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I understand that we never say that the null hypothesis is accepted, instead we say that the null hypothesis is not rejected since we can never prove an effect does not exist through empirical evidence.

But does the opposite hold? Is there a difference between rejecting the null hypothesis and not accepting the null hypothesis? In the same sense, how can we prove that an effect does exist? Surely we can only show that there is a low probability, say at 1% significance level, that an effect does not exist and thus we can only not accept it?

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  • $\begingroup$ It depends what you mean by reject and also by not. For a statistically significant effect it is usual to "not accept the null hypothesis" so for the opposite result you are going to "not not accept the null hypothesis" which you might say is "accept" or "not reject". Either way, you may still be making an error, and you may feel that justifies the "not reject" label $\endgroup$
    – Henry
    Mar 26, 2021 at 14:40
  • $\begingroup$ @Henry In a strict sense, I suppose the meaning of reject that would provide more use of not accepting would be that we have proven something to have a significant statistical effect beyond reasonable doubt. In which case “not” would not strictly be a logical negation but also comes as a result of inherent uncertainty in statistical arguments involving sample data and not considering all factors. In which case I feel that “not accept” would be justified in the same way as “not reject” given that there could be an error. $\endgroup$
    – Basekill
    Mar 26, 2021 at 15:22

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As a practical matter, there is no difference. It's "just" a semantic game.

On the other hand, for purposes of deeper understanding, the important philosophical idea is that we never "accept" anything. All of our knowledge is provisional, and has a degree of uncertainty.

We can certainly reject a hypothesis, as in, "well, that doesn't seem to have worked, let's try something else." But we don't want to accept something as true because it's a slippery slope to forgetting to doubt it or that it might not be true after all.

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