THE TRINITY

Premise: The Father, Son, and Holy Spirit are distinct persons.
Premise: The Father, Son, and Holy Spirit are united in one Godhead.
Inference: Therefore, the Father, Son, and Holy Spirit are distinct persons united in one Godhead.

We can use the predicates F(x) to represent the Father, S(x) to represent the Son, and HS(x) to represent the Holy Spirit. We can then use logical connectives and quantifiers to express the relationships between these predicates.

One way to express the idea that the Father, Son, and Holy Spirit are distinct persons within one Godhead is to use the following formula:

∀x(F(x) ∧ S(x) ∧ HS(x))

This formula states that for all x, x is both the Father and the Son and the Holy Spirit.

To express the idea that the Father, Son, and Holy Spirit are one God, we can use the following formula:

∀x(F(x) → S(x) ∧ HS(x)) ∧ ∀x(S(x) → F(x) ∧ HS(x)) ∧ ∀x(HS(x) → F(x) ∧ S(x))

This formula states that for all x, if x is the Father, then x is also the Son and the Holy Spirit, and if x is the Son, then x is also the Father and the Holy Spirit, and if x is the Holy Spirit, then x is also the Father and the Son.

Premise: God is one being.
Premise: The Father, Son, and Holy Spirit are distinct persons.
Inference: Therefore, the Father, Son, and Holy Spirit are distinct persons who share the same divine nature and attributes.
Premise: The Father, Son, and Holy Spirit are co-eternal and co-equal in their being and nature.
Inference: Therefore, the Father, Son, and Holy Spirit are ontologically equal, each fully God.
Premise: The Father, Son, and Holy Spirit are distinct in their relations, roles, and actions.
Inference: Therefore, the Father, Son, and Holy Spirit are distinct in their persons and relations, but united in their being and nature.
Inference: Therefore, the Father, Son, and Holy Spirit are ontologically distinct, but ontologically united as one being.

Premise: God is eternal, outside of time and space.

Biblical references: Psalm 90:2, 2 Peter 3:8

Premise: The Father, Son, and Holy Spirit are distinct persons.

Biblical references: Matthew 3:16-17, Matthew 28:19, 2 Corinthians 13:14

Inference: Therefore, the Father, Son, and Holy Spirit are distinct persons who share the same eternal nature and attributes.

Biblical references: Colossians 2:9, Hebrews 1:3

Premise: The Father, Son, and Holy Spirit are co-eternal and co-equal in their being and nature, existing eternally as one Godhead.

Biblical references: John 1:1, John 14:9, Philippians 2:6

Inference: Therefore, the Father, Son, and Holy Spirit are ontologically equal, each fully God, existing eternally and co-eternally.

Biblical references: John 10:30, John 14:9-11

Premise: The Father, Son, and Holy Spirit are distinct in their relations, roles, and actions within the eternal Godhead.

Biblical references: John 14:26, John 15:26, 1 Corinthians 12:4-6

Inference: Therefore, the Father, Son, and Holy Spirit are distinct in their persons and relations within the eternal Godhead, but united in their eternal being and nature.

Biblical references: John 10:30, John 17:11, 1 Corinthians 8:6

Inference: Therefore, the Father, Son, and Holy Spirit are ontologically distinct and united, existing eternally in a dynamic and intimate relationship within the eternal Godhead.

Biblical references: John 14:10, John 14:23, Ephesians 4:4-6

Premise: It is necessarily true that there is only one God.
Premise: It is necessarily true that the Father, Son, and Holy Spirit are distinct persons.
Inference: Therefore, it is necessarily true that the Father, Son, and Holy Spirit are distinct persons who share the same divine nature and attributes.
Premise: It is possibly true that the Father, Son, and Holy Spirit are co-eternal and co-equal in their being and nature, existing eternally as one Godhead.
Inference: Therefore, it is possibly true that the Father, Son, and Holy Spirit are ontologically equal, each fully God, existing eternally and co-eternally.
Premise: It is necessarily true that the Father, Son, and Holy Spirit are distinct in their relations, roles, and actions within the eternal Godhead.
Inference: Therefore, it is necessarily true that the Father, Son, and Holy Spirit are distinct in their persons and relations within the eternal Godhead, but united in their eternal being and nature.

