When looking at the hypothesis testing literature, I have noticed that there are different ways of phrasing your conclusions after having done a test.
When the test fails to reject, there seems to be strong consensus that we cannot say that we "accept the null" and there are very good reasons for this. This mainly stems from the idea that hypothesis tests are basically "proofs by contradiction given a particular significance level". If we fail to reject under the null (i.e. the data is not very "extreme" given the null, it still does not mean the null is true).
When the test rejects, we usually say something like "on an $\alpha$% significance level, we can say that $H_0$ is not true". My question is on what we can subsequently conclude about $H_1$. I know that all of these conclusions are not "proofs" and only hold under a particular significance level $\alpha$ which follows from restricting the Type I error of the test. I distinguish two cases:
$H_1$ is the complement of $H_0$ (this of course depends on the underlying parameter space). One example of this is the classical case where we test $H_0: \theta =0 $ vs $H_1: \theta \neq 0.$ I think that most people would say that when your test rejects here, you can say that $H_0$ is not true and thus $H_1$ has to be true. Note that the latter is not a very strong claim for the classical example, as this contains almost the whole parameter space. Nonetheless, when the null is also a composite hypothesis, I think we can still say that $H_1$ holds when it is the complement of $H_1$.
$H_1$ is not the complement of $H_0$. One classical example of this would be a simple null and simple alternative but the underlying parameter space also contains other values. Here, I would say that we cannot conclude the alternative when we reject, we can only say that the null is not true.
Does anyone have any objections to the above claims or any other thoughts? I think it all comes down to the fact that we are dealing with stochastic data and it is hard to make definitive claims about underlying DGPs. However, for policy recommendations etc., we still would like to make some kind of claim and that is why I am asking this question.
Edit: in the standard NHST framework, the alternative hypothesis is always the complement of the null. Therefore, point (2) is not well formulated. Thanks to all comments which provided useful insights.