You are asking if there is some sort of methodology applied to natural language that resembles that of the logician, mainly by building an abstract, symbolic system for examining the structure of language, and the answer is a resounding yes. Today such a method is broadly employed by some philosophers of language including those who are linguists, natural language processing experts in computer science, and natural language formal semanticists who examine common properties and mixed semantics in natural language.
While natural language grammar and logic go back thousands of years, attempts to marry those theories emerged after the popularization of computer systems to build formal models of natural language. Terri Winograd, for example developed a small, controlled natural language for controlling a robot arm for his PhD thesis in natural language understanding. But from a philosophical perspective, Richard Monatague and his semantic theory (SEP) is the paragon of logicians who helped give computational linguistics a solid academic footing. Today, there are many theories that go far above and beyond Montague semantics like the class of theories labeled Dynamic Semantics (SEP). Such theories use logical models and relations to structure such natural language phenomena as anaphora (SEP), indexicals (SEP), and discourse such as discourse representation theory (SEP).
To be sure, this is a very specialized form of philosophy that has emerged from the linguistic turn that requires understanding broadly how systems of logic function, how natural language functions, and is often developed to be automated. And within the field of natural language semantics, there are schools of thinking such as type-theoretic semantics which rely heavily on the notion of more recent type theories like intuitionistic type theory (SEP). Therefore, it's not something one might see at the undergraduate level.
The chief relationship to be understood is that natural language carries a variety of semantics. Some of those semantics are mathematical, some of them are logical, and some of them are computational. In fact, it's was a major discovery that these semantics all contain an overlap in semantics. That discovery was named the Curry-Howard Correspondence. A hundred years ago, as the logicians that looked to Frege and Tarski began seeing computation after converting logics to formal systems, logicians invented computer science (Turing's PhD thesis is a logical theory, for instance) and began applying it to the full breadth of natural language.
Thus under modern semantic theories as those listed above, theories include mathematical, logical, and linguistic structure married together. This sort of technical detail in language does not appeal to many philosophers who want to metaphysically speculate about the universe or discuss vice and virtue like the Ancient Greeks. It's tough to understand because type theory and computation, natural language grammars, and mathematical logic are all pillars to make it sensible, and its fair to say, that anyone who isn't familiar with the effects of logical positivism and ordinary language philosophy on the analytical tradition or anyone who draws a very narrow circle for defining philosophy itself with a metaphilosophical theory might not even view this sort of project as philosophy.
