Logic is the basis of science, maths, laws, and all of modern society and technology. It is hugely important. And it's not just human, all thinking animals apply at least basic logic. Without it, we simply wouldn't survive.
However logic can be misused when it is applied selectively or (in the case of false syllogisms, for example,) incorrectly. It is common human experience to have an emotional response to something, and then try to apply selective logic and selective facts in order to justify that response. Really the emotional response, and the determination of whether a thing is right or wrong, should come
after all pertinent facts have been considered, not
before. I'm not saying that we should be unfeeling robots and not have snap emotional responses, rather that we should be willing to amend that initial emotional reaction once we have considered the facts, not merely seek out only those facts that reinforce it.
We are the product of randomness, we are the children of irrationality
I would disagree. Whilst DNA mutations are random, the determination of which of those mutations survive through natural selection has much less to do with randomness. In general there are logical reasons that certain mutations survive long enough to pass into a next generation.