Negating a Conditional Statement

alexroark5

Hey guys,

I can't remember where this was in the course, but what is the proper way to negate a conditional statment? So for example:

~ (X-->Y)

How would we properly negate if X then Y?

alexroark5

If any one is curious the question that I'm looking at with regards to the above is PT 49, Section 2 Question 17

jdawg113

/Y->/X
http://7sage.com/lesson/the-contrapositive/

jdawg113

just looked at the question and dont see where your question comes from
eta: jk I got to the end and assuming trying to see how it destroys the argument... gotcha, yeah if theres no intelligence then there can be no complex, goal oriented behavior

Jonathan Wang

The contrapositive is not the same as the negation. Never forget that. The negation changes the meaning of your statement; the contrapositive's express purpose is to keep meaning the same. The negation is one step in taking a contrapositive, but they are not equivalent.

Long version: Put it into a simple example if you ever forget. If I told you "All apples are delicious" (A -> D), what would you have to do to prove me wrong? Negating that claim would give you "It's not true that all apples are delicious", meaning you only need one counterexample. Hence, the negation you are looking for is "A some ~D" - some apples are not delicious.

Short version: ~(X -> Y) = X some ~Y.

You can extend this to any variant. ~ (X -> ~Y) yields X some Y. ~ (~X -> ~Y) gives you ~X some Y. Remember also that negations work by chopping the world in two, so the negation of X some ~Y is X -> Y again.

alexroark5

ahhh that's right! Thanks Jonathan!

alexroark5

Jdawg,

Here is where I was coming from on that question

CGO some ~CA
------------------
~(I ---> CA)

(Notation key: CGO = complex goal oriented behavior, CA = conscious awareness, I = intelligence)

To prove the conlusion (now that Jonathan has filled me in) we would need

I some ~CA

But all we have is

CGO some ~CA

So we can bridget the gap like so:

CGO some ~CA
CGO ---> I
-----------------
I some ~CA

JY's explanation for the video is much simpler than this, but as he points out the rough way he arrived at the conclusion won't work in every case if the test writers decide to be more cruel. So this is the formal way you can work this question out every time.

jdawg113

oops lol #RCprobs
yeah I was working so kinda was thinking quickly and obviously not very efficiently

turnercm

The lessons are http://7sage.com/lesson/deny-the-relationship/ and http://7sage.com/lesson/how-to-negate-statements-in-english/ under "Some and Most Relationships." PT 30, Section 2, Q20 has an illustration of this reasoning as well: http://7sage.com/lesson/public-funding-mbt-question/

The proper way to negate conditional statements is by denying the conditional relationship; that is, X is not sufficient for Y.

~ (X -> Y) translates into "X some /Y", or, some things that are X are not Y. This can also be translated to X and /Y.

_FIDELIO_

@alexroark5

To negate a conditional statement you have two options:

A) "Some [of the sufficient] is not [the necessary]".

Thus,

X--->Y

Logical opposite: X SOME /Y

"One can be [the sufficient] and not be [the necessary]".

Thus,

X---Y

Logical opposite: X AND /Y

Use when form A) doesn't make sense.

Think of taking the logical opposite or negating a conditional statement exactly like you would for the word ALL. ALL is a group 1 sufficient indicator and used in conditional reasoning. The logical opposite of the word ALL is "SOME...NOT".

"ALL" means 100 (it is an exact point).

"SOME...NOT" is a range 0-99.

Therefore, you can find the logical opposite of a conditional statement by also using "SOME...NOT".

Hope this helps.

alexroark5

Thanks Turnercm! and Fedelio!