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delaenajoy on February 10, 2020

Missing Premise Questions

I'm working my way through the Missing Premise flashcards and I'm currently struggling to determine an approach to take when starting each questions. I'm getting some of the flashcards right, but when I do I don't always understand how I got to that point or why I'm right. Can anyone give me an idea of the approach I should take when attempting these questions?

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Skylar on February 11, 2020

@delaenajoy, happy to help!

Let's use an example to help clarify:
P: A -> B
P: ?
C: not B -> C

The first step you should take is to find the contrapositive of the statements you were given. To do this, you should reverse and negate. This gives us:
P: A -> B
not B -> not A
P: ?
C: not B -> C
not C -> B

Now, you should look for a way to connect the premise to the conclusion. Generally, you will find two routes (depending on whether you start from the original statement or the contrapositive). Either one is valid. In fact, one will be the contrapositive of the other.

Route 1: We know we want to be able to conclude that "not B -> C." However, we don't have anything about C mentioned in our premises currently. So, we should note that we need to incorporate the new variable C with something related to "not B." The contrapositive of our first premise tells us that "not B -> not A" so we should use "not A" as the related variable we want to connect to C. We can therefore answer: "not A -> C." This works as the missing premise because it allows us to use the transitive property to connect: not B -> not A -> C. This ultimately fills the gap and allows us to conclude that not B -> C.

Route 2: We could also connect the original first premise (A -> B) to the contrapositive of the conclusion (not C -> B). Again, we know that we need to introduce the new variable "not C" and we know that we need to make it related to "B." We can therefore say: "not C -> A." This allows us to use the transitive property on "not C -> A -> B" to ultimately conclude that "not C -> B."

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

delaenajoy on February 11, 2020

It makes a lot more sense now, thank you for your help!

Skylar on February 13, 2020

@delaenajoy, glad that it makes sense. Please don't hesitate to reach out with any other questions and best of luck with your studies!