Is there a quadrinity? Clearly, the Bible supports the triune nature of God. Moreover, the Scriptures demonstrate mutual love, honor, submission, and dependency between each part. I cannot help but notice that many of the Scriptures and logic we use to prove the Trinity could also be used if applied to us. In John 17:20-22 it seems as if Jesus invites us into the Oneness of God -- making a Quadrinity. I'm not saying we are equal to God, Jesus, or the Spirit... and I realize how all of this sounds. I guess my point is that Jesus has made us more than we think we are and we should conduct ourselves accordingly. What do you think about what I am saying? -- R.B.
Interesting question. The Trinity is the orthodox explanation for the interaction of the three members of the Deity. Those who reject the Spirit speak of a "binity." Yet I've never heard of a "quadrinity." Now you're right that Christians often miss the degree to which the Lord lets us share in his deity. 2 Peter 1:3-4 is a strong support of this idea:
His divine power has granted to us all things that pertain to life and godliness, through the knowledge of him who called us to his own glory and excellence, by which he has granted to us his precious and very great promises, so that through them you may become partakers of the divine nature, having escaped from the corruption that is in the world because of sinful desire. ESV
We are permitted to share in God's nature. This is emphasized fairly little by Protestants and Catholics, though much more by Orthodox. Peter's point is that we become more and more holy the more we (experientially) know God, and so there's a direct relationship between righteousness and righteous living and spirituality. Yet we are not invisible, omnipotent, providential, etc.--so it's not a sharing making us divine as we would normally think of divinity.
However, it is the Holy Spirit living in us that makes us holy. The Spirit does not make us divine (in the full sense -- as you were right to note), nor does it  elevate us to the level of the Spirit, and certainly not the level of the Son or the Father. I say "certainly not" because the Son submits to the Father, and the Spirit comes from the Father and Son. At any rate, there is no quadrinity.