Martin Luther King wasn't so much presenting an argument as an explanation, answering the criticisms by presenting the other side of the story in response to the concerns voiced by a group of white clergy members.
He first refutes their charge that the race demonstrations in Birmingham were being incited by outsiders. King points out he has a pre-existing connection to Birmingham through his involvement as president of the Southern Christian Leadership Conference (SCLC), which had been "invited...to be on call to engage in a nonviolent direct action program if such were deemed necessary."
Aside from his responsibility to be in Birmingham as part of the SCLC, King also states he needs to be in Birmingham "because injustice is here" and he is answering the need to deal with that situation. He explains his position that all communities in the United States are connected to each other and that problems in one place impact other places.
He lists the four stages of a nonviolent movement: "1) Collection of the facts to determine whether injustices are alive. 2) Negotiation. 3) Self-purification and 4) Direct action." and provides details of the research, attempts to work within the community in other ways, and the rebuffs that were received at each stage. In answer to the charge that the demonstrations were poorly timed, he responds,
I have never yet engaged in a direct action movement that was "well timed," according to the timetable of those who have not suffered unduly from the disease of segregation.
King suggests that there is "just law," which is in compliance with the decent moral standards of behavior and conduct in a God-fearing society, and "unjust law," which violates "moral law." Because segregation is based on a system of unjust laws, the demonstrations and other attempts to create change are justified. Indeed, he sees it as a recognition of the importance of honoring moral laws that individuals are working to change the system of unjust laws creating the segregated system then in existence.
an individual who breaks a law that conscience tells him is unjust, and willingly accepts the penalty by staying in jail to arouse the conscience of the community over its injustice, is in reality expressing the very highest respect for law.
In spelling out his position on these and other issues, Dr. King framed the civil rights movement as being based on the need for all persons to act in love toward every other person. His letter, which was published for all to read and consider in 1964, presented and supported the logical and intellectual basis validating the need for civil rights reform to a nationwide audience. In that sense, it was very successful.