Premise: □(God=1) (Necessarily, there is only one God)
Premise: □(P1, P2, P3) (Necessarily, the Father, Son, and Holy Spirit are distinct persons)
Inference: □(P1=P2=P3) (Necessarily, the Father, Son, and Holy Spirit are distinct persons who share the same divine nature and attributes)
Premise: ◇(P1=P2=P3) (Possibly, the Father, Son, and Holy Spirit are co-eternal and co-equal in their being and nature, existing eternally as one Godhead)
Inference: ◇(P1=P2=P3) (Possibly, the Father, Son, and Holy Spirit are ontologically equal, each fully God, existing eternally and co-eternally)
Premise: □(P1≠P2≠P3) (Necessarily, the Father, Son, and Holy Spirit are distinct in their relations, roles, and actions within the eternal Godhead)
Inference: □(P1=P2=P3) & □(P1≠P2≠P3) (Necessarily, the Father, Son, and Holy Spirit are distinct in their persons and relations within the eternal Godhead, but united in their eternal being and nature)

Premise: [□(God=1)] (It is necessarily the case that there is only one God)
Premise: [□(P1, P2, P3)] (It is necessarily the case that the Father, Son, and Holy Spirit are distinct persons)
Inference: [□(P1=P2=P3)] (It is necessarily the case that the Father, Son, and Holy Spirit are distinct persons who share the same divine nature and attributes)
Premise: [◊(P1=P2=P3)] (It is possibly the case that the Father, Son, and Holy Spirit are co-eternal and co-equal in their being and nature, existing eternally as one Godhead)
Inference: [◊(P1=P2=P3)] (It is possibly the case that the Father, Son, and Holy Spirit are ontologically equal, each fully God, existing eternally and co-eternally)
Premise: [□(P1≠P2≠P3)] (It is necessarily the case that the Father, Son, and Holy Spirit are distinct in their relations, roles, and actions within the eternal Godhead)
Inference: [□(P1=P2=P3) & □(P1≠P2≠P3)] (It is necessarily the case that the Father, Son, and Holy Spirit are distinct in their persons and relations within the eternal Godhead, but united in their eternal being and nature)

Axiom: There is only one God (1G)
Axiom: The Father, Son and Holy Spirit are distinct persons (P1, P2, P3)
Inference: The Father, Son, and Holy Spirit are distinct persons who share the same divine nature and attributes (P1=P2=P3)
Axiom: The Father, Son, and Holy Spirit are co-eternal and co-equal in their being and nature (P1=P2=P3)
Inference: The Father, Son, and Holy Spirit are ontologically equal, each fully God, existing eternally and co-eternally (P1=P2=P3)
Axiom: The Father, Son, and Holy Spirit are distinct in their relations, roles, and actions within the eternal Godhead (P1≠P2≠P3)
Inference: The Father, Son, and Holy Spirit are distinct in their persons and relations within the eternal Godhead, but united in their eternal being and nature (P1=P2=P3) & (P1≠P2≠P3)

