NHST is based on something akin to proof by contradiction. The best non-mathematical definition of the p-value I've ever seen is due to Nicholas Maxwell: "the degree to which the data are embarrassed by the null hypothesis." p-values provide evidence against something, never in favor of something, and are the basis for NHST. But proof by contradiction is only fully valid in the context of rules of logic where assertions are true or false without any uncertainty. The classic paper The Earth is Round (p<.05) by Jacob Cohen has a beautiful example pointing out the fallacy of combining probabilistic ideas with proof by contradiction in an attempt to make decisions about an effect.
The following is almost but not quite the reasononing of null hypothesis rejection:
If the null hypothesis is correct, then this datum (D) can not occur.
It has, however, occurred.
Therefore the null hypothesis is false.
If this were the reasoning of H0 testing, then it would be formally correct. … But this is not the reasoning of NHST. Instead, it makes this reasoning probabilistic, as follows:
If the null hypothesis is correct, then these data are highly unlikely.
These data have occurred.
Therefore, the null hypothesis is highly unlikely.
By making it probabilistic, it becomes invalid. … the syllogism becomes formally incorrect and leads to a conclusion that is not sensible:
If a person is an American, then he is probably not a member of Congress. (TRUE, RIGHT?)
This person is a member of Congress.
Therefore, he is probably not an American. (Pollard & Richardson, 1987)
… The illusion of attaining improbability or the Bayesian Id's wishful thinking error …
Induction has long been a problem in the philosophy of science. Meehl (1990) attributed to the distinguished philosopher Morris Raphael Cohen the saying "All logic texts are divided into two parts. In the first part, on deductive logic, the fallacies are explained; in the second part, on inductive logic, they are committed."Sometimes when an approach leads to numerous problems, the approach itself is OK and the problems can be repaired. But besides all the other problems caused by NHST (including need for arbitrary multiplicity adjustments, need for consideration of investigator intentions and not just her actions, rejecting H0 for trivial effects, incentivizing gaming, interpretation difficulties, etc.) it may be the case that the overall approach is defective and should not have been adopted.
With all of the amazing things Ronald Fisher gave us, and even though he recommended against the unthinking rejection of H0, his frequentist approach and dislike of the Bayesian approach did us all a disservice. He called the Bayesian method invalid and was possibly intellectually dishonest when he labeled it as "inverse probability." In fact the p-value is an indirect inverse probability and Bayesian posterior probabilities are direct forwards probabilities that do not condition on a hypothesis, and the Bayesian approach has not only been shown to be valid, but it actually delivers on its promise.