“The promise of the eternal inheritance” harkens back to the inheritance that God promised to Abraham, of which we become heirs through having the same faith as Abraham. It includes justification by faith, being part of a spiritual nation, and eternal life. As Paul writes in Galatians 3:29, “if you are Christ's, then you are Abraham's seed, and heirs according to the promise.”
Gilbert Wakefield offers an alternative translation of Hebrews 9:16-17 that brings out an important detail:
For where a covenant is, there must be necessarily introduced the death of that which establishe[s] the covenant; because a covenant is confirmed over dead things, and is of no force at all whil[e] that which establishe[s] the covenant is alive.
Similarly, Young's Literal Translation finds a commonality between the two covenants by using the term “covenant-victim” rather than “testator”:
. . . for where a covenant [is], the death of the covenant-victim to come in is necessary, for a covenant over dead victims [is] stedfast, since it is no force at all when the covenant-victim live[s].
In verses 16-17, most translations use “testament” and “testator,” which are indeed possible meanings of the Greek words. Like a “Last Will and Testament,” the New Covenant goes into effect only when the testator dies. This nuance, though, can apply only to the New Covenant, while the context of Hebrews 9 is both the Old and New Covenants. Both of them were sealed with “covenant-victims”—living beings that had their blood shed for the sake of establishing the respective covenants.
In the covenant with Israel, the covenant-victims were oxen and goats (see Exodus 24:5-8; Hebrews 9:19). The New Covenant, though, was confirmed with the bodily death of the Son of Man. Hebrews 10:5 says, “a body You have prepared for Me”—a body capable of having its blood drained out in sacrifice, both for the remission of sins and for the establishing of a covenant.
For Abraham, the covenant victims were mere animals. However, despite it not being explicitly stated, that covenant also required the life of the Creator. Paul explains in Galatians 3:8 that the promise that “all the families of the earth shall be blessed” indicates that God would justify the Gentiles by faith. Justification by faith is possible only through belief—trust—in a sacrifice of equal or greater value to the life forfeit due to sin. The blood of bulls and goats could never pay the life-debt of any human being; only the death of the sinless Creator could provide propitiation—justification—for all people. In this way, even though the Abrahamic covenant was confirmed only with slain animals, inherent within it was a promise of a future sacrifice so great that it would justify all those who believe in it.
David C. Grabbe
Why Was Jesus Not Crucified as Passover Began? (Part Two)