Self-study
I have in my notes written (A -> B) -> (A -> C) contrapositive = (A <-some-> /C) -> (A <-some-> /B) but I don't think that's right.
Shouldn't the contrapositive be (/C -> /A) -> (/B -> /A)? Is what I wrote in my notes maybe the negation of that embedded conditional that I was referring to?
Can anyone also point out where in the fundamentals this is referred to? Thank you!
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Your notes are right! When doing the contrapositive of the core conditional (the one that's not in parentheses), we switch both sides and negate both sides.
So "(A -> B)" goes to the right side and is negated. "(A -> C)" goes to the left side and negated.
What does it mean to negate "(A -> B)"? It means to say A does NOT guarantee B. Or in other words, it's possible to have A without B.
What does it mean to negate "(A -> C)"? It means to say A does NOT guarantee C. Or in other words, it's possible to have A without C. That's why you can think of the contrapositive as:
(A is possible without C) --> (A is possible without B)
Another way to visualize that:
(A some /C) --> (A some /B)
In English, the initial statement is saying if A require B, then it also requires C. The contrapositive is if A does NOT require C, then it also doesn't require B.
The problem with the version in your question is that you're doing the contrapositive of the parenthetical conditionals. But we want to negate those -- because we're doing the contrapositive of the conditional in the center.
By the way, what's going on here is far, far more difficult than what we'll be asked to do in almost any conditional-based question.