Briefly outline Plato's definition of justice and morality and compare it to Thrasymachus and conventional ideas of this concept.

Similar definition for justice and morality is for individuals. Plato assume that our soul has 3 parts (reason, spirit and desire) in witch reason is in control of two others. All 3 parts should work together without interfering with each other and if they all do their own "job" well person will be just, moral and with balanced soul. So, if the reason should be in control it is not justice if one of the other parts of soul tries to do reason's job and starts controlling person's life.

For Plato justice obviously is a good thing, when in Trasymachus theory justice is for the naive people who are controlled by the stronger, who are manipulated. Trasymachus is saying, that justice is in the interest of a stronger party, meaning that what is right and what is wrong is established by ruling elite and for everyone following these laws are namely in this ruler class benefits. Therefore, Tharsymachus says that for ordinary people only injustice pays and therefore it is a rational thing to ignore justice entirely. This is an obvious contradiction with Plato's idea that justice is a virtue, which is part of the soul and therefore is desirable in order to have a healthy soul.

Conventional morality is the one presented by Cephalus and means traditional view on morality. He had an idea that we need to live up to ours legal obligations and be honest. For example we need to give the man his due, treat friends and enemies differently. Plato had presented some objections to this view, such as returning weapon to a madman, however I thing some of the ideas of conventional morality are working within his own definition of this concept - "live up to yours legal obligations" sound very similar to "mind your own business", the only differences are that Plato presents more complex ideas of how this all works together and also he explaines separately what is justice of society and what is justice of an individual.