(I didn’t want to make this thread specifically about the following so I didn’t mention it in the title. It is however the best example of such a contradiction that I can think of. I’m sure there are others, maybe someone else can bring more examples for discussion)
Family purity is an extremely important thing in Jewish daily living. As is having a family and children.
Halakhic infertility: For people who follow Taharat Hamishpakah (rules of family purity/niddah), if a woman ovulates before they go to mikvah chances are high that they will have difficulty conceiving.
The mitzvah to be fruitful and multiply: Having children is high on the totem pole of Jewish values.
What happens when these two very important values come to head? This is an issue created by the Torah itself, something that is supposed to be a guide book for living a perfect Jewish life.
(Obviously anyone who is facing such an issue as specific as this should consult their own religious authority… I only brought this case as an example for illustrative purposes)
More generally: How do Rabbonim reconcile contradicts created by the Torah?