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The issue of New Testament quotations of the Old Testament is broad and well-studied, so it's well worth reading a book like John Wenham's Christ and the Bible to get a feel for the complexities.
For example, not only is there a diverse manuscript tradition for the gospels, there are also multiple versions of the Septuagint, and there's always the possibility that even a particular reading in the Masoretic text is not faithful to the original inspired text. And even more fundamentally, it is thought that Jesus spoke in Aramaic, so what appears in Matthew is a translation into Greek. This just scratches the surface – there are many ways to explain the divergence between the OT text and NT quotations.
However, in this case, the most common explanation Protestants give is that "praise" is a legitimate translation of the Hebrew word found in the Masoretic text in Ps 8:2. R. T. France is a respected Anglican NT scholar, and in New International Commentary on Matthew, 789, he writes:
The psalm speaks of how God the creator silences his enemies by means of "strength" (so the Hebrew) which comes from the mouths of children. "Strength" is often ascribed to God in a formula of praise (e.g., Pss 29:1; 59:16–17; 68:34–35), and when that "strength" issues from mouths it is not hard to see why LXX translated it as "praise." The LXX version makes the relevance of the text to Jesus' situation in the temple more explicit, but the underlying sense of the Hebrew also is of vindication by what children say, and it is that sense of which Jesus' quotation here depends.
He also cites other scholars, particularly R. H. Gundry and K. Stendahl, to defend the Septuagint's choice of translating the Hebrew word 'ōz as "praise" instead of "strength":
His mention of "instruments of strength" in 2 Chr 30:21 is particularly interesting – it demonstrates the close relationship between "strength" and "praise" when this Hebrew word is used in this context.
So there's no need, then, to assume that Jesus "misquoted" the inspired text. Instead it seems likely that he emphasized a legitimate, but not necessarily the only, meaning or sense of the word in question.
Upvote:1
The New International Version is in perfect harmony with Jesus' quoting from Psalm 8:2 in Matthew 21:16:
"Do you hear what these children are saying" they asked him. "Yes," replied Jesus, "have you never read, "'From the lips of children and infants you have ordained praise'?"
"From the lips of children and infants you have ordained praise" (Psalm 8:2)
As for the English Standard Version, it makes this comment against Psalm 8:2:
"The Greek translation of the Septuagint (see Matthew 21:16) rightly interprets "strength" as "strength attributed to God in song," or praise."
The children (in Matthew 21:16) were praising Jesus by cryinging out "Hosanna to the Son of David." The ESV comment says this:
"Jesus acknowledges the children's praise and links it to Psalm 8:2, which the religious leaders should have known applied such praise to God, thus confirming Jesus as the divine Messiah."
The New Living Translation also helps to shed some light on this. In Matthew 21:15 it says that the children were shouting, “Praise God for the Son of David.” Matthew 21:16 says:
”Haven’t you ever read the Scriptures? For they say, ‘You have taught children and infants to give you praise.” Psalm 8:2 in the NLT says:
”You have taught children and infants to tell of your strength, silencing your enemies and all who oppose you.”
The NLT Footnote explains that the Greek version reads “to give you praise” and refers to Matthew 21:16. The Hebrew root words for “Hosanna” are found in Psalm 118:25, which says, “Save us, we pray, O LORD!” (ESV). Literally, hosanna means “I beg you to save!” or “please deliver us!” In saying, “Hosanna!” the people were crying out for salvation, and that’s exactly why Jesus had come.
There is nothing here for Protestants to reconcile. Jesus did not misquote Psalm 8:2.