The logician Gerhard Gentzen proposed that the meanings of logical connectives could be given by the rules for introducing them into discourse.
For example, if one believes that the sky is blue and one also believes that grass is green, then one can introduce the connective and as follows: The sky is blue AND grass is green.
Prior suggested that this meant that inferential rules could not determine meaning.
He was answered by Nuel Belnap, that even though introduction and elimination rules can constitute meaning, not just any pair of such rules will determine a meaningful expression—they must meet certain constraints, such as not allowing us to deduce any new truths in the old vocabulary.
Harmony, then, refers to certain constraints that a proof system must require to hold between introduction and elimination rules in order for the proof system to be meaningful, or in other words, for its inference rules to be meaning-constituting.