P-value being defined as the probability of observing a result equally or more extreme under a model H0. So if you start with with assumption that H0 is true, there's not much you can say about alternative hypotheses.
The American Statistical Association actually put out a statement this year on the issue of p-value. You can find the whole article here http://dx.doi.org/10.1080/00031305.2016.1154108 but here are the main points:
1) P-values can indicate how incompatible the data are with a specified statistical model.
2) P-values do not measure the probability that the studied hypothesis is true, or the probability that the data were produced by random chance alone.
3) Scientific conclusions and business or policy decisions should not be based only on whether a p-value passes a specific threshold.
4) Proper inference requires full reporting and transparency.
5) A p-value, or statistical significance, does not measure the size of an effect or the importance of a result.
6) By itself, a p-value does not provide a good measure of evidence regarding a model or hypothesis.
As an additional curiosity, the group of writers was not completely unconflicted coming up those definitions and the article contains a number of supplemental articles by the individual authors to clarify/dispute some of the points made.
[edit] fixed formatting
The American Statistical Association actually put out a statement this year on the issue of p-value. You can find the whole article here http://dx.doi.org/10.1080/00031305.2016.1154108 but here are the main points:
1) P-values can indicate how incompatible the data are with a specified statistical model.
2) P-values do not measure the probability that the studied hypothesis is true, or the probability that the data were produced by random chance alone.
3) Scientific conclusions and business or policy decisions should not be based only on whether a p-value passes a specific threshold.
4) Proper inference requires full reporting and transparency.
5) A p-value, or statistical significance, does not measure the size of an effect or the importance of a result.
6) By itself, a p-value does not provide a good measure of evidence regarding a model or hypothesis.
As an additional curiosity, the group of writers was not completely unconflicted coming up those definitions and the article contains a number of supplemental articles by the individual authors to clarify/dispute some of the points made.
[edit] fixed formatting