Do Trinitarian denominations believe that the Holy Spirit is omniscient, and do any believe denying it is worthy of expulsion of members?

In Acts Chapter 5 we read the story of Annanias and Saphira.

It seems that prior to selling a piece of property, they dedicate the proceeds to God, however, deceitfully they bring only a portion of it to Peter.

God struck down both (Ananias first and later Saphira) because of the lie they told.

The crucial part of the story and its relevance as an answer to this question is found in verses 3&4.

Peter says to Ananias..."how is it that Satan has filled your heart to lie to the Holy Spirit and withhold some of the proceeds from the land....You have not lied to men, but to God!!"

We know that after Peter said this, Ananias fell down and died. The same fate became of Saphira when Peter questioned her.

Note what happens in verse 11...

"And great fear came over the whole church and all who heard about these events."

Peter's words to Ananias illustrate that indeed the same respect, power and authority are to be given to the Holy Spirit.

All three persons of the Godhead, Father, Son and Holy Spirit, as regards their divine nature, have all characteristics in common, each one always has been, is, and always will be omniscient.

The Son of God, in his divine nature as God was omniscient while he was on earth. But in his human nature he was not omniscient while on earth. So as a man he could say "But of that day and that hour knoweth no man, no, not the angels in heaven, neither the Son, but the Father (only)" Mark 13:32.

Nor is he, in his human nature, omniscient even now in heaven. The two natures are united in the person of Christ, but unmixed. They are unmixable. So when Christ was a babe lying in a manger, as God he was omnisicent, but as a babe he knew no more than any other babe.

It can be seen from this that the omniscience of the Son is quite a complex issue. It would take a young believer a while to appreciate the two natures of the Son, etc. But no such complexity exists for the Holy Spirit: He is God therefore he is omniscient.

If any church member insists the Holy Spirit is not omniscient then they are effectively denying his divine nature. If such a member refuses to believe in the absolute deity of the Holy Spirit then in any properly functioning (Trinitarian) Christian fellowship they would ultimately be excommunicated and removed from the membership. Belief in the Trinity has never been an optional extra in healthy Trinitarian fellowships.

It must be. In Trinitarian theology, the three persons have the same attributes even if they vary in their office (their interaction in time and space).

Answering your question: 1 Corinthians 2:10–12 (ESV)

¹⁰ ... these things God has revealed to us through the Spirit. For the Spirit searches everything, even the depths of God. ¹¹ For who knows a person's thoughts except the spirit of that person, which is in him? So also no one comprehends the thoughts of God except the Spirit of God. ¹² Now (we have received not the spirit of the world, but the Spirit who is from God, that we might understand the things freely given us by God.

John 14:26 (ESV)

But the Helper, the Holy Spirit, whom the Father will send in my name, he will teach you all things and bring to your remembrance all that I have said to you.

On the Trinity I,VI:XIII (St Augustine)

Similar evidence has been collected also concerning the Holy Spirit, of which those who have discussed the subject before ourselves have most fully availed themselves, that He too is God, and not a creature. But if not a creature, then not only God (for men likewise are called gods ), but also very God; and therefore absolutely equal with the Father and the Son, and in the unity of the Trinity consubstantial and co-eternal.

Why all 3 "persons" of the Trinity need to possess all the divine attributes

To deny that the Holy Spirit is omniscient forces us to drive a wedge within the single Being of God by separating which essential attributes are had by which "person" of the Trinity. But for Trinitarians a "person" is simply a code word for immanent and subsistent relations within the single being of God (per Aquinas who clarified Augustine).

Notes:

• I meant "essential" as a theological term, as belonging to the divine nature of the single being of God
• I meant "code word" in the sense of St. Anselm's labelling the 3 "persons" as tres nescio quid ("three I don't know what they are"). See Bishop Barron's interview The Holy Spirit as Person in the segment minutes 12:10-17:08. But Catholic theologians use the best approximation: Aquinas's "three subsistent relations", i.e. procession (analogy of knowing) and spiration (analogy of loving).

Thus, for Trinitarians, denying omniscience to any one "person" of the Trinity is to deny omniscience to God Himself, and we cannot have that. So yes, even though denominations don't explicitly state that denying omniscience to the Holy Spirit is heretical, the members who deny it will be expelled as denying omniscience to God.

