Compound contrapositives

lindsay.smith.lsmith

Hello,
I am struggling with compound contrapositives. I know that when negating, you also have to switch the and/or. However, what about the use or absence of "but not both"?

Example from February 1999, link provided at bottom:
Two conditions within the problem are listed below:
If yews are not in the park, then either laurels or oaks, but not both, are in the park.
Original: ~Y-----> L or O
Contrapositive: ~L and ~O -----> Y.

If it is not the case that the park contains both laurels and oaks, then it contains firs and spruces.
Original: ~L and ~O ----> F and S
Contrapositive: ~F and ~S. -----> L or O

This second rule doesn't include the "but not both" but it seems to diagram the contrapositive regardless of the "but not both" piece's inclusion.

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Thank you,
Lindsay

TheDeterminedC

The current conditional in the original doesn't visually represent the rule that L and O cannot both be in at the same time.

"Original: ~Y -----> L or O"

New: ~Y -----> 1(L ----> ~O)

For me, this is how I would write it on a timed go. It's important to remember that we write down rules in this abstract way to make the written rules visual, which should be easier to follow. As such, all that is important for writing a rule visually is that it quickly communicates the written form of the rule and that this visual representation is a complete and accurate representation of the English.

The (1) here is important because it visually tells me both of L and O cannot be out, or one of them must be in. The condition after this visually represents the "cannot both be in" rule. So from the abstracted rule, it's quickly told to me that if Y is not in, then I know I only need one of L or O, but I cannot have both of them.

I could also see "~Y -----> L or O" working to represent this same notion by putting a slash through the "or" in the necessary. A slash over the "or" can represent "not both." But as long as whatever notation you do to the visual rule communicates to you the entire rule, it doesn't matter how you represent it. You can even create another notation for this relationship altogether. For instance, a triangle or some other visual shape in place of the "or" could communicate visually that the rule is "I must have one, but I can't have both."

Contrapositive: ~O & ~L (neither is in) -----> Y or O & L (both are in) -----> Y.

EDIT:
Contrapositive: ~F OR ~S. -----> L or O

This is the correct contrapositive for the second rule. The negation of and is or.

I think it would help you to look over the OR lessons here on 7Sage. It explains in detail the implications of or statements and all the worlds that are created by it.