Jesus is not merely God’s son by virtue of creation or adoption. As it belongs to the nature of a fig tree to bear figs, so it belongs to the nature of God to beget a divine Son. In claiming to be the Father’s natural-born Son, Jesus claims to share the Father’s power to heal on the Sabbath and, by implication, the Father’s nature. His opponents well understood Jesus’ claim and thus accused him of “making himself equal with God.”
We sometimes miss significant aspects of biblical teaching on the Trinity because we are unfamiliar with philosophical terms and concepts common in the ancient world. One such example is the Greek term ἴδιος and the philosophical concept of a “natural property” to which the term sometimes refers.
The semantic range of ἴδιος is fairly broad. According to Friberg, the term can be used to refer to one’s “own” private (as opposed to public) property, it can be used as a simple possessive (Eph 5:22), or it can be used to refer to one’s family or nation (John 1:11). The term can also be used in a philosophical sense to refer to something’s natural property, i.e., some capacity or power that belongs to a thing because of the kind of thing it is.
The latter usage appears in Luke 6:44: “for each tree is known by its own [ἰδίου] fruit. For figs are not gathered from thornbushes, nor are grapes picked from a bramble bush.” It is the natural property or power of fig trees to produce figs as opposed to grapes.
Philo uses the term in a theological context to describe God’s power to create. According to Philo, God’s work of creating belongs to God’s natural power: “as it is the property of fire to burn, and of snow to chill, so also it is the property of God to be creating [ὥσπερ ἴδιον τὸ καίειν πυρὸς καὶ χιόνος τὸ ψύχειν, οὕτως καὶ θεοῦ τὸ ποιεῖν].” The work of creating is proper to God as God.
What does this have to do with trinitarian theology? John 5:18 says that Jesus’ Jewish opponents were seeking to kill him, not simply because Jesus broke the Sabbath by healing on the Sabbath, but also because of the justification Jesus provided for healing on the Sabbath: “he was even calling God his own [ἴδιον] Father, making himself equal with God.”
Contemporary interpreters tend to take the term ἴδιος here as a simple possessive. In the same way I might say, “Leigh Swain is my own wife,” Jesus is saying (on this interpretation), “God is his own Father.” However, such an interpretation seems to fall short of what is happening in this text. Jesus’ claim that God is “his own Father” (in the sense of a simple possessive) does not justify Jesus’ Sabbath healing. Nor does it entail the claim that Jesus is equal with God. By virtue of creation (Luke 3:38) and adoption (Gal 4:5-6), creatures can claim God as their own Father. But this does not entail that such creatures have the power to heal or that they are equal with God.