When someone asserts their Fifth Amendment privilege it basically means they feel that they might be subjected to prosecution for a crime if they give testimony. That right to remain silent goes back in the common law to England which allowed Defendants to remain silent and its such an important right that it permeates every area of the criminal law today in the United States and New York State. If it was not for this privilege, Defendants could be forced to testify against themselves or be held in contempt of court or any number of other remedies could be held against them as occurs in other countries. As mentioned in an earlier part it also has spread to the point of police enforcement procedures and limits the police from interrogating arrestees until they have given them their Miranda rights which specifically tell them they have the right to remain silent and that anything they may say can and will be held against them in a court of law. Then, beyond that the courts have added that they may consult with a lawyer before speaking with the police and then giving them an attorney if they cannot afford one on their own. All of these rights have grown out of the Fifth Amendment and the right to remain silent and not testify against oneself.
Criminal Defense Attorney