>You're entirely right about that specific case. In fact, this is the only case that matters. All other (valid) cases can be reduced to a single, null hypothes

>You're entirely right about that specific case. In fact, this is the only case that matters. All other (valid) cases can be reduced to a single, null hypothesis design. p-value is undefined for hypotheses that are not a null hypothesis. It is also undefined for hypotheses which do not hold. Sure, you can walk through the motions, put some numbers together, and eventually produce a number between 0 and 1. However that does not mean you have computed a p-value. If you are testing a non-null hypothesis you have not computed a p-value. If you are testing a null hypothesis that doesn't hold, you have not computed a p-value.