Can Research Hypotheses be Contratictory Among Them? I want to know how to proceed in an experimental research, where there is a conflict between different theories with different predictions. I.e. the theory A predicts X > Y, while the theory B predicts X ≠ Y or X &lt; Y.

So, can I, based on the conflicting theories, adopt and test both hypothesis in my research? Or do I really have to pick one?
From a statistical point of view, there is no reason you cannot have the
null hypothesis that X = Y and the alternative that X unequal to Y. Then when
you get data you can assess which is greater and by how much.

Before you start you should do a 'power computation' to find what the
sample sizes need to be in order to have a reasonable probability of detecting
a difference of a size that is of practical importance. (Technically,
'power' is the probability of rejecting the null hypothesis, at a particular
significance level, if the difference is at least D --- ideally, where D
is chosen to be a difference large enough to be worth talking about.)

If you fail to reject the null hypothesis, you will not have a "statictically significant" difference
to discuss, but you should be able to cite an updated version of the power
computation to say that the true difference, if any, is likely to be smaller
then D. If you reject the null hypothesis, you may want to present
confidence intervals (say 95% intervals) to give an idea how large the difference is and with
what margin of error.

From a political point of view, you may find your adviser or those who
are willing to support your work strongly on one side or the other of the
debate whether X &lt; Y or X > Y. Then you can write your proposal in terms
of testing the null hypothesis (that X = Y) against whichever alternative
(research) hypothesis is favored. But if possible, make sure your methods,
sample sizes, types of data, and so on, have anticipated the possibility that the
truth is in the other direction from the popular view, so that you will
still have something to publish, even if the prevailing view is wrong.