There could be a whole new Trinity: The Father, The Son, and their Prophet. No one really understands or cares for the idea of the holy ghost so no one would really notice.
Actually, the Holy Spirit/Holy Ghost is integral, and was very well understood by the 7th century. The Holy Ghost is the influence of the Father intervening in the lives of humans. Removing the Ghost/Spirit would mean that the Father or the Son have to personally intervene in the world, which would be the equivalent of a King dirtying his hands by dealing justice to individual peasants.