Well, it depends on what you mean by philosophy. Philosophy is practiced extensively in every field by those who seek to understand the field conceptually. Bertrand Russell is a living example that mathematics stopped being the study of arithmetic hundreds of years ago, and is largely an exercise in logic and philosophy at the graduate level. Computer science has also become largely infatuated in some quarters in inventing and automating logics.
If you love abstractions and formalisms in logic, then your studies in logic will likely take you across these three fields all of which are rather intertwined. Note these days the Aristotelian syllogism is just as "symbolic" as model theory, though less sophisticated.
Philosophical logic studied in Philosophy departments.
Mathematical logic studied in Mathematics departments.
Computational logic studied in Computer Science departments.
You'll find plenty of references on this site from philosophers who enjoy propositional, modal, temporal logics, etc., but if you want something much more abstract, I would start with books like:
Model Theory, 3rd Ed. by Chang Jerome
Computability and Logic, 5th Ed. by Boolos et al.
Both of these provide formalisms in abstractions of natural language logic replete with logic symbols.