In Book 3 of Plato's "Republic," what is the relationship between music and justice for Plato?
First, note that in many translations the term "music" is used to translate "mousike" , the arts governed by the Muses, which can include poetry, literature, and other arts as well as what English speakers call "music." Not all translations do this but the question does not specify a translation.
Justice for Plato was grounded in its effect on the soul of the individual. For Plato, this meant that achieving a just society depended on shaping a social system that would lead to justice residing in the souls of citizens. He considered (as is explained at length in Gorgias) that committing an injustice actually harmed the soul more than suffering one.
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He feel that music and justice is related. If people love art more, they would love music too, which leads to love more reasoning too, which is required for justice to happen