Fr. Thomas Joseph White in his Church Grammar episode Tough Trinity Questions said how in the history of theology there were attempts (by Duns Scotus, Bonaventure, and Richard of St. Victor) of formal distinction on which "person" of the Trinity has which attributes (min 30:10). But a better way is for a theologian to start with the single divine nature that is simple (Divine Simplicity), and THEN reconcile it with the Scriptural evidence of there being 3 "persons". Thus the Word needs to have the attribute of omniscience, and to be consistent with the seemingly non-fully omniscient Jesus (which we can only see in his human nature), the theologian can then postulate the distinction in terms of how each nature actuates the omniscience differently. So Jesus's human nature participates imperfectly (through the grace of prophecy) in the omniscience he fully possesses in his divine nature (see the next section below). This separation is possible because the Chalcedon definition posits two minds and two wills united in one person (Jesus as God-man, each nature has a mind & will) although both natures interact (see communicatio idiomatum), which Fr. Thomas White explained further in his Christology episode (see below).

The matter of Jesus's omniscience

As for your comment:

... but many Trinitarians seem to think the Son wasn't omniscient during his time on earth?

In the Church Grammar Tough Christology Questions episode (minute 25:28-45:54) Fr. Thomas White addresses the subtleties of how although Jesus in his human nature "participates by grace imperfectly ... in the knowledge he has otherwise as God" (26:02), Jesus possess omniscience by virtue of his divine nature. Jesus (in his human nature, in time) elected only to "actuate the knowledge" or to "turns toward it" when the Holy Spirit (his divine nature, from eternity) moved him to "reveal" that knowledge, which consists only the things we need for our salvation (which is in the NT).

For a more complete treatment, I refer you to the related topics Fr. White covered in the episodes:

• The relationship between Christ's divine and human knowledge (25:28)
• Christ's beatific vision and "self-awareness" (31:47)
• The Holy Spirit's role in Christ's human life (41:46)

as well as this 2020 blog post by Brandon D. Smith and R. Luke Stamps Was Jesus Omniscient? which navigates a tight space to avoid heresies such as Apollinarianism, Nestorianism, and modern Kenoticism.

This is consistent with another Trinitarian view that all things important to us related to Jesus is communicated to us through the human nature of God (named Jesus). Jesus models for us to be open to the Holy Spirit and receives whatever the Holy Spirit impresses us in our hearts as the OT prophets did too. We are not supposed to know everything (only the things we need for our salvation), and you can blame God for withholding information that we need to wait and discover as we live out the humanity's historical timeline until the end of the world.

Practical importance of the Trinity as possible reason why it is a dogma

In response to your comments

three I don't know what they are" I like Anselm's honesty. Goes well with St. Augustine's formulation of "whatever there are 3 of". :)

"You must assent to believing there are 3 I don't know what they are, or you will be expelled from the Church!" :)

We don't need to believe a specific conception of the "contents" of the 3 in 1 (the structure of the immanent the Trinity), but we need to believe God's two salvific works as the Trinity's outward mission (a.k.a. the divine economy) that are helpfully characterized by "knowing" and "loving" patterned after the immanent life of the Trinitarian God:

1. The immanent procession of knowing within His One Being (eternally begotten Word from the Father) manifests in time in the flesh as Jesus, who in turn provides us with the grace of the light of faith so we can truly know God as Jesus (in his human nature) knows the Father. It also helps that Jesus is the image of the invisible God (Col 1:15).

2. The immanent spiration of loving within His One Being (the Holy Spirit) manifests in time as the Pentecost where the Holy Spirit dwells in the church and in the saved individually so we can truly love God patterned after:

   • the Father loving the Son (Word) and the Son (Word) loving the Father immanently
   • the Father loving the human Jesus and the human Jesus loving the Father.

   This Trinitarian life of love helps unite us to Jesus and to obey the Great Commandments.

Salvation is knowing and loving God; having the above Trinitarian concept aids this process because to know and to love well we need to exercise our intellect fully in all our human acts and relations. The immanent Trinitarian life is our model for our soul as we govern our reason, desires, emotions, and wills (with the leading of the Holy Spirit, Rom 8:12-17):

• How do we form personal relationship with God in the aspect of "knowing"? Imitate Jesus in his immanent Trinitarian relation of knowing and being known.
• How do we form personal relationship with God in the aspect of "loving"? Be infused by the grace of the Holy Spirit to produce in us what is abundant in God's immanent Trinitarian relation of loving and being loved.

I think that is why Trinitarian denominations make the Trinity a dogma, i.e. not for being able to recite it like a parrot, but to chew on the partially revealed mystery of God and of His incarnation (Jesus) so the dogma helps us appreciate, know, and love God more.