When the economy is in a recession, overall demand for goods and services is low. If overall demand for goods and ser...

BenMingov November 28, 2019

Hi Mprezzy, thanks for the question.

I think we should first deal with the general ideas regarding combining conditional statements and how to deal with contrapositives before approaching this specific question.

I suggest that whenever you are dealing with conditional diagrams, you should take the contrapositive. This will more clearly and readily show you the complete array of possibilities that when triggered, guarantee a specific outcome.

E.g. A - > B on its own only tells me that when A we get B, but if we note the contrapositive, then we know that without B, we cannot have A. While this is implied in the original diagram, it is easier to see when you explicitly write out NOT B - > NOT A.

As for linking conditional statements, the only time we can link two statements is if the same condition is necessary in one statement and sufficient in another. Let's check out by far the simples example of this.

A - > B
B - > C

Note that B is necessary in the first statement and sufficient in the second. That is why we are able to link the two diagrams to create a longer chain.

This then becomes:

A - > B - > C and we can now deduce that A - > C (contrapositive: NOT C - > NOT A)

The test is usually not so friendly in terms of allowing us obvious linking opportunities such as this. And most often, it requires us to work with contrapositives in order to make these chains.

Here is a more "complex" example:

G - > Z
NOT X - > NOT Z

At first glance it doesn't seem like any single term is necessary in one statement and sufficient in another, however, if we take the contrapositive of these statements, we would see that we can make a chain.

Contrapositive 1: NOT Z - > NOT G
Contrapositive 2: Z - > X

Then, we can make the following chain: G - > Z - > X (plus its contrapositive; NOT X - > NOT Z - > NOT G)

Notice how if you had spotted that the common term between two statements was NOT Z, your initial diagram would be what I had written as contrapositive, and your contrapositive would have been my original diagram. So whichever one you choose as the "original" diagram is arbitrary, so long as you are linking correctly.

BenMingov November 28, 2019

Now let's look at the question specifically,

The reasoning structure in the stimulus is as such,

A - > B - > C
Therefore, NOT C - > NOT A

It is probably the case that the analysis started from the contrapositive just because the original argument used the contrapositive to trigger the condition.

I wouldn't worry too much about where you begin your diagramming or where you look first, so long as you are combining chains correctly and always noting that you need the sufficient condition to be satisfied in order for the condition to be triggered.

Let me know if this has helped and please let me know if you have any other questions.

mprezzy December 1, 2019

Thank you. That has helped me out a great deal. I do have another question as I practiced with the flash cards with missing premises.

This is the example:

A -> B

C-> not B

Contrapositives:

A -> B
not B -> not A

C -> not B
B -> not C

I came up with two chains. One is correct and one is not.
Can you please help me understand why the one that is incorrect, is indeed incorrect? I have noticed that I am able to double check my work in other cases because I end up at the same correct conclusion.


Incorrect chain:

A -> B -> not C
C -> not B -> not A

To conclude:
C-> not A

The correct chain:

C-> not B -> not A
A-> B -> not C

To conclude:
A -> not C

This is an example of how I end up with that same conclusion using different premises:

A-> B
not B -> not A

C -> A
not -> not C

C-> B
not B -> not C

C-> A -> B
not B -> not A -> not C
not B -> not C

not B -> not A -> not C
not B -> not C

Skylar December 1, 2019

@mprezzy, maybe I can help!

First, let's clarify something. The original S->N statement and the contrapositive are identical in meaning. So, if A -> B is the correct answer, NOT B -> NOT A is equally correct. If you use the contrapositive and the explanation given uses the original statement, you should still come to the same answer.

You asked about what chain can be drawn from the following premises:
Premise 1: A -> B
Contrapositive 1: NOT B -> NOT A
Premise 2: C -> NOT B
Contrapositive 2: B -> NOT C

We can combine Premise 1 and Contrapositive 2 to get: A -> B -> NOT C.
The contrapositive of this chain is: C -> NOT B -> NOT A.

We could also combine Premise 2 and Contrapositive 1 to get: C -> NOT B -> NOT A.
The contrapositive of this chain is A -> B -> NOT C.

You will notice that the chain we made from combining Premise 1 and Contrapositive 2 is the same as the contrapositive of the chain we made by combining Premise 2 and Contrapositive 1 (A -> B -> NOT C). Similarly, the chain we made from combining Premise 2 and Contrapositive 1 is the same as the contrapositive of the chain we made by combining Premise 1 and Contrapositive 2 (C -> NOT B -> NOT A). These two chains are contrapositives of each other.

This means that both of the chains you gave are correct! In fact, they are the contrapositives of each other. Remember, the original S->N statement and its contrapositive are synonymous in meaning and equally correct. Try not to get caught up in choosing which to use to reason to the same conclusion, as either choice is valid.

As for your second example with different premises, both the chain C -> A -> B and the chain NOT B -> NOT A -> NOT C are correct. Again, the contrapositive and the original S->N statement are both valid.

Does that make sense? Please let us know if you have any additional questions!

mprezzy December 3, 2019

That makes sense. Now that I pulled away and relooked this...I can see that it is the same thing, and I was concentrating on the wrong thing. Thank you for your help!