Are methods of p-value adjustment only necessary when a universal/overall hypothesis is the target of inference? I am trying to understand more clearly in which cases an adjustment of p-values is necessary. At the moment my reasoning about this can be summarized as follows:
Benjamini and Hochberg (1995) list three problems/scenarios where adjustment of the type I error is called for:

*

*multiple outcomes ("multiple end points problem"),

*multiple comparisons ("multiple-subgroups problem") and

*"screening problem" (e.g. screenin of multiple predictors).

Some, for instance Perneger (1998) have argued that such adjustment is necessary only in cases in which one wants make an inference about a "universal hypothesis" from multiple tests.
To me, all three of the above scenarios can be understood as such a case: In scenario 1) the overall (=univeral) hypothesis is that the treatment has an effect (on at least one of the outcomes). In scenario 2) the overall hypothesis is that there are (any) differences between conditions. In scenario 3) the overall hypothesis might be that any of the predictors affects the outcome (although that logic is less clear to me in this case).
So does that mean that drawing an inference about such an overall hypothesis (using several seperate tests) is the main/real/only reason for using p-value adjustment methods?
And does that mean that if I strictly avoid inferences about an overall hypotheses of any kind (directly/implicitly or otherwise), then there is no reason to use such adjustment methods?
Or am I misunderstanding something here (and if so, what)?

Usually I explain it this way: when you have p-value 0.05 it basically means you are wrong 1 in 20 times. On the other hand, our main consideration with hypothesis is to accept or reject them, not probability.
Hence, when you have 20 p-vals 0.05 "on average" you'll accept 1 false positive. This is why you introduce correction. You lose some accepted hypothesis, but you get more accurate results.
This is really also about how you read results. Sometimes (ie. while struggling with results) people will say that due to explanatory characteristics of analysis they omitted correction or they would give both values. This kind of results should be treated with caution. Mostly, while testing multiple hypothesis, you should use correction.
