Logic is the art of keeping thoughts in order, and beliefs are thoughts you would act on. Logic can help someone develop justified beliefs, but is insufficient in itself for acquiring a well-developed understanding of things.
so from what you say, logic comes after the beliefs. you organize your logic according to your beliefs-which means logic is included in beliefs...
though, is it possible for me to call illogical something i believe in?
Logic comes in two forms; inductive and deductive. A sound deductive argument MUST be true, by definition. Inductive logic is a guess based on available information (sometimes a flimsy guess, sometimes a very likely guess).
Logic is using past experiences or results to come to a conclusion about something. Belief is your moral stance on whether something is ethical or not.
In a situation you may have more than one logical action to take to resolve that situation, but your belief may force you to ignore one of those logical ways of resolving the situation because you don't believe that that action would be ethical, even if it is logical.
so from what you say, logic comes after the beliefs? you organize your logic according to your beliefs?-which means logic is included in beliefs...
is it possible for me to call illogical something i believe in?
You need something to apply logic to. That could be beliefs, ideas, hypotheticals, symbols, numbers, etc; but your logic is only as good as what you put into it. There is an old saying amongst computer programmers that goes, "garbage in, garbage out." Most beliefs in and of themselves are neither logical nor illogical. They only appear to be either in light of other beliefs.
As for whether you can honestly call something you believe 'illogical', I really can't say. I could argue either way on that one. It really depends on what precisely you would be intended to convey by saying that.
- This is actually an extremely deep question that isn't that apparent on the surface. This is precisely because logic is kind of like the background tool that we use to gain knowledge. For instance, one uses logic when studying physics. One uses logic when studying biology. One especially uses logic when studying math. But one necessarily uses logic when studying and inquiring about logic itself! So, inquiring about the nature of logic is kind of like a ruler trying to measure itself. The process of study (the inquiry) would have to be logical (otherwise it is pointless), which means you would have to inquire about the nature of the inquiry, and of that inquiry, ad infinitum.
That's the deep "answer". On the surface, one could simply say that logic is the study of, or tool for, sound argument and reasoning.
^^^^^ Can you see how that is reflexive? We all know what it is to believe something. A belief is a proposition that one holds to be true to the point that one lives according to that proposition. But, unfortunately, that does not truly capture the nature of belief, because one can easily believe something and not live accordingly. Even if you look up the word "believe" in a dictionary it will simply use synonyms like "faith" and "conviction" in its definition. It is doubtless, though, that we all know what the word means. Explaining it is superfluous, not to mention futile.
- I would say that one necessarily believes in that which one finds to be most logical. If something seemed absolutely ridiculous and in no way based on reality, then my mind would not be able to believe it. But, this is not to say that one believes in everything that one finds logical. I can find an argument to be very logical, but that doesn't mean I believe it. So the relationship is difficult to pinpoint exactly.
Edit: I think it also depends heavily on contradictions. If a statement is self-contradictory (i.e. illogical), I doubt anybody would believe that statement to be true. But if one can follow a pattern of truths and from those truths make a deduction that must be true, then there is every reason to believe in that deduction.