At one point fairly early on in his ministry, the synoptics record that Jesus was questioned about an apparent breach of the sabbath. This is Mark 2:25-26:
(I've used the NRSV here since it gives the problem in its most direct form, but we'll look at another way to translate it later.) Jesus is here referring to an incident described in 1 Samuel 21:1:
It certainly seems that either Jesus or Mark has made a mistake and named the wrong priest. However, since the Old Testament verse doesn't actaully say that Ahimelech is high priest, it's possible to argue that in fact Abiathar was high priest at that time, and Jesus is merely setting the date by mentioning him, even though he's not involved in the story.
However. The standard view of Bible scholarship is that Ahimelech was indeed high priest at the time (this is the view of the NIV Study Bible, the ESV Study Bible, and the NET Bible, for example). Abiathar, son of Ahimelech, is only introduced to the reader some time after the bread incident, at 1 Samuel 22:20. Bible scholars seem to agree that, although Abiathar was one of around 85 priests at this point, he was not yet a high priest. Abiathar is never described as "high priest" in the Old Testament, but Bible scholars seem to infer that he became a high priest at 1 Chronicles 15:11-12 when he and one other priest are specially summoned to serve David and carry the Ark of the Covenant.
Still, if you want to maintain that Abiathar truly was a high priest at this time, I'm not sure you can be proven wrong. But even the ESV Study Bible says "the incident with David actually occurred when Ahimelech, not his son Abiathar, was high priest." Inerrantists normally accept this and propose a different solution...
The ESV (and some other Bibles) translate Jesus' words like so:
The motive for this is to suggest that, since Abiathar had been born by the time of the bread incident, it can therefore be described as "the time of Abiathar the high priest". I would question this: it seems to me like using "the time of President Reagan" to refer to the 1960s, which you would never do. But perhaps 1,000 years from now, calling the 1960s the time of President Reagan will seem close enough, and maybe this sort of thing works better in ancient Greek. This seems to be the standard solution, and is not impossible.
Regardless of how the verse is translated, the fact remains that the high priest in the Old Testament story is Ahimelech. And so, if Jesus was going to mention a high priest, I'm sure it would be Ahimelech. There's very little reason for him to mention Abiathar.
Matthew and Luke, who both used Mark as a source, apparently both detected and eliminated Mark's mistake, as shown by this comparison of the three synoptic gospels. They are all basically the same, but Matthew and Luke wisely delete the clause "in the time of Abiathar the high priest". The fact that other gospel writers seem to think this is an error makes me think this is an error.
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