ἀββα is the Greek transliteration of the Aramaic אַבָּא. In both Hebrew and Aramaic, the vocative is often indicated by definitizing a noun.1 Hence, we can interpret אַבָּא into English as the nominative “the father” (e.g., as the subject of a sentence) or the vocative “father” (e.g., in an address).
In each of its three occurrences in the Greek NT,2 it is unequivocally being used as a vocative. Why, then, is the adjacent lemma πατήρ declined in the nominative, ὁ πατήρ, rather than in the vocative, πάτερ?
Robertson wrote, “Indeed the second member of the address is always in the nominative form.”3 He cites Rev. 15:3 as one example of this supposed law: «κύριε ὁ θεὸς ὁ παντοκράτωρ». There, we see the first member κύριε declined in the vocative, with the succeeding member ὁ θεὸς ὁ παντοκράτωρ declined in the nominative. Yet, a brief survey of the NT yielded «πάτερ κύριε» in Matt. 11:25. If Robertson’s assertion were true, ὁ κύριος should have been written instead of κύριε.
If we consider the three instances in question, we must ask ourselves if Jesus really said both words? Did Paul really think Greek speaking Christians would cry out both words? I am of the belief (among many commentators; e.g., Bengel, Lightfoot, Wesley, etc.) that the adjacent Greek was added by a later transcriber (or perhaps by the author himself) for the purpose of translating what was to some of its intended audience an unknown word (i.e., the Greek transliteration ἀββα). That being said, ὁ πατήρ is in the nominative because (1) it is a literal translation of the definite אַבָּא and (2) the nominative can function as a vocative.
1 Arnold, p. 10. To definitize a noun in Hebrew, the noun is preceded by the definite article ה, and in Aramaic, א is suffixed to the noun. Hence, Hebrew האב = Aramaic אבא.
2 Mark 14:36; Rom. 8:15; Gal. 4:6
3 p. 461
Arnold, Bill T.; Choi, John H. A Guide to Biblical Hebrew Syntax. 2nd ed., revised. Cambridge: Cambridge UP, 2018.
Robertson, Archibald Thomas. A Grammar of the Greek New Testament in the Light of Historical Research. Vol. 1. New York: Hodder, 1914.