Intellectual humility is often described as an intellectual virtue, along with other perceived virtues such as open-mindedness, intellectual courage and integrity, and in contrast to proposed intellectual vices, such as pride and arrogance. See "Doxastic Definition of Intellectual Humility."
Importance of intellectual humility
There is widespread agreement among philosophers and psychologists that intellectual humility is important and valuable in some way, especially if one is going to engage with deep disagreements in a productive way. However, there is little consensus about the precise nature of intellectual humility. For example, some philosophers emphasise the importance of a disposition to own one’s particular limitations (e.g. the limitations of one’s knowledge and perspective), while others focus on the connection between humility and a low concern for status. Meanwhile, psychologists such as Justin Barrett and Peter Hill are working on better understanding the science behind intellectual humility, and on developing accurate measures that can tell us more about how to quantify humility.
To this we must add that the participation of intellectual humility should not be prescribed, because when it is in a circle in which all the actors know the topic to be addressed, but for their private evidence are closed to the arguments to the contrary, intellectual arrogance (See "Intellectual Arrogance" defined in Section "A Virutous Mean" is unproductive and inappropriate because what is not being sought with this discussion is the exposure of all the evidence in a "game" to sudden death, according to the weight of the evidence that won the discussion; this allows the elimination of "Pilate's washing his hands", of not taking certain evidence as real and proven by mere caprice of evidence left in the deprivation. This leads to asperities within the discussion, but forces to set a clear and concise precedent, which is productive for the human advance within the religious discussions. That is, not to go to the intellectual humility to endure a discussion gives a stratum of understanding that can not continue to overlap business under the table of any kind.[further explanation needed] Endowing class and honor also puts the chest for its evidence defended and known true from the beginning.[further explanation needed]
Doxastic definition of intellectual humility
Drs. Ian M. Church and Peter L. Samuelson proposed a doxastic  account of intellectual humility. They considered intellectual humility as a virtue, one of valuing one’s own beliefs "as he or she ought.” 
A “Virtuous Mean”
With this “as he or she ought” as the guiding principle, Church and Samuelson proposed: “Prima facie, humility is the virtuous mean between arrogance, on the one hand, and self depreciation or diffidence on the other." 
People are intellectually arrogant when they erroneously evaluate their intellectual capacity higher than warranted (“smarter than average). This results in the intellectually arrogant person being more closed-minded and biased than the intellectually humble person.
People who are intellectually diffident are those who fail “to appropriately recognize or appreciate their intellectual achievements.”  Such a person is less inclined to speak out when he or she encounters wrong information.
Advantages to Doxastic Intellectual Humility
Drs. Church and Samuelson cite several advantages of this treatment of intellectual humility as a virtue. By adopting such an approach, one is better equipped to counter a natural tendency to overestimate one's strengths, and underestimate one’s weakness, thus countering confirmation bias. 2 Church and Samuelson further propose that by being open to the viewpoint of others, one positions oneself for growth and learning.
The doxastic definition of intellectual humility does not necessarily mean that one abandons one’s beliefs, only that one is open to new and alternative viewpoints.
1 Stanford Encyclopedia of Philosophy, “Self Knowledge” 2 Intellectual Humility: An Introduction to the Philosophy and Science, Bloomsbury Academic, Ian M. Church and Peter L. Samuelson