I'm working on some problems that demonstrate some simple implications. The logic seems to be very different from the way I'm used to using it in everyday language. I'm not sure what assumptions I am allowed to make to show that the statement is true or false. For example, can I consider hypothetical worlds in which a statement I know to be false is actually true? Below is an implication that I need to show can or cannot always be true.
Every good boy does fine $\Rightarrow$ Some bad boy doesn't do fine.
- True, assuming every good boy doesn't do fine.
- It could be false though, assuming every good boy does fine is true and some bad boy doesn't do fine is false.
If I'm allowed to assume anything I want how can an implication always be true? What am I missing?