And's & Or's question 2

[lorein21]

Hi I just posted a question and have a similar question about the same rule.

If I have A and B then I also have C
this means = A+B --> C
contrapositive = ~C ---> ~A and B
or is it = ~C ----> ~A or B


and to get this straight what the phrase is really saying is =

In order to have A and B together I need to have C. But if I just have B I dont need to have C. If I just have A I also don't need C. C only needs to be present when both A and B are together. This tells us nothing about the existence of C. This rule is only triggered when both A and B are placed in the game.


what if it was...

A or B --> C
how does the rule change?

thank you!

Lorein

[Jon Denning]

Hi Lorein,

Thanks for the question. The general rule for "and/or" contrapositives like the one you mention is that when you flip and negate an "and" becomes "or" in the contrapositive, and "or" becomes "and." So something like A and B --> C would yield the contrapositive Not C --> Not A or Not B. The logic behind that is that since we know A and B together give C, when C is not present it must mean that either A or B was also not present (they can't both be there). Of course, it COULD be the case that both A and B were gone, but all we know with certainty is that at least one of them was gone.

For the second example, A or B --> C, the contrapositive would be Not C --> Not A and Not B (where "or" becomes "and").

Hope that helps!