I'm sorry I sat on this so long, but I was really hoping that someone who studied formal logic would jump in. Since that didn't happen, here's me.
"So long as you're potty trained and don't make a mess, that's good enough."
Being potty trained might feel natural and get you to the bathroom, but it won't build you a bathroom or tell you why you're using it. Whenever someone touts common sense over logic, this is what I now think of: "I'M POTTY TRAINED, LEAVE ME ALONE!"
This kind of thinking only works if there's a structure in place that allows you to be this lazy. The bathroom analogy works because building one requires complex skills involving metalurgy and engineering, incorporating millenia of discoveries in physics ... but of course if you didn't have to figure out the local sewer system or install the toilet, being potty trained is good enough. Let the egg heads prevent colera and dysentary, if they really exist, right?
I would expound on the first three examples, but there are only two; because "your gut" and intuition are both the same background, unconscious, passive learning system. Pattern recognition is much the same but is the mistaken feeling that you recognize patterns when you see them. Socrates long ago recognized that the real danger in knowledge is the error in thinking that you already understand yourself and know as much as you need to know. The trouble is you just don't know what you don't know. This is evident int he statement "in an illogical world." The assumption that the world is illogical when people and objects both follow predictable patterns suggests you just don't undersand the world around you ... which means you probably shouldn't trust your gut or your intuition or your pattern recognition because it's not serving you well in understanding the big picture.
And the idea that formal logic somehow circles itself ignores the many ways that it serves us to create things like computer servers or alphabets and writing. I think what you're thinking of is some kind of philosphical rhetoric where someone sits in a corner and thinks himself into a circle of "but how do we really know?"
Perhaps the biggest danger to modern society is that other people's thinking has made our lives easy without having to understand why before using it and our natural curiosity only leads us as far as "I wonder who's going to be in the final episode of 'The Bachelor'."