The Logic of Double Procession
Double procession refers to how the Spirit relates to the Father and Son. It is said to be “double” because both the Father and the Son spirate (breathe out) the Spirit. The Spirit comes from both. Others find it better to say: the Spirit only proceeds from the Father in order to maintain the monarchy of the Father.
While both positions overlap, it is worthwhile to consider the logic of double procession since few discuss the relative (pun-intended) benefits of the doctrine today. The basic argument is that double procession ensures that each person in God has one relational difference from the other. The point is to ensure that we can distinguish the Father, Son, and Spirit.
Here is how this works.
Relations of Origin
The key issue in the early church was worship. Christians received the threefold name of God in baptism (Matt 28:19). They further worshipped the man Jesus Christ while confessing one God. So they had to think through how to explain their worship.
The first solution came by way of Logos theology. Following John (John 1:1), early Christians understood Jesus to be the Word of God, or the Father, as in the Reason in God. Just as our thoughts our in us but can be distinguished from us, so the Word is in God and can be distinguished from him.
And while the Logos analogy works, the primary metaphor for the Son and Father in Scripture is Son and Father. These names define their relationship in terms of origin.
The Son is Son because the Father begat him; the Father is Father because he begat the Son. So their distinguishing relationship according to the biblical idiom was to be one of paternity and filiation, of eternal begetting (because the Son always was the Son and was not begotten in time).
These relationships say very little about the inner-life of God but use biblical notions to describe the Father and Son as one divine being with distinguishable attributes. Christians coined these relations of origin, therefore, not to impute any sort of positive difference between the Father and Son beyond merely defining how they can be distinguished. Hence, they succeeded in giving words to their worship of the Son while remaining monotheists.
But later on, Christians realized they needed better language to descripte how the Spirit related to the Father and Son. Afterall, the Spirit too is in God (1 Cor 2:11) does the works of God, and comes from God (John 14:26; 15:26). So by analogy, the Spirit came to be understood like the Son to the Father. The Spirit spirates eternally of the Father (and of the Son).
The Logic of Double Procession
Double procession applies the relations of origin to the persons of God to explain how all three can be one God yet have relations of origin that distinguish the persons. It works like this:
The Spirit (X) proceeds from the Father (Y) and Son (Z). So X differs from Y and Z.
The Father (Y) begets the Son (Z) and spirates the Spirit (X). So Y differs from Z and X.
The Son (Z) proceeds from the Father (Y) and spirates the Spirit (x). So Z differs from Y and X.
From the perspective of each person, there are two distinguishing relations of origin. So the Father differs from both Son and Spirit, the Son from both Father and Spirit, the Spirit from both Father and Son.
Consider the alternate notion of single procession. If the Spirit proceeds from the Father alone, then how does the Spirit differ from the Son—at least from the Spirit’s perspective? If Son is begotten of the Father yet does not spirate the Spirit, then how does the Son differ from the Spirit—at least from the Son’s perspective? The answer seems less clear than it does on the double procession argument.
Also, if the Father sends both the Son and Spirit and the Spirit and Son do not have a clear way to distinguish the one from the other, then this implies a singularity of Son and Spirit according to person. But that cannot be case since they differ according to relations of origin.
Even on this single procession perspective, to be fair, the Son and Spirit differ from the Father’s perspective: one is begotten; the other is spirated. The problem really arises when considering the relations of origin from the Son and Spirit’s perspective. They both differ from the Father, but they do not seem to differ from each other.
So an implication here is di-personality instead of tri-personality in God. Now, single procession proponents obviously do not intend this. And they may defend their position by pointing to the Father’s monarchy and relation to the Son via begetting and Spirit via spiration to satisfactorily distinguish the persons in God according to origin.
Yet double procession does seem more logically satisfying, more rational. The major weakness of double procession is, however, that the Spirit spirates from both Father and Son. So the relational distinction here may collapse the differences between Father and Son. It may be beneficial to use another term than spirate to describe the Spirit in his relation either to the Father or Son.
In any case, my point is not to say one is better than the other. But simply to explain how double procession works. Having done that, make up your mind on the matter. I believe double procession makes the best sense.