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Wilhite, Shawn J. (2019). "Title of the Didache." In The Didache: A Commentary. (124-132). Eugene, OR: Cascade Books. (Personal Library).
Wilhite notes that the shorter title (inscriptio) and the longer title (incipit) of the Didache are found in the 11th century Codex Hierosolymitanus (H54) but not in other traditions, which are fragmentary in nature (Wilhite 2019, p. 124), although the no longer extant Georgian tradition may include both, but in an adapted form. However, the Georgian is considered dependent on the tradition from H54, so is not an independent witness.
Wilhite briefly explores the way the document could have received the two titles (Wilhite 2019, p. 125). One could be an early title, with the other added later. both titles could be additions or contain additions to the text. Wilhite observes that the work is referred to in antiquity by a variety of names. A third explanation could be an ancient tradition of a shorter and longer title, which was not uncommon in antiquity (Wilhite 2019, p. 126). After consideration, Wilhite considers it "quite plausible for the Didache to have both an inscriptio and incipit early in its formation" (Wilhite 2019, p. 127).
Because patristic references tend to use a brief title, Wilhite leans toward the short title as more likely to be authentic. Wilhite observes that the longer title is indicative of an interest in the handing on of teaching from Jesus, through the apostles, to other people. However, there is no indication in the text of specific apostolic authority (Wilhite 2019, p. 128). Therefore, he considers it likely to be pseudepigraphal.
The identity of "the lord" (κύριος) in the title is a frequent matter of debate, since it can refer to a human or divine master, and is used in Malachi of God, before the birth of Jesus, who is fairly regularly referred to as "the lord" (Wilhite 2019, p. 128).
The work is identified as instructing "the Gentiles" (Wilhite 2019, p. 128). The instructions are for all nations, which Wilhite takes to be the command of Jesus in Matthew 28:18-20.