Logic

In my opinion logic is not the set of basic facts/knowledge as you stated. Logic has nothing to do with the problem of dividing by zero or if a cow is a cow.
Logic is the way our minds work to come to conclusions from our perceptions. It's how we pass from concrete observation to abstract ideas.

Example: you see an object that reminds of something people have always been calling "cow". It resembles all the other objects you have been calling "cow" in your life. It smells like a cow, it sounds like a cow, it behaves like a cow.
Though you can't have any definite proof that it is, in fact, a cow (because even the definition of cow is not that solid), logic will still lead you to conclude it's a cow.

I figured as much, I'm just on the fence between the two. I in fact did list both.

Is logic a fact, a truth? Something most natural that constructs the world as we know it, and probably the only world out there whose scope we aren't yet fully aware of? (a cow is a cow, not two cows; it seems like a factual statement)

Or is logic just the way we infer things? This happens, it means this, then it makes sense for that to be like that, bla bla bla. In this way, clearly we could have gone off in a different directions (we see in daily lives that one event doesn't necessarily have just one implication, we tend to just pick up one inference). But if such were the case, what would it be like without logic? What if a cow were not just a cow?

The earlier just might come off as a rather strange notion to SP, so I elaborated more than warranted to demonstrate where I'm coming from.

 personal anecdote

Back when I went to university, majoring in Computer Science, we had to take some courses in the humanities or social sciences departments, "to keep our minds from becoming too narrow".
One favorite was the "Logic" course offered by the Philosophy department.
It was astounding to see how people who are not of a mathematical bent had a really hard time with what seemed trivial to us.
For example, the teacher would say:
"Given that (1) Everything green sprouts flowers and (2) alligators are green, what conclusion can we draw?"
The class would explode "But that's not true! Only plants have flowers! My marker is green and will never flower!" and so on.
Or the teacher would say, "I told my son that if he behaved himself I would take him to the beach. He misbehaved, and we went to the beach. Did I lie?" and everyone would say "yes!" and "you're spoiling the brat!"
They could not detach from real-world knowledge and "sense" enough to view a statement as just an example of "All A are B" or "If p then q" and deal with it at an abstract level.

Needless to say, the few Natural Sciences and Engineering majors in the room had absolutely no difficulty going GIGO mode...

I'm not totally sure which mode of thinking is better, to be honest.

Which is why I'm looking forward to taking that class. It's, like, the only good Humanities class. ;_;
Lol'd @ the examples, what a delightful class.

I'm also of the mind that humankind today is a result of natural selection and thus not the result of "truth" itself. What we do (logic) needs not be the fact of life, but merely the most appropriate that allowed our survival in a given setting of environment.
To be specific, take the emotion fear. We aren't afraid of something because "it is natural to be so", we host the fear emotion because we just do. And it is an emotional response to a perceived danger, thus it helped keep us surviving until today.
To stretch it, one could say logic is the truth in our known universe. One just has to be fed sufficient conditions/premises to make the "true" conclusion.

As you can probably tell from that point of view, I'm biased towards the "inference" definition of logic as well, which is why I'm wondering what it is like to be without logic.