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And a certain woman named Lydia, a seller of purple, of the city of Thyatira, which worshipped God, heard us: whose heart the Lord opened, that she attended unto the things which were spoken of Paul.
Arminian view: Lydia was only prompted or persuaded to believe. The grace upon her was not irresistible.
Calvinist view: The text clearly says that Lydia's heart was opened to respond to the message. Furthermore, if this is not irresistible grace, then why was Lydia saved when others were not? Was she better or more spiritual than other people?
The latter Calvinist appeal is one we have dealt with at the end of our item on unconditional election (link below). In terms of the former point, the word here "attended" means to pay attention to, adhere to, or attend to.
Does it mean "persuasion" or the making of a decision? Here is how Luke otherwise uses the word:
Luke 12:1 Beware ye of the leaven of the Pharisees, which is hypocrisy.
Luke 17:3 Take heed to yourselves: If thy brother trespass against thee, rebuke him; and if he repent, forgive him.
Luke 20:46 Beware of the scribes, which desire to walk in long robes, and love greetings in the markets, and the highest seats in the synagogues, and the chief rooms at feasts.
Acts 8:6 And the people with one accord gave heed unto those things which Philip spake, hearing and seeing the miracles which he did.
If anything this word can fit either a model of irresistible OR prevenient grace -- it tells us that the message got Lydia's attention because of the Lord's influence, but it does not attribute the actual decision to the Lord's influence. At best this verse is non-determining.