I'm having trouble applying contrapositives because in many instances they seem like bunk. Specifically, it seems like they require unsupported assumptions other than those that can be derived from game rules. Here is an example:
A group of three must be selected from the variables, A, B, C, D, & E:
1. If C is not selected, then B is not selected
2. D and E cannot both be selected
3. In order for A to be selected, B must be selected.
According to the study guide that I'm using, the contrapositive of rule 1 is, if B -> C. As I see it, although B -> C may be true there is nothing that requires it to be true. To my mind and contrary to the contrapositive of rule 1, it is also possible, although not required under the rules that if B is selected then C is still not selected. That being said, contrapositives are a proven concept and its unlikely that I'm right and my study book is wrong, so what am I missing, what systematic leap in logic am missing to make the concept of contrapositives useful on the LSAT? Thanks
Comments
Let's turn an abstract "A->B" statement into something concrete. I like geographical examples. Your specific case is "~C->~B," so let's say, "If you are not in Canada, then you are not in British Columbia." (You can see why I like geography--it's clear, and you can generally find something that starts with the proper letters!)
Assuming you agree with me that "if you're not in Canada, you're not in British Columbia," how do you feel about the contrapositive? "If you ARE in British Columbia, then you ARE in Canada." Does that sound like a far-fetched assumption, or does it seem inarguably true?
It's not like in real life, where if you have the rule
"If my little brother doesn't go to camp I can't go to camp" (if C is not selected B is not selected), you can still imagine a scenario where I go to camp and he doesn't, because he got sick the day of departure and I convinced my mom to let me go alone.
In LSAT world I have a really inflexible mom. If he's not there, there's no way in hell I'm there, cause that's her rule and it always applies. So if I am there, he must be there too. Because otherwise I'd be stuck at home.
Now, in more lawgical terms,
If your rule states
If C is not selected then B is not selected (/C-->/B)
The contrapositive B-->C (if B is selected, C must be selected), is absolutely, always true.
How would we construct a scenario that disregards the contrapositive, where B is selected, but C is not selected?
Let's say I select B. And I don't select C. Well, as soon as C is not selected, it triggers the original rule "if C is not selected then B is not selected" so B is forced out. B can't be both selected and not selected.
I'd bet your confusion lies in the fact that conditional statements are not inherently true. I can say "If I've got a glass of water, then I'm a spaceman living in orbit around Saturn." Well obviously that's nonsense and so the contrapositive would be as well. (Nonsense though they may be, they would still be logically equivalent nonsense.)
The thing about the LSAT is that the question stem will almost always ask you to assume that either the stimulus or the answer choices are true. So in a way, I guess you're right in that an assumption is required. But the content doesn't matter. The important thing is to understand why the underlying logical structure works. So if you see the water/spaceman example on the LSAT and it asks you to assume it is true, then yeah, if I'm not a spaceman living in orbit around Saturn, there's no way I've got a glass of water.
For example, we often use sentences like "If Trump is intelligent, then I'm a monkey's uncle!" to express a hyperbolic denial of Trump's intelligence. And the way that some linguists and philosophers account for this is hyperbole is precisely by way of contraposition! That is to say, it's outrageous to think that I'm actually a monkey's uncle, so it's obvious the consequent is false. And because the the consequent is so obviously false, by contraposition, so too must the antecedent be so obviously false. Thus, we "transfer", so to speak, the hyperbole from the denial of the consequent to the denial of the antecedent, which is how we use these expressions in ordinary language.
To bring this back to your example, imagine you and your friend are dying of thirst in a desert. Your friend asks you if you have a glass of water and you utter, "If I've got a glass of water, then I'm a spaceman living in orbit around Saturn." We would interpret you to mean "are you crazy? Of course I don't have a glass of water!" and we arrive at this implicature precisely because of reasoning by contraposition!
Some day. Some day.