How did Puritans determine who was “fit” to be admitted as members of their churches? 

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In order to become a full member of a Puritan congregation, a candidate had to give a convincing account of a conversion experience. They were also required to live their lives in a way that demonstrated godliness, which placed a great emphasis on reputation as a criteria for membership, especially in the tightly-knit communities in New England. Finally, they had to be literate, and knowledgeable about Scripture, which they held was the source of all religious truth. After the Massachusetts Bay Colony had been well-established, the so-called "Halfway Covenant" allowed for the children of church members, many of whom had not received a conversion experience, to become partial members of the church. Their children, in turn, could receive baptism.