1 Corinthians 12:4-6: This passage speaks of different spiritual gifts, such as wisdom, knowledge, faith, healing, and prophecy, being given by the same Spirit. It also mentions that Jesus Christ is Lord, implying the distinct persons of the Father, the Son, and the Holy Spirit.
1 John 5:7: "For there are three that bear record in heaven, the Father, the Word, and the Holy Ghost: and these three are one." This passage speaks of the Father, the Word (which is a title for Jesus Christ), and the Holy Ghost, implying the distinct persons of the Trinity.
Colossians 1:15-20: This passage speaks of Jesus Christ as the image of the invisible God, the firstborn over all creation, and the one in whom all things hold together. It also mentions that God was pleased to have all his fullness dwell in Jesus, and through him to reconcile to himself all things, whether things on earth or things in heaven. The passage implies the distinct persons of the Father, the Son, and the Holy Spirit.
Isaiah 48:16-17: "Come near me and listen to this: “From the first announcement I have not spoken in secret; at the time it happens, I am there.” And now the Sovereign Lord has sent me, with his Spirit. This is what the Lord says— your Redeemer, the Holy One of Israel: “I am the Lord your God, who teaches you what is best for you, who directs you in the way you should go." This passage speaks of the Lord, the Spirit of the Lord, and the Holy One of Israel, implying the distinct persons of the Trinity.
Matthew 28:19: "Go ye therefore, and teach all nations, baptizing them in the name of the Father, and of the Son, and of the Holy Ghost." In this passage, Jesus is instructing His disciples to baptize in the name of the three persons of the Trinity.
2 Corinthians 13:14: "The grace of the Lord Jesus Christ, and the love of God, and the communion of the Holy Ghost, be with you all. Amen." Here, Paul is wishing the Corinthians grace from Jesus Christ, love from God the Father, and fellowship with the Holy Spirit.
John 14:26: "But the Comforter, which is the Holy Ghost, whom the Father will send in my name, he shall teach you all things, and bring all things to your remembrance, whatsoever I have said unto you." In this passage, Jesus is speaking of the Holy Spirit, who will be sent by the Father in Jesus' name. This implies the distinct persons of the Father, the Son, and the Holy Spirit.
Ephesians 1:3-14: This passage speaks of the Father who chose us in Christ before the foundation of the world, the Son who redeemed us through his blood, and the Spirit who seals us as God's possession. This passage implies the distinct persons of the Father, the Son, and the Holy Spirit.
Matthew 3:16-17: "And Jesus, when he was baptized, went up straightway out of the water: and, lo, the heavens were opened unto him, and he saw the Spirit of God descending like a dove, and lighting upon him: And lo a voice from heaven, saying, This is my beloved Son, in whom I am well pleased." This passage speaks of the Father (the voice from heaven), the Son (Jesus), and the Holy Spirit (the Spirit descending like a dove) being present and active, implying the distinct persons of the Trinity.
John 15:26: "But when the Comforter is come, whom I will send unto you from the Father, even the Spirit of truth, which proceedeth from the Father, he shall testify of me" This verse speaks of the Spirit of truth, who will be sent by Jesus and proceeds from the Father, thus implying the distinct persons of the Father, the Son, and the Holy Spirit.
John 16:13-15: "However, when he, the Spirit of truth, is come, he will guide you into all truth: for he shall not speak of himself; but whatsoever he shall hear, that shall he speak: 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 hath are mine: therefore said I, that he shall take of mine, and shall show it unto you." This passage speaks of the Father, the Son, and the Holy Spirit, where the Spirit will glorify Jesus and receive from Jesus to show to the believers, implying the distinct persons of the Trinity.
1 Peter 1:2: "Elect according to the foreknowledge of God the Father, through sanctification of the Spirit, unto obedience and sprinkling of the blood of Jesus Christ: Grace unto you, and peace, be multiplied." This passage speaks of God the Father, the Spirit, and Jesus Christ, implying the distinct persons of the Trinity.
Romans 8:9-11: "But ye are not in the flesh, but in the Spirit, if so be that the Spirit of God dwell in you. Now if any man have not the Spirit of Christ, he is none of his. And if Christ be in you, the body is dead because of sin; but the Spirit is life because of righteousness. But if the Spirit of him that raised up Jesus from the dead dwell in you, he that raised up Christ from the dead shall also quicken your mortal bodies by his Spirit that dwelleth in you." This passage speaks of the Spirit of God, the Spirit of Christ, and the Spirit of him who raised Jesus from the dead, which implies the distinct persons of the Father, the Son, and the Holy Spirit.