I am having difficulty with the missing premise and argument completion drills. I can deduce the missing premise down to what two variables are missing but I always switch the variable that should be sufficient with the necessary one. Same thing in the Argument completion questions. How do you decide what is going to be sufficient and what will be necessary and how do you decide which one to negate?
Like for example in the argument completion one:
P1: A ---> not B
B ---> not A
P2: C ---> B
not B ---> not c
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The conclusion I got: not A ---> C
not C ---> A
The correct answer is: C —> not A
A —> not C
Or For the Premise Drills:
P1: not X —> not Y
Y ---> X
P2: ?
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C: not X —> not B
B —> X
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I'll start with your example for argument completion example. These drills are effectively asking us to form a chain by using common terms.
P1: A ---> not B P2: C ---> B
Right now, these premises have no terms in common. However, I can create one by using a contrapositive. Remember that a contrapositive has the exact same logic as its counterpart, they are interchangeable. I'll use the contrapositive of P1, even though either will work.
P1: B ---> not A P2: C ---> B
Because these premises have B in common, I can now create this chain:
C ---> B ---> not A
This is where we get the correct answer C ---> not A or A ---> not C.
Let's try the missing premise question.
P1: not X ---> not Y P2: ?
C: not X ---> not B
Take a look at the conclusion and ask yourself, "How can I get from not X to B?" You will then use the premise you are given to do so. We know that not X ----> not Y. How can I complete the chain to get to not B? Like this:
P1: not X ---> not Y P2: not Y --- not B
C: not X ---> not B
Combining these two premises gives me the answer. My guess is that you are misunderstanding contrapositives and how to use them. In these drills, they allow us to link sufficient and necessary statements to reach a conclusion or find a premise.
