is how I got over the uneasy feeling.