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On Vulgar and Platonic Justice in Plato

In Plato's Republic, Socrates attempts to give an account of justice in the soul by examining an ideal city. His reasoning is that if we can find justice in a city, which is large and easy to explore, we will be able to apply the same formula to find justice in the soul (368d-369a). I find this analogy problematic. I am willing to grant that the soul is composed of parts, each having a particular purpose. However,