9 In connection with this, we come also to some understanding of the further point, that is, so far as it can be understood by such beings as ourselves, why the Holy Spirit is not said to be born, but to proceed: since, if He also were called by the name of Son, He could not avoid being called the Son of both, which is utterly absurd. For no one is a son of two, unless of a father and mother. But it would be utterly abhorrent to entertain the suspicion of any such intervention between God the Father and God the Son.
For not even a son of human parents proceeds at the same time from father and from mother: but at the time that he proceeds from the father into the mother, it is not then that he proceeds from the mother; and when he comes forth from the mother into the light of day, it is not then that he proceeds from the father. But the Holy Spirit proceeds not from the Father into the Son, and then proceeds from the Son to the work of the creature's sanctification; but He proceeds at the same time from both: although this the Father has given unto the Son, that He should proceed from Him also, even as He proceeds from Himself.
And as little can we say that the Holy Spirit is not the life, seeing that the Father is the life, and the Son is the life. And in the same way as the Father, who has life in Himself, has given to the Son also to have life in Himself; so has He also given that life should proceed from Him, even as it also proceeds from Himself. But we come now to the words of our Lord that follow, when He says: “And He will show you things to come. He shall glorify me; for He shall receive of mine, and shall show it unto you. All things that the Father has are mine: therefore, said I, that He shall take of mine, and shall show it unto you.” But as the present discourse has already been protracted to some length, they must be left over for another.
Source: Tractates on the Gospel of John (New Advent)