(1) a Greek school of philosophy or
(2) a Jewish school.
Stern evaluates these options and reasons that Paul could not have been retreating from a Jewish environment (synagogue) to a Gentile environment (JNTC). Why? Because Stern says that the passage says that both Jews and Greeks heard the message. He concludes, "I am satisfied that Tyrannus was a Jewish rabbi, and that what he had was a yeshivah..." (ibid).
Hold on a second, Stern! What about Acts 18:1-8?
"1 After this, Paul left Athens and went to Corinth. 2 There he met a Jew named Aquila, a native of Pontus, who had recently come from Italy with his wife Priscilla, because Claudius had ordered all the Jews to leave Rome. Paul went to see them, 3 and because he was a tentmaker as they were, he stayed and worked with them. 4 Every Sabbath he reasoned in the synagogue, trying to persuade Jews and Greeks. 5 When Silas and Timothy came from Macedonia, Paul devoted himself exclusively to preaching, testifying to the Jews that Jesus was the Christ. 6 But when the Jews opposed Paul and became abusive, he shook out his clothes in protest and said to them, "Your blood be on your own heads! I am clear of my responsibility. From now on I will go to the Gentiles." 7 Then Paul left the synagogue and went next door to the house of Titius Justus, a worshiper of God. 8 Crispus, the synagogue ruler, and his entire household believed in the Lord; and many of the Corinthians who heard him believed and were baptized." (Acts 18:1-8)Says here that Paul went from a Jewish environment (synagogue in this case as well) and entered a Gentile environment (the house of Titius Justus).
It also says here, just as in the case of the Hall of Tyrannus, that both Jews and Greeks heard the message. Notice verse 8. Even though Paul was speaking at a Gentile's house, there was a synagogue ruler in attendance, by the name of Crispus, who believed the message he had heard at Justus' house! Thus, the audience included both Jews and Greeks.
So, in fairness, we should be open to the possibility that the "Hall of Tyrannus" was a bet midrash run by a Greek---a Greek turned Messianic!
I'd like to offer another possibility about the subject matter in this mixed hall.
In a Greek school, they used Socratic Circles, speaking dialogically in order to understand a given text. Note that the term used in Acts 19:9 is dialegomenos ("dispute"). So what might've been the text they were using? Acts records that Paul reasoned from the Scriptures. So it seems likely during the course of two whole years in the hall of Tyrannus that, at some point, Paul would've used the Scriptures to reason with the Jewish and Greek students there.
So what do you call it when you have a school that studies the Torah and the Prophets?
A bet midrash! (i.e. yeshivah).
Now, was the Greek school of Tyrannus Paul's first choice? No. He only went there because the Jews in the synagogue were defaming the Way. So he went to this Greek environment. But did he leave it as a Greek environment? No! He elevated it!
Under Paul's influence, the Hall of Tyrannus, whatever it may have been when it began, had become a first-century Bet Midrash that welcomed Jews and Gentiles, encouraging both to study together, learning the Torah and the Prophets.
P.S. Messianics around the world, keep in mind that we have a global Yeshiva. Please visit torahresource.com
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