Setup and Rule Diagrams

[SGD2021]

If you have a conditional statement and its contrapositive, the conditions in both of those statements cant exist at the same time right? So they each represent a mutually exclusive possibility for what can occur? Most importantly, if I have a chain like this for this game: not L-->R--> M-->T-->not V-->S (CP: not S--> V-->not T-->not M-->not R-->L), why do we know that at least one of S or L will always be in the group of volunteers? Can we look at a conditional statement and its contrapositive in a logic game and know that at least one of the statements will occur?

Also, why do we say that S and L are free variables here? Is it always the case that a variable at the end of a chain that isn’t negated is very free? If yes, is it because we can put that variable in the volunteer group and anything can happen with the game (since that variable is in the necessary condition only)? What if the chain above ended with not S and not L? would they be considered free then?

[Adam Tyson]

We can say that we always need either L or S or both because the chain and the contrapositive chain both start with one of them being out and end with the other way being in. If we start with L being out, we can leap over all the stuff in the middle and see that S must be in. If L is out, S is in, and if S is out, L must be in. Therefore, they can never both be out!

And it's not the variable at the end of the chain that is "free" - it's the one at the beginning. So while L being out is sufficient for all the stuff that comes after it in the chain, you still might have L IN and have the rest of that stuff all occur. If we said that L being IN triggered something, we would be making a Mistaken Negation. L and S can both be in the game; they just cannot both be out.