"In the philosophical tradition, such discourse was associated with the choice of a particular school, or with the choice of philosophy itself." He explains how there are three elements to writing in protreptic in which Paul uses: (1) [apelegmos] dissuasion or refutation, (2) [endeiktikos] demonstration, (3) [parainesis] personal appeal and exhortation (p.84).
The tradition of protreptic was used by philosophers to strengthen their arguments and convert outside followers. It is pure speculation, but it could be possible this was one of the argumentative methods Paul faced or embraced when he visited Athens preached the gospel in the Areopagus (cf. Acts 17:16-34).
Aristotle vs. Apostle Paul
Bryan briefly mentions how Aristotle's Protrepticus and other ancient philosophical literature was written in this tradition, so I studied this subject and found some interesting connections. Aristotle's Protrepticus was a very lengthy discourse about trying to find any plausible means for true happiness. Aristotle uses his greatest pupil, Eudemus of Rhodes, as an example and fellow philosopher seeking wisdom. However, the main protreptic argument revolves around intellectual wisdom as the chief virtue and only basis of reality for happiness. He boldly writes with dissuasion or refutation for philosophy [apelegmos], using many different metaphors of demonstration [endeiktikos], and presents a personal appeal and exhortation to conclude without philosophy, life and happiness are vanity [parainesis].
Paul uses a similar type of method. He argues from Scripture for faith in Christ [apelegmos] (1:1-4:25), demonstrates the out-workings of new life in Christ [endeiktikos] (5:1-11:36), and ends by encouraging his hears to live righteously in Christ [parainesis] (12:1-16:27).
In Romans 1:1-4:25, Paul uses dissuasion or refutation [apelegmos] through the use of Scripture. He uses the Scriptures to convince his hears of a correct perspective of the non-religious man (Gentile) and religious man (Jew) as equally guilty sinners facing judgment, and apart from the mercy and grace of God in Christ, no man could be justified. Therefore, he ends his scriptural argument by explaining how Abraham and David had faith in God, and righteousness was accounted to them; and whosoever has faith in Jesus Christ will have God's righteousness accounted to him through the work of the cross.
In Romans 5:1-11:36, Paul uses many examples to demonstrate [endeiktikos] how faith in Jesus Christ brings about justification, sanctification, and promised redemption. He especially demonstrates from Scripture how the Messiah initially came to save Israel, but Israel rejected the Messiah, and God fulfilled his promise to save the Gentiles as well. His use of metaphors and examples defend his message and provide a egalitarian gospel for all people groups to believe and partake in the out-workings of new life in Christ.
In Romans 12:1-16:27, Paul uses his last words with exhortatory measures. He encourages his hears with a personal touch and authoritative voice to live transformed and renewed in conformity to the image of Christ and not the image of the world. He encourages his hears to live pure, moral, and righteously with neighbors, governing authorities, and fellow believers. As Aristotle sought to protreptically explain the only real basis for true happiness and meaning was found in philosophy (i.e. through knowing how things work in the world), the Apostle Paul sought to protreptically explain the only real basis for true happiness and meaning was found in Jesus Christ (i.e. through knowing the work of Christ). Paul confirms this in 1 Corinthians 1:21, "For since, in the wisdom of God, the world did not know God through wisdom, it pleased God through the folly of what we preach to save those who believe." And continues to explain where true wisdom is found in 1 Corinthians 1:30-31, "And because of him you are in Christ Jesus, who became to us wisdom from God, righteousness and sanctification and redemption, so that, as it is written, 'Let the one who boasts, boast in the Lord.'"
This was a theological interpretation of the literary structure of Romans. I believe it is very plausible and most likely reasonable to believe Paul used a type of protreptic style to convince his readers that his gospel was the truth, but not entirely based his literary structure off of one literary device during the first century. Maybe this is why his doxological praise at the end of his letter is packed with theological realities of assurance in Romans 16:25-27, "Now to him who is able to strengthen you according to my gospel and the preaching of Jesus Christ, according to the revelation of the mystery that was kept secret for long ages but has now been disclosed and through the prophetic writings has been made known to all nations, according to the command of the eternal God, to bring about the obedience of faith—to the only wise God be glory forevermore through Jesus Christ! Amen."