How valid is the evidence used by those who permit a man to shake hands with a non-mahram woman, and how can it be refuted?

The Prophet, peace and blessings be upon him, did not shake hands with non-mahram (definition) women. A severe warning has been reported regarding touching them, and his refraining from shaking their hands is proof of its prohibition. As for the hadith (definition) about him taking the hand of a female slave, its meaning is that he would go with her to fulfill her need out of humility, not that he would touch her directly. As for the Prophet’s entry to Umm Haram (definition) and her delousing his head, there has been disagreement about her relationship to him. Some have firmly asserted a mahram relationship (a maternal aunt or a sister through suckling or lineage), while others have preferred that it was a specific privilege for the Prophet, peace and blessings be upon him, or that it occurred before the حجاب (veil), or in the presence of a mahram. As for the hadith of Abu Musa Al-Ash'ari with the woman who deloused his head, it is understood that the woman was a mahram to him.