I am having difficulties understanding the underlying logic in setting the null hypothesis. In this answer the obviously generally accepted proposition is stated that the null hypothesis is the hypothesis that there will be no effect, everything stays the same, i.e. nothing new under the sun, so to speak.
The alternative hypothesis is then what you try to prove, that e.g. a new drug delivers on its promises.
Now coming form science theory and general logic we know that we can only falsify propositions, we cannot prove something (no number of white swans can prove that all swans are white but one black swan can disprove it). This is why we try to disprove the null hypothesis, which is not equivalent to proving the alternative hypothesis - and this is where my skepticism starts - I will give an easy example:
Let's say I want to find out what kind of animal is behind a curtain. Unfortunately I cannot directly observe the animal but I have a test which gives me the number of legs of this animal. Now I have the following logical reasoning:
If the animal is a dog then it will have 4 legs.
If I conduct the test and find out that it has 4 legs this is no proof that it is a dog (it can be a horse, a rhino or any other 4-legged animal). But if I find out that it has not 4 legs this is a definite proof that it can not be a dog (assuming a healthy animal).
Translated into drug effectiveness I want to find out if the drug behind the curtain is effective. The only thing I will get is a number that gives me the effect. If the effect is positive, nothing is proved (4 legs). If there is no effect, I disprove the effectiveness of the drug.
Saying all this I think - contrary to common wisdom - the only valid null hypothesis must be
The drug is effective (i.e.: if the drug is effective you will see an effect).
because this is the only thing that I can disprove - up to the next round where I try to be more specific and so on. So it is the null hypothesis that states the effect and the alternative hypothesis is the default (no effect).
Why is it that statistical tests seem to have it backwards?
P.S.: You cannot even negate the above hypothesis to get a valid equivalent hypothesis, so you cannot say "The drug is not effective" as a null hypothesis because the only logically equivalent form would be "if you see no effect the drug will not be effective" which brings you nowhere because now the conclusion is what you want to find out!
P.P.S.: Just for clarification after reading the answers so far: If you accept scientific theory, that you can only falsify statements but not prove them, the only thing that is logically consistent is choosing the null hypothesis as the new theory - which can then be falsified. Because if you falsify the status quo you are left empty handed (the status quo is disproved but the new theory far from being proved!). And if you fail to falsify it you are in no better position either.