I had a very similar problem (and still do, in a way). I am actually an engineering student, so the idea that logic might not be real was quite troublesome for me as well. I emailed the professor of my philosophy class, who was able to help me a little. From this, I drew a different conclusion than the one you seem to be arriving at, so I'll only give you part of the conversation for now:

"Yet, you actually don't believe it could be wrong and you actually don't believe that logic could be unreal and you actually don't believe that it's not the case that concrete facts exist. That is clear through your behavior: you are arguing, which presupposes you believe in logic, which presupposes you believe in the existence of truth (whether or not you can access that truth).

Your beliefs are not merely expressed by your words. Beliefs are expressed through your deeds as well. In fact, I would argue that one's behavior is a much better indicator of one's genuinely held beliefs than the statements that person makes. Your behavior affirms your belief in logic.

Further, when questioning anything, we must have a clear idea of what is in question. I can question the mere words of 1 + 1 = 2. But, I'm not actually questioning the truth of the concept expressed which is nothing more than: . . .. To even attempt to form a real question of such a concept as that is to affirm the concept itself. It is the same with logic. Make p stand for 'a circle in my mind'

Now, try to conceptualize the following: simultaneously it is the case that both p and not-p.

You can't. There is no way to even begin to comprehend the concepts you are trying to express through your words. This is what it is to be meaningless. And, what then occurs is that the question becomes meaningless.

That is, there are some questions that we actually cannot ask. To combine the terms is to leave the concepts behind.

Underneath every question is an implicit, 'Well, it could be this way'. If you cannot actually comprehend any other possible way that it could be, then you cannot have a genuine question. Always try to conceptualize a possible way that something could be, if you cannot actually conceptualize any other possibility but you can conceptualize what is being asserted (that which you feel inclined to question), you might step back and ask, 'Why am I inclined to question this?'

So, say you are inclined to question logic. Why? What would it be for logic not to be real? Well, the world would not be logical. Physics and natural sciences would be meaningless and any supposedly scientific claims that have been tested would just appear as correct via chance. We couldn't predict anything, because causal connections are based on logic. It would be silly to spend time learning anything, because the belief that what I learn now I will remember later or will be useful later, etc., is based on logical induction and inference. In fact, the entire world would be an absurdity. What we would end up with is Lewis Carroll's Adventures of Alice in Wonderland.

That's correct: Carroll was a renowned logician. Alice in Wonderland is a barb against academics who ask too many 'why' questions, and lose their entire foundation. It's free online through Gutenberg Press. As an antidote to this uneasy state you are in, I would suggest reading it (and other short stories by Carroll) over the break."