Clarification on negation vs contrapositive

jpandl

I am in the Some and Most Relationships category of the curriculum and I am finding myself a little confused about negation vs the contrapositive. One of the quizzes gives this example: "All non-water breathing mammals have limbs". The task states to: 1. Translate all English statements into Lawgic. 2. Negate each statement in Lawgic. 3. Translate each statement back into English. So my translation looks like this: NWBM --> L. Now I believe that this is correct but the next step is where my confusion begins. I recognize that All is a group 1 logical indicator so the task for that involves finding the contrapositive which would be: /L --> /NWBM. This is not negation which would be in the english translation: Some NWBM do not have L. So I believe that my confusion is coming from not understanding when I am supposed to apply a negation rather than a contrapositive to the original statement in an LR question.

BinghamtonDave

The contrapositive is the logical equivalent of the original statement. I find it helpful with the contrapositive to think about what it is actually saying:
Original statement says:
If Water Breathing Mammal----->Limbs
What this means is that having limbs is necessary for Water Breathing Mammals that means Water Breathing Mammals need the condition of limbs.

The contrapositive states that in different language. Without limbs then we cannot have that Water Breathing Mammals
This is a way to tell us that that limbs is necessary for Water Breathing Mammals

So, lets think of an example that might be a bit more intuitive. Do you watch the walking dead? There is a child on the walking dead called Judith

A conditional statement that might help us understand what a contrapositive actually means would be:
If Judith is to survive, she needs caring adults to help her
JS---->CA

In the absence of caring adults, Judith cannot survive
CA---->JS

The second statement tells us what is necessary for Judith to survive, just in different language.

The above two example are examples of sufficient and necessary relationships
The conditions above exist in a relationship with each other. A very specific type of relationship that we deduce by taking apart the sentence as it is written in English.

Now a negation of a conditional statement is a denial of the relationship. The negation of a conditional statement is essentially saying: hey wait, that thing is not necessary for that other thing, I will show you how! This is not the contrapositive, the contrapositive of our conditional statement is a restatement of the conditional relationship in other words.

Logical equivalence:
Judith survive---->caring adults to help her
Caring adults to help her----->Judith cannot survive

Negation:
Judith survive---->caring adults to help her
Judith survive and caring adults to help her

You see what the negation is saying? The negation is saying: that conditional relationship you just posited, actually, it doesn't hold. The sufficient condition (Judith surviving) can hold without the necessary condition. This means (in the opinion of the negation) that the necessary condition isn't actually necessary! Because if it were necessary then its failure would mean that the sufficient condition would also fail.

Follow up to this post coming, I hope this helps.
David

jpandl

Thank you. That is good, but I guess my main question was trying to understand how I am to know when the contrapositive is need as opposed to the negation in an LR question.