How is the doctrine of the Trinity substantiated?

The New Testament did have these three: the Father, Son and Holy Spirit to be very necessary in every believer's life. There is no Christianity without these three. The Christian baptism proves exactly this.

A believer gets baptised into the name of the Father, Son and the Holy Spirit which means marking the believer as someone owned by the Father, Son and Holy Spirit (Matthew 28:19). In baptism, we are buried and raised with Christ to a new life (Romans 6:1-5). Thus, it is clear that the Trinity is very necessary in every believer's life.

Matthew 28:19 tells us that there is Trinity who are equal in respect to their relationship to every Christian. They all equally own the baptised Christian which logically means allegiance to them. And if every baptised Christian must be loyal to the Father, Son and Holy Spirit, these three must be higher and in status than their subjects (the baptised ones). They also must be worthy of receiving equal devotion since they own everyone who is united to the incarnate Christ.

The early church tried to understand why they have a Trinitarian formula in baptism if there is only one God. Jesus Christ didn't say, "baptise all nations in the name of the Father only" but rather, Jesus himself said, "baptise all nations in the name of the Father, Son and Holy Spirit" (Matthew 28:19). They also had prayers including these three (2 Corinthians 13:14). Since these three were very very important for every believer, every believer seek explanation to their worship practises. What was the exact relationship of these three whose name we got baptised into? What are they? The answer they got from the Bible and their bishops/ teachers was that the three were one, specifically, one God.

"God" in the "one God" points to the singular nature of deity. Thus, the Father, Son and Holy Spirit, being of the same nature (consubstantial) are all collectively the one God.

The gospel of John introduced that Jesus (the Son) was God (John 1:1, 10:33, 20:28) and that the Holy Spirit was a divine person (John 14:16, 16:14). If they are "God" and the Father is the "only true God" (John 17:3), there would be three true Gods if we count their divine nature (three "Gods") but we should not count their divine nature to be numerically three because in reality there is only one divine nature (which means numerically only one "God") and thus, logically speaking, the three persons: the Father, Son and Holy Spirit, exists as the one God. If we would count them collectively, we should count what is countable and that is their persons (which means they are numerically three "persons", not three Gods). However, we can absolutely describe them as "divine" persons (which means they are each God by nature) and all together they should be regarded as one God (due to their one nature). The same logic was used by the 4th century church fathers regarding the word "humans" (the fathers see this as inaccurate since human nature is not many but one. One should call a person with a human nature, well, a human person and not simply human because whilst human nature can be individuated (to John, Peter, and Paul), these does not mean that each has a different human nature. They all have the same human nature, that of one and only from Adam.

The other sense of θεος (God, god) is position of authority, not related to nature. (e.g. "You are gods", Psalm 82:6). The meaning of "god" in this case is "ruler", "judge" , similar to κυριος (master).

Jesus was not only God but also human. Jesus was the incarnate God (the Word [Jesus] "was God" and he "became flesh", John 1:1, 1:14). And Jesus was another divine person from the Father who is also a divine person. Hence, they are both ontologically θεος (God by nature). However, the incarnate God (Jesus) had his own Father as his God in the sense of position of authority. (the God of Jesus Christ, Eph. 1:17). The Father is positionally higher than Jesus because naturally Jesus is relationally subordinate to the Father (because Jesus is the Son of the Father). So now you would understand why in some cases the Father is equivalent to "God" (John 17:3, 1 Cor 8:6) and in other cases, he is not (John 1:1c, 1:18b, 10:35, 20:28). It is because θεος has more than one meaning and context alone determines its exact meaning in a biblical text.

Conclusion

Hence, the concept of consubstantiality was the logical explanation of the Christian church on how to understand the Trinity as one God.

The body of your question asks, What is the logical argumentation for the doctrine of Trinity?

If it can be assumed that God is infinite and eternal and perfect and if it can be assumed that God has internal integrity then it can be deduced that God has infinite, eternal, perfect integrity.

If God has infinite, eternal, perfect integrity then there is no ontological distinction between who God is, what God says, and what God does because He cannot deny Himself: He is an ontological unity in being, word, and deed and it is His modes of expression that are distinct.

God the Father is who He is:

and I will be a father to you, and you shall be sons and daughters to me, says the Lord Almighty.”  - 2 Corinthians 6:18

God the Son is what He says:

In the beginning was the Word, and the Word was with God, and the Word was God. - John 1:1

God the Holy Spirit is what He does:

“This is the word of the LORD to Zerubbabel: Not by might, nor by power, but by my Spirit, says the LORD of hosts. - Zechariah 4:6

Personhood (which is entirely internal) is known only by what persons say and do. Jesus (the Word made flesh) came in the power of the Holy Spirit to show us the Father. He is the Word of God sent out to invariably accomplish everything for which He was sent.

