Bultmann unleashed a set of criteria that were used to determine if what is attributed to Jesus in the Gospels really came from him. Now once again let’s remind ourselves of something: the historical Jesus quest is about discerning what the “real” Jesus was like in comparison to what the Gospels say about him. It is nearly always a deconstructive process (convering “red” letters into grey or black letters — see below) and then a reconstructive process (picturing a Jesus on the basis of what one finds to be historically-reliable). Everything depends on the “criteria.”
These criteria became sensationalized in the work of the Jesus Seminar led by Robert Funk, which led to the famous The Five Gospels: What Did Jesus Really Say? where we were treated to the red/pink/grey/black letter edition of the Gospels — red being “said by Jesus” and black being “not possibly said by Jesus.”
It operates on this assumption from Norman Perrin: “The early Church absolutely and completely identified the risen Lord of her experience with the earthly Jesus of Nazareth and created for her purposes, which she conceived to be his, the literary form of the gospel, in which words and deeds ascribed in her consciousness to both the earthly Jesus and the risen Lord were set down in terms of the former” (Rediscovering, 1).
Here is a brief statement of the major criteria, and these are then applied to the Gospels to see what floats to the top as “authentic.”
1. If a saying or event shows clear tension both with Judaism at the time of Jesus and with the early churches, then it neither came from Judaism nor from the early church, so therefore it came from Jesus. Jesus calling God “Abba” is typically used as an example since it is rare in Judaism and not used in Aramaic very often in the early churches. (Don’t pick on examples.)
2. If a saying or event is found in more than one of the Gospel sources then it is from Jesus. This like the “more than one witness” element in law. If it is found in Mark and in Q and in “M” (stuff only in Matthew) and “L” (stuff only in Luke) and in John and in Gospel of Thomas etc it is more likely to have been said or done by Jesus. The less sources, the less provable. (Jesus practiced table fellowship with sinners.)
3. If a saying or event coheres with what we know from the above then it is probable that it comes from Jesus. Jesus preached a kingdom theology of inclusion — or something like this.