P: not X → YP: X → not ZP: not Z → WC: ?

BenMingov January 14, 2020

Hi Lucas, thanks for bringing this up.

Before we hop into the actual problem at hand, let's talk about how we combine conditional diagrams in the first place and what we can conclude from a conditional diagram.

In order to combine two or more conditional statements, we need to find variables that serve as linking points. This means that a variable/term will be sufficient in one statement and necessary in another. This is the only way to make this happen.

E.g. A - > B
B - > C

We clearly see that B is necessary in the first conditional statement and sufficient in the second. This is the reason that we are able to combine these diagrams to make the following chain:

A - > B - > C

The second point I was talking about is what we can conclude from any given conditional statement.

There are only two things that we can gather from a conditional diagram:
1) The presence of the sufficient condition guarantees the presence of the necessary condition (We can think of the sufficient condition as "enough" information to conclude the necessary)
2) The lack of the necessary condition means we can't have the sufficient condition (think about it simply, if something is necessary for something else, then without what's needed, how can we have that something else?)

Going back to our example. We are given 3 statements:

1) Not X - > Y (Not Y - > X)
2) X - > Not Z (Z - > Not X)
3) Not Z - > W (Not W - > Z)

Looking for linking points (variables that are common in two statements, but as sufficient in one and necessary in the other), we find that X is the first such variable.

Not Y - > X - > Not Z

Looking for the next linking point, we find that "Not Z" plays that role. It is necessary in the second statement and sufficient in the third.

Not Y - > X - > Not Z - > W

This is the entire chain that we can deduce. Additionally, we also know the contrapositive.

Not W - > Z - > Not X - > Y

Referring back to your statement above, you mentioned you know that if W exists then Y does not. But this actually isn't the case. All we know is that when Y does not, then W does. Conversely, if W does not then Y does. Pay close attention to which variables are sufficient and necessary in your created chains.

Please let me know if this was helpful and understandable, if you have any other question, feel free to ask!