P: X → Y & not ZP: A → ZP: B → AP: Y and A exist.C: ?

kewheeler July 17 at 05:04PM

How to Solve

Hi! I don't understand what pattern/ process is being used to come up with the conclusion. It doesn't seem consistent from questions to question, whether you use the contrapostive or not. I also don't understand the consistency of the ordering of the answers as I feel like it is always different. Example, we take contrapostive on premise 1, but not premise 2 to get the conclusion. What is the best way to go about these problems?

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Mehran August 11 at 08:53PM

I think the confusion here comes from thinking there’s a “one-size-fits-all” rule about when to take contrapositives and when not to.

In reality, you only use the contrapositive when it helps you connect ideas that aren’t directly linked in their current form.

If the statement is already in a usable direction for your chain, there’s no need to flip it.

Let’s break this example down step-by-step:

Premises:

P1: X ==> Y & not Z
P2: A ==> Z
P3: B ==> A
P4: Y and A exist.

Step 1: See what’s already given as true.

We know Y and A exist from premise 4.

Y never appears as a sufficient condition though so we are unable to use this to draw any additional conclusions.

A, however, does appear as a sufficient condition in premise 2:

A ==> Z

We can immediately conclude that Z exists (no contrapositive needed).

Step 2: See if that conflicts with other conditions.

From premise 1, if X exists, then Z cannot exist.

But we just found that Z exists, so X cannot exist.

That’s where you use the contrapositive:

Z or not Y ==> not X

We plug in Z existing and we can conclude that X does not exist.

Step 3: Check if there’s more to connect.

Premise 3 says B ==> A but we already know A exists.

That doesn’t tell us whether B is true — remember, conditional statements don’t let you reverse the arrow (Don't Just Reverse!).

Step 4: Conclusions.

With the information given, the conclusions we can draw with 100% certainty are (1) Z exists and (2) X does not exist.

Big Picture Tip: Don’t think of contrapositive use as a “sometimes yes, sometimes no” rule — think of it as a tool you grab only when the arrow is pointing the wrong way for your chain.

If the direction already works, keep it as is. If not, flip it. That’s how you keep things consistent from question to question.