The separation of domains is the result of logic being a simple, finite mathematical tool which does not have limitless application. Take a tiny subset of the problem you propose for yourself: explain the unified logic theory that encompasses both formal plane geometry and peano's axioms of arithmetic in the same formal logic framework.
To finish your assignment, show me a crossover proof that uses lemmas derived from peano's axioms along with lemmas derived from plane geometry to formally demonstrate something valid--anything at all.
I'll save you some time. You can't even start to do it. The domain of discourse of plane geometry is is the ideal continuous plane. The domain of peano's axioms is discrete sets. This is like looking for recipes to bake an airplane.
Hmmm. If I look at Peano's axioms, and I look at the techniques used in plane geometry, I see the former as helping to form the fundamental basis for the validity of the latter. For example, Peano's axiom #3 is essentially a statement of the technique of inductive proofs, which would have some utility in plane geometry.
I'll save you some time. You can't even start to do it. The domain of discourse of plane geometry is is the ideal continuous plane. The domain of peano's axioms is discrete sets.
However, there are discrete entities in plane geometry, for which discrete treatments are possible.
This is like looking for recipes to bake an airplane.
You've no doubt heard of aircraft made from composites -- which are, in fact, baked..... ;-)
Your larger point remains, however. When we limit discussion to areas that can be addressed by logic, we exclude real things that the logic does not, and cannot touch: things like beauty, deliciousness, ugly, boring, interesting (all of which can vary even within the perceptions of a single observer!)
It's also difficult to applly it to something like Tolkein's books -- which are pure fantasy on one level (Tolkein essentially defined a logic in that world); and yet still have significant meaning to us here in the real world.