Yeah, this one annoys me to no end.  In the end, it comes down to this.  Logic is the description of the way we attain knowledge.  It is deductive, and so deductively certain.  We can't avoid using logic when we think.  It is literally a codified description of our thought.  The only thing we can do is use valid or invalid logic.  There is literally no other possible way to think.

Now, the trick to using logic is plugging in true data.  If you plug in true data, you get true conclusions.  As it turns out, there's only one way to get true data reliably, and that's the scientific method.

So, yeah, you can think outside the box of science if you want.  What you're going to get is invalid conclusions.  Anyone who's taken first year logic knows that invalid conclusions in logic are 100% useless, except for the fact that you've learned one way not to learn anything.