What are four different arguments in Dr. King's "Letter from Birmingham Jail?"

I would say that there are some distinct arguments that
Dr. King lays out beautifully in his letter.  The stages he outlines as justifying his
campaign would each constitute an argument leading to action.  Dr. King argues that
“there are four basic steps:”

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…collection of the facts to determine whether
injustices are alive,  negotiation, self- purification, and direct
action.

Each of these
represents a fundamental argument towards the situation that confronts Dr. King and his
followers.  The idea of collecting the facts that indicate injustice comes from the
overall laws that permit and allow segregation, and the police tactics used to
substantiate such edicts.  The negotiation argument is what King uses to reflect that he
and his followers have shown reasonability, yet have fallen as “victims of a broken
promise.”  Simply put, Dr. King quotes St. Augustine in arguing that there cannot be any
negotiation with laws that are substantively unfair and violate due process for “an
unjust law is no law at all.”  It is in the third phase of self- purification where King
achieves a moral transcendence.  It is this phase where King argues that non violent,
active resistance is the key to initiating social change.  The “self- purification” to
which King argues is essential for this moral transcendence is the resistance against
violence and the commitment to active resistance through peaceful means, showing the
moral inferiority of the opponent.  Finally, the direction action is what Dr. King ends
his letter with, as a call for those who believe in righteousness and justice to take a
stand against evil.  It is in these four phases where four distinct arguments are made
in order to initiate the call to action, the need to transform the world into what
should be as opposed to what is.