Formal Logic involving EITHER OR help!!!!!!

[kky215]

Hi Powerscore and Dave,

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!

[Dave Killoran]

Hi K,

You actually don't need any clarification, because you nailed each one I'll make a few comments on some of these, but overall, you did extremely well!

1 and 2. When you have a condition that relates to exactly one of the two occurring (such as "~B and C"), negating that term creates two scenarios: neither (~B and ~C) or both (B and C). While this is tricky when you first encounter it, and there is no single (easy) diagram to encompass both, once you seen it a few times (and it is somewhat rare in LSAT questions), it's relatively easy to handle. You got both of the contrapositives right for these problems.

In a sense, an alternate diagram for this (#1) is:

A exactly 1 of B/C

When viewed from that standpoint, anything aside from "exactly on" negates the term, and hence neither or both works in negating that term. The same logic works for #2.


3. This condition-- ~(A and B) -- encompasses three possible outcomes: ~A and B, A and ~B, and ~A and ~B. Any of those three scenarios forces C to occur. This wording is used infrequently in LSAT questions (but it does appear, so it's always good to know!)


4 and 5. In these two, 5 is just the contrapositive of 4 (if E is changed to A in #5). As you note correctly, the odd combination of conditions creates an "at least" scenario where you always have at least one of the three variables (and possibly two, or even all three). Apart from that, the range of possible outcomes is too varied to lock down anything else that absolutely must occur.

Please let me know if that helps, and great job!