"I'm not going to have him thumbing his nose at me every time I turn around," Brinkley said.
Cogan said he would file a so-called "king's bench" petition with the state Supreme Court on Tuesday asking it to take immediate jurisdiction of Williams' case.
Much of Cogan's argument was legal: Brinkley ruled the 27-year-old Williams violated his probation without the required two hearings: one to advise Williams he had violated probation and a second to hear Williams' lawyers' response and then rule.
Brinkley, however, said she had made it clear to Williams' lawyer Gary Silver on May 9 that the rapper would have to table his career from July 11 to Aug. 11 while she resolved his probationary issues.
Brinkley ordered Williams into custody on July 11, in part after learning he was to be in Washington that night for a concert. Williams did not make that concert - or any since.
Another member of Williams' legal team, Christopher Warren, directed Brinkley's attention to more than 50 supporters in court.
They included two members of the Philadelphia Prison System Board of Trustees - the Rev. Damone B. Jones Sr. and Chad Lassiter - and Charles "Charlie Mack" Alston, the Philadelphia-born producer who is a friend and aide of the actor Will Smith.
Assistant District Attorney Noelle Ann DeSantis urged Brinkley not to grant Williams early parole, saying "he put himself in prison."
Referring to Williams' supporters, DeSantis called it "disgraceful that he cannot get his act together for this room full of people."
Cogan told Brinkley that Williams' imprisonment has been a financial disaster for him and more than 30 employees, and warned that his career is in jeopardy.
Brinkley was unmoved.
In addition to denying early parole, the judge ordered Williams to complete prison courses in anger management, parenting skills - he has a son, 3 - and drug and alcohol counseling.
"If you don't complete them," Brinkley warned Williams, "you'll do the whole six months."
Williams protested that he was in protective custody - in a cell 23 hours a day - and not allowed to attend classes like general-population inmates.
When Brinkley said the warden assured her Williams could take classes, Williams insisted prison staff still refuse.
Cogan asked for Williams to be transferred into the general population and Brinkley agreed, but warned: "If something happens to him and he's not in protective custody, it's going to be on you - the legal team."