Why does Danforth allow Elizabeth to speak to John Proctor?

Danforth believes that Elizabeth, especially in her condition, can convince Protor to save his own life by confessing. In this act, we see contrasting considerations of self-interest lead Danforth and Parris to beg John Proctor for his confession. While Parris fears for his physical safety, Deputy Governor Danforth operates to defend the court from further attack. The change in Danforth's overt motivation is important. Previously, Danforth meant to uphold the integrity of the court, but here he suggests corruption to simply preserve the political stature of the government. Indeed, he even worries that postponing the executions would show the court's weakness. By prompting Proctor to give an obviously false confession, Danforth indicates that he likely believes that the witchcraft allegations are false. This fully demonstrates how the witch hunts have gained a life of their own; considerations of reputation and the political dynamic lead the court to continue with prosecutions and executions even when the original proponents of the trials are proven disreputable, and even when the political officials who run these trials show serious doubt in the validity of the charges.