Does a counterexample exist for the following argument?
If person A is not home, then person B is. But, if A is not home, then B isn’t. So, they are both home.
Translated to logical notation:
1) $\neg A \to B$
2) $\neg A \to \neg B$
3) $\therefore A \land B$
To my understanding, a counterexample is when the premises are true but the conclusion is false. I've equated them as such, but I'm stuck in proving whether or not there is a contradiction, since there are too many cases to deal with (e.g. $A \land B \equiv F$ has 3 cases). How would I find out whether there's a counterexample or not?
1) and 2) seem contradicting already, but they also have 3 cases each.