There are various types of covenants in Scripture, including a shoe covenant (Ruth 4:7), a salt covenant (Num. 18:19), and a blood covenant. The last example is the most solemn type of covenant.
Often, covenants were sealed by a covenant meal. That was true when the covenantal partners were people. An example is found in Genesis 26:30, where Isaac and Abimelech entered into a mutual covenant and the terms were sealed by a covenant meal. The principle also held true when the covenantal partners were God and men. This can be seen in Exodus 24, where, after establishing the Mosaic covenant, God shared a meal with the elders of Israel.
The various types of covenants had different elements. For example, a blood covenant always required the shedding of blood. The Abrahamic covenant is an example of such a covenant. Since there are different types of covenants, not every single element has to be true of every single covenant. For example, the principle of exchange was not always true for every covenant. It was certainly true with the salt and shoe covenants, but not with the blood covenant. Therefore, since not all the elements have to be present for each covenant, the lack of mention of these elements in the Bible may simply mean that they were not necessary. For example, the Adamic and Edenic covenants are covenants although they do not have all the elements of the other biblical covenants.
While all covenants contain certain promises and provisions, not all promises of God were put in the terms of a covenant. There are covenantal promises, but there are also promises which are not covenants. Many promises of God are not expressed as being a covenantal arrangement. For example, God promised the second coming and the Messianic kingdom. When He did so, there was no formal agreement, and no response was expected from the recipients of the promises. Yet, even without a covenantal arrangement, God will keep these promises.
Covenantal promises are always part of a legal arrangement. Therefore, the eight covenants make up all legal arrangements between God and the covenanted one.
In both cases, of course, God will fulfill both the promise and the covenant, but the covenant by nature is more solemn. Hence, in Hebrews 6:13, the author emphasizes both the promise and the oath.
You can find additional teaching about the nature of covenants, in Volume 1 of our Come and See series, titled The Word of God: Its Nature and Content, found in the Ariel Store.
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Arnold Answers is a bi-weekly Q & A with founder and director of Ariel Ministries, Dr. Arnold G. Fruchtenbaum.