The court in Salem works very differently from modern courts. Read Act 3 in The Crucible carefully and find out what counts as proof for Judges Danforth and Hathorne, and what does not.
Danforth and Hathorne are shown to accept hearsay and unsubstantiated testimony as evidence and proof. At the same time, they show a propensity to reject evidence that might challenge their presuppositions and belief systems.
Miller's construction of the Salem legal system in Act III is significant. He is able to show how fairness was violated in the name of personal agenda. The court system was used as an extension of individual power, rejecting basic legal tenets in the process.
One example of this would be the acceptance of evidence. For example, Giles Corey's questioning of Putnam is rejected as evidence. However, when Abigail "sees" a vision in court, the judges accept this as proof of witchcraft. The judges are able to accept Proctor's confession of adultery and Elizabeth's lying as evidence of guilt and yet are unable to see that Proctor's questioning of Mary reflects the fraudulent nature of witchcraft in Salem. In these settings, "proof" is only viewed through the lens of political advancement. Hathorne and Danforth have invested their personal reputations in the proceedings. They do not want to go where the evidence takes them if it is going to discredit what they are overseeing. Their acceptance of "proof" is whatever demonstrates witchcraft in Salem.