"Garbage out, garbage in."
I.e., logic is any reasoning procedure that can guarantee that whenever you get the wrong output from it, it's because you put the wrong input into it.
So any procedure that produces such a result is good enough to be called logic?
Do you have in mind any particular counterexample? I suppose a procedure that says nothing but "One is less than two.", regardless of what you put in, can make that guarantee. But I don't think it really counts as a reasoning procedure. For something to be a reasoning procedure there needs to be a causal connection between the input and the output.
And once you have settled on a particular procedure, how can you tell that it is correct? You can't possibly test all logical formulae to check that it works fine.
Oh, well, defining logic is a lot easier than explaining how to recognize it. In the immortal words of Supreme Court Justice John Paul Stevens, "That's a hard question. I don't answer hard questions."
That said, the basic approach logicians have taken for millennia is "divide and conquer". The theory is that an N-step reasoning procedure must satisfy the "garbage out garbage in" criterion, provided the (N-1)-step procedure it started with satisfies it, and the final step also satisfies it. There's no way for wrongness to get in except at the beginning, or in the first (N-1) steps, or in the final step. So if you can break down all the reasoning methods used in the procedure into a small number of elementary operations, you only need to test those elementary operations.
Also, this seems to suggest that (good) reasoning can't be logical unless you reason following such a procedure.
This is more a terminological issue. People use many reasoning procedures that usually work. Do we want to call them "logic", or do we want to give them other names such as "inductive reasoning" or "probabilistic reasoning"? If you prefer, we can call what I defined "deductive logic" and consider "logic" to be a broader category.
But there are nasty paradoxes lurking in non-deductive "logic". Consider the inductive reasoning principle, "Each time we see an instance satisfying a hypothesis, it helps confirm the hypothesis." See a six-foot man, it makes it more likely that all men are shorter than 100 feet. See an eight-foot man, that too makes it more likely that all men are shorter than 100 feet. But see a 99-foot man, that makes it less likely that all men are shorter than 100 feet.
Or is it good enough if I give the correct answer, i.e. my procedure in this case would be just to trust the output of my brain?
EB
When your brain gives a wrong answer, is it ever not because you were given incorrect data? For instance, have you ever made an arithmetic error?
