What aspect of the debate "prayer in school" may not be decided by logical analysis?
The main part of this debate that cannot be decided by logical analysis is the part about what is and is not desirable. We can logically discuss the extent to which prayer in school would violate the Establishment Clause of the First Amendment, but we cannot logically decide whether prayer in school is a good thing.
This aspect of the debate revolves around the relative importance of personal rights and public morality. Is it more important to teach people good morals (we will assume, for the moment, that prayer in schools can do this) or is it more important to cater to the rights of the individual? This is not something that we can logically determine. We cannot logically decide whether the need for public morality overrides the right of each student to pray or not pray as they wish. That issue is one of first principles, which are not something that logic can determine.