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Trevor on April 7, 2020

Missing Premise Drills

I am having trouble finding the missing premise when it is the first premise. For example: P1: ? P2: X--> Y P3: not A --> not B C: not A--> not B How do we know what letters we are supposed to be looking for?

Replies

Trevor on April 7, 2020

I made a mistake that conclusion should say B-->Y

Skylar on April 13, 2020

@trevor, happy to help!

The problem you described is:
P: ?
P: X -> Y
P: not A -> not B
C: B -> Y

We should start by first finding the contrapositives of the statements we were given.

P: ?
P: X -> Y
not Y -> not X
P: not A -> not B
B -> A
C: B -> Y
not Y -> not B

Now, we should look to our conclusion. We want to be able to conclude that B -> Y. What do we know about B? Well, the contrapositive of our third premise tell us that B -> A. What do we know about A? Nothing yet. Okay, let's move on to the Necessary condition of our conclusion, which is Y. What do we know about Y? Our second premise tells us that X -> Y. What do we know about X? Nothing yet.

So, we have B -> A and X -> Y. If we connect these two statements, we will have a chain that can be simplified into B -> Y, which is our conclusion. We can connect these statements by making "A -> X" our missing premise.

P: A -> X
not X -> not A
P: X -> Y
not Y -> not X
P: not A -> not B
B -> A
C: B -> Y
not Y -> not B

This allows us to say B -> A -> X -> Y.
The contrapositive of this- and another chain that we could make- is not Y -> not X -> not A -> not B.

Also, try not to get thrown off by which premise is missing. The order that the premises are introduced in does not matter, as this approach is the same regardless.

Does that make sense? Please let us know if you have any other questions and best of luck with your studies!

Trevor on April 19, 2020

Yes, that makes sense. Thank you!