One of the proofs presented by the Shi’ah that the Prophet صلى الله عليه وسلم appointed his ahl al-Bait (household) to the role of earthly leadership and administration of the affairs of the Ummah after him is the Hadith:
إِنِّي تَارِكٌ فِيكُمْ خَلِيفَتَيْنِ كِتَابُ اللهِ حَبْلٌ مَمْدُودٌ مَا بَيْنَ السَّمَاءِ وَالْأَرْضِ أَوْ مَا بَيْنَ السَّمَاءِ إِلَى الْأَرْضِ وَعِتْرَتِي أَهْلُ بَيْتِي وَإِنَّهُمَا لَنْ يَتَفَرَّقَا حَتَّى يَرِدَا عَلَيَّ الْحَوْضَ
I am leaving among you two successors [khalīfatayn]: the Book of Allah, the rope suspended between Heaven and Earth, and my progeny, the ahl al-Bait. And these two shall not be separated until they return to me at the Hawd
(Musnad Ahmad; v.35, p.456, #21578; v.35, p.512, #21654)
Firstly, the Hadith with this wording, describing the ahl al-Bait as one of the khalīfatayn or two “caliphs” of the Prophet, is weak due to the poor memory of the narrator Sharīk, who is Ibn Abd Allāh an-Nakha’i as Shu’aib Arnaut has explained:
وهذا إسناد ضعيف لسوء حفظ شريك، وهو ابن عبد الله النخعي
The first problem for the Shi’ah argument is that the term ahl al-Bait also includes the Prophet’s wives, mothers of the believers رضى الله عنهنّ as proven from the Qurān (33:33). Ahl al-Bait literally means “people of the House”. Common sense dictates that a man’s wives are an integral part of his household and included within his family.
Secondly, the narration describes ahl al-Bait as the Prophet’s successor collectively, whereas the Shi’ah argue that only select individuals from among the ahl al-Bait were invested with the role of Khilāfah, namely, the twelve Imams. However, this idea of only twelve individuals from among the ahl al-Bait as being the successors or caliphs of the Prophet عليه السلام cannot be extrapolated from the wording of this narration.
Thirdly, this narration uses the term itrati “my progeny”. The linguist Ibn al-A’rābi said:
العِتْرة ولدُ الرَّجُلِ وَذُرِّيَّتُهُ وعَقِبُه مِنْ صُلْبه قَالَ فعِتْرةُ النَّبِيِّ صَلَّى اللَّهُ عَلَيْهِ وَسَلَّمَ ولدُ فَاطِمَةَ البَتُول عَلَيْهَا السَّلَامُ
"The itrah are the children of a man and his descendants and offspring of his loins." He said: "So the irtrah of the Prophet sall Allahu alaihi wasallam are the children of Fatimah the virgin peace be upon her." (Lisān al-Arab; v.4, p.538):
So going by this narrow and restricted definition of itrah, which excludes the Prophet’s wives and his other relatives and their progeny, is still extremely problematic for the Shi’ah as it will necessarily exclude sayyidina Ali كرم الله وجهه too, whom they believe is the very first Imam, the Prophet’s immediate successor.
Finally, it is not necessary that the word khalīfah indicates a successor in earthly/political/administrative authority. We non-Shi’ite Muslims, at least myself, acknowledge the role of the ahl al-Bait as spiritual successors and heirs of the Prophet صلى الله عليه وآله وسلم, by virtue of their piety, knowledge and guidance.