Why there is no classification theorem for logics, if there are classification theorems for groups and algebras? Why there is no classification theorem for logics, if there are classification theorems for groups and algebras? Why the logics are different, if there are connections between logics and category theory and algebras (e.g. https://kwarc.info/people/frabe/Research/GMPRS_catlog_07.pdf). I guess that full classification of logics (universal logic) would involve handling of these aspects:


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*propositional approach vs quantification, higher order quantification

*modalities, modal operators that are applied to expressions

*non-classical logics - nonmonotonic, substructural, adaptive and paraconsistent logics.

 A: As Max states, the notion of "logic" is much more complicated than that of "group" or "algebra" - there is no generally accepted notion of what a "logic" is (e.g. related to your previous question, do we consider second-order logic with the standard semantics a "logic"? reasonable people disagree on this point - certainly I personally don't have a constant position on the question) although there are a few very common ones.
(Incidentally, an interesting question is why "logic" has not developed a precise mathematical meaning over time, given the important role the concept plays in the foundations of mathematics. I have some opinions on that, but I think the question is too vague (and my opinions too unjustified and subjective) to be appropriate here.)

That said, there are indeed theorems which I would call "classification theorems of logics." For example:


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*Lindstrom showed that first-order logic is the maximal regular logic satisfying the Downward Lowenheim-Skolem and Compactness properties, and also the maximal regular logic satisfying the Downward Lowenheim-Skolem property which admits a reasonable (= effective, sound, and complete) proof system. (A regular logic is closely related to first-order logic: sentences are identified with the classes of structures which satisfy them, and we demand closure under Boolean operations and "relativization.") There are a number of other similar results inspired by this, included a Lindstrom-style characterization of modal logic due to Benthem.

*Shelah showed that there are only four "nicely-definable" fragments of second-order logic: first-order logic, full second-order logic, monadic second-order logic (where we can quantify only over sets, that is, unary predicates), and "permutational" second-order logic (where we can quantify only over 1-1 functions).

*Abstract elementary classes constitute a kind of generalization of first-order logic, and there are many classification results and conjectures for AECs.
These and other similar results - about logics generally thought of semantically, where the structures involved are (usually) sets with functions, relations, and constants indicated (that is, the same kind of structure as in first-order logic) - belong to abstract model theory, and the collection Model-Theoretic Logics is an invaluable source in this regard. There are also classification results for logics syntactically defined, but I'm less familiar with that side of the subject.
