Learn English – Difference between “supposedly” and “supposably”

A theorem is a result that can be proven to be true from a set of axioms. The term is used especially in mathematics where the axioms are those of mathematical logic and the systems in question.

A theory is a set of ideas used to explain why something is true, or a set of rules on which a subject is based on.

You're close. To presume something is to make a reasonable assumption, usually based on logic or information. When Henry Morton Stanley said "Dr. Livingstone, I presume?", he was pretty sure he was talking to Livingstone (for one thing, he was the only other white man around).

To suppose something is to make an assumption or statement of belief, but it doesn't necessarily have to be a well grounded one. When someone says "presumably," therefore, it usually means they're pretty sure, or at least that they've got a good reason for making the statement. When someone say "supposedly," they may be pretty sure, or they may not--you don't have enough information to judge.

Now, this is a very general rule of thumb. In practice, people don't always make those fine distinctions while speaking or writing. But if you're looking for which word to use yourself, use "presumably" if you're fairly sure and "supposedly" if you're less sure.

As for "allegedly," it should really only be used when someone has made an explicit allegation, although that isn't always the case, unfortunately.