For as the rain and the snow come down from heaven and do not return there but water the earth, making it bring forth and sprout, giving seed to the sower and bread to the eater, so shall my word be that goes out from my mouth; it shall not return to me empty, but it shall accomplish that which I purpose, and shall succeed in the thing for which I sent it. - Isaiah 55:10-11

In John 14 when Phillip asked Jesus, "“Lord, show us the Father, and it is enough for us.” and it is astonishing that Jesus replied, "“Have I been with you so long, and you still do not know me,** Philip?

He answered ontologically as the Word of God, that is to say, as God.

This is a very good question. And let me say from the onset that I will not "dazzle" you with examples of the Trinity that deals with water, ice, eggs, four leaf clovers sun rays or anything else in like manner.

The definition of the doctrine of the Trinity is not an "assumption." It is the normative systematic theology of God in Christianity and is based on the fact that the Bible is EXPLICIT in telling us that there is, was and forever will be only ONE God AND the fact that the Bible identifies three (and only three) persons as God.

So, how is this "substantiated?" Now, please bear with me on this because it may appear complicated but it is not when you think about it. Also, the Trinity cannot be understood, argued or refuted by appealing to any single passage in the Bible. It is drawn from the whole Bible.

The Bible identifies God by: (1) His names. (2) His titles. (3) His unique attributes. (4) His unique actions. (5) His worship. If you will examine the Bible thoroughly you should be able to quickly discover that there are three and ONLY three "persons" who are identified as God by the 'COMBINATION" of the literary means I listed above.

These persons are each variously...CALLED by the NAMES of God. (YHWH and its variants) either directly or indirectly, usually both.

RECOGNIZED with the TITLES of God. (Lord, king, savior, redeemer etc.) For example God is identified at Isaiah 44:6, "Thus says the Lord, the King of Israel And his Redeemer, the Lord of host; I am the first and I am the last. And there is no God besides Me. At Isaiah 43:11, "I, even I; am the Lord; And there is no SAVIOR besides Me.

Jesus Christ is also described as the "first and the last in the book of Revelation. Jesus Christ is also identified as the "SAVIOR" at Luke 2:11, "for today in the city of David there has been born for you a "SAVIOR, who is Christ the Lord. Are you beginning to see the connection?

As a point of order let me say that when you see the word "God" in the Old Testament you "CANNOT" assume it refers to God the Father all the time. Let me pick another one! His unique actions. That is to say (Creation, origin of God's word, salvation of men etc).

Going back to Isaiah at Isaiah 44:24 the part that says, "I, the Lord am the maker of all things, Stretching out the heavens BY MYSELF, And spreading out the earth ALL ALONE."

And we have at John 1:3, "ALL things came into being by Him, and apart from Him (apart means without Him) nothing came into being that has come into being." Colossians 1:16 backs this up, "For by Him/Jesus Christ ALL things were CREATED, both in the heavens and on earth, visible and invisible, whether thrones or dominions or rulers or authorities ---all things have been created by Him and for Him." God the Father backs this up at Hebrews 1:10.

One more, God's worship. We already know God is worshiped in the Bible. Jesus also receives worship. The highest form of worship to Jesus was by Thomas at John 20:28 when he said literally to Jesus Himself, "The Lord of me and the God of me." At Hebrews 1:6 His Father says, "Let all the angels of God worship Him." The Greek word for worship in this text is "proskuneo" which is the same word used for worshiping the Father.

At this point I'm not going go over His unique attributes/characteristics which are easy to figure out on your own, i.e omnipresence, omnipotence, eternality, etc.

Keep in mind that I am saying that each person of the Trinity receives some COMBINATION of the 5 means of identifying and distinguishing God listed above. Remember, theology is not salvation. It is in Christ that we find God fully and sufficiently manifested to us human beings, AS a human being and for His own glory. And it is only through His Holy Spirit that we can know Him and be saved by His grace.

Finally, at Matthew 16:13 Jesus said, "Who do people say that the Son of Man is? At verse 16 Peter replied, "Thou art the Christ, the Son of the living God." Jesus replies to Peter, "Blessed are you Simon Barjona, because flesh and blood did not reveal this to you, but My Father who is in heaven."