Formal Logic involving EITHER OR BUT NOT BOTH help please!!

[karin.yoo]

Hi Manhattan LSAT and Matt,

Please check if my interpretation of formal logic statements below are correct. If they are not, please correct me! (A friend of mine corrected some of my diagrams below, but I am in need of clarification/explanations.)

I seem to have a hard time in drawing conditional statement diagrams with statements involving "either or but not both". Are there any useful tricks or rules that I should keep in mind?

1. If A then either B or C but not both
My original diagram: A --> ~B and C AND A --> B and ~C
Modified diagram (needs explanation): (B and C) or (~B and ~C) --> ~A

2. If A or B but not both, then C
My original diagram: A and ~ B --> C AND ~A and B ---> C
Modified diagram (needs explanation): ~C --> (A and B) or (~A and ~B)

3. If it is not the case that both A and B are present, then C
~(A and B) --> C THUS ~A or ~B ---> C

4. ~A and ~B --> C
If there is a statement like this above, then are there any useful deductions/inferences that I should be making, such as
"ATL 1 of the A,B,C must be there?"

5. ~C --> B or E
Same question as above. If there is a statement like this above, then are there any useful deductions/inferences that I should be making, such as
"ATL 1 of the C,B,E must be there?"

I understand that this is rather a long list of very specific questions, but I seriously need some clarifications.

Please help. I would greatly appreciate any comment!
Thank you a lot in advance!

[timmydoeslsat]

Hi Manhattan LSAT and Matt,

Please check if my interpretation of formal logic statements below are correct. If they are not, please correct me! (A friend of mine corrected some of my diagrams below, but I am in need of clarification/explanations.)

I seem to have a hard time in drawing conditional statement diagrams with statements involving "either or but not both". Are there any useful tricks or rules that I should keep in mind?

You ask great questions and you have come to the right place. Below are my responses to each question you have.

1. If A then either B or C but not both
My original diagram: A --> ~B and C AND A --> B and ~C
Modified diagram (needs explanation): (B and C) or (~B and ~C) --> ~A

We need to really simplify this thinking.

I would diagram that statement like this:

A ---> 1 B/C in, 1 B/C out

This way is much simpler and easy to read. We know when the contrapositive triggers as well. If it is the case that both B and C are out...or both B and C are in, we know that A will not be occurring.

2. If A or B but not both, then C
My original diagram: A and ~ B --> C AND ~A and B ---> C
Modified diagram (needs explanation): ~C --> (A and B) or (~A and ~B)

1 A/B in, 1 A/B out ---> C

Same line of reasoning as the first situation. We know the contrapostive would trigger with ~C. The consequences of this action are such that A and B will either be both in or both out. In other words, if C is out, we know that A and B will both be joined together either in or out...they cannot be split amongst themselves.

3. If it is not the case that both A and B are present, then C
~(A and B) --> C THUS ~A or ~B ---> C

This is the classic structure of what an "at least 1" rule is. For instance, if you do not have A, you have B. According to this rule, I could never have both A and B both be out at the same time. As having even one variable out triggers the other to be in.

So for this situation, your statement can be notated as:

~ [A and B] ---> C

We must have it be true that one of [A and B] or C is always in. Think of [A and B] as a single entity. That coupled variable set or C must always be in.

4. ~A and ~B --> C
If there is a statement like this above, then are there any useful deductions/inferences that I should be making, such as
"ATL 1 of the A,B,C must be there?"

There are inferences to be made when we have that ~A ---> B structure. That inference is that one of those guys must always be in. The same idea holds true in terms of A ---> ~B, which means at least one variable must be out. Like you have noted, there is no possible way that A, B, and C could all be out because the sufficient condition would have been met and C must be in.

5. ~C --> B or E
Same question as above. If there is a statement like this above, then are there any useful deductions/inferences that I should be making, such as
"ATL 1 of the C,B,E must be there?"

This point has been addressed above coincidentally.