When we find clear and specific instructions in the Qur'an...
When we find clear and specific instructions in the Qur'an about a certain matter our knowledge reaches the degree of certitude whether we are Islamic scholars or lay- men. When we do not have a clear Qur'anic instruction we solicit that from the hadiths of the Messenger. Some of the hadiths are clear in their indication and reported by numerous companions. Again our knowledge through this type of hadith attains certitude.
The difficulty is that hadiths of this type are not numerous and the majority of them are reported by one or two or a very few companions. Through such hadiths our knowledge concerning the rules which we try to know never reaches the level of certainty because the conveying companion did not report it to us directly because he is not living in our time nor did he record it in a book. A person received a hadith from a companion. He in turn reported it to another and so on.
Later the hadiths were recorded in a book after they passed through many hands. Thus our knowledge through this type of hadith would be at best conjectural. There are other places at which instructions of the Messenger have been reported unclearly or in two opposite ways. The conclusion can be drawn in such cases only by professional scholars or "mujtahids." The conclusion reached and the opinions formed by the scholars in any of the above mentioned cases are mostly conjectural.
They do not usually reach the height of certainty nor do they certainly agree with the Book of God. The probability of disagreement with it is very high considering only one of the opinions in each case. If we consider two opposite opinions of two scholars we would be certain that one of them is in disagreement with the Holy Qur'an because the two opinions contradict one another and the Qur'an cannot agree with two contradictory views.
From this it becomes clear that the Mujtahids whether Hashimites or non-Hashimites are not included in the particular membership of the House of Mohammad. This is because the knowledge of Mujtahids is mostly conjectural and in many cases does not agree with actual teaching of the Qur'an while the knowledge of the members of the House is securely in agreement with the Book of God.
This is because the aforementioned hadiths clearly indicate that the knowledge of the members of the House of Mohammad is a knowledge of certainty rather than a knowledge of conjecture; otherwise they would have parted in many cases with the Holy Qur'an.