Twice in this chapter, it is stated that Paul "reasoned" with his hearers, trying to persuade the Jews of the truth of the gospel of Jesus Christ. Late in the chapter, Priscilla and Aquila take aside another preacher of the gospel and instruct him more carefully, so that what he states will be more accurate. All the persuading and refutation going on here is based upon reason and an accurate understanding of scripture and historical facts.
There is no touchy-feely, loosey-goosey gospel here, but sober, careful, prolonged reasoning in which accuracy is paramount. Paul does not rely on tricks, or emotional appeals, but seems to rely upon careful discourse with educated men and women. Earlier in the book of Acts, a possessed slave girl known for divination follows Paul around and proclaims that he is telling the truth. One might think that Paul would welcome this testimony, but he does not. He is annoyed with it, and eventually casts out the demon, even though the demon was "supporting his message." He is content with the gospel message, carefully and reasonably preached and discussed, and needs and wants no dramatic sideshows, even if they support his message.
Today, the gospel, taken seriously, learned carefully, and discussed faithfully and accurately, is equally persuasive. The reason of man may not be sufficient to discover the gospel or attain salvation, but it is nevertheless a gift to man from God which is to be used in all our discourse about the gospel.