Did Shlomo expect that the liar will consent to kill the live baby. which reveals that she is not the mother?!
Ri ibn Shu'iv citing a Midrash, Me'iri (Yevamos 17b): The mother of the live baby (let us call her Rivkah) was the mother-in-law of the other (we will call her Basmas). Basmas' husband died 1 after his brother was born 2 , and the live baby was her husband's only brother. The dead baby did not exempt Basmas from Yibum or Chalitzah; since he died within 30 days, he is a Safek Nefel. Basmas would be an Agunah (unable to remarry) for 13 years, until the baby matures and can do Chalitzah. Switching the babies 'permitted' her in two ways - she says that her husband left a live baby, and there is no Yavam. She hoped to be believed, and transgress Safek Yevamah l'Shuk (remarry without Chalitzah). She was happy to hear Shlomo command to kill the live baby, for now she can remarry b'Heter 3 !
Abarvenel #1: When he heard that one lied on her baby and killed him, he thought that her nature is Ma'adim (to spill blood), and she will consent to kill the live baby.
Abarvenel #2: He recognized the truth via his great Chachmah, through the women's faces, words and movements. He said to bring the sword? so others would recognize the truth of his ruling. 4
Malbim (22): He inferred from the first one's words that she wanted that neither of them have a live baby. Refer to 3:22:1:1, 3:22:1:1 5 . (PF)
Me'iri says that both husbands died. Perhaps he explains why neither husband came to claim. To answer our question, it suffices that Basmas' husband died. (PF)
Ri ibn Shu'iv: If not, Basmas would be Eshes Achiv she'Lo Hayah b'Olamo, and exempt from Yibum or Chalitzah!
Also, even if Shlomo would give the live baby to her, perhaps it will be known that Rivkah says that the live baby is hers, and men will fear to marry Basmas, lest she lied (for she had no proof, and all the more so if the baby will resemble his father or his dead brother when he grows up)! If she hated her mother-in-law, like often occurs (Gitin 24b), she is more prone to agree to kill Rivkah's baby for her own benefit. (PF)
Perhaps indeed, he recognized that she would consent to kill the live baby! Or, it would be recognized that the real mother sincerely cried or screamed to save him, and the other was not so emotional about this. (PF)
