This is a complex question though a good one. Many Bible readers point to Genesis 9:6 insisting "See God demands the death penalty here." Yet while the New Testament recognizes that the governments of man may enforce capital punishment (see Romans 13:4) it is difficult to build a NT doctrine on an Old Testament law. Though I am wary of supporting the death penalty -- because of inconsistency and slowness of application not to mention the irrevocability of a wrong decision -- I am afraid this issue will remain in the minds of most believers a matter of opinion.
I would however like to point out a major inconsistency among proponents of the death penalty. The same OT law that stipulates execution for murder also requires execution for adultery. But who favors that? Certainly not most death penalty advocates. The OT required the death penalty for many crimes and offences. Following is a list of capital crimes I have found in the OT. (Some lists you might find will be longer or shorter though substantially the same.)
Adultery (Leviticus 20:11)
Attacking one's parents (Exodus 21:15)
Bestiality (Exodus 22:19 Leviticus 20:15)
Blasphemy (Leviticus 24:16)
Bull goring (Exodus 21:29)
Contempt of court (Deuteronomy 17:12)
Cursing parents (Exodus 21:17 Leviticus 20:9)
Being a rebellious disobedient son (Deuteronomy 21:21)
Female promiscuity (Deuteronomy 22:21)
Idolatry (Exodus 22:20 Deuteronomy 13:5 17:2)
Incest (Leviticus 20:11+)
Kidnapping (Exodus 21:16 Deuteronomy 24:7)
Being a malicious witness in a capital case (Deuteronomy 19:16)
Manslaughter (Genesis 9:6 Exodus 21:12 Leviticus 24:17 Numbers 35:16+)
Priestly arrogation (Numbers 3:10 18:7)
Sabbath breaking (Numbers 15:35)
Sodomy (Leviticus 20:13)
Sorcery (Exodus 22:18 Leviticus 20:17)
Do you see the difficulty here? Where do we draw the line? Why favor a death penalty for murder (based on the OT law) while rejecting a death penalty for other capital crimes in OT law? For whatever position we assume must be consistent -- both internally consistent and consistent with the letter and spirit of the scriptures.
For further consideration, please listen to my podcast on this topic. Click here.
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