Do Contrapositives Require Assumptions???

phrogfella

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

attorneysomerville

Three cheers for being brave enough to challenge the contrapositives! I hope I can demonstrate why they work to your satisfaction. If I can't, I'm pretty sure someone else will be able to do so.

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?

Aidoe

First, instead of thinking about it as "if C is not, then B is not," think about it the other way. I presume you would see that it would be logical to to take "if B is selected, then C is selected" and get "if C is not selected, then B is not selected"? Both statements are in fact logical equivalents. Because we know that C is necessary for the selection of B, if it were not selected then B could not. Same thing for "C is not, then B is not." What is absolutely necessary for the condition that "C is not" selected? "B is not" selected. When you deny "B is not" selected (which is equivalent to "B is" selected) then you're denying the necessary condition for "C is not" selected (which is equivalent to "C is" selected). It's the same thing. On the surface it may seem different but it is the same application of logic and it requires no leap by any means on your part.

runiggyrun

I think for getting how ironclad contrapositives are, you need to really internalize the fact that in a game (or LR stim) the stated rules ALWAYS apply.
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.

Cant Get Right

@jjfheintz It's great that you're taking the time to truly grasp the concepts instead of simply memorizing rules and mechanically apply them. There've already been some great examples, so I'll try to expand the conversation without being redundant.

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.

quinnxzhang

@"Cant Get Right" said:
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.)

I wouldn't go so far as to say that your example is nonsensical. In fact, sentences like these can be very good illustrations of how we subconsciously reason by contraposition in our everyday lives! I like to colloquially refer to sentences like these as "monkey's uncle" sentences, and there's some interesting work on these in pragmatics and philosophy of language.

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!

Cant Get Right

Haha, That makes sense. I was more just thinking that I want to be a spaceman living in orbit around Saturn. I looked around and happened to have a glass of water. So a fulfilled sufficient condition with a tragically unfulfilled necessary condition:'(

Some day. Some day.

phrogfella

Thanks for all of the help - the light bulb finally came on! To whoever may find this discussion in the future, the concept of "logical equivalence" was the assumption that I was missing, which led me to learning to test it via a truth table.

J.Y. Ping

@jjfheintz said:
the concept of "logical equivalence" was the assumption that I was missing, which led me to learning to test it via a truth table.

And this is why 7Sage teaches the LSAT better. Which book are you using?

Cant Get Right

@"J.Y. Ping" said:
And this is why 7Sage teaches the LSAT better.

Indeed!

nye8870

I would say contrapositive is in fact a Law not an assumption