What is the motivation behind " Descriptive Complexity "?
If you're interested in logic, "what problems can you express in this logic?" is a completely natural question.

If you're interested in computational complexity, descriptive complexity gives another tool that you might be able to use to prove things. For example, the fact that existential second-order logic exactly corresponds to&nbsp;NP means that, if you can prove either (a) no logic corresponds exactly to&nbsp;P or (b) there is a logic that corresponds exactly to&nbsp;P but it has different expressive power to existential second-order, then you've proven that P$\,\neq\,$NP. And it's turned out that trying to prove it via logic has been exactly as fruitful as every other approach!