What do derivation systems in a formal logical language tell us about logic? Or

What do derivation systems in a formal logical language tell us about logic? Or

What do derivation systems in a formal logical language tell us about logic? Or about the propositions in the proof? Are their purpose only to show us that a particular proof or argument can be demonstrated using that particular language? IN other words, why do we have derivations in formal logic ... what is their grand purpose?

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