For example, if one believes that A is true, whereas A is true because of some unspoken assumption B. Is there a word to describe such B?
You already used one such name, implicit, when you called it implicit assumption. Another word that is used is tacit, as in tacit assumption.
You can also say that B is a premise for A. Although the word premise doesn't imply the meaning of being unspoken. That would need to be added by another word, like implicit premise.
You might find some of the language you need in this description of the Gettier Problem on the Wikipedia.
In 1963, Edmund Gettier published a short paper in Analysis, questioning the conventional wisdom that knowledge could be defined as “justified true belief”. There are now a vast number of “Gettier type counterexamples”, some of which might align with what you are looking for.
The question you are asking is actually quite deep. There are quite a few words that could be used to describe your hypothetical assumption B. In the Responses to Gettier section of the Wikipedia article, you’ll see that many noted philosophers have tackled the problem in their own way.
For a single word, though, putative might be a reasonable choice.
Looking for a word that means an implicit assumption?
- apparent; seeming
A is ostensibly true, an unspoken assumption.
B = ostensibility
It also potentially tees up a Supreme Court battle that will test the conservative majority’s ostensible commitment to religious freedom. Slate Feb 5, 2019