In his situation, he killed Irvin's friend and will kill Irvin because his dog died and he was nearly killed.
His reaction is completely normal, so it is just right to expect that Irvin wouldn't be lenient with him after he killed Irvin's friends and also came after Irvin himself.
Besides, Irvin has killed people for less transgressions. He has killed someone for looking at his woman in a lustful way.
So there was no way he would have let Arthur go if Arthur had proven to be weak. But he didn't call Irvin out on his lie.
He liked to see Irvin lie in order to survive. It made him chuckle.
Then he asked, "Who asked you to kill the Arthur boy?"
Irvin was stunned. He asked, "Didn't I already answer that…"
Arthur interrupted him with a kick to his stomach and said, "I ask, you answer. Not the other way around."
Irvin nodded. "Yes, sir."
Then he replied, "I was asked by the R brothers of the society of thieves to kill him."