Some Bibles are very small, containing only New Testament and Psalm. Some have Proverbs also.
What is the reason for making such a Bible which has just 28 or 29 books?
Can we really call this a Bible?
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The NT+Psalms Bibles are typically distributed by the Gideons, Navigators, and other groups that are seeking to get people to read the Scriptures. They are intentionally introductory, and designed with the premise that new believers should begin by reading "the Good News" rather than needing to slog through a whole bunch of background and ceremony first.
It is a tactical decision that says, "People have a limited attention span. Let us encourage them to focus their efforts on our distinctives." Put another way, if you are stuck in an elevator with the CEO you don't spend the entire time talking about the problem - you show him the solution.
As to whether or not such a thing is a "Bible," the answer, OF COURSE. A "Bible," from the root biblia, means book. In Christian circles, it has also taken on the meaning of library. If, at my local library, they have only 3 of the Harry Potter books, does that library cease to be a library?
Likewise, having a complete collection of the Scriptures is a modern innovation. For years, devout church fathers and monastics had to make do with whatever Scriptures they had. The sheer cost of a full codex or set of scriptures would have been prohibitive! And yet, they made do! In our day, the cost of materials is less important than the cost of time, but there is still a scarcity!
Finally, Christians understand that "In the beginning was the Word." And that Word was Jesus! It is very easy to mistake the Scripture for the Word - and this comes from one who loves the Scripture. But remember - the books merely point us to the true Word. To mistake the printed artifact for the one of whom it speaks is Bibliolatry - an idolatry that I myself have suffered - but idolatry nonetheless.
Would I love to see a mature Christian read all the Scriptures? Of course. But until then, I'd rather they had the Word than a complete library.
I have never seen a NT+Ps that is called a Bible. Also, NT+Ps's are (please correct me if I'm wrong) rarely sold, but usually meant to be given away for free. In that sense the purpose of NT+Ps is clear: to be a cheap mass-printable package containing the essential Gospel, suitable to be spread wide in a large quantity. I think that is a very good purpose.