After some thought I will restate.
Ancient Zog said to his cavemates 'If it is raing 0ut tomorrow and I go outside it wI will get wet'. Zog's mates grunt in understanding.
Logic developed as part of language without any formal structure.
Along came Aristotle who gets credit for a formalization, a sxcintifc consistent quantification.
Today all the expressions of a formal system like formal logic symbology and Bollean Algebra ARE the result of sientic investigations into logic.
Before digital logic elevators were controlled by what is a called relay logic.
Scientific investigation spurred by complex relay systems led to formalizations. Quine band McClusky are well known names in electronics.
The nature of logic is philosophy. The wiring of the brain that gives rise to logic is science. Animals clearly have the ability to reason and apply logical thinking. Brains are different, but the are the same general model.
There was Turing and his algorithm machine, logic but not classical logic.
Old Zog used informal logic. The simple truth table for a logic function like AND or OR is a scientific quantification of log in a sense.
So, you claim there is no scientific approach to logic is unfounded.
https://en.wikipedia.org/wiki/Relay_logic
https://en.wikipedia.org/wiki/Willard_Van_Orman_Quine#Logic
https://en.wikipedia.org/wiki/Quine–McCluskey_algorithm
If you were a scince professor writing a gtant proposal to investigate logic scinetificaly how wold you do it.
Imagine I'm a wealthy guy who funds research projects for starving professors. I have a staff of scintisrs on call to evaluate proposals. Convince us there is a problem to address and how you will investigat
1. What is the problem in detail you want to address. What are the issues, being specific.
2. Outline your reaerch proposal. XChart the course of your investigation.