In a theological dictionary, one the possesses infallibility would be defined as an infallible.  An infallible is one who, by the grace of God, is free from all forms of mistake and error, and does not commit even the smallest of sins.  However, one who is infallible isn't forced to be so, he/she chooses to.  What causes the infallible to voluntarily prevent making mistakes and committing sins, is the ownership of certain knowledge and traits in that person.  Such possessions include: a profound understanding of the reality of this world as well as the good and bad, unique approval through the grace of Allah (swt), the ability to make use of the Holy Spirit (a pure and holy spirit that always accompanies the infallible, and other possession as well.

Besides from the Prophets sent by Allah and the fourteen infallibles that have been presented in Islam, no other person is considered infallible.  As a result, there is no reason to attempt proving that others too are in possession of the Holy Spirit.

However, it should be noted that the lack of infallibility in righteous figures such as Abalfazl Abbas, Abu Talib (father of Imam Ali (as)), Ali Akbar, the mothers of the infallibles, and many other figures does not mean that they were wrongdoers or sinners.  They are undoubtedly considered to be righteous, and at a level not far below infallibility.

Generally speaking, in Islam, when evaluating people, one of three terms can be used [of course if the person has any value Islamically]:

1- Idalah: A spiritual state which comes as a result of performing the wajibaat and not committing sins.

A person who performs the wajibaat and does not commit sins, meaning that he/she does not commit any great (kabirah) sins or regularly commit small (saghirah) sins, is considered an “adel” person.

2.  Greatness: Greatness is a level higher than “Idalah” and is acquired when one reaches a considerable spiritual level.  This level is reached only by pious servants of Allah.

3.  Infallibility:  A person who is infallible, does not commit or even think of a committing even the smallest sins.